Subgroups — #19212084

Order 5888.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#15671597	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×D₃₂	C₂	1	1	yes	maximal, characteristic
4	#13092243	—	2	1	no	—
5	C₂₃⋊C₆₄	—	2	1	no	—
6	C₂×C₂₃×D₁₆	—	2	1	no	—
7	C₂₃×D₃₂	—	2	1	no	—
8	C₂₃×D₃₂	C₄	1	2	yes	complemented
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₂₃	—	4	1	no	—
11	C₂₃×D₁₆	—	2	2	no	—
12	C₂₃×D₁₆	—	4	1	no	—
13	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
14	#10625000	—	4	1	no	—
15	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂₃×C₃₂	D₄	1	1	yes	characteristic
17	C₂×D₄×C₂₃	—	8	1	no	—
18	D₈×C₂₃	—	4	2	no	—
19	D₈×C₂₃	—	8	1	no	—
20	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
21	C₄⋊Dic₂₃	—	8	1	no	—
22	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
24	#7537	—	23	1	no	maximal, Sylow
25	C₂³×C₂₃	—	16	1	no	—
26	D₄×C₂₃	—	8	2	no	—
27	D₄×C₂₃	—	16	1	no	—
28	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
29	C₂×Dic₂₃	—	16	1	no	—
30	C₈×C₂₃	#102	1	1	yes	characteristic
31	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
32	C₂×D₃₂	D₂₃	1	1	yes	characteristic
33	C₂×C₆₄	—	23	1	no	—
34	#1423	—	23	1	no	—
35	C₂²×C₂₃	—	16	2	no	—
36	C₂²×C₂₃	—	32	1	no	—
37	C₂²×C₂₃	—	32	1	no	—
38	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
39	C₄×C₂₃	#357	1	1	yes	characteristic
40	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
41	Dic₂₃	—	32	1	no	—
42	C₂×D₁₆	—	2	1	no	—
43	D₃₂	—	2	1	no	—
44	D₃₂	Dic₂₃	1	2	yes	complemented
45	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
46	#366	—	46	1	no	—
47	C₆₄	—	46	1	no	—
48	C₂×C₂₃	—	32	2	no	—
49	C₂×C₂₃	#1415	1	1	yes	characteristic
50	C₂×C₂₃	D₆₄	1	1	yes	characteristic
51	C₂×C₂₃	SD₁₂₈	1	1	yes	characteristic
52	C₂×D₈	—	4	1	no	—
53	D₁₆	—	2	2	no	—
54	D₁₆	—	4	1	no	—
55	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
56	#107	—	92	1	no	—
57	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
58	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂₃	#7537	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₄	—	8	1	no	—
61	D₈	—	4	2	no	—
62	D₈	—	8	1	no	—
63	C₂×C₈	#67272	1	1	yes	characteristic
64	C₄⋊C₄	—	184	1	no	—
65	C₁₆	#67273	1	1	yes	characteristic
66	C₁₆	C₂²⋊Dic₂₃	1	1	yes	characteristic
67	C₂³	—	16	1	no	—
68	D₄	—	8	2	no	—
69	D₄	—	16	1	no	—
70	C₂×C₄	—	368	1	no	—
71	C₂×C₄	#10625008	1	1	yes	characteristic
72	C₈	#10625009	1	1	yes	characteristic
73	C₈	#10625014	1	1	yes	characteristic
74	C₂²	—	16	2	no	—
75	C₂²	—	32	1	no	—
76	C₂²	—	32	1	no	—
77	C₂²	#13092252	1	1	yes	characteristic
78	C₄	—	736	1	no	—
79	C₄	#13092218	1	1	yes	characteristic
80	C₄	#13092255	1	1	yes	characteristic
81	C₂	—	32	2	no	—
82	C₂	#15671575	1	1	yes	characteristic
83	C₂	#15671609	1	1	yes	characteristic
84	C₂	#15671612	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13092252
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂×C₂₃×D₃₂	C₂
socle	C₂²×C₂₃	D₃₂
hypercenter	C₂×D₃₂	D₂₃
hypocenter	C₂₃	#7537
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×D₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃⋊C₆₄	C₂	semidirect
2. #15671597	C₂	semidirect
8. C₂₃×D₃₂	C₄	semidirect
44. D₃₂	Dic₂₃	semidirect
59. C₂₃	#7537	semidirect