Subgroups — #19212093

Order 5888.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671608	C₂	1	1	yes	maximal, characteristic, complemented
2	#15671599	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2262	C₂	1	1	yes	maximal, characteristic
4	#13092243	—	2	1	no	—
5	C₂₃⋊C₆₄	—	2	1	no	—
6	C₂₃×#508	—	2	1	no	—
7	C₂₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
8	C₂₃×SD₆₄	—	2	1	no	—
9	C₂₃×D₃₂	C₄	1	1	yes	characteristic, complemented
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	#10625000	—	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	C₄⋊Dic₂₃	—	8	1	no	—
25	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
27	#7540	—	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	C₂×Dic₂₃	—	16	1	no	—
34	C₈×C₂₃	#102	1	1	yes	characteristic
35	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
36	#2262	D₂₃	1	1	yes	characteristic
37	M₁₂₈	—	23	1	no	—
38	#1424	—	23	1	no	—
39	C₂²×C₂₃	—	16	1	no	—
40	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
41	C₄×C₂₃	—	16	1	no	—
42	C₄×C₂₃	#357	1	1	yes	characteristic
43	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
44	Dic₂₃	—	32	1	no	—
45	#508	—	2	1	no	—
46	Q₆₄	Dic₂₃	1	1	yes	characteristic, complemented
47	SD₆₄	—	2	1	no	—
48	D₃₂	Dic₂₃	1	1	yes	characteristic, complemented
49	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
50	#366	—	46	1	no	—
51	C₆₄	—	46	1	no	—
52	C₂×C₂₃	—	2	1	no	—
53	C₂×C₂₃	—	32	1	no	—
54	C₂×C₂₃	#1415	1	1	yes	characteristic
55	#135	—	4	1	no	—
56	Q₃₂	—	2	1	no	—
57	SD₃₂	—	4	1	no	—
58	D₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
60	#107	—	92	1	no	—
61	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
63	C₂₃	#7540	1	1	yes	characteristic, complemented, Sylow
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	#67272	1	1	yes	characteristic
69	C₄⋊C₄	—	184	1	no	—
70	C₁₆	#67273	1	1	yes	characteristic
71	C₁₆	C₂²⋊Dic₂₃	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	368	1	no	—
77	C₂×C₄	#10625008	1	1	yes	characteristic
78	C₈	#10625009	1	1	yes	characteristic
79	C₈	#10625014	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	#13092252	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	736	1	no	—
84	C₄	#13092218	1	1	yes	characteristic
85	C₄	#13092255	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#15671575	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15671575
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×#2262	C₂
socle	C₂×C₂₃	#1415
hypercenter	#2262	D₂₃
hypocenter	C₂₃	#7540
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ 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. #15671608	C₂	semidirect
2. #15671599	C₂	semidirect
7. C₂₃×Q₆₄	C₄	semidirect
9. C₂₃×D₃₂	C₄	semidirect
46. Q₆₄	Dic₂₃	semidirect
48. D₃₂	Dic₂₃	semidirect
63. C₂₃	#7540	semidirect