Subgroups — #15675273

Order 2944.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
2	#13092245	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092218	C₂	1	2	yes	maximal, complemented
4	#13092192	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×D₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×D₈×C₂₃	C₂²	1	2	yes	—
7	D₄×Dic₂₃	—	2	2	no	—
8	C₂₃×D₁₆	C₄	1	4	yes	complemented
9	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	#10625014	—	2	2	no	—
11	#10625000	C₂²	1	2	yes	—
12	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₃	—	2	2	no	—
15	C₂²×Dic₂₃	—	4	2	no	—
16	D₈×C₂₃	—	2	2	no	—
17	D₈×C₂₃	C₂×C₄	1	4	yes	complemented
18	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²⋊Dic₂₃	—	4	2	no	—
20	C₄⋊Dic₂₃	—	2	2	no	—
21	C₄×Dic₂₃	D₄	1	1	yes	characteristic
22	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
23	C₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
24	C₂₃⋊C₁₆	—	2	1	no	—
25	C₂³×C₂₃	—	4	2	no	—
26	D₄×C₂₃	—	2	4	no	—
27	D₄×C₂₃	—	4	2	no	—
28	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₂₃	—	4	2	no	—
30	C₂×Dic₂₃	—	8	4	no	—
31	C₂×Dic₂₃	D₈	1	2	yes	—
32	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₃	#41	1	1	yes	characteristic
34	C₂₃⋊C₈	—	2	1	no	—
35	#2177	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	—	4	4	no	—
37	C₂²×C₂₃	—	8	2	no	—
38	C₂²×C₂₃	—	8	2	no	—
39	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
40	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₂₃	#135	1	1	yes	characteristic
42	Dic₂₃	—	2	2	no	—
43	Dic₂₃	—	8	2	no	—
44	C₂×D₁₆	D₂₃	1	1	yes	characteristic
45	C₄×D₈	—	23	2	no	—
46	#367	—	23	1	no	—
47	#357	—	23	2	no	—
48	#346	—	23	1	no	—
49	C₂×C₂₃	—	8	4	no	—
50	C₂×C₂₃	C₄×D₈	1	1	yes	characteristic
51	C₂×C₂₃	#509	1	2	yes	—
52	C₂×D₈	C₂×D₂₃	1	2	yes	—
53	C₄×D₄	—	46	2	no	—
54	D₁₆	Dic₂₃	1	4	yes	complemented
55	C₂×C₁₆	—	23	1	no	—
56	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
57	#107	—	23	2	no	—
58	#102	—	46	2	no	—
59	C₄×C₈	—	23	1	no	—
60	C₂₃	#2177	1	1	yes	characteristic, complemented, Sylow
61	C₂×D₄	—	2	2	no	—
62	C₂²×C₄	—	92	2	no	—
63	D₈	—	2	2	no	—
64	D₈	C₂×Dic₂₃	1	4	yes	complemented
65	C₂×C₈	—	23	1	no	—
66	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
67	C₄⋊C₄	—	46	2	no	—
68	C₂²⋊C₄	—	92	2	no	—
69	C₄²	—	23	1	no	—
70	C₁₆	—	46	1	no	—
71	C₁₆	C₂×Dic₂₃	1	2	yes	complemented
72	C₂³	—	4	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	4	2	no	—
75	C₂×C₄	—	23	2	no	—
76	C₂×C₄	—	92	2	no	—
77	C₂×C₄	—	184	4	no	—
78	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
79	C₈	—	46	1	no	—
80	C₈	#67290	1	1	yes	characteristic
81	C₈	C₂²×Dic₂₃	1	1	yes	characteristic
82	C₂²	—	4	4	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	D₈×D₂₃	1	1	yes	characteristic
86	C₄	—	46	2	no	—
87	C₄	—	184	2	no	—
88	C₄	#10625083	1	1	yes	characteristic
89	C₄	D₄×Dic₂₃	1	1	yes	characteristic
90	C₂	—	8	4	no	—
91	C₂	D₈×Dic₂₃	1	1	yes	characteristic
92	C₂	#13092851	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₂₃
commutator	C₈×C₂₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃
Fitting, prime radical	C₂×C₂₃×D₁₆	C₂
socle	C₂²×C₂₃	C₂×D₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂×D₁₆	D₂₃
hypocenter	C₂₃	#2177
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₂₃	C₂	semidirect
2. #13092245	C₂	semidirect
3. #13092218	C₂	semidirect
4. #13092192	C₂	semidirect
8. C₂₃×D₁₆	C₄	semidirect
17. D₈×C₂₃	C₂×C₄	semidirect
23. C₁₆×C₂₃	C₂×C₄	semidirect
54. D₁₆	Dic₂₃	semidirect
60. C₂₃	#2177	semidirect
64. D₈	C₂×Dic₂₃	semidirect
71. C₁₆	C₂×Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₂₃	2
#13092245	2
#13092218	2
#13092192	2
C₂×C₂₃×D₁₆	2
#2177	23