Finite Group Databasebeta

← C₅×ΓL(1,𝔽₈)×C₂₂₇ #37558600

Subgroups — C₅×ΓL(1,𝔽₈)×C₂₂₇ #37558600

Order 23835.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₂₂₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×C₂₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₂₂₇	—	7	1	no	maximal, Hall
4	C₇×C₂₂₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
5	C₅×C₂₂₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
6	C₃×C₂₂₇	—	7	1	no	Hall
7	C₂₂₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
8	C₅×ΓL(1,𝔽₈)	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇	C₃×C₂₂₇	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)	C₅×C₂₂₇	1	1	yes	characteristic, direct product factor, Hall
11	C₃×C₅	—	7	1	no	Hall
12	C₇	C₃×C₅×C₂₂₇	1	1	yes	characteristic, complemented, Sylow
13	C₅	ΓL(1,𝔽₈)×C₂₂₇	1	1	yes	characteristic, direct product factor, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₂₂₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₅×C₂₂₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₂₂₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₅×C₂₂₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₂₂₇	C₃	semidirect
2. ΓL(1,𝔽₈)×C₂₂₇	C₅	direct (N × H)
4. C₇×C₂₂₇	C₃×C₅	semidirect
5. C₅×C₂₂₇	ΓL(1,𝔽₈)	direct (N × H)
7. C₂₂₇	C₅×ΓL(1,𝔽₈)	direct (N × H)
8. C₅×ΓL(1,𝔽₈)	C₂₂₇	direct (N × H)
9. C₅×C₇	C₃×C₂₂₇	semidirect
10. ΓL(1,𝔽₈)	C₅×C₂₂₇	direct (N × H)
12. C₇	C₃×C₅×C₂₂₇	semidirect
13. C₅	ΓL(1,𝔽₈)×C₂₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₇×C₂₂₇	3
ΓL(1,𝔽₈)×C₂₂₇	5
C₃×C₅×C₂₂₇	7
C₅×ΓL(1,𝔽₈)	227