Subgroups — C₂×D₃×C₃₇×C₅²⋊C₃ #45686343

Order 33300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₃₇×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₃₇×C₅²⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅²×D₃×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃₇×C₅²⋊C₃	—	3	1	no	maximal
5	C₃×C₃₇×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅²×D₃×C₃₇	C₂×C₃	1	2	yes	complemented
7	C₂×C₃₇×C₅²⋊C₃	—	2	1	no	—
8	C₂×C₃₇×C₅²⋊C₃	—	3	2	no	—
9	C₂×C₃₇×C₅²⋊C₃	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂²×C₅²×C₃₇	—	3	1	no	Hall
12	C₃₇×C₅²⋊C₃	—	2	1	no	—
13	C₃₇×C₅²⋊C₃	C₂×D₃	1	1	yes	characteristic, direct product factor
14	C₃×C₅²×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₅×D₃×C₃₇	—	3	2	no	—
16	C₂×C₅²×C₃₇	—	3	2	no	—
17	C₂×C₅²×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented
18	C₂×C₃×D₃×C₃₇	—	25	1	no	maximal, Hall
19	C₂×C₃×C₅×C₃₇	—	3	2	no	—
20	C₅×D₃×C₃₇	—	3	4	no	—
21	C₅²×C₃₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
22	C₂×D₃×C₅²⋊C₃	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₅×C₃₇	—	9	2	no	—
24	C₂×C₃²×C₃₇	—	25	1	no	—
25	C₃×D₃×C₃₇	—	25	2	no	—
26	C₃×C₅×C₃₇	—	3	2	no	—
27	C₂×C₃×C₅²⋊C₃	C₂×C₃₇	1	1	yes	characteristic, complemented
28	D₃×C₅²⋊C₃	C₂×C₃₇	1	2	yes	direct product factor
29	C₂²×C₃×C₃₇	—	75	1	no	—
30	C₂×D₃×C₃₇	C₅²⋊C₃	1	1	yes	characteristic, direct product factor
31	C₂×C₅×C₃₇	—	3	2	no	—
32	C₂×C₅×C₃₇	—	9	4	no	—
33	C₃²×C₃₇	—	25	1	no	Hall
34	C₂×C₅²×D₃	C₃×C₃₇	1	1	yes	characteristic, complemented
35	C₂²×C₅²⋊C₃	—	3	1	no	—
36	C₃×C₅²⋊C₃	C₂²×C₃₇	1	1	yes	characteristic, complemented, Hall
37	C₂×C₃×C₃₇	—	25	1	no	—
38	C₂×C₃×C₃₇	—	50	1	no	—
39	C₂×C₃×C₃₇	—	75	2	no	—
40	C₂×C₃×C₃₇	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
41	D₃×C₃₇	C₂×C₅²⋊C₃	1	2	yes	direct product factor
42	C₅×C₃₇	—	3	2	no	—
43	C₂×C₃×C₅²	C₂×C₃×C₃₇	1	1	yes	characteristic, complemented
44	C₅²×D₃	C₂×C₃×C₃₇	1	2	yes	complemented
45	C₂×C₅²⋊C₃	—	2	1	no	—
46	C₂×C₅²⋊C₃	—	3	2	no	—
47	C₂×C₅²⋊C₃	D₃×C₃₇	1	1	yes	characteristic, direct product factor
48	C₂²×C₃₇	—	3	1	no	Hall
49	C₃×C₃₇	—	25	1	no	—
50	C₃×C₃₇	—	50	1	no	—
51	C₃×C₃₇	C₂²×C₅²⋊C₃	1	1	yes	characteristic, complemented
52	C₂²×C₅²	—	3	1	no	Hall
53	C₃×C₅²	C₂²×C₃×C₃₇	1	1	yes	characteristic, complemented
54	C₅²⋊C₃	—	2	1	no	—
55	C₅²⋊C₃	C₂×D₃×C₃₇	1	1	yes	characteristic, direct product factor
56	C₂×C₃₇	—	3	2	no	—
57	C₂×C₃₇	D₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
58	C₂×C₅×D₃	—	3	2	no	—
59	C₂×C₅²	—	3	2	no	—
60	C₂×C₅²	C₃×D₃×C₃₇	1	1	yes	characteristic, complemented
61	C₃₇	C₂×D₃×C₅²⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₃×D₃	—	25	1	no	Hall
63	C₂×C₃×C₅	—	3	2	no	—
64	C₅×D₃	—	3	4	no	—
65	C₅²	C₂×C₃×D₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₅	—	9	2	no	—
67	C₂×C₃²	—	25	1	no	—
68	C₃×D₃	—	25	2	no	—
69	C₃×C₅	—	3	2	no	—
70	C₂²×C₃	—	75	1	no	—
71	C₂×D₃	C₃₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor
72	C₂×C₅	—	3	2	no	—
73	C₂×C₅	—	9	4	no	—
74	C₃²	—	25	1	no	Sylow
75	C₂×C₃	—	25	1	no	—
76	C₂×C₃	—	50	1	no	—
77	C₂×C₃	—	75	2	no	—
78	C₂×C₃	C₂×C₃₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
79	D₃	C₂×C₃₇×C₅²⋊C₃	1	2	yes	direct product factor
80	C₅	—	3	2	no	—
81	C₂²	—	3	1	no	Sylow
82	C₃	—	25	1	no	—
83	C₃	—	50	1	no	—
84	C₃	C₂²×C₃₇×C₅²⋊C₃	1	1	yes	characteristic, complemented
85	C₂	—	3	2	no	—
86	C₂	D₃×C₃₇×C₅²⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃₇	D₃×C₅²⋊C₃
commutator, cosocle, hypocenter	C₃×C₅²	C₂²×C₃×C₃₇
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₃₇	C₂×C₃
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₂×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₃₇×C₅²⋊C₃	C₂	semidirect
2. D₃×C₃₇×C₅²⋊C₃	C₂	direct (N × H)
3. C₂×C₅²×D₃×C₃₇	C₃	semidirect
5. C₃×C₃₇×C₅²⋊C₃	C₂²	semidirect
6. C₅²×D₃×C₃₇	C₂×C₃	semidirect
9. C₂×C₃₇×C₅²⋊C₃	D₃	direct (N × H)
10. C₂×C₃×C₅²×C₃₇	C₂×C₃	semidirect
13. C₃₇×C₅²⋊C₃	C₂×D₃	direct (N × H)
14. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
17. C₂×C₅²×C₃₇	C₃×D₃	semidirect
21. C₅²×C₃₇	C₂×C₃×D₃	semidirect
22. C₂×D₃×C₅²⋊C₃	C₃₇	direct (N × H)
27. C₂×C₃×C₅²⋊C₃	C₂×C₃₇	semidirect
28. D₃×C₅²⋊C₃	C₂×C₃₇	direct (N × H)
30. C₂×D₃×C₃₇	C₅²⋊C₃	direct (N × H)
34. C₂×C₅²×D₃	C₃×C₃₇	semidirect
36. C₃×C₅²⋊C₃	C₂²×C₃₇	semidirect
40. C₂×C₃×C₃₇	C₂×C₅²⋊C₃	semidirect
41. D₃×C₃₇	C₂×C₅²⋊C₃	direct (N × H)
43. C₂×C₃×C₅²	C₂×C₃×C₃₇	semidirect
44. C₅²×D₃	C₂×C₃×C₃₇	semidirect
47. C₂×C₅²⋊C₃	D₃×C₃₇	direct (N × H)
51. C₃×C₃₇	C₂²×C₅²⋊C₃	semidirect
53. C₃×C₅²	C₂²×C₃×C₃₇	semidirect
55. C₅²⋊C₃	C₂×D₃×C₃₇	direct (N × H)
57. C₂×C₃₇	D₃×C₅²⋊C₃	direct (N × H)
60. C₂×C₅²	C₃×D₃×C₃₇	semidirect
61. C₃₇	C₂×D₃×C₅²⋊C₃	direct (N × H)
65. C₅²	C₂×C₃×D₃×C₃₇	semidirect
71. C₂×D₃	C₃₇×C₅²⋊C₃	direct (N × H)
78. C₂×C₃	C₂×C₃₇×C₅²⋊C₃	semidirect
79. D₃	C₂×C₃₇×C₅²⋊C₃	direct (N × H)
84. C₃	C₂²×C₃₇×C₅²⋊C₃	semidirect
86. C₂	D₃×C₃₇×C₅²⋊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×C₃₇×C₅²⋊C₃	2
D₃×C₃₇×C₅²⋊C₃	2
C₂×C₅²×D₃×C₃₇	3
C₂²×C₃₇×C₅²⋊C₃	3
C₂×C₃×D₃×C₃₇	25
C₂×D₃×C₅²⋊C₃	37

Subgroups — C2×D3×C37×C52⋊C3 #45686343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₃₇×C₅²⋊C₃ #45686343