Subgroups — #15671443

Order 2944.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₂×D₈×C₂₃	—	2	1	no	—
5	C₂×D₈×C₂₃	C₄	1	2	yes	—
6	C₂₃×#127	—	2	1	no	—
7	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃⋊C₁₆	—	2	1	no	—
9	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
10	C₂×D₄×C₂₃	—	2	2	no	—
11	C₂×D₄×C₂₃	—	4	1	no	—
12	C₂×D₄×C₂₃	—	4	1	no	—
13	D₈×C₂₃	—	4	2	no	—
14	D₈×C₂₃	—	4	1	no	—
15	D₈×C₂₃	—	4	1	no	—
16	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₄²×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₂₃⋊C₈	—	4	1	no	—
20	C₂₃⋊C₁₆	—	4	1	no	—
21	C₂³×C₂₃	—	4	2	no	—
22	D₄×C₂₃	—	4	2	no	—
23	D₄×C₂₃	—	4	2	no	—
24	D₄×C₂₃	—	8	2	no	—
25	D₄×C₂₃	—	8	1	no	—
26	D₄×C₂₃	—	8	1	no	—
27	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₂₃	—	2	1	no	—
31	C₈×C₂₃	D₈	1	1	yes	characteristic
32	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
33	C₂₃⋊C₈	—	8	1	no	—
34	#1361	—	23	1	no	maximal, Sylow
35	C₂²×C₂₃	—	8	2	no	—
36	C₂²×C₂₃	—	8	2	no	—
37	C₂²×C₂₃	—	8	2	no	—
38	C₂²×C₂₃	#102	1	1	yes	characteristic
39	C₄×C₂₃	—	2	1	no	—
40	C₄×C₂₃	#100	1	1	yes	characteristic
41	C₄×C₂₃	#102	1	1	yes	characteristic
42	C₄×C₂₃	D₁₆	1	1	yes	characteristic
43	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
44	#493	D₂₃	1	1	yes	characteristic
45	C₄⋊C₁₆	—	23	1	no	—
46	#335	—	23	1	no	—
47	C₂×C₂₃	—	16	2	no	—
48	C₂×C₂₃	#331	1	1	yes	characteristic
49	C₂×C₂₃	#357	1	1	yes	characteristic
50	C₂×C₂₃	#361	1	1	yes	characteristic
51	C₂×D₈	—	2	1	no	—
52	C₂×D₈	Dic₂₃	1	2	yes	—
53	#127	—	2	1	no	—
54	C₂×C₁₆	—	46	1	no	—
55	C₄⋊C₈	—	46	1	no	—
56	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂₃	#1361	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₄	—	2	2	no	—
59	C₂×D₄	—	4	1	no	—
60	C₂×D₄	—	4	1	no	—
61	D₈	—	4	2	no	—
62	D₈	—	4	1	no	—
63	D₈	—	4	1	no	—
64	C₂×C₈	—	92	1	no	—
65	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
67	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
68	C₁₆	—	92	1	no	—
69	C₂³	—	4	2	no	—
70	D₄	—	4	2	no	—
71	D₄	—	4	2	no	—
72	D₄	—	8	2	no	—
73	D₄	—	8	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	#67272	1	1	yes	characteristic
76	C₂×C₄	#67273	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	184	1	no	—
80	C₈	#67272	1	1	yes	characteristic
81	C₈	#67273	1	1	yes	characteristic
82	C₂²	—	8	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	#10625014	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#10625008	1	1	yes	characteristic
88	C₄	#10625009	1	1	yes	characteristic
89	C₄	#10625014	1	1	yes	characteristic
90	C₄	#10625015	1	1	yes	characteristic
91	C₂	—	16	2	no	—
92	C₂	#13092161	1	1	yes	characteristic
93	C₂	#13092218	1	1	yes	characteristic
94	C₂	#13092230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625014
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₃
Fitting, prime radical	C₂₃×#493	C₂
socle	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#493	D₂₃
hypocenter	C₂₃	#1361
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₂₃⋊C₁₆)	C₂	semidirect
2. #13092170	C₂	semidirect
57. C₂₃	#1361	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₃⋊C₁₆)	2
#13092170	2
C₂₃×#493	2
#1361	23