Subgroups — C₃²×D₅²×C₃₇ #45686394

Order 33300.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×D₅×C₃₇	C₂	1	2	yes	maximal, complemented
2	C₃²×C₃₇×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅²×C₃₇	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₅²×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃²×D₅×C₃₇	—	5	2	no	maximal
6	C₃×C₃₇×C₅⋊D₅	C₂×C₃	1	4	yes	complemented
7	C₃×C₅×D₅×C₃₇	C₂×C₃	1	8	yes	complemented
8	D₅²×C₃₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₃²×D₅×C₃₇	—	5	2	no	—
10	C₃²×D₅×C₃₇	—	10	2	no	—
11	C₃²×D₅×C₃₇	D₅	1	2	yes	direct product factor
12	C₂×C₃²×C₅×C₃₇	—	5	2	no	—
13	C₃×C₅²×C₃₇	C₂²×C₃	1	4	yes	complemented
14	C₂×C₃×D₅×C₃₇	—	5	8	no	—
15	C₃₇×C₅⋊D₅	C₂×C₃²	1	1	yes	characteristic, complemented
16	C₅×D₅×C₃₇	C₂×C₃²	1	2	yes	complemented
17	C₃²×C₅×C₃₇	—	2	2	no	—
18	C₃²×C₅×C₃₇	C₂×D₅	1	2	yes	complemented
19	C₂²×C₃²×C₃₇	—	25	1	no	Hall
20	C₂×C₃×C₅×C₃₇	—	5	8	no	—
21	C₃×D₅×C₃₇	—	5	8	no	—
22	C₃×D₅×C₃₇	—	10	8	no	—
23	C₃×D₅×C₃₇	C₃×D₅	1	8	yes	direct product factor
24	C₅²×C₃₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
25	C₃²×D₅²	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂×D₅×C₃₇	—	5	2	no	—
27	C₂×C₃²×C₃₇	—	5	2	no	—
28	C₂×C₃²×C₃₇	—	25	1	no	—
29	C₃×C₅×C₃₇	—	2	8	no	—
30	C₃×C₅×C₃₇	C₂×C₃×D₅	1	8	yes	complemented
31	C₃²×C₅⋊D₅	C₂×C₃₇	1	1	yes	characteristic, complemented
32	C₃²×C₅×D₅	C₂×C₃₇	1	2	yes	complemented
33	C₂²×C₃×C₃₇	—	25	4	no	—
34	C₂×C₅×C₃₇	—	5	2	no	—
35	D₅×C₃₇	—	5	2	no	—
36	D₅×C₃₇	—	10	2	no	—
37	D₅×C₃₇	C₃²×D₅	1	2	yes	direct product factor
38	C₃²×C₃₇	D₅²	1	1	yes	characteristic, direct product factor, Hall
39	C₃×D₅²	C₃×C₃₇	1	4	yes	direct product factor
40	C₃²×C₅²	C₂²×C₃₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₃₇	—	5	8	no	—
42	C₂×C₃×C₃₇	—	25	4	no	—
43	C₅×C₃₇	—	2	2	no	—
44	C₅×C₃₇	C₂×C₃²×D₅	1	2	yes	complemented
45	C₂×C₃²×D₅	—	5	2	no	—
46	C₃×C₅⋊D₅	C₂×C₃×C₃₇	1	4	yes	complemented
47	C₃×C₅×D₅	C₂×C₃×C₃₇	1	8	yes	complemented
48	C₂²×C₃₇	—	25	1	no	Hall
49	C₃×C₃₇	C₃×D₅²	1	4	yes	direct product factor
50	D₅²	C₃²×C₃₇	1	1	yes	characteristic, direct product factor, Hall
51	C₂×C₃²×C₅	—	5	2	no	—
52	C₃²×D₅	—	5	2	no	—
53	C₃²×D₅	—	10	2	no	—
54	C₃²×D₅	D₅×C₃₇	1	2	yes	direct product factor
55	C₃×C₅²	C₂²×C₃×C₃₇	1	4	yes	complemented
56	C₂×C₃₇	—	5	2	no	—
57	C₂×C₃₇	—	25	1	no	—
58	C₂×C₃×D₅	—	5	8	no	—
59	C₅⋊D₅	C₂×C₃²×C₃₇	1	1	yes	characteristic, complemented
60	C₅×D₅	C₂×C₃²×C₃₇	1	2	yes	complemented
61	C₃²×C₅	—	2	2	no	—
62	C₃²×C₅	C₂×D₅×C₃₇	1	2	yes	complemented
63	C₃₇	C₃²×D₅²	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₃²	—	25	1	no	Hall
65	C₂×C₃×C₅	—	5	8	no	—
66	C₃×D₅	—	5	8	no	—
67	C₃×D₅	—	10	8	no	—
68	C₃×D₅	C₃×D₅×C₃₇	1	8	yes	direct product factor
69	C₅²	C₂²×C₃²×C₃₇	1	1	yes	characteristic, complemented, Sylow
70	C₂×D₅	—	5	2	no	—
71	C₂×C₃²	—	5	2	no	—
72	C₂×C₃²	—	25	1	no	—
73	C₃×C₅	—	2	8	no	—
74	C₃×C₅	C₂×C₃×D₅×C₃₇	1	8	yes	complemented
75	C₂²×C₃	—	25	4	no	—
76	C₂×C₅	—	5	2	no	—
77	D₅	—	5	2	no	—
78	D₅	—	10	2	no	—
79	D₅	C₃²×D₅×C₃₇	1	2	yes	direct product factor
80	C₃²	D₅²×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₃	—	5	8	no	—
82	C₂×C₃	—	25	4	no	—
83	C₅	—	2	2	no	—
84	C₅	C₂×C₃²×D₅×C₃₇	1	2	yes	complemented
85	C₂²	—	25	1	no	Sylow
86	C₃	C₃×D₅²×C₃₇	1	4	yes	direct product factor
87	C₂	—	5	2	no	—
88	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₃₇	D₅²
commutator, cosocle, hypocenter	C₅²	C₂²×C₃²×C₃₇
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₃²×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×D₅×C₃₇	C₂	semidirect
2. C₃²×C₃₇×C₅⋊D₅	C₂	semidirect
3. C₃×D₅²×C₃₇	C₃	direct (N × H)
4. C₃²×C₅²×C₃₇	C₂²	semidirect
6. C₃×C₃₇×C₅⋊D₅	C₂×C₃	semidirect
7. C₃×C₅×D₅×C₃₇	C₂×C₃	semidirect
8. D₅²×C₃₇	C₃²	direct (N × H)
11. C₃²×D₅×C₃₇	D₅	direct (N × H)
13. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
15. C₃₇×C₅⋊D₅	C₂×C₃²	semidirect
16. C₅×D₅×C₃₇	C₂×C₃²	semidirect
18. C₃²×C₅×C₃₇	C₂×D₅	semidirect
23. C₃×D₅×C₃₇	C₃×D₅	direct (N × H)
24. C₅²×C₃₇	C₂²×C₃²	semidirect
25. C₃²×D₅²	C₃₇	direct (N × H)
30. C₃×C₅×C₃₇	C₂×C₃×D₅	semidirect
31. C₃²×C₅⋊D₅	C₂×C₃₇	semidirect
32. C₃²×C₅×D₅	C₂×C₃₇	semidirect
37. D₅×C₃₇	C₃²×D₅	direct (N × H)
38. C₃²×C₃₇	D₅²	direct (N × H)
39. C₃×D₅²	C₃×C₃₇	direct (N × H)
40. C₃²×C₅²	C₂²×C₃₇	semidirect
44. C₅×C₃₇	C₂×C₃²×D₅	semidirect
46. C₃×C₅⋊D₅	C₂×C₃×C₃₇	semidirect
47. C₃×C₅×D₅	C₂×C₃×C₃₇	semidirect
49. C₃×C₃₇	C₃×D₅²	direct (N × H)
50. D₅²	C₃²×C₃₇	direct (N × H)
54. C₃²×D₅	D₅×C₃₇	direct (N × H)
55. C₃×C₅²	C₂²×C₃×C₃₇	semidirect
59. C₅⋊D₅	C₂×C₃²×C₃₇	semidirect
60. C₅×D₅	C₂×C₃²×C₃₇	semidirect
62. C₃²×C₅	C₂×D₅×C₃₇	semidirect
63. C₃₇	C₃²×D₅²	direct (N × H)
68. C₃×D₅	C₃×D₅×C₃₇	direct (N × H)
69. C₅²	C₂²×C₃²×C₃₇	semidirect
74. C₃×C₅	C₂×C₃×D₅×C₃₇	semidirect
79. D₅	C₃²×D₅×C₃₇	direct (N × H)
80. C₃²	D₅²×C₃₇	direct (N × H)
84. C₅	C₂×C₃²×D₅×C₃₇	semidirect
86. C₃	C₃×D₅²×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×D₅×C₃₇	2
C₃²×C₃₇×C₅⋊D₅	2
C₃×D₅²×C₃₇	3
C₂×C₃²×D₅×C₃₇	5
C₃²×D₅²	37

Subgroups — C32×D52×C37 #45686394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₅²×C₃₇ #45686394