Finite Group Databasebeta

← C₂₃×#13064105 #44569008

Subgroups — C₂₃×#13064105 #44569008

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	C₂₃×C₆₇⋊C₃	—	7	1	no	maximal, Hall
3	C₂₃×C₆₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
4	#13064105	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	ΓL(1,𝔽₈)×C₂₃	—	67	1	no	maximal, Hall
6	C₇×C₆₇	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₆₇⋊C₃	—	7	1	no	Hall
8	C₇×C₂₃	C₆₇⋊C₃	1	1	yes	characteristic, complemented, Hall
9	C₃×C₂₃	—	469	1	no	Hall
10	C₆₇	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented, Sylow
11	C₂₃	#13064105	1	1	yes	characteristic, direct product factor, Sylow
12	ΓL(1,𝔽₈)	—	67	1	no	Hall
13	C₇	C₂₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	469	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃	#13064105
commutator, cosocle, hypocenter	C₇×C₆₇	C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₇×C₂₃×C₆₇	C₃
LCS	G ▷ C₇×C₆₇
UCS	C₁ ◁ C₂₃

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)	semidirect
4. #13064105	C₂₃	direct (N × H)
6. C₇×C₆₇	C₃×C₂₃	semidirect
8. C₇×C₂₃	C₆₇⋊C₃	semidirect
10. C₆₇	ΓL(1,𝔽₈)×C₂₃	semidirect
11. C₂₃	#13064105	direct (N × H)
13. C₇	C₂₃×C₆₇⋊C₃	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
C₂₃×C₆₇⋊C₃	7
#13064105	23
ΓL(1,𝔽₈)×C₂₃	67