Finite Group Databasebeta

← C₅×A₄×C₂₅₁ #28418921

Subgroups — C₅×A₄×C₂₅₁ #28418921

Order 15060.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂₅₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×C₂₅₁	—	4	1	no	maximal, Hall
3	A₄×C₂₅₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₅×C₂₅₁	—	3	1	no	—
5	C₅×C₂₅₁	A₄	1	1	yes	characteristic, direct product factor, Hall
6	C₂²×C₂₅₁	C₃×C₅	1	1	yes	characteristic, complemented, Hall
7	C₃×C₂₅₁	—	4	1	no	Hall
8	C₂×C₂₅₁	—	3	1	no	—
9	C₂₅₁	C₅×A₄	1	1	yes	characteristic, direct product factor, Sylow
10	C₅×A₄	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂²×C₅	C₃×C₂₅₁	1	1	yes	characteristic, complemented, Hall
12	C₃×C₅	—	4	1	no	Hall
13	A₄	C₅×C₂₅₁	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₅	—	3	1	no	—
15	C₅	A₄×C₂₅₁	1	1	yes	characteristic, direct product factor, Sylow
16	C₂²	C₃×C₅×C₂₅₁	1	1	yes	characteristic, complemented, Sylow
17	C₃	—	4	1	no	Sylow
18	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅×C₂₅₁	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₅×C₂₅₁
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₂₅₁	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₅×C₂₅₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₂₅₁	C₃	semidirect
3. A₄×C₂₅₁	C₅	direct (N × H)
5. C₅×C₂₅₁	A₄	direct (N × H)
6. C₂²×C₂₅₁	C₃×C₅	semidirect
9. C₂₅₁	C₅×A₄	direct (N × H)
10. C₅×A₄	C₂₅₁	direct (N × H)
11. C₂²×C₅	C₃×C₂₅₁	semidirect
13. A₄	C₅×C₂₅₁	direct (N × H)
15. C₅	A₄×C₂₅₁	direct (N × H)
16. C₂²	C₃×C₅×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₂₅₁	3
C₃×C₅×C₂₅₁	4
A₄×C₂₅₁	5
C₅×A₄	251