Finite Group Databasebeta

← C₂²⋊ΓL(1,𝔽₈)×C₁₃₁ #24927009

Subgroups — C₂²⋊ΓL(1,𝔽₈)×C₁₃₁ #24927009

Order 11004.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×C₁₃₁	—	4	1	no	maximal, Hall
3	C₂×C₇×C₁₃₁	—	3	1	no	—
4	A₄×C₁₃₁	—	7	1	no	maximal, Hall
5	C₇×C₁₃₁	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₁₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
7	C₃×C₁₃₁	—	28	1	no	Hall
8	C₂×C₁₃₁	—	3	1	no	—
9	C₁₃₁	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
10	C₂²⋊ΓL(1,𝔽₈)	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₇	C₃×C₁₃₁	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)	—	4	1	no	Hall
13	C₂×C₇	—	3	1	no	—
14	A₄	—	7	1	no	Hall
15	C₇	A₄×C₁₃₁	1	1	yes	characteristic, complemented, Sylow
16	C₂²	ΓL(1,𝔽₈)×C₁₃₁	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	28	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃₁	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₁₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₃₁	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₃₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₃₁	C₃	semidirect
5. C₇×C₁₃₁	A₄	semidirect
6. C₂²×C₁₃₁	ΓL(1,𝔽₈)	semidirect
9. C₁₃₁	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
10. C₂²⋊ΓL(1,𝔽₈)	C₁₃₁	direct (N × H)
11. C₂²×C₇	C₃×C₁₃₁	semidirect
15. C₇	A₄×C₁₃₁	semidirect
16. C₂²	ΓL(1,𝔽₈)×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₃₁	3
ΓL(1,𝔽₈)×C₁₃₁	4
A₄×C₁₃₁	7
C₂²⋊ΓL(1,𝔽₈)	131