Finite Group Databasebeta

← C₂×C₅×C₆₃₁⋊C₅ #44538792

Subgroups — C₂×C₅×C₆₃₁⋊C₅ #44538792

Order 31550.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₅×C₆₃₁⋊C₅	C₂	direct (N × H)
2. C₂×C₅×C₆₃₁	C₅	semidirect
3. C₂×C₆₃₁⋊C₅	C₅	direct (N × H)
4. C₅×C₆₃₁	C₂×C₅	semidirect
5. C₆₃₁⋊C₅	C₂×C₅	direct (N × H)
6. C₂×C₆₃₁	C₅²	semidirect
7. C₆₃₁	C₂×C₅²	semidirect
11. C₂×C₅	C₆₃₁⋊C₅	direct (N × H)
13. C₅	C₂×C₆₃₁⋊C₅	direct (N × H)
14. C₂	C₅×C₆₃₁⋊C₅	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×C₆₃₁⋊C₅	2
C₂×C₅×C₆₃₁	5
C₂×C₆₃₁⋊C₅	5
C₂×C₅²	631