Finite Group Databasebeta

Subgroups — #36347366

Order 22010.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24927021	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₂₀₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
3	C₃₁×C₇₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
4	D₇₁⋊C₅	—	31	1	no	maximal, Hall
5	C₇₁⋊C₅	—	31	1	no	Hall
6	D₃₁⋊C₅	—	71	1	no	maximal, Hall
7	ΓL(1,𝔽₃₂)	—	71	1	no	Hall
8	D₇₁	—	155	1	no	Hall
9	C₇₁	D₃₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
10	D₃₁	—	355	1	no	Hall
11	C₃₁	D₇₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₅	—	2201	1	no	Hall
13	C₅	—	2201	1	no	Sylow
14	C₂	—	11005	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃₁×C₇₁	C₂×C₅
LCS	G ▷ C₃₁×C₇₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #24927021	C₂	semidirect
2. D₂₂₀₁	C₅	semidirect
3. C₃₁×C₇₁	C₂×C₅	semidirect
9. C₇₁	D₃₁⋊C₅	semidirect
11. C₃₁	D₇₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
#24927021	2
D₂₂₀₁	5
D₇₁⋊C₅	31
D₃₁⋊C₅	71