Subgroups — C₃×C₇²×C₃⋊(C₅⋊D₅) #36348074

Order 22050.

720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₇²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×C₃⋊(C₅⋊D₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₇²×C₅⋊D₅	—	3	1	no	maximal
4	C₃×C₇²×D₁₅	—	5	6	no	maximal
5	C₃×C₅²×C₇²	—	2	1	no	—
6	C₃×C₅²×C₇²	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×C₇×C₃⋊(C₅⋊D₅)	C₇	1	8	yes	maximal, direct product factor
9	C₇²×C₅⋊D₅	—	3	1	no	Hall
10	C₃²×C₅×C₇²	D₅	1	6	yes	complemented
11	C₃²×C₅²×C₇	C₂×C₇	1	8	yes	complemented
12	C₇²×D₁₅	—	5	6	no	—
13	C₃×C₇²×D₅	—	15	6	no	—
14	C₅²×C₇²	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₇×C₃⋊(C₅⋊D₅)	C₃×C₇	1	8	yes	direct product factor
16	C₃×C₇×C₅⋊D₅	—	3	8	no	—
17	C₃×D₃×C₇²	—	25	1	no	Hall
18	C₃×C₅×C₇²	—	2	6	no	—
19	C₃×C₅×C₇²	C₃×D₅	1	6	yes	complemented
20	C₃×C₅×C₇²	D₁₅	1	6	yes	complemented
21	C₃×C₇×D₁₅	—	5	48	no	—
22	C₃×C₅²×C₇	—	2	8	no	—
23	C₃×C₅²×C₇	D₃×C₇	1	8	yes	complemented
24	C₃×C₅²×C₇	C₂×C₃×C₇	1	8	yes	complemented
25	C₇²×D₅	—	15	6	no	—
26	C₃×C₃⋊(C₅⋊D₅)	C₇²	1	1	yes	characteristic, direct product factor, Hall
27	C₃²×C₇²	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
28	C₇×C₅⋊D₅	—	3	8	no	—
29	C₃²×C₅×C₇	C₇×D₅	1	48	yes	complemented
30	C₂×C₃×C₇²	—	75	1	no	—
31	D₃×C₇²	—	25	1	no	—
32	C₅×C₇²	C₃×D₁₅	1	6	yes	complemented
33	C₃²×C₅²	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
34	C₇×D₁₅	—	5	48	no	—
35	C₃×C₇×D₅	—	15	48	no	—
36	C₅²×C₇	C₃×D₃×C₇	1	8	yes	complemented
37	C₃⋊(C₅⋊D₅)	C₃×C₇²	1	1	yes	characteristic, direct product factor
38	C₃×C₅⋊D₅	—	3	1	no	—
39	C₃×C₇²	—	2	1	no	—
40	C₃×C₇²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
41	C₃×C₇²	C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
42	C₃×D₃×C₇	—	25	8	no	—
43	C₃×C₅×C₇	—	2	48	no	—
44	C₃×C₅×C₇	C₃×C₇×D₅	1	48	yes	complemented
45	C₃×C₅×C₇	C₇×D₁₅	1	48	yes	complemented
46	C₂×C₇²	—	75	1	no	Hall
47	C₃×D₁₅	—	5	6	no	—
48	C₃×C₅²	—	2	1	no	—
49	C₃×C₅²	D₃×C₇²	1	1	yes	characteristic, complemented
50	C₃×C₅²	C₂×C₃×C₇²	1	1	yes	characteristic, complemented
51	C₇×D₅	—	15	48	no	—
52	C₃²×C₇	C₇×C₅⋊D₅	1	8	yes	complemented
53	C₅⋊D₅	—	3	1	no	Hall
54	C₇²	C₃×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor, Sylow
55	C₃²×C₅	C₇²×D₅	1	6	yes	complemented
56	C₂×C₃×C₇	—	75	8	no	—
57	D₃×C₇	—	25	8	no	—
58	C₅×C₇	C₃×C₇×D₁₅	1	48	yes	complemented
59	D₁₅	—	5	6	no	—
60	C₃×D₅	—	15	6	no	—
61	C₅²	C₃×D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
62	C₃×C₇	—	2	8	no	—
63	C₃×C₇	C₃×C₇×C₅⋊D₅	1	8	yes	complemented
64	C₃×C₇	C₇×C₃⋊(C₅⋊D₅)	1	8	yes	direct product factor
65	C₃×D₃	—	25	1	no	Hall
66	C₃×C₅	—	2	6	no	—
67	C₃×C₅	C₃×C₇²×D₅	1	6	yes	complemented
68	C₃×C₅	C₇²×D₁₅	1	6	yes	complemented
69	C₂×C₇	—	75	8	no	—
70	D₅	—	15	6	no	—
71	C₃²	C₇²×C₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
72	C₇	C₃×C₇×C₃⋊(C₅⋊D₅)	1	8	yes	direct product factor
73	C₂×C₃	—	75	1	no	—
74	D₃	—	25	1	no	—
75	C₅	C₃×C₇²×D₁₅	1	6	yes	complemented
76	C₃	—	2	1	no	—
77	C₃	C₃×C₇²×C₅⋊D₅	1	1	yes	characteristic, complemented
78	C₃	C₇²×C₃⋊(C₅⋊D₅)	1	1	yes	characteristic, direct product factor
79	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇²	C₃⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₃×C₅²	C₂×C₃×C₇²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₃×C₇²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇²	C₂	semidirect
2. C₇²×C₃⋊(C₅⋊D₅)	C₃	direct (N × H)
6. C₃×C₅²×C₇²	D₃	semidirect
7. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₃×C₇×C₃⋊(C₅⋊D₅)	C₇	direct (N × H)
10. C₃²×C₅×C₇²	D₅	semidirect
11. C₃²×C₅²×C₇	C₂×C₇	semidirect
14. C₅²×C₇²	C₃×D₃	semidirect
15. C₇×C₃⋊(C₅⋊D₅)	C₃×C₇	direct (N × H)
19. C₃×C₅×C₇²	C₃×D₅	semidirect
20. C₃×C₅×C₇²	D₁₅	semidirect
23. C₃×C₅²×C₇	D₃×C₇	semidirect
24. C₃×C₅²×C₇	C₂×C₃×C₇	semidirect
26. C₃×C₃⋊(C₅⋊D₅)	C₇²	direct (N × H)
27. C₃²×C₇²	C₅⋊D₅	semidirect
29. C₃²×C₅×C₇	C₇×D₅	semidirect
32. C₅×C₇²	C₃×D₁₅	semidirect
33. C₃²×C₅²	C₂×C₇²	semidirect
36. C₅²×C₇	C₃×D₃×C₇	semidirect
37. C₃⋊(C₅⋊D₅)	C₃×C₇²	direct (N × H)
40. C₃×C₇²	C₃×C₅⋊D₅	semidirect
41. C₃×C₇²	C₃⋊(C₅⋊D₅)	direct (N × H)
44. C₃×C₅×C₇	C₃×C₇×D₅	semidirect
45. C₃×C₅×C₇	C₇×D₁₅	semidirect
49. C₃×C₅²	D₃×C₇²	semidirect
50. C₃×C₅²	C₂×C₃×C₇²	semidirect
52. C₃²×C₇	C₇×C₅⋊D₅	semidirect
54. C₇²	C₃×C₃⋊(C₅⋊D₅)	direct (N × H)
55. C₃²×C₅	C₇²×D₅	semidirect
58. C₅×C₇	C₃×C₇×D₁₅	semidirect
61. C₅²	C₃×D₃×C₇²	semidirect
63. C₃×C₇	C₃×C₇×C₅⋊D₅	semidirect
64. C₃×C₇	C₇×C₃⋊(C₅⋊D₅)	direct (N × H)
67. C₃×C₅	C₃×C₇²×D₅	semidirect
68. C₃×C₅	C₇²×D₁₅	semidirect
71. C₃²	C₇²×C₅⋊D₅	semidirect
72. C₇	C₃×C₇×C₃⋊(C₅⋊D₅)	direct (N × H)
75. C₅	C₃×C₇²×D₁₅	semidirect
77. C₃	C₃×C₇²×C₅⋊D₅	semidirect
78. C₃	C₇²×C₃⋊(C₅⋊D₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅²×C₇²	2
C₇²×C₃⋊(C₅⋊D₅)	3
C₃×C₇²×C₅⋊D₅	3
C₃×C₇²×D₁₅	5
C₃×C₇×C₃⋊(C₅⋊D₅)	7

Subgroups — C3×C72×C3⋊(C5⋊D5) #36348074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×C₃⋊(C₅⋊D₅) #36348074