Finite Group Databasebeta

← C₂₂₃⋊C₆₄ #28383299

Subgroups — C₂₂₃⋊C₆₄ #28383299

Order 14272.

14 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
3	C₈×C₂₂₃	C₈	1	1	yes	characteristic
4	C₄×C₂₂₃	C₁₆	1	1	yes	characteristic
5	C₂×C₂₂₃	C₃₂	1	1	yes	characteristic
6	C₂₂₃	C₆₄	1	1	yes	characteristic, complemented, Sylow
7	C₆₄	—	223	1	no	maximal, Sylow
8	C₃₂	D₂₂₃	1	1	yes	characteristic
9	C₁₆	Dic₂₂₃	1	1	yes	characteristic
10	C₈	C₂₂₃⋊C₈	1	1	yes	characteristic
11	C₄	C₂₂₃⋊C₁₆	1	1	yes	characteristic
12	C₂	C₂₂₃⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₃₂	D₂₂₃
commutator, hypocenter	C₂₂₃	C₆₄
Fitting, cosocle	C₃₂×C₂₂₃	C₂
socle, prime radical	C₂×C₂₂₃	C₃₂
LCS	G ▷ C₂₂₃
UCS	C₁ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
6. C₂₂₃	C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₃₂×C₂₂₃	2
C₆₄	223