Subgroups — C₅²×(C₃⋊D₃)⋊D₃₇ #45686444

Order 33300.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×D₁₁₁	C₂	1	2	yes	maximal, complemented
2	C₅²×C₃⋊D₃×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×D₃×D₃₇	—	3	2	no	maximal
4	C₃²×C₅²×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅×(C₃⋊D₃)⋊D₃₇	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₁₁₁	D₃	1	2	yes	complemented
7	C₅²×D₃×C₃₇	—	3	2	no	—
8	C₅²×D₃×C₃₇	—	6	1	no	—
9	C₃×C₅²×D₃₇	—	3	2	no	—
10	C₂×C₅²×D₃₇	—	9	1	no	Hall
11	C₅×C₃⋊D₃×C₃₇	C₂×C₅	1	6	yes	complemented
12	C₃×C₅×D₁₁₁	C₂×C₅	1	12	yes	complemented
13	C₃×C₅²×C₃₇	—	2	1	no	—
14	C₃×C₅²×C₃₇	C₂×D₃	1	2	yes	complemented
15	C₅×D₃×D₃₇	—	3	12	no	—
16	C₂×C₅²×C₃₇	—	9	1	no	—
17	C₅²×D₃₇	—	3	2	no	—
18	C₃²×C₅×C₃₇	C₂²×C₅	1	6	yes	complemented
19	(C₃⋊D₃)⋊D₃₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
20	C₅×D₁₁₁	C₅×D₃	1	12	yes	complemented
21	C₅×D₃×C₃₇	—	3	12	no	—
22	C₅×D₃×C₃₇	—	6	6	no	—
23	C₃×C₅×D₃₇	—	3	12	no	—
24	C₅²×C₃₇	D₃²	1	1	yes	characteristic, complemented, Hall
25	C₅²×D₃²	—	37	1	no	maximal, Hall
26	C₂×C₅×D₃₇	—	9	6	no	—
27	C₃⋊D₃×C₃₇	C₂×C₅²	1	1	yes	characteristic, complemented
28	C₃×D₁₁₁	C₂×C₅²	1	2	yes	complemented
29	C₃×C₅×C₃₇	—	2	6	no	—
30	C₃×C₅×C₃₇	C₂×C₅×D₃	1	12	yes	complemented
31	C₅²×C₃⋊D₃	D₃₇	1	1	yes	characteristic, complemented
32	C₃×C₅²×D₃	—	37	2	no	—
33	D₃×D₃₇	—	3	2	no	—
34	C₂×C₅×C₃₇	—	9	6	no	—
35	C₅×D₃₇	—	3	12	no	—
36	C₃²×C₃₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
37	C₂×C₅²×D₃	—	111	2	no	—
38	C₃²×C₅²	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
39	D₁₁₁	C₅²×D₃	1	2	yes	complemented
40	C₃×D₃₇	—	3	2	no	—
41	D₃×C₃₇	—	3	2	no	—
42	D₃×C₃₇	—	6	1	no	—
43	C₅×C₃₇	C₅×D₃²	1	6	yes	complemented
44	C₅×D₃²	—	37	6	no	—
45	C₂×C₃×C₅²	—	111	2	no	—
46	C₅²×D₃	—	3	2	no	—
47	C₅²×D₃	—	6	1	no	—
48	C₅²×D₃	—	37	2	no	—
49	C₂×D₃₇	—	9	1	no	Hall
50	C₃×C₃₇	—	2	1	no	—
51	C₃×C₃₇	C₂×C₅²×D₃	1	2	yes	complemented
52	C₂²×C₅²	—	333	1	no	Hall
53	C₅×C₃⋊D₃	C₅×D₃₇	1	6	yes	complemented
54	C₃×C₅×D₃	—	37	12	no	—
55	C₃×C₅²	—	2	1	no	—
56	C₃×C₅²	D₃×D₃₇	1	2	yes	complemented
57	C₂×C₃₇	—	9	1	no	—
58	D₃₇	—	3	2	no	—
59	C₂×C₅×D₃	—	111	12	no	—
60	C₂×C₅²	—	9	1	no	—
61	C₂×C₅²	—	111	2	no	—
62	C₃²×C₅	C₂×C₅×D₃₇	1	6	yes	complemented
63	C₃₇	C₅²×D₃²	1	1	yes	characteristic, complemented, Sylow
64	D₃²	—	37	1	no	Hall
65	C₂×C₃×C₅	—	111	12	no	—
66	C₅×D₃	—	3	12	no	—
67	C₅×D₃	—	6	6	no	—
68	C₅×D₃	—	37	12	no	—
69	C₅²	(C₃⋊D₃)⋊D₃₇	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₅	—	333	6	no	—
71	C₃⋊D₃	C₅²×D₃₇	1	1	yes	characteristic, complemented
72	C₃×D₃	—	37	2	no	—
73	C₃×C₅	—	2	6	no	—
74	C₃×C₅	C₅×D₃×D₃₇	1	12	yes	complemented
75	C₂×D₃	—	111	2	no	—
76	C₂×C₅	—	9	6	no	—
77	C₂×C₅	—	111	12	no	—
78	C₃²	C₂×C₅²×D₃₇	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃	—	111	2	no	—
80	D₃	—	3	2	no	—
81	D₃	—	6	1	no	—
82	D₃	—	37	2	no	—
83	C₅	C₅×(C₃⋊D₃)⋊D₃₇	1	6	yes	direct product factor
84	C₂²	—	333	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	C₅²×D₃×D₃₇	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	111	2	no	—

	subgroup	G/H
center, hypercenter	C₅²	(C₃⋊D₃)⋊D₃₇
commutator, cosocle, hypocenter	C₃²×C₃₇	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₂²
LCS	G ▷ C₃²×C₃₇
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×D₁₁₁	C₂	semidirect
2. C₅²×C₃⋊D₃×C₃₇	C₂	semidirect
4. C₃²×C₅²×C₃₇	C₂²	semidirect
5. C₅×(C₃⋊D₃)⋊D₃₇	C₅	direct (N × H)
6. C₅²×D₁₁₁	D₃	semidirect
11. C₅×C₃⋊D₃×C₃₇	C₂×C₅	semidirect
12. C₃×C₅×D₁₁₁	C₂×C₅	semidirect
14. C₃×C₅²×C₃₇	C₂×D₃	semidirect
18. C₃²×C₅×C₃₇	C₂²×C₅	semidirect
19. (C₃⋊D₃)⋊D₃₇	C₅²	direct (N × H)
20. C₅×D₁₁₁	C₅×D₃	semidirect
24. C₅²×C₃₇	D₃²	semidirect
27. C₃⋊D₃×C₃₇	C₂×C₅²	semidirect
28. C₃×D₁₁₁	C₂×C₅²	semidirect
30. C₃×C₅×C₃₇	C₂×C₅×D₃	semidirect
31. C₅²×C₃⋊D₃	D₃₇	semidirect
36. C₃²×C₃₇	C₂²×C₅²	semidirect
38. C₃²×C₅²	C₂×D₃₇	semidirect
39. D₁₁₁	C₅²×D₃	semidirect
43. C₅×C₃₇	C₅×D₃²	semidirect
51. C₃×C₃₇	C₂×C₅²×D₃	semidirect
53. C₅×C₃⋊D₃	C₅×D₃₇	semidirect
56. C₃×C₅²	D₃×D₃₇	semidirect
62. C₃²×C₅	C₂×C₅×D₃₇	semidirect
63. C₃₇	C₅²×D₃²	semidirect
69. C₅²	(C₃⋊D₃)⋊D₃₇	direct (N × H)
71. C₃⋊D₃	C₅²×D₃₇	semidirect
74. C₃×C₅	C₅×D₃×D₃₇	semidirect
78. C₃²	C₂×C₅²×D₃₇	semidirect
83. C₅	C₅×(C₃⋊D₃)⋊D₃₇	direct (N × H)
86. C₃	C₅²×D₃×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×D₁₁₁	2
C₅²×C₃⋊D₃×C₃₇	2
C₅²×D₃×D₃₇	3
C₅×(C₃⋊D₃)⋊D₃₇	5
C₅²×D₃²	37

Subgroups — C52×(C3⋊D3)⋊D37 #45686444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×(C₃⋊D₃)⋊D₃₇ #45686444