Subgroups — C₃₇⋊C₃×#3874⋊C₂ #45686161

Order 33300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇⋊C₃×(C₅⋊D₅)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₇⋊C₃×#3874	C₂	1	2	yes	maximal, complemented
3	C₃₇×#3874⋊C₂	C₃	1	1	yes	maximal, characteristic, complemented
4	D₅²×C₃₇⋊C₃	—	3	1	no	maximal
5	C₅²⋊C₃×C₃₇⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	(C₅⋊D₅)⋊(C₃₇⋊C₃)	—	2	1	no	—
7	C₃₇×(C₅⋊D₅)⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃₇×#3874	C₂×C₃	1	2	yes	complemented
9	C₅⋊D₅×C₃₇⋊C₃	D₃	1	1	yes	characteristic, complemented
10	C₅×D₅×C₃₇⋊C₃	—	3	2	no	—
11	D₅²×C₃₇	—	3	1	no	Hall
12	C₅²⋊(C₃₇⋊C₃)	—	2	1	no	—
13	C₃₇×C₅²⋊C₃	C₂²×C₃	1	1	yes	characteristic, complemented
14	C₅²×C₃₇⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
15	C₂×D₅×C₃₇⋊C₃	—	15	2	no	—
16	C₃₇×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic, complemented
17	C₅×D₅×C₃₇	—	3	2	no	—
18	C₂×D₃×C₃₇⋊C₃	—	25	1	no	maximal, Hall
19	C₂×C₅×C₃₇⋊C₃	—	15	2	no	—
20	D₅×C₃₇⋊C₃	—	3	2	no	—
21	D₅×C₃₇⋊C₃	—	15	2	no	—
22	C₅²×C₃₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
23	C₃×#3874⋊C₂	—	37	1	no	maximal, Hall
24	C₂×D₅×C₃₇	—	15	2	no	—
25	C₂×C₃×C₃₇⋊C₃	—	25	1	no	—
26	D₃×C₃₇⋊C₃	—	25	2	no	—
27	C₅×C₃₇⋊C₃	—	3	2	no	—
28	C₃×(C₅⋊D₅)⋊C₃	—	37	1	no	—
29	C₃×#3874	—	37	2	no	—
30	C₂×D₃×C₃₇	—	25	1	no	—
31	C₂²×C₃₇⋊C₃	—	75	1	no	—
32	C₂×C₅×C₃₇	—	15	2	no	—
33	D₅×C₃₇	—	3	2	no	—
34	D₅×C₃₇	—	15	2	no	—
35	C₃×C₃₇⋊C₃	—	25	1	no	Hall
36	C₃×D₅²	—	111	1	no	—
37	#3874⋊C₂	C₃₇⋊C₃	1	1	yes	characteristic, direct product factor
38	C₃×C₅²⋊C₃	—	37	1	no	Hall
39	C₂×C₃×C₃₇	—	25	1	no	—
40	D₃×C₃₇	—	25	2	no	—
41	C₂×C₃₇⋊C₃	—	15	2	no	—
42	C₂×C₃₇⋊C₃	—	25	1	no	—
43	C₂×C₃₇⋊C₃	—	50	1	no	—
44	C₅×C₃₇	—	3	2	no	—
45	C₃×C₅⋊D₅	—	37	1	no	—
46	C₃×C₅×D₅	—	111	2	no	—
47	(C₅⋊D₅)⋊C₃	—	74	1	no	—
48	(C₅⋊D₅)⋊C₃	C₂×C₃₇⋊C₃	1	1	yes	characteristic, complemented
49	#3874	C₂×C₃₇⋊C₃	1	2	yes	complemented
50	C₂²×C₃₇	—	75	1	no	Hall
51	C₃×C₃₇	—	25	1	no	—
52	C₃₇⋊C₃	—	50	1	no	—
53	C₃₇⋊C₃	#3874⋊C₂	1	1	yes	characteristic, direct product factor
54	D₅²	—	3	1	no	Hall
55	C₃×C₅²	—	37	1	no	—
56	C₅²⋊C₃	—	74	1	no	—
57	C₅²⋊C₃	C₂²×C₃₇⋊C₃	1	1	yes	characteristic, complemented
58	C₂×C₃₇	—	15	2	no	—
59	C₂×C₃₇	—	25	1	no	—
60	C₂×C₃×D₅	—	555	2	no	—
61	C₅⋊D₅	D₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
62	C₅×D₅	—	3	2	no	—
63	C₃₇	C₃×#3874⋊C₂	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃×D₃	—	925	1	no	Hall
65	C₂×C₃×C₅	—	555	2	no	—
66	C₃×D₅	—	555	2	no	—
67	C₃×D₅	—	111	2	no	—
68	C₅²	C₂×D₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₅	—	15	2	no	—
70	C₂×C₃²	—	925	1	no	—
71	C₃×D₃	—	925	2	no	—
72	C₃×C₅	—	111	2	no	—
73	C₂²×C₃	—	2775	1	no	—
74	C₂×D₃	—	25	1	no	—
75	C₂×C₅	—	15	2	no	—
76	D₅	—	3	2	no	—
77	D₅	—	15	2	no	—
78	C₃²	—	925	1	no	Sylow
79	C₂×C₃	—	25	1	no	—
80	C₂×C₃	—	555	2	no	—
81	C₂×C₃	—	1850	1	no	—
82	C₂×C₃	—	925	1	no	—
83	D₃	—	25	2	no	—
84	C₅	—	3	2	no	—
85	C₂²	—	75	1	no	Sylow
86	C₃	—	25	1	no	—
87	C₃	—	37	1	no	—
88	C₃	—	1850	1	no	—
89	C₂	—	15	2	no	—
90	C₂	—	25	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₃₇⋊C₃×(C₅⋊D₅)⋊C₃	C₂	semidirect
2. C₃₇⋊C₃×#3874	C₂	semidirect
3. C₃₇×#3874⋊C₂	C₃	semidirect
5. C₅²⋊C₃×C₃₇⋊C₃	C₂²	semidirect
7. C₃₇×(C₅⋊D₅)⋊C₃	C₂×C₃	semidirect
8. C₃₇×#3874	C₂×C₃	semidirect
9. C₅⋊D₅×C₃₇⋊C₃	D₃	semidirect
13. C₃₇×C₅²⋊C₃	C₂²×C₃	semidirect
14. C₅²×C₃₇⋊C₃	C₂×D₃	semidirect
16. C₃₇×C₅⋊D₅	C₃×D₃	semidirect
22. C₅²×C₃₇	C₂×C₃×D₃	semidirect
37. #3874⋊C₂	C₃₇⋊C₃	direct (N × H)
48. (C₅⋊D₅)⋊C₃	C₂×C₃₇⋊C₃	semidirect
49. #3874	C₂×C₃₇⋊C₃	semidirect
53. C₃₇⋊C₃	#3874⋊C₂	direct (N × H)
57. C₅²⋊C₃	C₂²×C₃₇⋊C₃	semidirect
61. C₅⋊D₅	D₃×C₃₇⋊C₃	semidirect
63. C₃₇	C₃×#3874⋊C₂	semidirect
68. C₅²	C₂×D₃×C₃₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃₇⋊C₃×(C₅⋊D₅)⋊C₃	2
C₃₇⋊C₃×#3874	2
C₃₇×#3874⋊C₂	3
D₅²×C₃₇⋊C₃	3
C₂×D₃×C₃₇⋊C₃	25
C₃×#3874⋊C₂	37

Subgroups — C37⋊C3×#3874⋊C2 #45686161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇⋊C₃×#3874⋊C₂ #45686161