Finite Group Databasebeta

← C₂₃×#11744748 #35209342

Subgroups — C₂₃×#11744748 #35209342

Order 20769.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂₃×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂₃×C₄₃⋊C₃	—	7	1	no	maximal, Hall
3	C₂₃×C₄₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
4	#11744748	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	ΓL(1,𝔽₈)×C₂₃	—	43	1	no	maximal, Hall
6	C₇×C₄₃	C₃×C₂₃	1	1	yes	characteristic, complemented, Hall
7	C₇×C₂₃	C₄₃⋊C₃	1	1	yes	characteristic, complemented, Hall
8	C₄₃⋊C₃	—	7	1	no	Hall
9	C₃×C₂₃	—	301	1	no	Hall
10	C₄₃	ΓL(1,𝔽₈)×C₂₃	1	1	yes	characteristic, complemented, Sylow
11	C₂₃	#11744748	1	1	yes	characteristic, direct product factor, Sylow
12	ΓL(1,𝔽₈)	—	43	1	no	Hall
13	C₇	C₂₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	301	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₃	#11744748
commutator, cosocle, hypocenter	C₇×C₄₃	C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₇×C₂₃×C₄₃	C₃
LCS	G ▷ C₇×C₄₃
UCS	C₁ ◁ C₂₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₂₃×C₄₃	C₃	semidirect
3. C₂₃×C₄₃	ΓL(1,𝔽₈)	semidirect
4. #11744748	C₂₃	direct (N × H)
6. C₇×C₄₃	C₃×C₂₃	semidirect
7. C₇×C₂₃	C₄₃⋊C₃	semidirect
10. C₄₃	ΓL(1,𝔽₈)×C₂₃	semidirect
11. C₂₃	#11744748	direct (N × H)
13. C₇	C₂₃×C₄₃⋊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×C₄₃	3
C₂₃×C₄₃⋊C₃	7
#11744748	23
ΓL(1,𝔽₈)×C₂₃	43