Finite Group Databasebeta

← C₅×D₂₃₅₉ #37556174

Subgroups — C₅×D₂₃₅₉ #37556174

Order 23590.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₇×C₃₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₂₃₅₉	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₅×D₃₃₇	—	7	1	no	maximal, Hall
4	C₇×C₃₃₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
5	C₅×C₃₃₇	D₇	1	1	yes	characteristic, complemented, Hall
6	D₃₃₇	—	7	1	no	Hall
7	C₃₃₇	C₅×D₇	1	1	yes	characteristic, complemented, Sylow
8	C₅×D₇	—	337	1	no	maximal, Hall
9	C₅×C₇	D₃₃₇	1	1	yes	characteristic, complemented, Hall
10	D₇	—	337	1	no	Hall
11	C₂×C₅	—	2359	1	no	Hall
12	C₇	C₅×D₃₃₇	1	1	yes	characteristic, complemented, Sylow
13	C₅	D₂₃₅₉	1	1	yes	characteristic, direct product factor, Sylow
14	C₂	—	2359	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	D₂₃₅₉
commutator, cosocle, hypocenter	C₇×C₃₃₇	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₃₇	C₂
LCS	G ▷ C₇×C₃₃₇
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₃₃₇	C₂	semidirect
2. D₂₃₅₉	C₅	direct (N × H)
4. C₇×C₃₃₇	C₂×C₅	semidirect
5. C₅×C₃₃₇	D₇	semidirect
7. C₃₃₇	C₅×D₇	semidirect
9. C₅×C₇	D₃₃₇	semidirect
12. C₇	C₅×D₃₃₇	semidirect
13. C₅	D₂₃₅₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₇×C₃₃₇	2
D₂₃₅₉	5
C₅×D₃₃₇	7
C₅×D₇	337