Finite Group Databasebeta

Subgroups — #40997207

Order 26510.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27210424	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₆₅₁	C₅	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₁×C₂₄₁	C₂×C₅	1	1	yes	characteristic, complemented, Hall
4	D₂₄₁⋊C₅	—	11	1	no	maximal, Hall
5	C₂₄₁⋊C₅	—	11	1	no	Hall
6	D₂₄₁	—	55	1	no	Hall
7	C₂₄₁	AGL(1,𝔽₁₁)	1	1	yes	characteristic, complemented, Sylow
8	AGL(1,𝔽₁₁)	—	241	1	no	maximal, Hall
9	C₁₁⋊C₅	—	241	1	no	Hall
10	D₁₁	—	1205	1	no	Hall
11	C₁₁	D₂₄₁⋊C₅	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₅	—	2651	1	no	Hall
13	C₅	—	2651	1	no	Sylow
14	C₂	—	13255	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₂₄₁	C₂×C₅
LCS	G ▷ C₁₁×C₂₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #27210424	C₂	semidirect
2. D₂₆₅₁	C₅	semidirect
3. C₁₁×C₂₄₁	C₂×C₅	semidirect
7. C₂₄₁	AGL(1,𝔽₁₁)	semidirect
11. C₁₁	D₂₄₁⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
#27210424	2
D₂₆₅₁	5
D₂₄₁⋊C₅	11
AGL(1,𝔽₁₁)	241