Subgroups — C₃×C₅×C₃₇×D₂₁ #37552136

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	C₅×C₃₇×D₂₁	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅×D₇×C₃₇	—	3	1	no	maximal
4	C₃×C₃₇×D₂₁	C₅	1	1	yes	maximal, characteristic, direct product factor, 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×C₅×D₃×C₃₇	—	7	1	no	maximal, Hall
9	C₅×D₇×C₃₇	—	3	1	no	Hall
10	C₃²×C₇×C₃₇	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃²×C₅×C₃₇	D₇	1	1	yes	characteristic, complemented, Hall
12	C₃₇×D₂₁	C₃×C₅	1	1	yes	characteristic, direct product factor
13	C₃×D₇×C₃₇	—	3	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₅×C₃₇	—	21	1	no	—
16	C₅×D₃×C₃₇	—	7	1	no	—
17	C₃×C₇×C₃₇	—	2	1	no	—
18	C₃×C₇×C₃₇	C₅×D₃	1	1	yes	characteristic, complemented
19	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
20	C₃×D₃×C₃₇	—	7	1	no	Hall
21	C₃×C₅×D₂₁	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₃×C₅×C₃₇	—	2	1	no	—
23	C₃×C₅×C₃₇	C₃×D₇	1	1	yes	characteristic, complemented
24	C₃×C₅×C₃₇	D₂₁	1	1	yes	characteristic, direct product factor
25	D₇×C₃₇	—	3	1	no	Hall
26	C₂×C₅×C₃₇	—	21	1	no	Hall
27	C₃²×C₃₇	C₅×D₇	1	1	yes	characteristic, complemented, Hall
28	C₃²×C₅×C₇	C₂×C₃₇	1	1	yes	characteristic, complemented, Hall
29	C₇×C₃₇	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₃₇	—	21	1	no	—
31	D₃×C₃₇	—	7	1	no	—
32	C₅×D₂₁	C₃×C₃₇	1	1	yes	characteristic, direct product factor
33	C₃×C₅×D₇	—	3	1	no	—
34	C₅×C₃₇	C₃×D₂₁	1	1	yes	characteristic, direct product factor, Hall
35	C₃×D₂₁	C₅×C₃₇	1	1	yes	characteristic, direct product factor, Hall
36	C₃×C₃₇	—	2	1	no	—
37	C₃×C₃₇	C₃×C₅×D₇	1	1	yes	characteristic, complemented
38	C₃×C₃₇	C₅×D₂₁	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×C₅×D₃	—	7	1	no	Hall
43	C₂×C₃₇	—	21	1	no	Hall
44	C₅×D₇	—	3	1	no	Hall
45	C₃²×C₇	C₂×C₅×C₃₇	1	1	yes	characteristic, complemented, Hall
46	C₃²×C₅	D₇×C₃₇	1	1	yes	characteristic, complemented, Hall
47	D₂₁	C₃×C₅×C₃₇	1	1	yes	characteristic, direct product factor
48	C₃×D₇	—	3	1	no	—
49	C₃₇	C₃×C₅×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
50	C₅×C₇	C₃×D₃×C₃₇	1	1	yes	characteristic, complemented, Hall
51	C₂×C₃×C₅	—	21	1	no	—
52	C₅×D₃	—	7	1	no	—
53	C₃×C₇	—	2	1	no	—
54	C₃×C₇	C₅×D₃×C₃₇	1	1	yes	characteristic, complemented
55	C₃×C₇	C₂×C₃×C₅×C₃₇	1	1	yes	characteristic, complemented
56	C₃×D₃	—	7	1	no	Hall
57	C₃×C₅	—	2	1	no	—
58	C₃×C₅	C₃×D₇×C₃₇	1	1	yes	characteristic, complemented
59	C₃×C₅	C₃₇×D₂₁	1	1	yes	characteristic, direct product factor
60	D₇	—	3	1	no	Hall
61	C₂×C₅	—	21	1	no	Hall
62	C₃²	C₅×D₇×C₃₇	1	1	yes	characteristic, complemented, Sylow
63	C₇	C₃×C₅×D₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	21	1	no	—
65	D₃	—	7	1	no	—
66	C₅	C₃×C₃₇×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	2	1	no	—
68	C₃	C₃×C₅×D₇×C₃₇	1	1	yes	characteristic, complemented
69	C₃	C₅×C₃₇×D₂₁	1	1	yes	characteristic, direct product factor
70	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅×C₃₇	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃×C₅×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₇×C₃₇	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃×C₅×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. C₅×C₃₇×D₂₁	C₃	direct (N × H)
4. C₃×C₃₇×D₂₁	C₅	direct (N × H)
6. C₃×C₅×C₇×C₃₇	D₃	semidirect
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
10. C₃²×C₇×C₃₇	C₂×C₅	semidirect
11. C₃²×C₅×C₃₇	D₇	semidirect
12. C₃₇×D₂₁	C₃×C₅	direct (N × H)
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
18. C₃×C₇×C₃₇	C₅×D₃	semidirect
19. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
21. C₃×C₅×D₂₁	C₃₇	direct (N × H)
23. C₃×C₅×C₃₇	C₃×D₇	semidirect
24. C₃×C₅×C₃₇	D₂₁	direct (N × H)
27. C₃²×C₃₇	C₅×D₇	semidirect
28. C₃²×C₅×C₇	C₂×C₃₇	semidirect
29. C₇×C₃₇	C₃×C₅×D₃	semidirect
32. C₅×D₂₁	C₃×C₃₇	direct (N × H)
34. C₅×C₃₇	C₃×D₂₁	direct (N × H)
35. C₃×D₂₁	C₅×C₃₇	direct (N × H)
37. C₃×C₃₇	C₃×C₅×D₇	semidirect
38. C₃×C₃₇	C₅×D₂₁	direct (N × H)
40. C₃×C₅×C₇	D₃×C₃₇	semidirect
41. C₃×C₅×C₇	C₂×C₃×C₃₇	semidirect
45. C₃²×C₇	C₂×C₅×C₃₇	semidirect
46. C₃²×C₅	D₇×C₃₇	semidirect
47. D₂₁	C₃×C₅×C₃₇	direct (N × H)
49. C₃₇	C₃×C₅×D₂₁	direct (N × H)
50. C₅×C₇	C₃×D₃×C₃₇	semidirect
54. C₃×C₇	C₅×D₃×C₃₇	semidirect
55. C₃×C₇	C₂×C₃×C₅×C₃₇	semidirect
58. C₃×C₅	C₃×D₇×C₃₇	semidirect
59. C₃×C₅	C₃₇×D₂₁	direct (N × H)
62. C₃²	C₅×D₇×C₃₇	semidirect
63. C₇	C₃×C₅×D₃×C₃₇	semidirect
66. C₅	C₃×C₃₇×D₂₁	direct (N × H)
68. C₃	C₃×C₅×D₇×C₃₇	semidirect
69. C₃	C₅×C₃₇×D₂₁	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
C₅×C₃₇×D₂₁	3
C₃×C₅×D₇×C₃₇	3
C₃×C₃₇×D₂₁	5
C₃×C₅×D₃×C₃₇	7
C₃×C₅×D₂₁	37

Subgroups — C3×C5×C37×D21 #37552136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₃₇×D₂₁ #37552136