Finite Group Databasebeta

Subgroups — #28358024

Order 13902.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294330	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₃₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₃₃₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	D₃₃₁⋊C₃	—	7	1	no	maximal, Hall
5	C₃₃₁⋊C₃	—	7	1	no	Hall
6	D₃₃₁	—	7	1	no	Hall
7	C₃₃₁	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
8	AGL(1,𝔽₇)	—	331	1	no	maximal, Hall
9	ΓL(1,𝔽₈)	—	331	1	no	Hall
10	D₇	—	993	1	no	Hall
11	C₇	D₃₃₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	2317	1	no	Hall
13	C₃	—	2317	1	no	Sylow
14	C₂	—	6951	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₃₁	C₂×C₃
LCS	G ▷ C₇×C₃₃₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #20294330	C₂	semidirect
2. D₂₃₁₇	C₃	semidirect
3. C₇×C₃₃₁	C₂×C₃	semidirect
7. C₃₃₁	AGL(1,𝔽₇)	semidirect
11. C₇	D₃₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#20294330	2
D₂₃₁₇	3
D₃₃₁⋊C₃	7
AGL(1,𝔽₇)	331