Subgroups — C₃×C₇×D₁₅×C₃₇ #37552138

Order 23310.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₇×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₁₅×C₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₇×D₅×C₃₇	—	3	1	no	maximal
4	C₃×D₃×C₇×C₃₇	—	5	1	no	maximal, Hall
5	C₃×C₅×C₇×C₃₇	—	2	1	no	—
6	C₃×C₅×C₇×C₃₇	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×D₁₅×C₃₇	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×D₅×C₃₇	—	3	1	no	Hall
10	C₃²×C₇×C₃₇	D₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₃₇	C₂×C₇	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₇×C₃₇	—	15	1	no	—
13	D₃×C₇×C₃₇	—	5	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	D₁₅×C₃₇	C₃×C₇	1	1	yes	characteristic, direct product factor
16	C₃×D₅×C₃₇	—	3	1	no	—
17	C₃×C₇×C₃₇	—	2	1	no	—
18	C₃×C₇×C₃₇	C₃×D₅	1	1	yes	characteristic, complemented
19	C₃×C₇×C₃₇	D₁₅	1	1	yes	characteristic, direct product factor
20	C₃×D₃×C₃₇	—	5	1	no	Hall
21	C₃×C₇×D₁₅	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₅×C₃₇	—	2	1	no	—
23	C₃×C₅×C₃₇	D₃×C₇	1	1	yes	characteristic, complemented
24	C₃×C₅×C₃₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
25	C₂×C₇×C₃₇	—	15	1	no	Hall
26	D₅×C₃₇	—	3	1	no	Hall
27	C₃²×C₃₇	C₇×D₅	1	1	yes	characteristic, complemented, Hall
28	C₃²×C₅×C₇	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
29	C₇×C₃₇	C₃×D₁₅	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₃×C₃₇	—	15	1	no	—
31	D₃×C₃₇	—	5	1	no	—
32	C₇×D₁₅	C₃×C₃₇	1	1	yes	characteristic, direct product factor
33	C₃×C₇×D₅	—	3	1	no	—
34	C₅×C₃₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented, Hall
35	C₃×D₃×C₇	—	5	1	no	Hall
36	C₃×C₃₇	—	2	1	no	—
37	C₃×C₃₇	C₃×C₇×D₅	1	1	yes	characteristic, complemented
38	C₃×C₃₇	C₇×D₁₅	1	1	yes	characteristic, direct product factor
39	C₃×C₅×C₇	—	2	1	no	—
40	C₃×C₅×C₇	D₃×C₃₇	1	1	yes	characteristic, complemented
41	C₃×C₅×C₇	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
42	C₃×D₁₅	C₇×C₃₇	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃₇	—	15	1	no	Hall
44	C₇×D₅	—	3	1	no	Hall
45	C₃²×C₇	D₅×C₃₇	1	1	yes	characteristic, complemented, Hall
46	C₃²×C₅	C₂×C₇×C₃₇	1	1	yes	characteristic, complemented, Hall
47	C₂×C₃×C₇	—	15	1	no	—
48	D₃×C₇	—	5	1	no	—
49	C₃₇	C₃×C₇×D₁₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₅×C₇	C₃×D₃×C₃₇	1	1	yes	characteristic, complemented, Hall
51	D₁₅	C₃×C₇×C₃₇	1	1	yes	characteristic, direct product factor
52	C₃×D₅	—	3	1	no	—
53	C₃×C₇	—	2	1	no	—
54	C₃×C₇	C₃×D₅×C₃₇	1	1	yes	characteristic, complemented
55	C₃×C₇	D₁₅×C₃₇	1	1	yes	characteristic, direct product factor
56	C₃×D₃	—	5	1	no	Hall
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	D₃×C₇×C₃₇	1	1	yes	characteristic, complemented
59	C₃×C₅	C₂×C₃×C₇×C₃₇	1	1	yes	characteristic, complemented
60	C₂×C₇	—	15	1	no	Hall
61	D₅	—	3	1	no	Hall
62	C₃²	C₇×D₅×C₃₇	1	1	yes	characteristic, complemented, Sylow
63	C₇	C₃×D₁₅×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	—	15	1	no	—
65	D₃	—	5	1	no	—
66	C₅	C₃×D₃×C₇×C₃₇	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	2	1	no	—
68	C₃	C₃×C₇×D₅×C₃₇	1	1	yes	characteristic, complemented
69	C₃	C₇×D₁₅×C₃₇	1	1	yes	characteristic, direct product factor
70	C₂	—	15	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₇×C₃₇	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₃×C₇×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₃₇	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₃×C₇×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×C₇×C₃₇	C₂	semidirect
2. C₇×D₁₅×C₃₇	C₃	direct (N × H)
6. C₃×C₅×C₇×C₃₇	D₃	semidirect
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
8. C₃×D₁₅×C₃₇	C₇	direct (N × H)
10. C₃²×C₇×C₃₇	D₅	semidirect
11. C₃²×C₅×C₃₇	C₂×C₇	semidirect
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
15. D₁₅×C₃₇	C₃×C₇	direct (N × H)
18. C₃×C₇×C₃₇	C₃×D₅	semidirect
19. C₃×C₇×C₃₇	D₁₅	direct (N × H)
21. C₃×C₇×D₁₅	C₃₇	direct (N × H)
23. C₃×C₅×C₃₇	D₃×C₇	semidirect
24. C₃×C₅×C₃₇	C₂×C₃×C₇	semidirect
27. C₃²×C₃₇	C₇×D₅	semidirect
28. C₃²×C₅×C₇	C₂×C₃₇	semidirect
29. C₇×C₃₇	C₃×D₁₅	direct (N × H)
32. C₇×D₁₅	C₃×C₃₇	direct (N × H)
34. C₅×C₃₇	C₃×D₃×C₇	semidirect
37. C₃×C₃₇	C₃×C₇×D₅	semidirect
38. C₃×C₃₇	C₇×D₁₅	direct (N × H)
40. C₃×C₅×C₇	D₃×C₃₇	semidirect
41. C₃×C₅×C₇	C₂×C₃×C₃₇	semidirect
42. C₃×D₁₅	C₇×C₃₇	direct (N × H)
45. C₃²×C₇	D₅×C₃₇	semidirect
46. C₃²×C₅	C₂×C₇×C₃₇	semidirect
49. C₃₇	C₃×C₇×D₁₅	direct (N × H)
50. C₅×C₇	C₃×D₃×C₃₇	semidirect
51. D₁₅	C₃×C₇×C₃₇	direct (N × H)
54. C₃×C₇	C₃×D₅×C₃₇	semidirect
55. C₃×C₇	D₁₅×C₃₇	direct (N × H)
58. C₃×C₅	D₃×C₇×C₃₇	semidirect
59. C₃×C₅	C₂×C₃×C₇×C₃₇	semidirect
62. C₃²	C₇×D₅×C₃₇	semidirect
63. C₇	C₃×D₁₅×C₃₇	direct (N × H)
66. C₅	C₃×D₃×C₇×C₃₇	semidirect
68. C₃	C₃×C₇×D₅×C₃₇	semidirect
69. C₃	C₇×D₁₅×C₃₇	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×C₅×C₇×C₃₇	2
C₇×D₁₅×C₃₇	3
C₃×C₇×D₅×C₃₇	3
C₃×D₃×C₇×C₃₇	5
C₃×D₁₅×C₃₇	7
C₃×C₇×D₁₅	37

Subgroups — C3×C7×D15×C37 #37552138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×D₁₅×C₃₇ #37552138