Employee service share based compensation tax benefit realized from exercise of stock option.

No definition available.

Name:	dgica_EmployeeServiceShareBasedCompensationTaxBenefitRealizedFromExerciseOfStockOption
Namespace Prefix:	dgica_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Number of periods considered to arrived at average price to issue common stock.

+ References

No definition available.

+ Details

Name:	dgica_NumberOfPeriodsConsideredToArrivedAtAveragePriceToIssueCommonStock
Namespace Prefix:	dgica_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Share based compensation arrangement by share based payment award maximum amount of subscription per period.

+ References

No definition available.

+ Details

Name:	dgica_ShareBasedCompensationArrangementByShareBasedPaymentAwardMaximumAmountOfSubscriptionPerPeriod
Namespace Prefix:	dgica_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Share based compensation arrangement by share based payment award number of restricted shares to issue to all directors.

+ References

No definition available.

+ Details

Name:	dgica_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfRestrictedSharesToIssueToAllDirectors
Namespace Prefix:	dgica_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Share based compensation arrangement by share based payment award number of shares authorized to issue to employees of subsidiary and affiliates.

+ References

No definition available.

+ Details

Name:	dgica_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorizedToIssueToEmployeesOfSubsidiaryAndAffiliates
Namespace Prefix:	dgica_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Share based compensation arrangement by share based payment award number of shares authorized to issue to our directors and of subsidiaries and affiliates.

+ References

No definition available.

+ Details

Name:	dgica_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorizedToIssueToOurDirectorsAndOfSubsidiariesAndAffiliates
Namespace Prefix:	dgica_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

Share based compensation arrangement by share based payment award shares issued in period price per share.

+ References

No definition available.

+ Details

Name:	dgica_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesIssuedInPeriodPricePerShare
Namespace Prefix:	dgica_
Data Type:	num:perShareItemType
Balance Type:	na
Period Type:	duration

- Definition

Amount of expense for award under share-based payment arrangement. Excludes amount capitalized.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 1
-Subparagraph (b)
-URI http://asc.fasb.org/extlink&oid=116856206&loc=d3e5047-113901

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SAB Topic 14.F)
-URI http://asc.fasb.org/extlink&oid=115993241&loc=d3e301413-122809

Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (h)(1)(i)
-URI http://asc.fasb.org/extlink&oid=116856206&loc=d3e5070-113901

+ Details

Name:	us-gaap_AllocatedShareBasedCompensationExpense
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of cost not yet recognized for nonvested award under share-based payment arrangement.

+ References

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognized
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Weighted-average period over which cost not yet recognized is expected to be recognized for award under share-based payment arrangement, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationNonvestedAwardsTotalCompensationCostNotYetRecognizedPeriodForRecognition1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Amount of tax benefit for recognition of expense of award under share-based payment arrangement.

+ References

+ Details

Name:	us-gaap_EmployeeServiceShareBasedCompensationTaxBenefitFromCompensationExpense
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Amount of cash inflow from exercise of option under share-based payment arrangement.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2A
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=116856206&loc=SL79508275-113901

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 230
-SubTopic 10
-Section 45
-Paragraph 14
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=118932676&loc=d3e3255-108585

+ Details

Name:	us-gaap_ProceedsFromStockOptionsExercised
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Period over which grantee's right to exercise award under share-based payment arrangement is no longer contingent on satisfaction of service or performance condition, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days. Includes, but is not limited to, combination of market, performance or service condition.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardAwardVestingPeriod1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

The estimated dividend rate (a percentage of the share price) to be paid (expected dividends) to holders of the underlying shares over the option's term.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedDividendRate
Namespace Prefix:	us-gaap_
Data Type:	num:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

The estimated measure of the percentage by which a share price is expected to fluctuate during a period. Volatility also may be defined as a probability-weighted measure of the dispersion of returns about the mean. The volatility of a share price is the standard deviation of the continuously compounded rates of return on the share over a specified period. That is the same as the standard deviation of the differences in the natural logarithms of the stock prices plus dividends, if any, over the period.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsExpectedVolatilityRate
Namespace Prefix:	us-gaap_
Data Type:	num:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

The risk-free interest rate assumption that is used in valuing an option on its own shares.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardFairValueAssumptionsRiskFreeInterestRate
Namespace Prefix:	us-gaap_
Data Type:	num:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

No definition available.

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Number of shares authorized for issuance under share-based payment arrangement.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAuthorized
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

The difference between the maximum number of shares (or other type of equity) authorized for issuance under the plan (including the effects of amendments and adjustments), and the sum of: 1) the number of shares (or other type of equity) already issued upon exercise of options or other equity-based awards under the plan; and 2) shares (or other type of equity) reserved for issuance on granting of outstanding awards, net of cancellations and forfeitures, if applicable.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-URI http://asc.fasb.org/extlink&oid=116856206&loc=d3e5070-113901

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardNumberOfSharesAvailableForGrant
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	instant

- Definition

The weighted average grant-date fair value of options granted during the reporting period as calculated by applying the disclosed option pricing methodology.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardOptionsGrantsInPeriodWeightedAverageGrantDateFairValue
Namespace Prefix:	us-gaap_
Data Type:	num:perShareItemType
Balance Type:	na
Period Type:	duration

- Definition

Number of shares issued under share-based payment arrangement.

+ References

+ Details

Name:	us-gaap_ShareBasedCompensationArrangementByShareBasedPaymentAwardSharesIssuedInPeriod
Namespace Prefix:	us-gaap_
Data Type:	xbrli:sharesItemType
Balance Type:	na
Period Type:	duration

- Definition

Expected term of award under share-based payment arrangement, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents reported fact of one year, five months, and thirteen days.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section 50
-Paragraph 2
-Subparagraph (f)(2)(i)
-URI http://asc.fasb.org/extlink&oid=116856206&loc=d3e5070-113901

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 718
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SAB Topic 14.D.2)
-URI http://asc.fasb.org/extlink&oid=115993241&loc=d3e301413-122809

+ Details

Name:	us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardFairValueAssumptionsExpectedTerm1
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Purchase price of common stock expressed as a percentage of its fair value.

+ References

No definition available.

+ Details

Name:	us-gaap_SharebasedCompensationArrangementBySharebasedPaymentAwardPurchasePriceOfCommonStockPercent
Namespace Prefix:	us-gaap_
Data Type:	num:percentItemType
Balance Type:	na
Period Type:	duration

- Details

Name:	us-gaap_PlanNameAxis=dgica_EquityIncentivePlanMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_StatementClassOfStockAxis=us-gaap_CommonClassAMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_AwardTypeAxis=us-gaap_RestrictedStockMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_PlanNameAxis=dgica_EmployeeStockPurchasePlanMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_SubsequentEventTypeAxis=us-gaap_SubsequentEventMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_PlanNameAxis=dgica_AgencyStockPurchasePlanMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_PlanNameAxis=dgica_TwoThousandNineteenAmendmentPlanMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

Stock Compensation Plans - Additional Information (Detail) - USD ($)					12 Months Ended
	Jan. 01, 2020	Jan. 02, 2019	Jan. 02, 2018	Jan. 03, 2017	Dec. 31, 2019	Dec. 31, 2018	Dec. 31, 2017	Dec. 31, 2015	Dec. 31, 2011
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Compensation expense in stock compensation plans					$ 1,400,000	$ 1,700,000	$ 2,000,000
Income tax benefit of stock compensation plans					288,901	354,412	692,164
Unrecognized compensation expense related to nonvested share-based compensation granted under the plan					$ 2,000,000	2,500,000
Weighted average period of unrecognized compensation expense					1 year 10 months 24 days
Cash from option exercises					$ 2,900,000	1,100,000	13,400,000
Tax benefit for tax deductions from option exercises					$ 64,765	$ 25,938	$ 873,515
Equity Incentive Plan [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Exercisable period of stock options					5 years
Restricted shares issued to directors					500
Weighted-average grant date fair value of options granted					$ 1.15	$ 1.66	$ 1.81
Fair value of risk-free interest rate					1.64%	2.68%	2.01%
Expected life					3 years	3 years	3 years
Expected volatility					17.00%	22.00%	19.00%
Expected dividend yield					4.00%	4.00%	3.00%
Shares available for future option grants					4.0
Equity Incentive Plan [Member] \| Class A Common Stock [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Common stock available for issuance to employees of subsidiaries and affiliates under identical plan					4,500,000
Common stock available for issuance to directors and those of the subsidiaries and affiliates under identical plan					500,000
Equity Incentive Plan [Member] \| Restricted Stock [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Additional issued shares		8,500	8,500	9,000
Employee Stock Purchase Plan [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Exercisable period of stock options					10 years
Common stock available for issuance to employees/directors and those of the subsidiaries and affiliates									300,000
Percentage of purchase price of shares					85.00%	85.00%
Employee Stock Purchase Plan [Member] \| Subsequent Event [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Additional issued shares	20,424
Shares issued during period price per share	$ 12.28
Agency Stock Purchase Plan [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Additional issued shares					110,836	117,935	104,418
Common stock available for issuance to employees/directors and those of the subsidiaries and affiliates								350,000
Percentage of purchase price of shares					90.00%
Investment					$ 12,000
Number of trading days of subscription period					10 days
2019 Amendment Plan [Member]
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]
Common stock available for issuance to employees/directors and those of the subsidiaries and affiliates					500,000