Subgroups — C₃×D₅×D₆₁⋊C₅ #32913005

Order 18300.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₆₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅⋊(D₆₁⋊C₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×D₆₁⋊C₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×D₆₁⋊C₅	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₅×C₆₁⋊C₅	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×D₅×D₆₁	C₅	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×D₆₁⋊C₅	—	5	1	no	maximal
8	D₅×C₆₁⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅⋊(D₆₁⋊C₅)	C₂×C₃	1	1	yes	characteristic, complemented
10	C₅×D₆₁⋊C₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×D₃₀₅	C₂×C₅	1	1	yes	characteristic, complemented
12	C₃×D₅×C₆₁	C₂×C₅	1	1	yes	characteristic, complemented
13	C₃×C₅×D₆₁	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂×C₃×C₆₁⋊C₅	—	5	1	no	—
15	C₃×D₆₁⋊C₅	—	2	2	no	—
16	C₃×D₆₁⋊C₅	—	5	1	no	—
17	C₃×D₆₁⋊C₅	D₅	1	1	yes	characteristic, direct product factor
18	C₅×C₆₁⋊C₅	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
19	D₅×D₆₁	C₃×C₅	1	1	yes	characteristic, complemented
20	C₂×D₆₁⋊C₅	—	5	1	no	—
21	C₃×C₅×C₆₁	C₂²×C₅	1	1	yes	characteristic, complemented
22	C₃×C₆₁⋊C₅	—	2	2	no	—
23	C₃×C₆₁⋊C₅	C₂×D₅	1	1	yes	characteristic, complemented
24	C₂×C₃×D₆₁	—	5	1	no	Hall
25	D₃₀₅	C₂×C₃×C₅	1	1	yes	characteristic, complemented
26	C₅×D₆₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
27	D₅×C₆₁	C₂×C₃×C₅	1	1	yes	characteristic, complemented
28	C₂×C₆₁⋊C₅	—	5	1	no	—
29	D₆₁⋊C₅	—	2	2	no	—
30	D₆₁⋊C₅	—	5	1	no	—
31	D₆₁⋊C₅	C₃×D₅	1	1	yes	characteristic, direct product factor
32	C₂×C₃×C₆₁	—	5	1	no	—
33	C₃×D₆₁	—	5	1	no	—
34	C₃×D₆₁	C₅×D₅	1	1	yes	characteristic, complemented
35	C₅×C₆₁	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
36	C₆₁⋊C₅	—	2	2	no	—
37	C₆₁⋊C₅	C₂×C₃×D₅	1	1	yes	characteristic, complemented
38	C₂×C₃×C₅×D₅	—	61	1	no	maximal, Hall
39	C₂×D₆₁	—	5	1	no	Hall
40	C₃×C₆₁	C₂×C₅×D₅	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₅²	—	61	1	no	—
42	C₃×C₅×D₅	—	61	1	no	—
43	C₃×C₅×D₅	—	122	1	no	—
44	C₂×C₆₁	—	5	1	no	—
45	D₆₁	—	5	1	no	—
46	D₆₁	C₃×C₅×D₅	1	1	yes	characteristic, complemented
47	C₂×C₅×D₅	—	61	1	no	Hall
48	C₃×C₅²	—	122	1	no	Hall
49	C₆₁	C₂×C₃×C₅×D₅	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₃×C₅	—	305	1	no	—
51	C₂×C₃×D₅	—	61	1	no	—
52	C₂×C₅²	—	61	1	no	—
53	C₅×D₅	—	61	1	no	—
54	C₅×D₅	—	122	1	no	—
55	C₂×C₃×C₅	—	305	1	no	—
56	C₂×C₃×C₅	—	61	1	no	—
57	C₂×C₃×C₅	—	61	1	no	—
58	C₂×C₃×C₅	—	610	1	no	—
59	C₂×C₃×C₅	—	122	2	no	—
60	C₃×D₅	—	61	1	no	—
61	C₃×D₅	D₆₁⋊C₅	1	1	yes	characteristic, direct product factor
62	C₅²	—	122	1	no	Sylow
63	C₂²×C₅	—	305	1	no	—
64	C₂×D₅	—	61	1	no	—
65	C₃×C₅	—	122	1	no	—
66	C₃×C₅	—	244	2	no	—
67	C₃×C₅	C₂×D₆₁⋊C₅	1	1	yes	characteristic, complemented
68	C₂²×C₃	—	305	1	no	Hall
69	C₂×C₅	—	305	1	no	—
70	C₂×C₅	—	61	1	no	—
71	C₂×C₅	—	61	1	no	—
72	C₂×C₅	—	610	1	no	—
73	C₂×C₅	—	122	2	no	—
74	D₅	—	61	1	no	—
75	D₅	C₃×D₆₁⋊C₅	1	1	yes	characteristic, direct product factor
76	C₂×C₃	—	5	1	no	—
77	C₂×C₃	—	305	1	no	—
78	C₂×C₃	—	61	1	no	—
79	C₅	—	122	1	no	—
80	C₅	—	244	2	no	—
81	C₅	C₂×C₃×D₆₁⋊C₅	1	1	yes	characteristic, complemented
82	C₂²	—	305	1	no	Sylow
83	C₃	D₅×D₆₁⋊C₅	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	5	1	no	—
85	C₂	—	305	1	no	—
86	C₂	—	61	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₅×D₆₁⋊C₅
commutator, cosocle, hypocenter	C₅×C₆₁	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₆₁	C₂²×C₅
LCS	G ▷ C₅×C₆₁
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×D₅×C₆₁⋊C₅	C₂	semidirect
2. C₃×C₅⋊(D₆₁⋊C₅)	C₂	semidirect
3. C₃×C₅×D₆₁⋊C₅	C₂	semidirect
4. D₅×D₆₁⋊C₅	C₃	direct (N × H)
5. C₃×C₅×C₆₁⋊C₅	C₂²	semidirect
6. C₃×D₅×D₆₁	C₅	semidirect
8. D₅×C₆₁⋊C₅	C₂×C₃	semidirect
9. C₅⋊(D₆₁⋊C₅)	C₂×C₃	semidirect
10. C₅×D₆₁⋊C₅	C₂×C₃	semidirect
11. C₃×D₃₀₅	C₂×C₅	semidirect
12. C₃×D₅×C₆₁	C₂×C₅	semidirect
13. C₃×C₅×D₆₁	C₂×C₅	semidirect
17. C₃×D₆₁⋊C₅	D₅	direct (N × H)
18. C₅×C₆₁⋊C₅	C₂²×C₃	semidirect
19. D₅×D₆₁	C₃×C₅	semidirect
21. C₃×C₅×C₆₁	C₂²×C₅	semidirect
23. C₃×C₆₁⋊C₅	C₂×D₅	semidirect
25. D₃₀₅	C₂×C₃×C₅	semidirect
26. C₅×D₆₁	C₂×C₃×C₅	semidirect
27. D₅×C₆₁	C₂×C₃×C₅	semidirect
31. D₆₁⋊C₅	C₃×D₅	direct (N × H)
34. C₃×D₆₁	C₅×D₅	semidirect
35. C₅×C₆₁	C₂²×C₃×C₅	semidirect
37. C₆₁⋊C₅	C₂×C₃×D₅	semidirect
40. C₃×C₆₁	C₂×C₅×D₅	semidirect
46. D₆₁	C₃×C₅×D₅	semidirect
49. C₆₁	C₂×C₃×C₅×D₅	semidirect
61. C₃×D₅	D₆₁⋊C₅	direct (N × H)
67. C₃×C₅	C₂×D₆₁⋊C₅	semidirect
75. D₅	C₃×D₆₁⋊C₅	direct (N × H)
81. C₅	C₂×C₃×D₆₁⋊C₅	semidirect
83. C₃	D₅×D₆₁⋊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
C₃×D₅×C₆₁⋊C₅	2
C₃×C₅⋊(D₆₁⋊C₅)	2
C₃×C₅×D₆₁⋊C₅	2
D₅×D₆₁⋊C₅	3
C₃×D₅×D₆₁	5
C₂×C₃×D₆₁⋊C₅	5
C₂×C₃×C₅×D₅	61

Subgroups — C3×D5×D61⋊C5 #32913005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅×D₆₁⋊C₅ #32913005