Finite Group Databasebeta

Subgroups — #44541199

Order 31794.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30629659	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₅₂₉₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₇₅₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	D₇₅₇⋊C₃	—	7	1	no	maximal, Hall
5	C₇₅₇⋊C₃	—	7	1	no	Hall
6	D₇₅₇	—	7	1	no	Hall
7	C₇₅₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
8	AGL(1,𝔽₇)	—	757	1	no	maximal, Hall
9	ΓL(1,𝔽₈)	—	757	1	no	Hall
10	D₇	—	757	1	no	Hall
11	C₇	D₇₅₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	5299	1	no	Hall
13	C₃	—	5299	1	no	Sylow
14	C₂	—	5299	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₇₅₇	C₂×C₃
LCS	G ▷ C₇×C₇₅₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #30629659	C₂	semidirect
2. D₅₂₉₉	C₃	semidirect
3. C₇×C₇₅₇	C₂×C₃	semidirect
7. C₇₅₇	AGL(1,𝔽₇)	semidirect
11. C₇	D₇₅₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#30629659	2
D₅₂₉₉	3
D₇₅₇⋊C₃	7
AGL(1,𝔽₇)	757