Subgroups — C₅×C₇×C₃⋊(D₃₇⋊C₃) #37552091

Order 23310.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₇×C₃₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅×C₇×D₃₇⋊C₃	—	3	1	no	maximal
3	C₅×C₇×D₁₁₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×C₃⋊(D₃₇⋊C₃)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₅×C₇×C₃₇⋊C₃	—	2	1	no	—
6	C₅×C₇×C₃₇⋊C₃	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅×C₃⋊(D₃₇⋊C₃)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₇×D₃₇	—	3	1	no	Hall
10	C₃×C₇×C₃₇⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅×C₃₇⋊C₃	C₂×C₇	1	1	yes	characteristic, complemented, Hall
12	C₇×D₁₁₁	C₃×C₅	1	1	yes	characteristic, complemented
13	C₇×D₃₇⋊C₃	—	3	1	no	—
14	C₅×C₇×C₃₇	C₃×D₃	1	1	yes	characteristic, complemented, Hall
15	C₅×D₁₁₁	C₃×C₇	1	1	yes	characteristic, complemented
16	C₅×D₃₇⋊C₃	—	3	1	no	—
17	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
18	C₇×C₃₇⋊C₃	—	2	1	no	—
19	C₇×C₃₇⋊C₃	C₅×D₃	1	1	yes	characteristic, complemented
20	C₃⋊(D₃₇⋊C₃)	C₅×C₇	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₅×D₃×C₇	—	37	1	no	maximal, Hall
22	C₃×C₅×C₃₇	C₂×C₃×C₇	1	1	yes	characteristic, complemented
23	C₅×C₃₇⋊C₃	—	2	1	no	—
24	C₅×C₃₇⋊C₃	D₃×C₇	1	1	yes	characteristic, complemented
25	C₇×D₃₇	—	3	1	no	Hall
26	C₅×D₃₇	—	3	1	no	Hall
27	C₃×C₃₇⋊C₃	C₂×C₅×C₇	1	1	yes	characteristic, complemented, Hall
28	C₃²×C₅×C₇	—	37	1	no	Hall
29	C₇×C₃₇	C₃×C₅×D₃	1	1	yes	characteristic, complemented, Hall
30	D₁₁₁	C₃×C₅×C₇	1	1	yes	characteristic, complemented
31	D₃₇⋊C₃	—	3	1	no	—
32	C₂×C₃×C₅×C₇	—	111	1	no	—
33	C₅×D₃×C₇	—	37	1	no	—
34	C₅×C₃₇	C₃×D₃×C₇	1	1	yes	characteristic, complemented, Hall
35	C₃×D₃×C₇	—	37	1	no	Hall
36	C₃×C₃₇	C₂×C₃×C₅×C₇	1	1	yes	characteristic, complemented
37	C₃₇⋊C₃	—	2	1	no	—
38	C₃₇⋊C₃	C₅×D₃×C₇	1	1	yes	characteristic, complemented
39	C₃×C₅×C₇	—	37	1	no	—
40	C₃×C₅×C₇	—	74	1	no	—
41	C₃×C₅×C₇	D₃₇⋊C₃	1	1	yes	characteristic, complemented
42	C₃×C₅×D₃	—	37	1	no	Hall
43	D₃₇	—	3	1	no	Hall
44	C₂×C₅×C₇	—	111	1	no	Hall
45	C₃²×C₇	—	37	1	no	Hall
46	C₃²×C₅	—	37	1	no	Hall
47	C₂×C₃×C₇	—	111	1	no	—
48	D₃×C₇	—	37	1	no	—
49	C₃₇	C₃×C₅×D₃×C₇	1	1	yes	characteristic, complemented, Sylow
50	C₅×C₇	C₃⋊(D₃₇⋊C₃)	1	1	yes	characteristic, direct product factor, Hall
51	C₂×C₃×C₅	—	111	1	no	—
52	C₅×D₃	—	37	1	no	—
53	C₃×C₇	—	37	1	no	—
54	C₃×C₇	—	74	1	no	—
55	C₃×C₇	C₅×D₃₇⋊C₃	1	1	yes	characteristic, complemented
56	C₃×D₃	—	37	1	no	Hall
57	C₃×C₅	—	37	1	no	—
58	C₃×C₅	—	74	1	no	—
59	C₃×C₅	C₇×D₃₇⋊C₃	1	1	yes	characteristic, complemented
60	C₂×C₇	—	111	1	no	Hall
61	C₂×C₅	—	111	1	no	Hall
62	C₃²	—	37	1	no	Sylow
63	C₇	C₅×C₃⋊(D₃₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₃	—	111	1	no	—
65	D₃	—	37	1	no	—
66	C₅	C₇×C₃⋊(D₃₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
67	C₃	—	37	1	no	—
68	C₃	—	74	1	no	—
69	C₃	C₅×C₇×D₃₇⋊C₃	1	1	yes	characteristic, complemented
70	C₂	—	111	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅×C₇	C₃⋊(D₃₇⋊C₃)
commutator, cosocle, hypocenter	C₃×C₃₇	C₂×C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
LCS	G ▷ C₃×C₃₇
UCS	C₁ ◁ C₅×C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₇×C₃₇⋊C₃	C₂	semidirect
3. C₅×C₇×D₁₁₁	C₃	semidirect
4. C₇×C₃⋊(D₃₇⋊C₃)	C₅	direct (N × H)
6. C₅×C₇×C₃₇⋊C₃	D₃	semidirect
7. C₃×C₅×C₇×C₃₇	C₂×C₃	semidirect
8. C₅×C₃⋊(D₃₇⋊C₃)	C₇	direct (N × H)
10. C₃×C₇×C₃₇⋊C₃	C₂×C₅	semidirect
11. C₃×C₅×C₃₇⋊C₃	C₂×C₇	semidirect
12. C₇×D₁₁₁	C₃×C₅	semidirect
14. C₅×C₇×C₃₇	C₃×D₃	semidirect
15. C₅×D₁₁₁	C₃×C₇	semidirect
17. C₃×C₇×C₃₇	C₂×C₃×C₅	semidirect
19. C₇×C₃₇⋊C₃	C₅×D₃	semidirect
20. C₃⋊(D₃₇⋊C₃)	C₅×C₇	direct (N × H)
22. C₃×C₅×C₃₇	C₂×C₃×C₇	semidirect
24. C₅×C₃₇⋊C₃	D₃×C₇	semidirect
27. C₃×C₃₇⋊C₃	C₂×C₅×C₇	semidirect
29. C₇×C₃₇	C₃×C₅×D₃	semidirect
30. D₁₁₁	C₃×C₅×C₇	semidirect
34. C₅×C₃₇	C₃×D₃×C₇	semidirect
36. C₃×C₃₇	C₂×C₃×C₅×C₇	semidirect
38. C₃₇⋊C₃	C₅×D₃×C₇	semidirect
41. C₃×C₅×C₇	D₃₇⋊C₃	semidirect
49. C₃₇	C₃×C₅×D₃×C₇	semidirect
50. C₅×C₇	C₃⋊(D₃₇⋊C₃)	direct (N × H)
55. C₃×C₇	C₅×D₃₇⋊C₃	semidirect
59. C₃×C₅	C₇×D₃₇⋊C₃	semidirect
63. C₇	C₅×C₃⋊(D₃₇⋊C₃)	direct (N × H)
66. C₅	C₇×C₃⋊(D₃₇⋊C₃)	direct (N × H)
69. C₃	C₅×C₇×D₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₇×C₃₇⋊C₃	2
C₅×C₇×D₃₇⋊C₃	3
C₅×C₇×D₁₁₁	3
C₇×C₃⋊(D₃₇⋊C₃)	5
C₅×C₃⋊(D₃₇⋊C₃)	7
C₃×C₅×D₃×C₇	37

Subgroups — C5×C7×C3⋊(D37⋊C3) #37552091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇×C₃⋊(D₃₇⋊C₃) #37552091