Subgroups — #19212087

Order 5888.

90 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
2	#15671597	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	#13092243	—	2	1	no	—
5	C₂₃⋊C₆₄	—	2	1	no	—
6	C₂×C₂₃×Q₃₂	—	2	1	no	—
7	C₂₃×Q₆₄	—	2	1	no	—
8	C₂₃×Q₆₄	C₄	1	2	yes	complemented
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₂₃	—	4	1	no	—
11	C₂₃×Q₃₂	—	2	2	no	—
12	C₂₃×Q₃₂	—	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×Q₈×C₂₃	—	8	1	no	—
18	Q₁₆×C₂₃	—	4	2	no	—
19	Q₁₆×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	#7538	—	23	1	no	maximal, Sylow
25	Q₈×C₂₃	—	8	2	no	—
26	Q₈×C₂₃	—	16	1	no	—
27	C₂×C₄×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×Q₆₄	D₂₃	1	1	yes	characteristic
33	C₂×C₆₄	—	23	1	no	—
34	#1423	—	23	1	no	—
35	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
36	C₄×C₂₃	—	16	2	no	—
37	C₄×C₂₃	#357	1	1	yes	characteristic
38	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
39	Dic₂₃	—	32	1	no	—
40	C₂×Q₃₂	—	2	1	no	—
41	Q₆₄	—	2	1	no	—
42	Q₆₄	Dic₂₃	1	2	yes	complemented
43	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
44	#366	—	46	1	no	—
45	C₆₄	—	46	1	no	—
46	C₂×C₂₃	#1415	1	1	yes	characteristic
47	C₂×C₂₃	SD₁₂₈	1	1	yes	characteristic
48	C₂×C₂₃	Q₁₂₈	1	1	yes	characteristic
49	C₂×Q₁₆	—	4	1	no	—
50	Q₃₂	—	2	2	no	—
51	Q₃₂	—	4	1	no	—
52	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
53	#107	—	92	1	no	—
54	C₃₂	C₂×Dic₂₃	1	1	yes	characteristic
55	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
56	C₂₃	#7538	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	#67272	1	1	yes	characteristic
61	C₄⋊C₄	—	184	1	no	—
62	C₁₆	#67273	1	1	yes	characteristic
63	C₁₆	C₂²⋊Dic₂₃	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	368	1	no	—
68	C₂×C₄	#10625008	1	1	yes	characteristic
69	C₈	#10625009	1	1	yes	characteristic
70	C₈	#10625014	1	1	yes	characteristic
71	C₂²	#13092252	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	736	1	no	—
74	C₄	#13092218	1	1	yes	characteristic
75	C₄	#13092255	1	1	yes	characteristic
76	C₂	#15671575	1	1	yes	characteristic
77	C₂	#15671611	1	1	yes	characteristic
78	C₂	#15671614	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13092252
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂×C₂₃×Q₆₄	C₂
socle	C₂²×C₂₃	D₃₂
hypercenter	C₂×Q₆₄	D₂₃
hypocenter	C₂₃	#7538
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×Q₆₄

Complements

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

N	H ≅ G/N	type
8. C₂₃×Q₆₄	C₄	semidirect
42. Q₆₄	Dic₂₃	semidirect
56. C₂₃	#7538	semidirect