Subgroups — C₂×C₃×C₂₅×D₃₇⋊C₃ #45685865

Order 33300.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂₅×C₃₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂₅×D₃₇⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₂₅×D₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₅×D₃₇⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₂₅×C₃₇⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×C₃×C₅×D₃₇⋊C₃	C₅	1	1	yes	maximal, characteristic
7	C₃×C₂₅×D₃₇	C₂×C₃	1	2	yes	complemented
8	C₂×C₂₅×C₃₇⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂₅×D₃₇⋊C₃	C₂×C₃	1	6	yes	direct product factor
10	C₂×C₃×C₂₅×C₃₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₂₅×D₃₇	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×C₃×C₅×C₃₇⋊C₃	C₂×C₅	1	1	yes	characteristic
13	C₃×C₅×D₃₇⋊C₃	C₂×C₅	1	2	yes	—
14	C₂₅×C₃₇⋊C₃	C₂²×C₃	1	3	yes	complemented
15	C₃×C₂₅×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₅×D₃₇	C₃×C₅	1	1	yes	characteristic
17	C₂×C₅×D₃₇⋊C₃	C₃×C₅	1	3	yes	—
18	C₂×C₂₅×C₃₇	C₂×C₃²	1	1	yes	characteristic, complemented
19	C₂₅×D₃₇	C₂×C₃²	1	2	yes	complemented
20	C₃×C₅×C₃₇⋊C₃	C₂²×C₅	1	1	yes	characteristic
21	C₂×C₃×D₃₇⋊C₃	C₂₅	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₃×C₅×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic
23	C₃×C₅×D₃₇	C₂×C₃×C₅	1	2	yes	—
24	C₂×C₅×C₃₇⋊C₃	C₂×C₃×C₅	1	3	yes	—
25	C₅×D₃₇⋊C₃	C₂×C₃×C₅	1	6	yes	—
26	C₂₅×C₃₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
27	C₂²×C₃²×C₂₅	—	37	1	no	maximal, Hall
28	C₂×C₅×D₃₇	C₃²×C₅	1	1	yes	characteristic
29	C₂×C₃×C₃₇⋊C₃	C₂×C₂₅	1	1	yes	characteristic, complemented
30	C₃×D₃₇⋊C₃	C₂×C₂₅	1	2	yes	direct product factor
31	C₃×C₅×C₃₇	C₂²×C₃×C₅	1	1	yes	characteristic
32	C₅×C₃₇⋊C₃	C₂²×C₃×C₅	1	3	yes	—
33	C₂×C₃²×C₂₅	—	37	2	no	—
34	C₂×C₃²×C₂₅	—	37	1	no	—
35	C₂×C₃×D₃₇	C₃×C₂₅	1	1	yes	characteristic, complemented
36	C₂×D₃₇⋊C₃	C₃×C₂₅	1	3	yes	direct product factor
37	C₂×C₅×C₃₇	C₂×C₃²×C₅	1	1	yes	characteristic
38	C₅×D₃₇	C₂×C₃²×C₅	1	2	yes	—
39	C₃×C₃₇⋊C₃	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
40	C₂²×C₃×C₂₅	—	37	3	no	—
41	C₂²×C₃×C₂₅	—	37	1	no	—
42	C₃²×C₂₅	—	37	1	no	Hall
43	C₂×C₃×C₃₇	C₂×C₃×C₂₅	1	1	yes	characteristic, complemented
44	C₃×D₃₇	C₂×C₃×C₂₅	1	2	yes	complemented
45	C₂×C₃₇⋊C₃	C₂×C₃×C₂₅	1	3	yes	complemented
46	D₃₇⋊C₃	C₂×C₃×C₂₅	1	6	yes	direct product factor
47	C₅×C₃₇	C₂²×C₃²×C₅	1	1	yes	characteristic
48	C₂²×C₃²×C₅	—	37	1	no	—
49	C₂×C₃×C₂₅	—	37	6	no	—
50	C₂×C₃×C₂₅	—	37	3	no	—
51	C₂×C₃×C₂₅	—	37	2	no	—
52	C₂×C₃×C₂₅	D₃₇⋊C₃	1	1	yes	characteristic, direct product factor
53	C₂×D₃₇	C₃²×C₂₅	1	1	yes	characteristic, complemented, Hall
54	C₃×C₃₇	C₂²×C₃×C₂₅	1	1	yes	characteristic, complemented
55	C₃₇⋊C₃	C₂²×C₃×C₂₅	1	3	yes	complemented
56	C₂²×C₂₅	—	37	1	no	Hall
57	C₂×C₃²×C₅	—	37	2	no	—
58	C₂×C₃²×C₅	—	37	1	no	—
59	C₃×C₂₅	—	37	3	no	—
60	C₃×C₂₅	C₂×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂×C₃₇	C₂×C₃²×C₂₅	1	1	yes	characteristic, complemented
62	D₃₇	C₂×C₃²×C₂₅	1	2	yes	complemented
63	C₂²×C₃×C₅	—	37	3	no	—
64	C₂²×C₃×C₅	—	37	1	no	—
65	C₂×C₂₅	—	37	2	no	—
66	C₂×C₂₅	C₃×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor
67	C₃²×C₅	—	37	1	no	—
68	C₃₇	C₂²×C₃²×C₂₅	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₃²	—	37	1	no	Hall
70	C₂×C₃×C₅	—	37	6	no	—
71	C₂×C₃×C₅	—	37	2	no	—
72	C₂×C₃×C₅	—	37	3	no	—
73	C₂×C₃×C₅	C₅×D₃₇⋊C₃	1	1	yes	characteristic
74	C₂₅	C₂×C₃×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₅	—	37	1	no	—
76	C₂×C₃²	—	37	2	no	—
77	C₂×C₃²	—	37	1	no	—
78	C₃×C₅	—	37	3	no	—
79	C₃×C₅	C₂×C₅×D₃₇⋊C₃	1	1	yes	characteristic
80	C₂²×C₃	—	37	3	no	—
81	C₂²×C₃	—	37	1	no	—
82	C₂×C₅	—	37	2	no	—
83	C₂×C₅	C₃×C₅×D₃₇⋊C₃	1	1	yes	characteristic
84	C₃²	—	37	1	no	Sylow
85	C₂×C₃	—	37	6	no	—
86	C₂×C₃	—	37	2	no	—
87	C₂×C₃	—	37	3	no	—
88	C₂×C₃	C₂₅×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor
89	C₅	C₂×C₃×C₅×D₃₇⋊C₃	1	1	yes	characteristic
90	C₂²	—	37	1	no	Sylow
91	C₃	—	37	3	no	—
92	C₃	C₂×C₂₅×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor
93	C₂	—	37	2	no	—
94	C₂	C₃×C₂₅×D₃₇⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₂₅	D₃₇⋊C₃
commutator, hypocenter	C₃₇	C₂²×C₃²×C₂₅
Frattini	C₅	C₂×C₃×C₅×D₃₇⋊C₃
Fitting	C₂×C₃×C₂₅×C₃₇	C₂×C₃
socle	C₂×C₃×C₅×C₃₇	C₂×C₃×C₅
prime radical	C₂×C₃×C₅×D₃₇⋊C₃	C₅
cosocle	C₅×C₃₇	C₂²×C₃²×C₅
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₃×C₂₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₂₅×C₃₇⋊C₃	C₂	semidirect
2. C₃×C₂₅×D₃₇⋊C₃	C₂	direct (N × H)
3. C₂×C₃×C₂₅×D₃₇	C₃	semidirect
4. C₂×C₂₅×D₃₇⋊C₃	C₃	direct (N × H)
5. C₃×C₂₅×C₃₇⋊C₃	C₂²	semidirect
7. C₃×C₂₅×D₃₇	C₂×C₃	semidirect
8. C₂×C₂₅×C₃₇⋊C₃	C₂×C₃	semidirect
9. C₂₅×D₃₇⋊C₃	C₂×C₃	direct (N × H)
10. C₂×C₃×C₂₅×C₃₇	C₂×C₃	semidirect
11. C₂×C₂₅×D₃₇	C₃²	semidirect
14. C₂₅×C₃₇⋊C₃	C₂²×C₃	semidirect
15. C₃×C₂₅×C₃₇	C₂²×C₃	semidirect
18. C₂×C₂₅×C₃₇	C₂×C₃²	semidirect
19. C₂₅×D₃₇	C₂×C₃²	semidirect
21. C₂×C₃×D₃₇⋊C₃	C₂₅	direct (N × H)
26. C₂₅×C₃₇	C₂²×C₃²	semidirect
29. C₂×C₃×C₃₇⋊C₃	C₂×C₂₅	semidirect
30. C₃×D₃₇⋊C₃	C₂×C₂₅	direct (N × H)
35. C₂×C₃×D₃₇	C₃×C₂₅	semidirect
36. C₂×D₃₇⋊C₃	C₃×C₂₅	direct (N × H)
39. C₃×C₃₇⋊C₃	C₂²×C₂₅	semidirect
43. C₂×C₃×C₃₇	C₂×C₃×C₂₅	semidirect
44. C₃×D₃₇	C₂×C₃×C₂₅	semidirect
45. C₂×C₃₇⋊C₃	C₂×C₃×C₂₅	semidirect
46. D₃₇⋊C₃	C₂×C₃×C₂₅	direct (N × H)
52. C₂×C₃×C₂₅	D₃₇⋊C₃	direct (N × H)
53. C₂×D₃₇	C₃²×C₂₅	semidirect
54. C₃×C₃₇	C₂²×C₃×C₂₅	semidirect
55. C₃₇⋊C₃	C₂²×C₃×C₂₅	semidirect
60. C₃×C₂₅	C₂×D₃₇⋊C₃	direct (N × H)
61. C₂×C₃₇	C₂×C₃²×C₂₅	semidirect
62. D₃₇	C₂×C₃²×C₂₅	semidirect
66. C₂×C₂₅	C₃×D₃₇⋊C₃	direct (N × H)
68. C₃₇	C₂²×C₃²×C₂₅	semidirect
74. C₂₅	C₂×C₃×D₃₇⋊C₃	direct (N × H)
88. C₂×C₃	C₂₅×D₃₇⋊C₃	direct (N × H)
92. C₃	C₂×C₂₅×D₃₇⋊C₃	direct (N × H)
94. C₂	C₃×C₂₅×D₃₇⋊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×C₂₅×C₃₇⋊C₃	2
C₃×C₂₅×D₃₇⋊C₃	2
C₂×C₃×C₂₅×D₃₇	3
C₂×C₂₅×D₃₇⋊C₃	3
C₂×C₃×C₅×D₃₇⋊C₃	5
C₂²×C₃²×C₂₅	37

Subgroups — C2×C3×C25×D37⋊C3 #45685865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂₅×D₃₇⋊C₃ #45685865