Subgroups — C₂×C₅²×D₂₃₁ #37528751

Order 23100.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×D₂₃₁	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×C₅²×C₇×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₇₇	—	3	1	no	maximal, Hall
4	C₃×C₅²×C₇×C₁₁	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×D₂₃₁	C₅	1	6	yes	maximal, direct product factor
6	C₅²×D₇₇	—	3	2	no	—
7	C₂×C₅²×C₇×C₁₁	D₃	1	1	yes	characteristic, complemented
8	C₂×C₅²×D₃₃	—	7	1	no	maximal, Hall
9	C₅×D₂₃₁	C₂×C₅	1	12	yes	direct product factor
10	C₂×C₃×C₅×C₇×C₁₁	C₂×C₅	1	6	yes	complemented
11	C₂×C₅²×D₂₁	—	11	1	no	maximal, Hall
12	C₅²×C₇×C₁₁	C₂×D₃	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₅²×C₁₁	D₇	1	1	yes	characteristic, complemented
14	C₅²×D₃₃	—	7	2	no	—
15	C₂×C₅×D₇₇	—	3	6	no	—
16	C₃×C₅×C₇×C₁₁	C₂²×C₅	1	6	yes	complemented
17	C₂×C₅²×D₁₁	—	21	1	no	Hall
18	C₂×C₃×C₅²×C₇	D₁₁	1	1	yes	characteristic, complemented
19	C₅²×D₂₁	—	11	2	no	—
20	C₂×D₂₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
21	C₃×C₅²×C₁₁	C₂×D₇	1	1	yes	characteristic, complemented, Hall
22	C₂×C₅×C₇×C₁₁	C₅×D₃	1	6	yes	complemented
23	C₅×D₇₇	—	3	12	no	—
24	C₂×C₅²×D₇	—	33	1	no	Hall
25	C₂×C₅×D₃₃	—	7	6	no	—
26	C₂×C₅²×C₁₁	D₂₁	1	1	yes	characteristic, complemented
27	C₅²×D₁₁	—	21	2	no	—
28	C₃×C₅²×C₇	C₂×D₁₁	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₇×C₁₁	C₂×C₅²	1	1	yes	characteristic, complemented
30	D₂₃₁	C₂×C₅²	1	2	yes	direct product factor
31	C₂×C₅×D₂₁	—	11	6	no	—
32	C₅×C₇×C₁₁	C₂×C₅×D₃	1	6	yes	complemented
33	C₂×C₅²×C₇	D₃₃	1	1	yes	characteristic, complemented
34	C₅²×D₇	—	33	2	no	—
35	C₂×C₃×C₅×C₁₁	C₅×D₇	1	6	yes	complemented
36	C₅×D₃₃	—	7	12	no	—
37	C₂×D₇₇	—	3	1	no	Hall
38	C₂×C₅²×D₃	—	77	1	no	Hall
39	C₅²×C₁₁	C₂×D₂₁	1	1	yes	characteristic, complemented, Hall
40	C₃×C₇×C₁₁	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
41	C₂×C₅×D₁₁	—	21	6	no	—
42	C₂×C₃×C₅×C₇	C₅×D₁₁	1	6	yes	complemented
43	C₅×D₂₁	—	11	12	no	—
44	C₅²×C₇	C₂×D₃₃	1	1	yes	characteristic, complemented, Hall
45	C₃×C₅×C₁₁	C₂×C₅×D₇	1	6	yes	complemented
46	C₂×C₇×C₁₁	C₅²×D₃	1	1	yes	characteristic, complemented
47	D₇₇	—	3	2	no	—
48	C₂×C₃×C₅²	D₇₇	1	1	yes	characteristic, complemented
49	C₅²×D₃	—	77	2	no	—
50	C₂×C₅×D₇	—	33	6	no	—
51	C₂×D₃₃	—	7	1	no	Hall
52	C₂×C₅×C₁₁	C₅×D₂₁	1	6	yes	complemented
53	C₅×D₁₁	—	21	12	no	—
54	C₃×C₅×C₇	C₂×C₅×D₁₁	1	6	yes	complemented
55	C₂²×C₅²	—	231	1	no	Hall
56	C₂×D₂₁	—	11	1	no	Hall
57	C₇×C₁₁	C₂×C₅²×D₃	1	1	yes	characteristic, complemented, Hall
58	C₃×C₅²	C₂×D₇₇	1	1	yes	characteristic, complemented, Hall
59	C₂×C₅×C₇	C₅×D₃₃	1	6	yes	complemented
60	C₅×D₇	—	33	12	no	—
61	C₂×C₃×C₁₁	C₅²×D₇	1	1	yes	characteristic, complemented
62	D₃₃	—	7	2	no	—
63	C₂×C₅×D₃	—	77	6	no	—
64	C₅×C₁₁	C₂×C₅×D₂₁	1	6	yes	complemented
65	C₂×C₅²	—	231	2	no	—
66	C₂×C₅²	D₂₃₁	1	1	yes	characteristic, direct product factor
67	C₂×D₁₁	—	21	1	no	Hall
68	C₂×C₃×C₇	C₅²×D₁₁	1	1	yes	characteristic, complemented
69	D₂₁	—	11	2	no	—
70	C₅×C₇	C₂×C₅×D₃₃	1	6	yes	complemented
71	C₃×C₁₁	C₂×C₅²×D₇	1	1	yes	characteristic, complemented, Hall
72	C₂×C₃×C₅	C₅×D₇₇	1	6	yes	complemented
73	C₅×D₃	—	77	12	no	—
74	C₂×D₇	—	33	1	no	Hall
75	C₅²	C₂×D₂₃₁	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₁₁	C₅²×D₂₁	1	1	yes	characteristic, complemented
77	D₁₁	—	21	2	no	—
78	C₃×C₇	C₂×C₅²×D₁₁	1	1	yes	characteristic, complemented, Hall
79	C₂²×C₅	—	231	6	no	—
80	C₃×C₅	C₂×C₅×D₇₇	1	6	yes	complemented
81	C₂×C₇	C₅²×D₃₃	1	1	yes	characteristic, complemented
82	D₇	—	33	2	no	—
83	C₂×D₃	—	77	1	no	Hall
84	C₁₁	C₂×C₅²×D₂₁	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₅	—	231	12	no	—
86	C₂×C₅	C₅×D₂₃₁	1	6	yes	direct product factor
87	C₇	C₂×C₅²×D₃₃	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	C₅²×D₇₇	1	1	yes	characteristic, complemented
89	D₃	—	77	2	no	—
90	C₅	C₂×C₅×D₂₃₁	1	6	yes	direct product factor
91	C₂²	—	231	1	no	Sylow
92	C₃	C₂×C₅²×D₇₇	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	231	2	no	—
94	C₂	C₅²×D₂₃₁	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅²	D₂₃₁
commutator, cosocle, hypocenter	C₃×C₇×C₁₁	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₇×C₁₁	C₂
LCS	G ▷ C₃×C₇×C₁₁
UCS	C₁ ◁ C₂×C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×D₂₃₁	C₂	direct (N × H)
2. C₂×C₃×C₅²×C₇×C₁₁	C₂	semidirect
4. C₃×C₅²×C₇×C₁₁	C₂²	semidirect
5. C₂×C₅×D₂₃₁	C₅	direct (N × H)
7. C₂×C₅²×C₇×C₁₁	D₃	semidirect
9. C₅×D₂₃₁	C₂×C₅	direct (N × H)
10. C₂×C₃×C₅×C₇×C₁₁	C₂×C₅	semidirect
12. C₅²×C₇×C₁₁	C₂×D₃	semidirect
13. C₂×C₃×C₅²×C₁₁	D₇	semidirect
16. C₃×C₅×C₇×C₁₁	C₂²×C₅	semidirect
18. C₂×C₃×C₅²×C₇	D₁₁	semidirect
20. C₂×D₂₃₁	C₅²	direct (N × H)
21. C₃×C₅²×C₁₁	C₂×D₇	semidirect
22. C₂×C₅×C₇×C₁₁	C₅×D₃	semidirect
26. C₂×C₅²×C₁₁	D₂₁	semidirect
28. C₃×C₅²×C₇	C₂×D₁₁	semidirect
29. C₂×C₃×C₇×C₁₁	C₂×C₅²	semidirect
30. D₂₃₁	C₂×C₅²	direct (N × H)
32. C₅×C₇×C₁₁	C₂×C₅×D₃	semidirect
33. C₂×C₅²×C₇	D₃₃	semidirect
35. C₂×C₃×C₅×C₁₁	C₅×D₇	semidirect
39. C₅²×C₁₁	C₂×D₂₁	semidirect
40. C₃×C₇×C₁₁	C₂²×C₅²	semidirect
42. C₂×C₃×C₅×C₇	C₅×D₁₁	semidirect
44. C₅²×C₇	C₂×D₃₃	semidirect
45. C₃×C₅×C₁₁	C₂×C₅×D₇	semidirect
46. C₂×C₇×C₁₁	C₅²×D₃	semidirect
48. C₂×C₃×C₅²	D₇₇	semidirect
52. C₂×C₅×C₁₁	C₅×D₂₁	semidirect
54. C₃×C₅×C₇	C₂×C₅×D₁₁	semidirect
57. C₇×C₁₁	C₂×C₅²×D₃	semidirect
58. C₃×C₅²	C₂×D₇₇	semidirect
59. C₂×C₅×C₇	C₅×D₃₃	semidirect
61. C₂×C₃×C₁₁	C₅²×D₇	semidirect
64. C₅×C₁₁	C₂×C₅×D₂₁	semidirect
66. C₂×C₅²	D₂₃₁	direct (N × H)
68. C₂×C₃×C₇	C₅²×D₁₁	semidirect
70. C₅×C₇	C₂×C₅×D₃₃	semidirect
71. C₃×C₁₁	C₂×C₅²×D₇	semidirect
72. C₂×C₃×C₅	C₅×D₇₇	semidirect
75. C₅²	C₂×D₂₃₁	direct (N × H)
76. C₂×C₁₁	C₅²×D₂₁	semidirect
78. C₃×C₇	C₂×C₅²×D₁₁	semidirect
80. C₃×C₅	C₂×C₅×D₇₇	semidirect
81. C₂×C₇	C₅²×D₃₃	semidirect
84. C₁₁	C₂×C₅²×D₂₁	semidirect
86. C₂×C₅	C₅×D₂₃₁	direct (N × H)
87. C₇	C₂×C₅²×D₃₃	semidirect
88. C₂×C₃	C₅²×D₇₇	semidirect
90. C₅	C₂×C₅×D₂₃₁	direct (N × H)
92. C₃	C₂×C₅²×D₇₇	semidirect
94. C₂	C₅²×D₂₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×D₂₃₁	2
C₂×C₃×C₅²×C₇×C₁₁	2
C₂×C₅²×D₇₇	3
C₂×C₅×D₂₃₁	5
C₂×C₅²×D₃₃	7
C₂×C₅²×D₂₁	11

Subgroups — C2×C52×D231 #37528751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅²×D₂₃₁ #37528751