Subgroups — #120×C₂₃⋊C₁₁ #72247828

Order 8096.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊C₄×C₂₃⋊C₁₁	C₂	1	3	yes	maximal, complemented
2	C₂⁴×C₂₃⋊C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×C₂₃⋊C₁₁	C₂	1	3	yes	maximal, complemented
4	C₂³×C₂₃⋊C₁₁	—	2	6	no	—
5	C₂³×C₂₃⋊C₁₁	C₂²	1	3	yes	—
6	C₂³×C₂₃⋊C₁₁	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₄×C₂₃⋊C₁₁	C₂²	1	3	yes	complemented
8	D₄×C₂₃⋊C₁₁	—	2	6	no	—
9	C₂²×C₂₃⋊C₁₁	—	2	8	no	—
10	C₂²×C₂₃⋊C₁₁	—	2	6	no	—
11	C₂²×C₂₃⋊C₁₁	—	2	3	no	—
12	C₂²×C₂₃⋊C₁₁	D₄	1	6	yes	complemented
13	C₂²×C₂₃⋊C₁₁	C₂³	1	1	yes	characteristic
14	C₄×C₂₃⋊C₁₁	—	2	3	no	—
15	C₂₃×#120	C₁₁	1	1	yes	maximal, characteristic, complemented, Hall
16	C₂×C₂₃⋊C₁₁	—	2	6	no	—
17	C₂×C₂₃⋊C₁₁	—	4	1	no	—
18	C₂×C₂₃⋊C₁₁	C₂×D₄	1	3	yes	—
19	C₂⁴×C₂₃	C₂×C₁₁	1	1	yes	characteristic, complemented
20	C₂×D₄×C₂₃	C₂×C₁₁	1	3	yes	complemented
21	C₂²⋊C₄×C₂₃	C₂×C₁₁	1	3	yes	complemented
22	C₁₁×#120	—	23	1	no	maximal, Hall
23	C₂₃⋊C₁₁	#120	1	1	yes	characteristic, direct product factor, Hall
24	C₂³×C₂₃	—	2	6	no	—
25	C₂³×C₂₃	C₂²×C₁₁	1	3	yes	—
26	C₂³×C₂₃	C₂²×C₁₁	1	1	yes	characteristic, complemented
27	D₄×C₂₃	—	2	6	no	—
28	C₂×C₄×C₂₃	C₂²×C₁₁	1	3	yes	complemented
29	C₂⁴×C₁₁	—	23	1	no	—
30	C₂×D₄×C₁₁	—	23	3	no	—
31	C₁₁×C₂²⋊C₄	—	23	3	no	—
32	C₂²×C₂₃	—	2	8	no	—
33	C₂²×C₂₃	—	2	6	no	—
34	C₂²×C₂₃	—	2	3	no	—
35	C₂²×C₂₃	D₄×C₁₁	1	6	yes	complemented
36	C₂²×C₂₃	C₂³×C₁₁	1	1	yes	characteristic
37	C₄×C₂₃	—	2	3	no	—
38	C₂³×C₁₁	—	23	3	no	—
39	C₂³×C₁₁	—	23	1	no	—
40	C₂³×C₁₁	—	46	6	no	—
41	D₄×C₁₁	—	46	6	no	—
42	C₂×C₄×C₁₁	—	23	3	no	—
43	C₂×C₂₃	—	2	6	no	—
44	C₂×C₂₃	—	4	1	no	—
45	C₂×C₂₃	C₂×D₄×C₁₁	1	3	yes	—
46	C₂²×C₁₁	—	23	6	no	—
47	C₂²×C₁₁	—	23	1	no	—
48	C₂²×C₁₁	—	46	6	no	—
49	C₂²×C₁₁	—	46	3	no	—
50	C₂²×C₁₁	—	46	8	no	—
51	C₄×C₁₁	—	46	3	no	—
52	#120	C₂₃⋊C₁₁	1	1	yes	characteristic, direct product factor, Sylow
53	C₂₃	C₁₁×#120	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₁₁	—	23	3	no	—
55	C₂×C₁₁	—	46	6	no	—
56	C₂×C₁₁	—	92	1	no	—
57	C₂⁴	C₂×C₂₃⋊C₁₁	1	1	yes	characteristic, complemented
58	C₂×D₄	C₂×C₂₃⋊C₁₁	1	3	yes	complemented
59	C₂²⋊C₄	C₂×C₂₃⋊C₁₁	1	3	yes	complemented
60	C₁₁	—	23	1	no	Sylow
61	C₂³	—	2	6	no	—
62	C₂³	C₂²×C₂₃⋊C₁₁	1	3	yes	—
63	C₂³	C₂²×C₂₃⋊C₁₁	1	1	yes	characteristic, complemented
64	D₄	—	2	6	no	—
65	C₂×C₄	C₂²×C₂₃⋊C₁₁	1	3	yes	complemented
66	C₂²	—	2	8	no	—
67	C₂²	—	2	6	no	—
68	C₂²	—	2	3	no	—
69	C₂²	D₄×C₂₃⋊C₁₁	1	6	yes	complemented
70	C₂²	C₂³×C₂₃⋊C₁₁	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₂	—	2	6	no	—
73	C₂	—	4	1	no	—
74	C₂	C₂×D₄×C₂₃⋊C₁₁	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₂₃⋊C₁₁
commutator, socle, cosocle	C₂²×C₂₃	C₂³×C₁₁
Fitting	C₂₃×#120	C₁₁
hypercenter	#120	C₂₃⋊C₁₁
hypocenter	C₂₃	C₁₁×#120
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₂²⋊C₄×C₂₃⋊C₁₁	C₂	semidirect
2. C₂⁴×C₂₃⋊C₁₁	C₂	semidirect
3. C₂×D₄×C₂₃⋊C₁₁	C₂	semidirect
6. C₂³×C₂₃⋊C₁₁	C₂²	semidirect
7. C₂×C₄×C₂₃⋊C₁₁	C₂²	semidirect
12. C₂²×C₂₃⋊C₁₁	D₄	semidirect
15. C₂₃×#120	C₁₁	semidirect
19. C₂⁴×C₂₃	C₂×C₁₁	semidirect
20. C₂×D₄×C₂₃	C₂×C₁₁	semidirect
21. C₂²⋊C₄×C₂₃	C₂×C₁₁	semidirect
23. C₂₃⋊C₁₁	#120	direct (N × H)
26. C₂³×C₂₃	C₂²×C₁₁	semidirect
28. C₂×C₄×C₂₃	C₂²×C₁₁	semidirect
35. C₂²×C₂₃	D₄×C₁₁	semidirect
52. #120	C₂₃⋊C₁₁	direct (N × H)
53. C₂₃	C₁₁×#120	semidirect
57. C₂⁴	C₂×C₂₃⋊C₁₁	semidirect
58. C₂×D₄	C₂×C₂₃⋊C₁₁	semidirect
59. C₂²⋊C₄	C₂×C₂₃⋊C₁₁	semidirect
63. C₂³	C₂²×C₂₃⋊C₁₁	semidirect
65. C₂×C₄	C₂²×C₂₃⋊C₁₁	semidirect
69. C₂²	D₄×C₂₃⋊C₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊C₄×C₂₃⋊C₁₁	2
C₂⁴×C₂₃⋊C₁₁	2
C₂×D₄×C₂₃⋊C₁₁	2
C₂₃×#120	11
C₁₁×#120	23

Subgroups — #120×C23⋊C11 #72247828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #120×C₂₃⋊C₁₁ #72247828