Subgroups — C₃×C₅²×Dic₃×C₃₇ #45686133

Order 33300.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₅²×Dic₃×C₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₄×C₅²×C₃₇	—	3	1	no	maximal
4	C₃²×C₅²×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×Dic₃×C₃₇	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₃×C₅²×C₃₇	—	2	1	no	—
7	C₂×C₃×C₅²×C₃₇	D₃	1	1	yes	characteristic
8	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	1	yes	characteristic
9	C₄×C₅²×C₃₇	—	3	1	no	Hall
10	C₂×C₃²×C₅×C₃₇	C₂×C₅	1	6	yes	—
11	C₃×C₅²×C₃₇	—	2	1	no	—
12	C₃×C₅²×C₃₇	Dic₃	1	1	yes	characteristic, direct product factor
13	C₃×C₅²×C₃₇	C₃×C₄	1	1	yes	characteristic, complemented
14	C₅×Dic₃×C₃₇	C₃×C₅	1	6	yes	direct product factor
15	C₃×C₄×C₅×C₃₇	—	3	6	no	—
16	C₂×C₅²×C₃₇	C₃×D₃	1	1	yes	characteristic
17	C₃²×C₅×C₃₇	C₄×C₅	1	6	yes	complemented
18	C₃×Dic₃×C₃₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃×C₅×C₃₇	—	2	6	no	—
20	C₂×C₃×C₅×C₃₇	C₅×D₃	1	6	yes	—
21	C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	1	6	yes	—
22	C₅²×C₃₇	C₃×Dic₃	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₅²×Dic₃	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₄×C₅×C₃₇	—	3	6	no	—
25	C₂×C₃²×C₃₇	C₂×C₅²	1	1	yes	characteristic
26	C₃×C₅×C₃₇	—	2	6	no	—
27	C₃×C₅×C₃₇	C₅×Dic₃	1	6	yes	direct product factor
28	C₃×C₅×C₃₇	C₃×C₄×C₅	1	6	yes	complemented
29	C₂×C₃²×C₅²	C₂×C₃₇	1	1	yes	characteristic
30	C₃×C₄×C₃₇	—	3	1	no	—
31	Dic₃×C₃₇	C₃×C₅²	1	1	yes	characteristic, direct product factor
32	C₂×C₅×C₃₇	C₃×C₅×D₃	1	6	yes	—
33	C₃²×C₃₇	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
34	C₃×C₄×C₅²	—	3	1	no	—
35	C₅²×Dic₃	C₃×C₃₇	1	1	yes	characteristic, direct product factor
36	C₃²×C₅²	C₄×C₃₇	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₇	—	2	1	no	—
38	C₂×C₃×C₃₇	C₅²×D₃	1	1	yes	characteristic
39	C₂×C₃×C₃₇	C₂×C₃×C₅²	1	1	yes	characteristic
40	C₅×C₃₇	C₃×C₅×Dic₃	1	6	yes	direct product factor
41	C₃×C₅×Dic₃	C₅×C₃₇	1	6	yes	direct product factor
42	C₂×C₃×C₅²	—	2	1	no	—
43	C₂×C₃×C₅²	D₃×C₃₇	1	1	yes	characteristic
44	C₂×C₃×C₅²	C₂×C₃×C₃₇	1	1	yes	characteristic
45	C₄×C₃₇	—	3	1	no	Hall
46	C₃×C₃₇	—	2	1	no	—
47	C₃×C₃₇	C₅²×Dic₃	1	1	yes	characteristic, direct product factor
48	C₃×C₃₇	C₃×C₄×C₅²	1	1	yes	characteristic, complemented
49	C₄×C₅²	—	3	1	no	Hall
50	C₂×C₃²×C₅	C₂×C₅×C₃₇	1	6	yes	—
51	C₃×C₅²	—	2	1	no	—
52	C₃×C₅²	Dic₃×C₃₇	1	1	yes	characteristic, direct product factor
53	C₃×C₅²	C₃×C₄×C₃₇	1	1	yes	characteristic, complemented
54	C₂×C₃₇	C₃×C₅²×D₃	1	1	yes	characteristic
55	C₃×C₄×C₅	—	3	6	no	—
56	C₅×Dic₃	C₃×C₅×C₃₇	1	6	yes	direct product factor
57	C₂×C₅²	C₃×D₃×C₃₇	1	1	yes	characteristic
58	C₃²×C₅	C₄×C₅×C₃₇	1	6	yes	complemented
59	C₃₇	C₃×C₅²×Dic₃	1	1	yes	characteristic, direct product factor, Sylow
60	C₃×Dic₃	C₅²×C₃₇	1	1	yes	characteristic, direct product factor, Hall
61	C₂×C₃×C₅	—	2	6	no	—
62	C₂×C₃×C₅	C₅×D₃×C₃₇	1	6	yes	—
63	C₂×C₃×C₅	C₂×C₃×C₅×C₃₇	1	6	yes	—
64	C₅²	C₃×Dic₃×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
65	C₄×C₅	—	3	6	no	—
66	C₂×C₃²	C₂×C₅²×C₃₇	1	1	yes	characteristic
67	C₃×C₅	—	2	6	no	—
68	C₃×C₅	C₃×C₄×C₅×C₃₇	1	6	yes	complemented
69	C₃×C₅	C₅×Dic₃×C₃₇	1	6	yes	direct product factor
70	C₃×C₄	—	3	1	no	—
71	Dic₃	C₃×C₅²×C₃₇	1	1	yes	characteristic, direct product factor
72	C₂×C₅	C₃×C₅×D₃×C₃₇	1	6	yes	—
73	C₃²	C₄×C₅²×C₃₇	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	2	1	no	—
75	C₂×C₃	C₂×C₃×C₅²×C₃₇	1	1	yes	characteristic
76	C₂×C₃	C₅²×D₃×C₃₇	1	1	yes	characteristic
77	C₅	C₃×C₅×Dic₃×C₃₇	1	6	yes	direct product factor
78	C₄	—	3	1	no	Sylow
79	C₃	—	2	1	no	—
80	C₃	C₃×C₄×C₅²×C₃₇	1	1	yes	characteristic, complemented
81	C₃	C₅²×Dic₃×C₃₇	1	1	yes	characteristic, direct product factor
82	C₂	C₃×C₅²×D₃×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²×C₃₇	D₃
commutator, hypocenter	C₃	C₃×C₄×C₅²×C₃₇
Frattini	C₂	C₃×C₅²×D₃×C₃₇
Fitting, socle, prime radical	C₂×C₃²×C₅²×C₃₇	C₂
cosocle	C₂×C₃	C₂×C₃×C₅²×C₃₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅²×C₃₇

Complements

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

N	H ≅ G/N	type
2. C₅²×Dic₃×C₃₇	C₃	direct (N × H)
4. C₃²×C₅²×C₃₇	C₄	semidirect
5. C₃×C₅×Dic₃×C₃₇	C₅	direct (N × H)
12. C₃×C₅²×C₃₇	Dic₃	direct (N × H)
13. C₃×C₅²×C₃₇	C₃×C₄	semidirect
14. C₅×Dic₃×C₃₇	C₃×C₅	direct (N × H)
17. C₃²×C₅×C₃₇	C₄×C₅	semidirect
18. C₃×Dic₃×C₃₇	C₅²	direct (N × H)
22. C₅²×C₃₇	C₃×Dic₃	direct (N × H)
23. C₃×C₅²×Dic₃	C₃₇	direct (N × H)
27. C₃×C₅×C₃₇	C₅×Dic₃	direct (N × H)
28. C₃×C₅×C₃₇	C₃×C₄×C₅	semidirect
31. Dic₃×C₃₇	C₃×C₅²	direct (N × H)
33. C₃²×C₃₇	C₄×C₅²	semidirect
35. C₅²×Dic₃	C₃×C₃₇	direct (N × H)
36. C₃²×C₅²	C₄×C₃₇	semidirect
40. C₅×C₃₇	C₃×C₅×Dic₃	direct (N × H)
41. C₃×C₅×Dic₃	C₅×C₃₇	direct (N × H)
47. C₃×C₃₇	C₅²×Dic₃	direct (N × H)
48. C₃×C₃₇	C₃×C₄×C₅²	semidirect
52. C₃×C₅²	Dic₃×C₃₇	direct (N × H)
53. C₃×C₅²	C₃×C₄×C₃₇	semidirect
56. C₅×Dic₃	C₃×C₅×C₃₇	direct (N × H)
58. C₃²×C₅	C₄×C₅×C₃₇	semidirect
59. C₃₇	C₃×C₅²×Dic₃	direct (N × H)
60. C₃×Dic₃	C₅²×C₃₇	direct (N × H)
64. C₅²	C₃×Dic₃×C₃₇	direct (N × H)
68. C₃×C₅	C₃×C₄×C₅×C₃₇	semidirect
69. C₃×C₅	C₅×Dic₃×C₃₇	direct (N × H)
71. Dic₃	C₃×C₅²×C₃₇	direct (N × H)
73. C₃²	C₄×C₅²×C₃₇	semidirect
77. C₅	C₃×C₅×Dic₃×C₃₇	direct (N × H)
80. C₃	C₃×C₄×C₅²×C₃₇	semidirect
81. C₃	C₅²×Dic₃×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
C₂×C₃²×C₅²×C₃₇	2
C₅²×Dic₃×C₃₇	3
C₃×C₄×C₅²×C₃₇	3
C₃×C₅×Dic₃×C₃₇	5
C₃×C₅²×Dic₃	37

Subgroups — C3×C52×Dic3×C37 #45686133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×Dic₃×C₃₇ #45686133