Finite Group Databasebeta

← ΓL(1,𝔽₈)×C₂₃×C₆₇ #44569007

Subgroups — ΓL(1,𝔽₈)×C₂₃×C₆₇ #44569007

Order 32361.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₂₃×C₆₇	—	7	1	no	maximal, Hall
3	C₂₃×C₆₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
4	ΓL(1,𝔽₈)×C₆₇	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	ΓL(1,𝔽₈)×C₂₃	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₇×C₆₇	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₃×C₆₇	—	7	1	no	Hall
8	C₇×C₂₃	C₃×C₆₇	1	1	yes	characteristic, complemented, Hall
9	C₃×C₂₃	—	7	1	no	Hall
10	C₆₇	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
11	C₂₃	ΓL(1,𝔽₈)×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
12	ΓL(1,𝔽₈)	C₂₃×C₆₇	1	1	yes	characteristic, direct product factor, Hall
13	C₇	C₃×C₂₃×C₆₇	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃×C₆₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₂₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₇×C₂₃×C₆₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂₃×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₃×C₆₇	C₃	semidirect
3. C₂₃×C₆₇	ΓL(1,𝔽₈)	direct (N × H)
4. ΓL(1,𝔽₈)×C₆₇	C₂₃	direct (N × H)
5. ΓL(1,𝔽₈)×C₂₃	C₆₇	direct (N × H)
6. C₇×C₆₇	C₃×C₂₃	semidirect
8. C₇×C₂₃	C₃×C₆₇	semidirect
10. C₆₇	ΓL(1,𝔽₈)×C₂₃	direct (N × H)
11. C₂₃	ΓL(1,𝔽₈)×C₆₇	direct (N × H)
12. ΓL(1,𝔽₈)	C₂₃×C₆₇	direct (N × H)
13. C₇	C₃×C₂₃×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₂₃×C₆₇	3
C₃×C₂₃×C₆₇	7
ΓL(1,𝔽₈)×C₆₇	23
ΓL(1,𝔽₈)×C₂₃	67