Finite Group Databasebeta

← C₅×D₅×C₆₄₁ #44545335

Subgroups — C₅×D₅×C₆₄₁ #44545335

Order 32050.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₆₄₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₅×C₆₄₁	—	5	1	no	maximal
3	D₅×C₆₄₁	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×C₆₄₁	—	2	2	no	—
5	C₅×C₆₄₁	D₅	1	1	yes	characteristic, direct product factor
6	C₅×C₆₄₁	C₂×C₅	1	1	yes	characteristic, complemented
7	C₂×C₆₄₁	—	5	1	no	Hall
8	C₆₄₁	C₅×D₅	1	1	yes	characteristic, direct product factor, Sylow
9	C₅×D₅	C₆₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₅²	C₂×C₆₄₁	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₅	—	5	1	no	—
12	D₅	C₅×C₆₄₁	1	1	yes	characteristic, direct product factor
13	C₅	—	2	2	no	—
14	C₅	D₅×C₆₄₁	1	1	yes	characteristic, direct product factor
15	C₅	C₂×C₅×C₆₄₁	1	1	yes	characteristic, complemented
16	C₂	—	5	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₆₄₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₅×C₆₄₁
Frattini	C₁	G
Fitting, socle	C₅²×C₆₄₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₅×C₆₄₁

Complements

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

N	H ≅ G/N	type
1. C₅²×C₆₄₁	C₂	semidirect
3. D₅×C₆₄₁	C₅	direct (N × H)
5. C₅×C₆₄₁	D₅	direct (N × H)
6. C₅×C₆₄₁	C₂×C₅	semidirect
8. C₆₄₁	C₅×D₅	direct (N × H)
9. C₅×D₅	C₆₄₁	direct (N × H)
10. C₅²	C₂×C₆₄₁	semidirect
12. D₅	C₅×C₆₄₁	direct (N × H)
14. C₅	D₅×C₆₄₁	direct (N × H)
15. C₅	C₂×C₅×C₆₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₆₄₁	2
C₂×C₅×C₆₄₁	5
D₅×C₆₄₁	5
C₅×D₅	641