Subgroups — #19212090

Order 5888.

88 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	#15671601	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2262	C₂	1	1	yes	maximal, characteristic
4	#13092247	—	2	1	no	—
5	C₂₃⋊C₆₄	—	2	1	no	—
6	C₂₃×#508	—	2	1	no	—
7	C₂₃×Q₆₄	C₄	1	1	yes	characteristic
8	C₂₃×SD₆₄	—	2	1	no	—
9	C₂₃×D₃₂	C₄	1	1	yes	characteristic
10	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
11	C₂₃×#135	—	4	1	no	—
12	C₂₃×Q₃₂	—	2	1	no	—
13	C₂₃×SD₃₂	—	4	1	no	—
14	C₂₃×D₁₆	—	2	1	no	—
15	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
16	#10625001	—	4	1	no	—
17	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	1	yes	characteristic
19	#41×C₂₃	—	8	1	no	—
20	Q₁₆×C₂₃	—	4	1	no	—
21	SD₁₆×C₂₃	—	8	1	no	—
22	D₈×C₂₃	—	4	1	no	—
23	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
24	#67267	—	8	1	no	—
25	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
27	#7539	—	23	1	no	maximal, Sylow
28	Q₈×C₂₃	—	8	1	no	—
29	D₄×C₂₃	—	8	1	no	—
30	D₄×C₂₃	—	16	1	no	—
31	C₂×C₄×C₂₃	—	16	1	no	—
32	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
33	C₈×C₂₃	#102	1	1	yes	characteristic
34	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
35	C₂₃⋊C₈	—	16	1	no	—
36	#2262	D₂₃	1	1	yes	characteristic
37	C₂×C₆₄	—	23	1	no	—
38	#1425	—	23	1	no	—
39	C₂²×C₂₃	—	16	1	no	—
40	C₂²×C₂₃	SD₆₄	1	1	yes	characteristic
41	C₄×C₂₃	—	16	1	no	—
42	C₄×C₂₃	#357	1	1	yes	characteristic
43	C₄×C₂₃	D₃₂	1	1	yes	characteristic
44	#508	—	2	1	no	—
45	Q₆₄	Dic₂₃	1	1	yes	characteristic
46	SD₆₄	—	2	1	no	—
47	D₃₂	Dic₂₃	1	1	yes	characteristic
48	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
49	#368	—	46	1	no	—
50	C₆₄	—	46	1	no	—
51	C₂×C₂₃	—	2	1	no	—
52	C₂×C₂₃	—	32	1	no	—
53	C₂×C₂₃	#1415	1	1	yes	characteristic
54	#135	—	4	1	no	—
55	Q₃₂	—	2	1	no	—
56	SD₃₂	—	4	1	no	—
57	D₁₆	—	2	1	no	—
58	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
59	#108	—	92	1	no	—
60	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
61	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
62	C₂₃	#7539	1	1	yes	characteristic, complemented, Sylow
63	#41	—	8	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	8	1	no	—
66	D₈	—	4	1	no	—
67	M₁₆	—	184	1	no	—
68	C₂×C₈	#67272	1	1	yes	characteristic
69	C₁₆	#67273	1	1	yes	characteristic
70	C₁₆	C₂²⋊Dic₂₃	1	1	yes	characteristic
71	Q₈	—	8	1	no	—
72	D₄	—	8	1	no	—
73	D₄	—	16	1	no	—
74	C₂×C₄	—	16	1	no	—
75	C₂×C₄	#10625008	1	1	yes	characteristic
76	C₈	—	368	1	no	—
77	C₈	#10625009	1	1	yes	characteristic
78	C₈	#10625014	1	1	yes	characteristic
79	C₂²	—	16	1	no	—
80	C₂²	#13092255	1	1	yes	characteristic
81	C₄	—	16	1	no	—
82	C₄	#13092218	1	1	yes	characteristic
83	C₄	#13092252	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	32	1	no	—
86	C₂	#15671575	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13092252
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×#2262	C₂
socle	C₂×C₂₃	#1415
hypercenter	#2262	D₂₃
hypocenter	C₂₃	#7539
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃⋊C₆₄	C₂	semidirect
2. #15671601	C₂	semidirect
62. C₂₃	#7539	semidirect