Subgroups — C₆₁×(C₃⋊D₃)⋊D₅ #24926844

Order 10980.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊D₃×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₁₅×C₆₁	C₂	1	2	yes	maximal, complemented
3	D₃×D₅×C₆₁	—	3	2	no	maximal
4	C₃²×C₅×C₆₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₃²×C₆₁	—	5	1	no	maximal, Hall
6	D₁₅×C₆₁	D₃	1	2	yes	complemented
7	C₅×D₃×C₆₁	—	3	2	no	—
8	C₅×D₃×C₆₁	—	6	1	no	—
9	C₃×D₅×C₆₁	—	3	2	no	—
10	C₂×D₅×C₆₁	—	9	1	no	Hall
11	C₃⋊D₃×C₆₁	D₅	1	1	yes	characteristic, complemented
12	C₃×D₃×C₆₁	—	5	2	no	—
13	C₃×C₅×C₆₁	—	2	1	no	—
14	C₃×C₅×C₆₁	C₂×D₃	1	2	yes	complemented
15	C₂×D₃×C₆₁	—	15	2	no	—
16	C₂×C₅×C₆₁	—	9	1	no	—
17	D₅×C₆₁	—	3	2	no	—
18	C₃²×C₆₁	C₂×D₅	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×C₆₁	—	15	2	no	—
20	D₃×C₆₁	—	3	2	no	—
21	D₃×C₆₁	—	5	2	no	—
22	D₃×C₆₁	—	6	1	no	—
23	C₅×C₆₁	D₃²	1	1	yes	characteristic, complemented, Hall
24	C₂²×C₆₁	—	45	1	no	Hall
25	C₃×C₆₁	—	2	1	no	—
26	C₃×C₆₁	D₃×D₅	1	2	yes	complemented
27	(C₃⋊D₃)⋊D₅	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₆₁	—	9	1	no	—
29	C₂×C₆₁	—	15	2	no	—
30	C₅×C₃⋊D₃	C₂×C₆₁	1	1	yes	characteristic, complemented
31	C₃×D₁₅	C₂×C₆₁	1	2	yes	complemented
32	C₆₁	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic, direct product factor, Sylow
33	D₃×D₅	—	3	2	no	—
34	C₃²×C₅	C₂²×C₆₁	1	1	yes	characteristic, complemented, Hall
35	D₃²	—	5	1	no	Hall
36	D₁₅	D₃×C₆₁	1	2	yes	complemented
37	C₃×D₅	—	3	2	no	—
38	C₅×D₃	—	3	2	no	—
39	C₅×D₃	—	6	1	no	—
40	C₂×D₅	—	9	1	no	Hall
41	C₃⋊D₃	D₅×C₆₁	1	1	yes	characteristic, complemented
42	C₃×D₃	—	5	2	no	—
43	C₃×C₅	—	2	1	no	—
44	C₃×C₅	C₂×D₃×C₆₁	1	2	yes	complemented
45	C₂×D₃	—	15	2	no	—
46	C₂×C₅	—	9	1	no	—
47	D₅	—	3	2	no	—
48	C₃²	C₂×D₅×C₆₁	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	15	2	no	—
50	D₃	—	3	2	no	—
51	D₃	—	5	2	no	—
52	D₃	—	6	1	no	—
53	C₅	D₃²×C₆₁	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	45	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	D₃×D₅×C₆₁	1	2	yes	complemented
57	C₂	—	9	1	no	—
58	C₂	—	15	2	no	—

	subgroup	G/H
center, hypercenter	C₆₁	(C₃⋊D₃)⋊D₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂²×C₆₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₆₁	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₆₁

Complements

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

N	H ≅ G/N	type
1. C₅×C₃⋊D₃×C₆₁	C₂	semidirect
2. C₃×D₁₅×C₆₁	C₂	semidirect
4. C₃²×C₅×C₆₁	C₂²	semidirect
6. D₁₅×C₆₁	D₃	semidirect
11. C₃⋊D₃×C₆₁	D₅	semidirect
14. C₃×C₅×C₆₁	C₂×D₃	semidirect
18. C₃²×C₆₁	C₂×D₅	semidirect
23. C₅×C₆₁	D₃²	semidirect
26. C₃×C₆₁	D₃×D₅	semidirect
27. (C₃⋊D₃)⋊D₅	C₆₁	direct (N × H)
30. C₅×C₃⋊D₃	C₂×C₆₁	semidirect
31. C₃×D₁₅	C₂×C₆₁	semidirect
32. C₆₁	(C₃⋊D₃)⋊D₅	direct (N × H)
34. C₃²×C₅	C₂²×C₆₁	semidirect
36. D₁₅	D₃×C₆₁	semidirect
41. C₃⋊D₃	D₅×C₆₁	semidirect
44. C₃×C₅	C₂×D₃×C₆₁	semidirect
48. C₃²	C₂×D₅×C₆₁	semidirect
53. C₅	D₃²×C₆₁	semidirect
56. C₃	D₃×D₅×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊D₃×C₆₁	2
C₃×D₁₅×C₆₁	2
D₃×D₅×C₆₁	3
D₃²×C₆₁	5
(C₃⋊D₃)⋊D₅	61

Subgroups — C61×(C3⋊D3)⋊D5 #24926844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×(C₃⋊D₃)⋊D₅ #24926844