Finite Group Databasebeta

← C₇×D₅×C₃₃₇ #37556175

Subgroups — C₇×D₅×C₃₃₇ #37556175

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	C₂×C₇×C₃₃₇	—	5	1	no	maximal, Hall
3	D₅×C₃₃₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₇×C₃₃₇	D₅	1	1	yes	characteristic, direct product factor, Hall
5	C₅×C₃₃₇	C₂×C₇	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃₃₇	—	5	1	no	Hall
7	C₃₃₇	C₇×D₅	1	1	yes	characteristic, direct product factor, Sylow
8	C₇×D₅	C₃₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇	C₂×C₃₃₇	1	1	yes	characteristic, complemented, Hall
10	C₂×C₇	—	5	1	no	Hall
11	D₅	C₇×C₃₃₇	1	1	yes	characteristic, direct product factor, Hall
12	C₇	D₅×C₃₃₇	1	1	yes	characteristic, direct product factor, Sylow
13	C₅	C₂×C₇×C₃₃₇	1	1	yes	characteristic, complemented, Sylow
14	C₂	—	5	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₃₃₇	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₇×C₃₃₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₃₃₇	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₇×C₃₃₇

Complements

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

N	H ≅ G/N	type
1. C₅×C₇×C₃₃₇	C₂	semidirect
3. D₅×C₃₃₇	C₇	direct (N × H)
4. C₇×C₃₃₇	D₅	direct (N × H)
5. C₅×C₃₃₇	C₂×C₇	semidirect
7. C₃₃₇	C₇×D₅	direct (N × H)
8. C₇×D₅	C₃₃₇	direct (N × H)
9. C₅×C₇	C₂×C₃₃₇	semidirect
11. D₅	C₇×C₃₃₇	direct (N × H)
12. C₇	D₅×C₃₃₇	direct (N × H)
13. C₅	C₂×C₇×C₃₃₇	semidirect

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
C₂×C₇×C₃₃₇	5
D₅×C₃₃₇	7
C₇×D₅	337