Finite Group Databasebeta

← C₁₁×ΓL(1,𝔽₈)×C₉₇ #36375709

Subgroups — C₁₁×ΓL(1,𝔽₈)×C₉₇ #36375709

Order 22407.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₁×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₁₁×C₉₇	—	7	1	no	maximal, Hall
3	ΓL(1,𝔽₈)×C₉₇	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₁₁×C₉₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
5	C₇×C₉₇	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
6	C₃×C₉₇	—	7	1	no	Hall
7	C₁₁×ΓL(1,𝔽₈)	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₉₇	C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
9	C₇×C₁₁	C₃×C₉₇	1	1	yes	characteristic, complemented, Hall
10	C₃×C₁₁	—	7	1	no	Hall
11	ΓL(1,𝔽₈)	C₁₁×C₉₇	1	1	yes	characteristic, direct product factor, Hall
12	C₁₁	ΓL(1,𝔽₈)×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
13	C₇	C₃×C₁₁×C₉₇	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁×C₉₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₁₁×C₉₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₁×C₉₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₁×C₉₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₁×C₉₇	C₃	semidirect
3. ΓL(1,𝔽₈)×C₉₇	C₁₁	direct (N × H)
4. C₁₁×C₉₇	ΓL(1,𝔽₈)	direct (N × H)
5. C₇×C₉₇	C₃×C₁₁	semidirect
7. C₁₁×ΓL(1,𝔽₈)	C₉₇	direct (N × H)
8. C₉₇	C₁₁×ΓL(1,𝔽₈)	direct (N × H)
9. C₇×C₁₁	C₃×C₉₇	semidirect
11. ΓL(1,𝔽₈)	C₁₁×C₉₇	direct (N × H)
12. C₁₁	ΓL(1,𝔽₈)×C₉₇	direct (N × H)
13. C₇	C₃×C₁₁×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₁×C₉₇	3
C₃×C₁₁×C₉₇	7
ΓL(1,𝔽₈)×C₉₇	11
C₁₁×ΓL(1,𝔽₈)	97