Subgroups — #19212091

Order 5888.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15675706	C₂	1	1	yes	maximal, characteristic
2	#15671601	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092849	—	2	1	no	—
5	Dic₃₆₈	C₄	1	1	yes	characteristic
6	#13092256	—	2	1	no	—
7	D₇₃₆	C₄	1	1	yes	characteristic
8	#13092247	—	2	1	no	—
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂₃×C₆₄	—	2	1	no	—
11	#10625075	—	4	1	no	—
12	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
13	#10625001	—	4	1	no	—
14	Dic₁₈₄	—	2	1	no	—
15	#10624982	—	4	1	no	—
16	D₃₆₈	—	2	1	no	—
17	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	1	yes	characteristic
19	#67288	—	8	1	no	—
20	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
21	#67267	—	8	1	no	—
22	Dic₉₂	—	4	1	no	—
23	D₁₈₄	—	4	1	no	—
24	#67263	—	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	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
29	C₂²⋊D₂₃	—	16	1	no	—
30	D₉₂	—	8	1	no	—
31	C₄×D₂₃	—	16	1	no	—
32	Dic₄₆	—	8	1	no	—
33	C₈×C₂₃	#102	1	1	yes	characteristic
34	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
35	C₂₃⋊C₈	—	16	1	no	—
36	#2262	—	23	1	no	—
37	C₂×C₆₄	D₂₃	1	1	yes	characteristic, complemented
38	#1425	—	23	1	no	—
39	C₂²×C₂₃	SD₆₄	1	1	yes	characteristic
40	C₂×D₂₃	—	16	1	no	—
41	C₄×C₂₃	#357	1	1	yes	characteristic
42	C₄×C₂₃	D₃₂	1	1	yes	characteristic
43	Dic₂₃	—	16	1	no	—
44	#508	—	46	1	no	—
45	Q₆₄	—	23	1	no	—
46	SD₆₄	—	46	1	no	—
47	D₃₂	—	23	1	no	—
48	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
49	#368	—	46	1	no	—
50	C₆₄	—	2	1	no	—
51	C₂×C₂₃	—	2	1	no	—
52	C₂×C₂₃	#1415	1	1	yes	characteristic
53	D₂₃	—	32	1	no	—
54	#135	—	92	1	no	—
55	Q₃₂	—	46	1	no	—
56	SD₃₂	—	92	1	no	—
57	D₁₆	—	46	1	no	—
58	C₂×C₁₆	D₉₂	1	1	yes	characteristic
59	#108	—	92	1	no	—
60	C₃₂	C₄×D₂₃	1	1	yes	characteristic
61	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
62	C₂₃	#7539	1	1	yes	characteristic, complemented, Sylow
63	#41	—	184	1	no	—
64	Q₁₆	—	92	1	no	—
65	SD₁₆	—	184	1	no	—
66	D₈	—	92	1	no	—
67	M₁₆	—	184	1	no	—
68	C₂×C₈	D₁₈₄	1	1	yes	characteristic
69	C₁₆	#67263	1	1	yes	characteristic
70	C₁₆	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
71	Q₈	—	184	1	no	—
72	D₄	—	368	1	no	—
73	D₄	—	184	1	no	—
74	C₂×C₄	—	368	1	no	—
75	C₂×C₄	D₃₆₈	1	1	yes	characteristic
76	C₈	—	368	1	no	—
77	C₈	#10624982	1	1	yes	characteristic
78	C₈	#10625003	1	1	yes	characteristic
79	C₂²	—	368	1	no	—
80	C₂²	#13092256	1	1	yes	characteristic
81	C₄	—	368	1	no	—
82	C₄	#13092219	1	1	yes	characteristic
83	C₄	D₇₃₆	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	736	1	no	—
86	C₂	#15671576	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₇₃₆
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂×C₂₃×C₆₄	C₂
socle	C₂×C₂₃	#1415
hypercenter	C₂×C₆₄	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 ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #15671601	C₂	semidirect
3. C₂×C₂₃×C₆₄	C₂	semidirect
37. C₂×C₆₄	D₂₃	semidirect
62. C₂₃	#7539	semidirect