Subgroups — C₂×C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃) #45686336

Order 33300.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²⋊(C₃₇⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃₇×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₅⋊D₅)⋊(C₃₇⋊C₃)	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₅²⋊(C₃₇⋊C₃)	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₃₇×C₅⋊D₅	C₂×C₃	1	2	yes	complemented
7	C₂×C₅²⋊(C₃₇⋊C₃)	C₂×C₃	1	3	yes	complemented
8	(C₅⋊D₅)⋊(C₃₇⋊C₃)	C₂×C₃	1	6	yes	direct product factor
9	C₂×C₃×C₅²×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃₇×C₅⋊D₅	C₃²	1	1	yes	characteristic, complemented, Hall
11	C₅²⋊(C₃₇⋊C₃)	C₂²×C₃	1	3	yes	complemented
12	C₃×C₅²×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₅×C₃₇	—	15	2	no	—
14	C₂×C₅²×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented
15	C₃₇×C₅⋊D₅	C₂×C₃²	1	2	yes	complemented
16	C₂²×C₃×C₃₇⋊C₃	—	25	1	no	maximal, Hall
17	C₂×C₃×C₅×C₃₇	—	3	2	no	—
18	C₃×D₅×C₃₇	—	15	4	no	—
19	C₅²×C₃₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×(C₅⋊D₅)⋊C₃	—	37	1	no	maximal, Hall
21	C₂×D₅×C₃₇	—	15	2	no	—
22	C₂×C₃×C₃₇⋊C₃	—	25	2	no	—
23	C₂×C₃×C₃₇⋊C₃	—	25	1	no	—
24	C₃×C₅×C₃₇	—	3	2	no	—
25	C₂×C₃×C₅²⋊C₃	—	37	1	no	—
26	C₃×(C₅⋊D₅)⋊C₃	—	37	2	no	—
27	C₂²×C₃×C₃₇	—	25	1	no	—
28	C₂²×C₃₇⋊C₃	—	25	3	no	—
29	C₂×C₅×C₃₇	—	3	2	no	—
30	D₅×C₃₇	—	15	4	no	—
31	C₃×C₃₇⋊C₃	—	25	1	no	Hall
32	C₂×C₃×C₅⋊D₅	C₃₇⋊C₃	1	1	yes	characteristic, complemented
33	C₂×(C₅⋊D₅)⋊C₃	—	37	3	no	—
34	C₃×C₅²⋊C₃	—	37	1	no	Hall
35	C₂×C₃×C₃₇	—	25	2	no	—
36	C₂×C₃×C₃₇	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
37	C₂×C₃₇⋊C₃	—	25	6	no	—
38	C₂×C₃₇⋊C₃	—	25	3	no	—
39	C₅×C₃₇	—	3	2	no	—
40	C₂×C₃×C₅²	C₂×C₃₇⋊C₃	1	1	yes	characteristic, complemented
41	C₃×C₅⋊D₅	C₂×C₃₇⋊C₃	1	2	yes	complemented
42	C₂×C₅²⋊C₃	—	37	3	no	—
43	(C₅⋊D₅)⋊C₃	—	37	6	no	—
44	C₂²×C₃₇	—	25	1	no	Hall
45	C₃×C₃₇	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
46	C₃₇⋊C₃	—	25	3	no	—
47	C₂×C₅⋊D₅	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Hall
48	C₃×C₅²	C₂²×C₃₇⋊C₃	1	1	yes	characteristic, complemented
49	C₅²⋊C₃	—	37	3	no	—
50	C₂×C₃₇	—	25	2	no	—
51	C₂×C₃₇	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
52	C₂×C₃×D₅	—	15	2	no	—
53	C₂×C₅²	C₂×C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
54	C₅⋊D₅	C₂×C₃×C₃₇⋊C₃	1	2	yes	complemented
55	C₃₇	C₂×C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₃²	—	925	1	no	Hall
57	C₂×C₃×C₅	—	3	2	no	—
58	C₃×D₅	—	15	4	no	—
59	C₅²	C₂²×C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₅	—	15	2	no	—
61	C₂×C₃²	—	925	2	no	—
62	C₂×C₃²	—	925	1	no	—
63	C₃×C₅	—	3	2	no	—
64	C₂²×C₃	—	25	1	no	—
65	C₂²×C₃	—	925	3	no	—
66	C₂×C₅	—	3	2	no	—
67	D₅	—	15	4	no	—
68	C₃²	—	925	1	no	Sylow
69	C₂×C₃	—	25	2	no	—
70	C₂×C₃	—	925	6	no	—
71	C₂×C₃	—	925	3	no	—
72	C₂×C₃	(C₅⋊D₅)⋊(C₃₇⋊C₃)	1	1	yes	characteristic, direct product factor
73	C₅	—	3	2	no	—
74	C₂²	—	25	1	no	Sylow
75	C₃	—	925	3	no	—
76	C₃	C₂×(C₅⋊D₅)⋊(C₃₇⋊C₃)	1	1	yes	characteristic, direct product factor
77	C₂	—	25	2	no	—
78	C₂	C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₅⋊D₅)⋊(C₃₇⋊C₃)
commutator, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₃₇	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²⋊(C₃₇⋊C₃)	C₂	semidirect
2. C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃)	C₂	direct (N × H)
3. C₂×C₃×C₃₇×C₅⋊D₅	C₃	semidirect
4. C₂×(C₅⋊D₅)⋊(C₃₇⋊C₃)	C₃	direct (N × H)
5. C₃×C₅²⋊(C₃₇⋊C₃)	C₂²	semidirect
6. C₃×C₃₇×C₅⋊D₅	C₂×C₃	semidirect
7. C₂×C₅²⋊(C₃₇⋊C₃)	C₂×C₃	semidirect
8. (C₅⋊D₅)⋊(C₃₇⋊C₃)	C₂×C₃	direct (N × H)
9. C₂×C₃×C₅²×C₃₇	C₂×C₃	semidirect
10. C₂×C₃₇×C₅⋊D₅	C₃²	semidirect
11. C₅²⋊(C₃₇⋊C₃)	C₂²×C₃	semidirect
12. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
14. C₂×C₅²×C₃₇	C₂×C₃²	semidirect
15. C₃₇×C₅⋊D₅	C₂×C₃²	semidirect
19. C₅²×C₃₇	C₂²×C₃²	semidirect
32. C₂×C₃×C₅⋊D₅	C₃₇⋊C₃	semidirect
36. C₂×C₃×C₃₇	(C₅⋊D₅)⋊C₃	semidirect
40. C₂×C₃×C₅²	C₂×C₃₇⋊C₃	semidirect
41. C₃×C₅⋊D₅	C₂×C₃₇⋊C₃	semidirect
45. C₃×C₃₇	C₂×(C₅⋊D₅)⋊C₃	semidirect
47. C₂×C₅⋊D₅	C₃×C₃₇⋊C₃	semidirect
48. C₃×C₅²	C₂²×C₃₇⋊C₃	semidirect
51. C₂×C₃₇	C₃×(C₅⋊D₅)⋊C₃	semidirect
53. C₂×C₅²	C₂×C₃×C₃₇⋊C₃	semidirect
54. C₅⋊D₅	C₂×C₃×C₃₇⋊C₃	semidirect
55. C₃₇	C₂×C₃×(C₅⋊D₅)⋊C₃	semidirect
59. C₅²	C₂²×C₃×C₃₇⋊C₃	semidirect
72. C₂×C₃	(C₅⋊D₅)⋊(C₃₇⋊C₃)	direct (N × H)
76. C₃	C₂×(C₅⋊D₅)⋊(C₃₇⋊C₃)	direct (N × H)
78. C₂	C₃×(C₅⋊D₅)⋊(C₃₇⋊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×C₅²⋊(C₃₇⋊C₃)	2
C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃)	2
C₂×C₃×C₃₇×C₅⋊D₅	3
C₂×(C₅⋊D₅)⋊(C₃₇⋊C₃)	3
C₂²×C₃×C₃₇⋊C₃	25
C₂×C₃×(C₅⋊D₅)⋊C₃	37

Subgroups — C2×C3×(C5⋊D5)⋊(C37⋊C3) #45686336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×(C₅⋊D₅)⋊(C₃₇⋊C₃) #45686336