Subgroups — A₅×C₁₂₁⋊C₅ #49137834

Order 36300.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₅×C₁₂₁	C₅	1	1	yes	maximal, characteristic, complemented
2	A₄×C₁₂₁⋊C₅	—	5	1	no	maximal
3	D₅×C₁₂₁⋊C₅	—	6	1	no	maximal
4	D₃×C₁₂₁⋊C₅	—	10	1	no	maximal
5	C₁₁⋊C₅×A₅	—	11	1	no	maximal
6	C₅×C₁₂₁⋊C₅	—	6	1	no	Hall
7	C₂²×C₁₂₁⋊C₅	—	5	1	no	—
8	C₃×C₁₂₁⋊C₅	—	10	1	no	—
9	A₄×C₁₂₁	—	5	1	no	Hall
10	D₅×C₁₂₁	—	6	1	no	—
11	C₂×C₁₂₁⋊C₅	—	15	1	no	—
12	D₃×C₁₂₁	—	10	1	no	—
13	A₄×C₁₁⋊C₅	—	55	1	no	—
14	C₁₁×A₅	C₁₁⋊C₅	1	1	yes	characteristic
15	C₅×C₁₂₁	—	6	1	no	—
16	C₁₂₁⋊C₅	—	12	2	no	—
17	C₁₂₁⋊C₅	A₅	1	1	yes	maximal normal, characteristic, direct product factor
18	D₅×C₁₁⋊C₅	—	66	1	no	—
19	C₂²×C₁₂₁	—	5	1	no	Hall
20	C₃×C₁₂₁	—	10	1	no	Hall
21	D₃×C₁₁⋊C₅	—	110	1	no	—
22	C₅×A₅	—	121	1	no	Hall
23	C₅×C₁₁⋊C₅	—	66	1	no	—
24	C₂×C₁₂₁	—	15	1	no	—
25	C₂²×C₁₁⋊C₅	—	55	1	no	—
26	C₃×C₁₁⋊C₅	—	110	1	no	—
27	C₁₁×A₄	—	5	1	no	—
28	C₁₂₁	C₅×A₅	1	1	yes	characteristic, complemented, Sylow
29	D₅×C₁₁	—	6	1	no	—
30	C₂×C₁₁⋊C₅	—	165	1	no	—
31	D₃×C₁₁	—	10	1	no	—
32	C₅×A₄	—	605	1	no	—
33	A₅	C₁₂₁⋊C₅	1	1	yes	characteristic, direct product factor
34	C₅×C₁₁	—	6	1	no	—
35	C₁₁⋊C₅	—	11	1	no	—
36	C₁₁⋊C₅	—	132	2	no	—
37	C₅×D₅	—	726	1	no	—
38	C₂²×C₁₁	—	5	1	no	—
39	C₃×C₁₁	—	10	1	no	—
40	C₅×D₃	—	1210	1	no	—
41	C₅²	—	726	1	no	Sylow
42	C₂×C₁₁	—	15	1	no	—
43	C₂²×C₅	—	605	1	no	—
44	C₃×C₅	—	1210	1	no	—
45	A₄	—	5	1	no	Hall
46	C₁₁	C₁₁⋊C₅×A₅	1	1	yes	characteristic
47	C₂×C₅	—	1815	1	no	—
48	D₅	—	6	1	no	—
49	D₃	—	10	1	no	—
50	C₅	—	6	1	no	—
51	C₅	—	121	1	no	—
52	C₅	—	1452	2	no	—
53	C₂²	—	5	1	no	Sylow
54	C₃	—	10	1	no	Sylow
55	C₂	—	15	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	A₅×C₁₂₁	C₅
Frattini	C₁₁	C₁₁⋊C₅×A₅
Fitting, cosocle	C₁₂₁	C₅×A₅
socle	C₁₁×A₅	C₁₁⋊C₅
LCS	G ▷ A₅×C₁₂₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. A₅×C₁₂₁	C₅	semidirect
17. C₁₂₁⋊C₅	A₅	direct (N × H)
28. C₁₂₁	C₅×A₅	semidirect
33. A₅	C₁₂₁⋊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
A₅×C₁₂₁	5
A₄×C₁₂₁⋊C₅	5
D₅×C₁₂₁⋊C₅	6
D₃×C₁₂₁⋊C₅	10
C₁₁⋊C₅×A₅	11

Subgroups — A5×C121⋊C5 #49137834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₅×C₁₂₁⋊C₅ #49137834