Subgroups — C₂²×C₃×C₅²×C₃₇⋊C₃ #45686508

Order 33300.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×C₃₇⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₃₇⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₅²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅²×C₃₇⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₃₇⋊C₃	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₃₇⋊C₃	C₂×C₃	1	9	yes	direct product factor
7	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	3	yes	complemented
8	C₂²×C₅²×C₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅×C₃₇⋊C₃	C₂×C₅	1	18	yes	direct product factor
10	C₅²×C₃₇⋊C₃	C₂²×C₃	1	3	yes	direct product factor
11	C₃×C₅²×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₂²×C₅×C₃₇⋊C₃	C₃×C₅	1	18	yes	direct product factor
13	C₂²×C₃×C₅×C₃₇	C₃×C₅	1	6	yes	complemented
14	C₂×C₅²×C₃₇	C₂×C₃²	1	3	yes	complemented
15	C₃×C₅×C₃₇⋊C₃	C₂²×C₅	1	6	yes	direct product factor
16	C₂²×C₃×C₃₇⋊C₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	1	18	yes	complemented
18	C₂×C₅×C₃₇⋊C₃	C₂×C₃×C₅	1	54	yes	direct product factor
19	C₅²×C₃₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₃²×C₅²	—	37	1	no	maximal, Hall
21	C₂²×C₅×C₃₇	C₃²×C₅	1	6	yes	complemented
22	C₂×C₃×C₃₇⋊C₃	C₂×C₅²	1	3	yes	direct product factor
23	C₃×C₅×C₃₇	C₂²×C₃×C₅	1	6	yes	complemented
24	C₅×C₃₇⋊C₃	C₂²×C₃×C₅	1	18	yes	direct product factor
25	C₂×C₃²×C₅²	—	37	3	no	—
26	C₂²×C₃×C₃₇	C₃×C₅²	1	1	yes	characteristic, complemented
27	C₂²×C₃₇⋊C₃	C₃×C₅²	1	3	yes	direct product factor
28	C₂×C₅×C₃₇	C₂×C₃²×C₅	1	18	yes	complemented
29	C₃×C₃₇⋊C₃	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
30	C₂²×C₃×C₅²	—	37	3	no	—
31	C₂²×C₃×C₅²	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
32	C₃²×C₅²	—	37	1	no	Hall
33	C₂×C₃×C₃₇	C₂×C₃×C₅²	1	3	yes	complemented
34	C₂×C₃₇⋊C₃	C₂×C₃×C₅²	1	9	yes	direct product factor
35	C₅×C₃₇	C₂²×C₃²×C₅	1	6	yes	complemented
36	C₂²×C₃²×C₅	—	37	6	no	—
37	C₂×C₃×C₅²	—	37	9	no	—
38	C₂×C₃×C₅²	C₂×C₃₇⋊C₃	1	3	yes	direct product factor
39	C₂²×C₃₇	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
40	C₃×C₃₇	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented
41	C₃₇⋊C₃	C₂²×C₃×C₅²	1	3	yes	direct product factor
42	C₂²×C₅²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₃²×C₅	—	37	18	no	—
44	C₃×C₅²	—	37	3	no	—
45	C₃×C₅²	C₂²×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
46	C₂×C₃₇	C₂×C₃²×C₅²	1	3	yes	complemented
47	C₂²×C₃×C₅	—	37	18	no	—
48	C₂²×C₃×C₅	C₅×C₃₇⋊C₃	1	6	yes	direct product factor
49	C₂×C₅²	C₂×C₃×C₃₇⋊C₃	1	3	yes	direct product factor
50	C₃²×C₅	—	37	6	no	—
51	C₃₇	C₂²×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₃²	—	37	1	no	Hall
53	C₂×C₃×C₅	—	37	54	no	—
54	C₂×C₃×C₅	C₂×C₅×C₃₇⋊C₃	1	18	yes	direct product factor
55	C₅²	C₂²×C₃×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	C₃×C₅×C₃₇⋊C₃	1	6	yes	direct product factor
57	C₂×C₃²	—	37	3	no	—
58	C₃×C₅	—	37	18	no	—
59	C₃×C₅	C₂²×C₅×C₃₇⋊C₃	1	6	yes	direct product factor
60	C₂²×C₃	—	37	3	no	—
61	C₂²×C₃	C₅²×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
62	C₂×C₅	C₂×C₃×C₅×C₃₇⋊C₃	1	18	yes	direct product factor
63	C₃²	—	37	1	no	Sylow
64	C₂×C₃	—	37	9	no	—
65	C₂×C₃	C₂×C₅²×C₃₇⋊C₃	1	3	yes	direct product factor
66	C₅	C₂²×C₃×C₅×C₃₇⋊C₃	1	6	yes	direct product factor
67	C₂²	C₃×C₅²×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	37	3	no	—
69	C₃	C₂²×C₅²×C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×C₅²×C₃₇⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅²	C₃₇⋊C₃
commutator, cosocle, hypocenter	C₃₇	C₂²×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₃₇	C₃
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂²×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²×C₃₇⋊C₃	C₂	direct (N × H)
2. C₂²×C₅²×C₃₇⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₃₇	C₃	semidirect
4. C₃×C₅²×C₃₇⋊C₃	C₂²	direct (N × H)
5. C₂²×C₃×C₅×C₃₇⋊C₃	C₅	direct (N × H)
6. C₂×C₅²×C₃₇⋊C₃	C₂×C₃	direct (N × H)
7. C₂×C₃×C₅²×C₃₇	C₂×C₃	semidirect
8. C₂²×C₅²×C₃₇	C₃²	semidirect
9. C₂×C₃×C₅×C₃₇⋊C₃	C₂×C₅	direct (N × H)
10. C₅²×C₃₇⋊C₃	C₂²×C₃	direct (N × H)
11. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
12. C₂²×C₅×C₃₇⋊C₃	C₃×C₅	direct (N × H)
13. C₂²×C₃×C₅×C₃₇	C₃×C₅	semidirect
14. C₂×C₅²×C₃₇	C₂×C₃²	semidirect
15. C₃×C₅×C₃₇⋊C₃	C₂²×C₅	direct (N × H)
16. C₂²×C₃×C₃₇⋊C₃	C₅²	direct (N × H)
17. C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	semidirect
18. C₂×C₅×C₃₇⋊C₃	C₂×C₃×C₅	direct (N × H)
19. C₅²×C₃₇	C₂²×C₃²	semidirect
21. C₂²×C₅×C₃₇	C₃²×C₅	semidirect
22. C₂×C₃×C₃₇⋊C₃	C₂×C₅²	direct (N × H)
23. C₃×C₅×C₃₇	C₂²×C₃×C₅	semidirect
24. C₅×C₃₇⋊C₃	C₂²×C₃×C₅	direct (N × H)
26. C₂²×C₃×C₃₇	C₃×C₅²	semidirect
27. C₂²×C₃₇⋊C₃	C₃×C₅²	direct (N × H)
28. C₂×C₅×C₃₇	C₂×C₃²×C₅	semidirect
29. C₃×C₃₇⋊C₃	C₂²×C₅²	direct (N × H)
31. C₂²×C₃×C₅²	C₃₇⋊C₃	direct (N × H)
33. C₂×C₃×C₃₇	C₂×C₃×C₅²	semidirect
34. C₂×C₃₇⋊C₃	C₂×C₃×C₅²	direct (N × H)
35. C₅×C₃₇	C₂²×C₃²×C₅	semidirect
38. C₂×C₃×C₅²	C₂×C₃₇⋊C₃	direct (N × H)
39. C₂²×C₃₇	C₃²×C₅²	semidirect
40. C₃×C₃₇	C₂²×C₃×C₅²	semidirect
41. C₃₇⋊C₃	C₂²×C₃×C₅²	direct (N × H)
42. C₂²×C₅²	C₃×C₃₇⋊C₃	direct (N × H)
45. C₃×C₅²	C₂²×C₃₇⋊C₃	direct (N × H)
46. C₂×C₃₇	C₂×C₃²×C₅²	semidirect
48. C₂²×C₃×C₅	C₅×C₃₇⋊C₃	direct (N × H)
49. C₂×C₅²	C₂×C₃×C₃₇⋊C₃	direct (N × H)
51. C₃₇	C₂²×C₃²×C₅²	semidirect
54. C₂×C₃×C₅	C₂×C₅×C₃₇⋊C₃	direct (N × H)
55. C₅²	C₂²×C₃×C₃₇⋊C₃	direct (N × H)
56. C₂²×C₅	C₃×C₅×C₃₇⋊C₃	direct (N × H)
59. C₃×C₅	C₂²×C₅×C₃₇⋊C₃	direct (N × H)
61. C₂²×C₃	C₅²×C₃₇⋊C₃	direct (N × H)
62. C₂×C₅	C₂×C₃×C₅×C₃₇⋊C₃	direct (N × H)
65. C₂×C₃	C₂×C₅²×C₃₇⋊C₃	direct (N × H)
66. C₅	C₂²×C₃×C₅×C₃₇⋊C₃	direct (N × H)
67. C₂²	C₃×C₅²×C₃₇⋊C₃	direct (N × H)
69. C₃	C₂²×C₅²×C₃₇⋊C₃	direct (N × H)
70. C₂	C₂×C₃×C₅²×C₃₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅²×C₃₇⋊C₃	2
C₂²×C₅²×C₃₇⋊C₃	3
C₂²×C₃×C₅²×C₃₇	3
C₂²×C₃×C₅×C₃₇⋊C₃	5
C₂²×C₃²×C₅²	37

Subgroups — C22×C3×C52×C37⋊C3 #45686508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅²×C₃₇⋊C₃ #45686508