Finite Group Databasebeta

Subgroups — #41002013

Order 26901.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₆₁	C₃	1	1	yes	maximal, characteristic
2	#16838635	—	7	8	no	maximal
3	C₇²×C₆₁	C₉	1	1	yes	characteristic, complemented, Hall
4	C₃×C₇×C₆₁	ΓL(1,𝔽₈)	1	8	yes	—
5	C₆₁⋊C₉	—	49	1	no	Hall
6	#89204	—	61	1	no	maximal, Hall
7	C₇×C₆₁	C₇⋊C₉	1	8	yes	complemented
8	C₃×C₆₁	#3861	1	1	yes	characteristic
9	C₃×C₇²	C₆₁⋊C₃	1	1	yes	characteristic
10	C₇⋊C₉	—	427	8	no	—
11	C₆₁	#89204	1	1	yes	characteristic, complemented, Sylow
12	C₇²	C₆₁⋊C₉	1	1	yes	characteristic, complemented, Sylow
13	C₃×C₇	#13046196	1	8	yes	—
14	C₉	—	2989	1	no	Sylow
15	C₇	#16838635	1	8	yes	complemented
16	C₃	#22582226	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃	#22582226
commutator, hypocenter	C₇²×C₆₁	C₉
Fitting, socle, prime radical, cosocle	C₃×C₇²×C₆₁	C₃
LCS	G ▷ C₇²×C₆₁
UCS	C₁ ◁ C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₇²×C₆₁	C₉	semidirect
7. C₇×C₆₁	C₇⋊C₉	semidirect
11. C₆₁	#89204	semidirect
12. C₇²	C₆₁⋊C₉	semidirect
15. C₇	#16838635	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₇²×C₆₁	3
#16838635	7
#89204	61