Subgroups — C₂×C₃²×C₅²×D₃₇ #45686509

Order 33300.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×D₃₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₅²×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅²×D₃₇	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₅²×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×C₅×D₃₇	C₅	1	6	yes	maximal, direct product factor
6	C₃×C₅²×D₃₇	C₂×C₃	1	8	yes	direct product factor
7	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	4	yes	complemented
8	C₂×C₅²×D₃₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₃²×C₅×D₃₇	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₃²×C₅×C₃₇	C₂×C₅	1	6	yes	complemented
11	C₃×C₅²×C₃₇	C₂²×C₃	1	4	yes	complemented
12	C₂×C₃×C₅×D₃₇	C₃×C₅	1	24	yes	direct product factor
13	C₂×C₅²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented
14	C₅²×D₃₇	C₂×C₃²	1	2	yes	direct product factor
15	C₃²×C₅×C₃₇	C₂²×C₅	1	6	yes	complemented
16	C₂×C₃²×D₃₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	1	24	yes	complemented
18	C₃×C₅×D₃₇	C₂×C₃×C₅	1	48	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	C₂×C₅×D₃₇	C₃²×C₅	1	6	yes	direct product factor
22	C₂×C₃²×C₃₇	C₂×C₅²	1	1	yes	characteristic, complemented
23	C₃²×D₃₇	C₂×C₅²	1	2	yes	direct product factor
24	C₃×C₅×C₃₇	C₂²×C₃×C₅	1	24	yes	complemented
25	C₂×C₃²×C₅²	—	37	2	no	—
26	C₂×C₃²×C₅²	D₃₇	1	1	yes	characteristic, direct product factor
27	C₂×C₃×D₃₇	C₃×C₅²	1	4	yes	direct product factor
28	C₂×C₅×C₃₇	C₂×C₃²×C₅	1	6	yes	complemented
29	C₅×D₃₇	C₂×C₃²×C₅	1	12	yes	direct product factor
30	C₃²×C₃₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₃×C₅²	—	37	4	no	—
32	C₃²×C₅²	C₂×D₃₇	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₃×C₃₇	C₂×C₃×C₅²	1	4	yes	complemented
34	C₃×D₃₇	C₂×C₃×C₅²	1	8	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	8	no	—
38	C₂×C₃×C₅²	C₃×D₃₇	1	4	yes	direct product factor
39	C₂×D₃₇	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Hall
40	C₃×C₃₇	C₂²×C₃×C₅²	1	4	yes	complemented
41	C₂²×C₅²	—	37	1	no	Hall
42	C₂×C₃²×C₅	—	37	12	no	—
43	C₂×C₃²×C₅	C₅×D₃₇	1	6	yes	direct product factor
44	C₃×C₅²	C₂×C₃×D₃₇	1	4	yes	direct product factor
45	C₂×C₃₇	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented
46	D₃₇	C₂×C₃²×C₅²	1	2	yes	direct product factor
47	C₂²×C₃×C₅	—	37	24	no	—
48	C₂×C₅²	—	37	2	no	—
49	C₂×C₅²	C₃²×D₃₇	1	1	yes	characteristic, direct product factor
50	C₃²×C₅	C₂×C₅×D₃₇	1	6	yes	direct product factor
51	C₃₇	C₂²×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₃²	—	37	1	no	Hall
53	C₂×C₃×C₅	—	37	48	no	—
54	C₂×C₃×C₅	C₃×C₅×D₃₇	1	24	yes	direct product factor
55	C₅²	C₂×C₃²×D₃₇	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₅	—	37	6	no	—
57	C₂×C₃²	—	37	2	no	—
58	C₂×C₃²	C₅²×D₃₇	1	1	yes	characteristic, direct product factor
59	C₃×C₅	C₂×C₃×C₅×D₃₇	1	24	yes	direct product factor
60	C₂²×C₃	—	37	4	no	—
61	C₂×C₅	—	37	12	no	—
62	C₂×C₅	C₃²×C₅×D₃₇	1	6	yes	direct product factor
63	C₃²	C₂×C₅²×D₃₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	—	37	8	no	—
65	C₂×C₃	C₃×C₅²×D₃₇	1	4	yes	direct product factor
66	C₅	C₂×C₃²×C₅×D₃₇	1	6	yes	direct product factor
67	C₂²	—	37	1	no	Sylow
68	C₃	C₂×C₃×C₅²×D₃₇	1	4	yes	direct product factor
69	C₂	—	37	2	no	—
70	C₂	C₃²×C₅²×D₃₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₅²	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂²×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅²×C₃₇	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₃²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×D₃₇	C₂	direct (N × H)
2. C₂×C₃²×C₅²×C₃₇	C₂	semidirect
3. C₂×C₃×C₅²×D₃₇	C₃	direct (N × H)
4. C₃²×C₅²×C₃₇	C₂²	semidirect
5. C₂×C₃²×C₅×D₃₇	C₅	direct (N × H)
6. C₃×C₅²×D₃₇	C₂×C₃	direct (N × H)
7. C₂×C₃×C₅²×C₃₇	C₂×C₃	semidirect
8. C₂×C₅²×D₃₇	C₃²	direct (N × H)
9. C₃²×C₅×D₃₇	C₂×C₅	direct (N × H)
10. C₂×C₃²×C₅×C₃₇	C₂×C₅	semidirect
11. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
12. C₂×C₃×C₅×D₃₇	C₃×C₅	direct (N × H)
13. C₂×C₅²×C₃₇	C₂×C₃²	semidirect
14. C₅²×D₃₇	C₂×C₃²	direct (N × H)
15. C₃²×C₅×C₃₇	C₂²×C₅	semidirect
16. C₂×C₃²×D₃₇	C₅²	direct (N × H)
17. C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	semidirect
18. C₃×C₅×D₃₇	C₂×C₃×C₅	direct (N × H)
19. C₅²×C₃₇	C₂²×C₃²	semidirect
21. C₂×C₅×D₃₇	C₃²×C₅	direct (N × H)
22. C₂×C₃²×C₃₇	C₂×C₅²	semidirect
23. C₃²×D₃₇	C₂×C₅²	direct (N × H)
24. C₃×C₅×C₃₇	C₂²×C₃×C₅	semidirect
26. C₂×C₃²×C₅²	D₃₇	direct (N × H)
27. C₂×C₃×D₃₇	C₃×C₅²	direct (N × H)
28. C₂×C₅×C₃₇	C₂×C₃²×C₅	semidirect
29. C₅×D₃₇	C₂×C₃²×C₅	direct (N × H)
30. C₃²×C₃₇	C₂²×C₅²	semidirect
32. C₃²×C₅²	C₂×D₃₇	direct (N × H)
33. C₂×C₃×C₃₇	C₂×C₃×C₅²	semidirect
34. C₃×D₃₇	C₂×C₃×C₅²	direct (N × H)
35. C₅×C₃₇	C₂²×C₃²×C₅	semidirect
38. C₂×C₃×C₅²	C₃×D₃₇	direct (N × H)
39. C₂×D₃₇	C₃²×C₅²	direct (N × H)
40. C₃×C₃₇	C₂²×C₃×C₅²	semidirect
43. C₂×C₃²×C₅	C₅×D₃₇	direct (N × H)
44. C₃×C₅²	C₂×C₃×D₃₇	direct (N × H)
45. C₂×C₃₇	C₂×C₃²×C₅²	semidirect
46. D₃₇	C₂×C₃²×C₅²	direct (N × H)
49. C₂×C₅²	C₃²×D₃₇	direct (N × H)
50. C₃²×C₅	C₂×C₅×D₃₇	direct (N × H)
51. C₃₇	C₂²×C₃²×C₅²	semidirect
54. C₂×C₃×C₅	C₃×C₅×D₃₇	direct (N × H)
55. C₅²	C₂×C₃²×D₃₇	direct (N × H)
58. C₂×C₃²	C₅²×D₃₇	direct (N × H)
59. C₃×C₅	C₂×C₃×C₅×D₃₇	direct (N × H)
62. C₂×C₅	C₃²×C₅×D₃₇	direct (N × H)
63. C₃²	C₂×C₅²×D₃₇	direct (N × H)
65. C₂×C₃	C₃×C₅²×D₃₇	direct (N × H)
66. C₅	C₂×C₃²×C₅×D₃₇	direct (N × H)
68. C₃	C₂×C₃×C₅²×D₃₇	direct (N × H)
70. C₂	C₃²×C₅²×D₃₇	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×D₃₇	2
C₂×C₃²×C₅²×C₃₇	2
C₂×C₃×C₅²×D₃₇	3
C₂×C₃²×C₅×D₃₇	5
C₂²×C₃²×C₅²	37

Subgroups — C2×C32×C52×D37 #45686509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅²×D₃₇ #45686509