Finite Group Databasebeta

← C₅×D₃₁₅₅ #44538795

Subgroups — C₅×D₃₁₅₅ #44538795

Order 31550.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₆₃₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₃₁₅₅	C₅	1	1	yes	maximal, characteristic, direct product factor
3	C₅×D₆₃₁	—	5	1	no	maximal
4	C₅×C₆₃₁	—	2	2	no	—
5	C₅×C₆₃₁	D₅	1	1	yes	characteristic, complemented
6	C₅×C₆₃₁	C₂×C₅	1	1	yes	characteristic, complemented
7	D₆₃₁	—	5	1	no	Hall
8	C₆₃₁	C₅×D₅	1	1	yes	characteristic, complemented, Sylow
9	C₅×D₅	—	631	1	no	maximal, Hall
10	C₅²	D₆₃₁	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₅	—	3155	1	no	—
12	D₅	—	631	1	no	—
13	C₅	—	2	2	no	—
14	C₅	C₅×D₆₃₁	1	1	yes	characteristic, complemented
15	C₅	D₃₁₅₅	1	1	yes	characteristic, direct product factor
16	C₂	—	3155	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅	D₃₁₅₅
commutator, cosocle, hypocenter	C₅×C₆₃₁	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅²×C₆₃₁	C₂
LCS	G ▷ C₅×C₆₃₁
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅²×C₆₃₁	C₂	semidirect
2. D₃₁₅₅	C₅	direct (N × H)
5. C₅×C₆₃₁	D₅	semidirect
6. C₅×C₆₃₁	C₂×C₅	semidirect
8. C₆₃₁	C₅×D₅	semidirect
10. C₅²	D₆₃₁	semidirect
14. C₅	C₅×D₆₃₁	semidirect
15. C₅	D₃₁₅₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₆₃₁	2
D₃₁₅₅	5
C₅×D₆₃₁	5
C₅×D₅	631