Subgroups — #19212083

Order 5888.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₆₈	—	2	1	no	—
5	D₇₃₆	—	2	1	no	—
6	D₇₃₆	C₄	1	2	yes	complemented
7	C₂₃⋊C₆₄	—	2	1	no	—
8	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
9	C₂₃×#366	—	2	1	no	—
10	C₂×D₁₈₄	—	4	1	no	—
11	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
12	C₂₃×#107	—	4	1	no	—
13	D₃₆₈	—	2	2	no	—
14	D₃₆₈	—	4	1	no	—
15	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂₃×C₃₂	D₄	1	1	yes	characteristic
17	C₂×D₉₂	—	8	1	no	—
18	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	—	8	1	no	—
20	D₁₈₄	—	4	2	no	—
21	D₁₈₄	—	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	C₂²×D₂₃	—	16	1	no	—
26	C₂×C₄×C₂₃	—	16	1	no	—
27	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
28	D₉₂	—	8	2	no	—
29	D₉₂	—	16	1	no	—
30	C₈×C₂₃	#102	1	1	yes	characteristic
31	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
32	C₂×D₃₂	—	23	1	no	—
33	C₂×C₆₄	—	23	1	no	—
34	#1423	D₂₃	1	1	yes	characteristic, complemented
35	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
36	C₂×D₂₃	—	16	2	no	—
37	C₂×D₂₃	—	32	1	no	—
38	C₂×D₂₃	—	32	1	no	—
39	C₄×C₂₃	—	32	1	no	—
40	C₄×C₂₃	#357	1	1	yes	characteristic
41	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
42	C₂×D₁₆	—	46	1	no	—
43	D₃₂	—	23	2	no	—
44	D₃₂	—	46	1	no	—
45	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
46	#366	—	2	1	no	—
47	C₆₄	—	46	1	no	—
48	C₂×C₂₃	#1415	1	1	yes	characteristic
49	C₂×C₂₃	D₆₄	1	1	yes	characteristic
50	C₂×C₂₃	SD₁₂₈	1	1	yes	characteristic
51	D₂₃	—	32	2	no	—
52	C₂×D₈	—	92	1	no	—
53	D₁₆	—	46	2	no	—
54	D₁₆	—	92	1	no	—
55	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
56	#107	—	4	1	no	—
57	C₃₂	C₄×D₂₃	1	1	yes	characteristic
58	C₃₂	D₉₂	1	1	yes	characteristic
59	C₂₃	#7537	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₄	—	184	1	no	—
61	D₈	—	92	2	no	—
62	D₈	—	184	1	no	—
63	C₂×C₈	#67272	1	1	yes	characteristic
64	C₄⋊C₄	—	8	1	no	—
65	C₁₆	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
66	C₁₆	#67274	1	1	yes	characteristic
67	C₂³	—	368	1	no	—
68	D₄	—	368	1	no	—
69	D₄	—	184	2	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	#10625008	1	1	yes	characteristic
72	C₈	#10624991	1	1	yes	characteristic
73	C₈	#10625010	1	1	yes	characteristic
74	C₂²	—	368	2	no	—
75	C₂²	—	736	1	no	—
76	C₂²	—	736	1	no	—
77	C₂²	#13092252	1	1	yes	characteristic
78	C₄	—	32	1	no	—
79	C₄	#13092217	1	1	yes	characteristic
80	C₄	#13092254	1	1	yes	characteristic
81	C₂	—	736	2	no	—
82	C₂	#15671574	1	1	yes	characteristic
83	C₂	#15671609	1	1	yes	characteristic
84	C₂	#15671611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13092252
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×#1423	C₂
socle	C₂²×C₂₃	D₃₂
hypercenter	#1423	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 ◁ #1423

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃⋊C₆₄	C₂	semidirect
3. C₂₃×#1423	C₂	semidirect
6. D₇₃₆	C₄	semidirect
34. #1423	D₂₃	semidirect
59. C₂₃	#7537	semidirect