Subgroups — #19212016

Order 5888.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13092250	C₂	1	1	yes	maximal, characteristic
2	#15671608	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₂₃	C₄	1	1	yes	characteristic
5	#13092250	—	2	1	no	—
6	#13092250	—	2	1	no	—
7	C₂×C₂₃⋊C₃₂	C₄	1	1	yes	characteristic
8	C₂₃⋊C₆₄	—	2	1	no	—
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂₃×C₆₄	—	2	1	no	—
11	C₂×C₈×D₂₃	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₂₃⋊C₁₆	C₈	1	1	yes	characteristic
14	C₁₆×D₂₃	—	2	1	no	—
15	C₁₆×D₂₃	—	2	1	no	—
16	C₂₃×C₃₂	D₄	1	1	yes	characteristic
17	C₂₃×C₃₂	D₄	1	1	yes	characteristic
18	C₂₃⋊C₃₂	—	2	1	no	—
19	C₂×C₄×D₂₃	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
21	C₂×C₂₃⋊C₈	C₁₆	1	1	yes	characteristic
22	C₈×D₂₃	—	2	1	no	—
23	C₈×D₂₃	—	2	1	no	—
24	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₂₃	M₁₆	1	1	yes	characteristic
26	C₂₃⋊C₁₆	—	2	1	no	—
27	#7513	—	23	1	no	maximal, Sylow
28	C₂²×D₂₃	C₃₂	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
30	C₂×Dic₂₃	C₃₂	1	1	yes	characteristic
31	C₄×D₂₃	—	2	1	no	—
32	C₄×D₂₃	—	2	1	no	—
33	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₂₃	M₃₂	1	1	yes	characteristic
35	C₂₃⋊C₈	—	2	1	no	—
36	C₂×M₆₄	—	23	1	no	—
37	M₁₂₈	—	23	1	no	—
38	M₁₂₈	D₂₃	1	1	yes	characteristic, complemented
39	C₂²×C₂₃	C₂×C₃₂	1	1	yes	characteristic
40	C₂×D₂₃	—	2	1	no	—
41	C₂×D₂₃	—	4	1	no	—
42	C₄×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₄×C₂₃	M₆₄	1	1	yes	characteristic
44	Dic₂₃	—	2	1	no	—
45	C₂²×C₁₆	—	23	1	no	—
46	M₆₄	—	46	1	no	—
47	M₆₄	—	46	1	no	—
48	C₂×C₃₂	—	23	1	no	—
49	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
50	C₆₄	—	2	1	no	—
51	C₆₄	—	46	1	no	—
52	C₂×C₂₃	—	2	1	no	—
53	C₂×C₂₃	C₂²⋊C₃₂	1	1	yes	characteristic
54	D₂₃	—	4	1	no	—
55	C₂²×C₈	—	23	1	no	—
56	C₂×C₁₆	—	23	1	no	—
57	C₂×C₁₆	—	46	1	no	—
58	C₂×C₁₆	—	46	1	no	—
59	C₂×C₁₆	C₄×D₂₃	1	1	yes	characteristic
60	C₃₂	—	46	1	no	—
61	C₃₂	D₉₂	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
63	C₂₃	#7513	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₄	—	23	1	no	—
65	C₂×C₈	—	23	1	no	—
66	C₂×C₈	—	46	1	no	—
67	C₂×C₈	—	46	1	no	—
68	C₂×C₈	C₈×D₂₃	1	1	yes	characteristic
69	C₁₆	—	46	1	no	—
70	C₁₆	#67262	1	1	yes	characteristic
71	C₁₆	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
72	C₂³	—	23	1	no	—
73	C₂×C₄	—	23	1	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	—	46	1	no	—
76	C₂×C₄	C₁₆×D₂₃	1	1	yes	characteristic
77	C₈	—	46	1	no	—
78	C₈	#10624980	1	1	yes	characteristic
79	C₈	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
80	C₂²	—	46	1	no	—
81	C₂²	—	92	1	no	—
82	C₂²	C₃₂×D₂₃	1	1	yes	characteristic
83	C₄	—	46	1	no	—
84	C₄	(C₂×D₂₃)⋊C₁₆	1	1	yes	characteristic
85	C₄	#13092250	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	92	1	no	—
88	C₂	(C₂×D₂₃)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	(C₂×D₂₃)⋊C₄
commutator	C₂²×C₂₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×M₁₂₈	C₂
socle	C₂×C₂₃	C₂²⋊C₃₂
hypercenter	M₁₂₈	D₂₃
hypocenter	C₂₃	#7513
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #15671608	C₂	semidirect
3. C₂₃×M₁₂₈	C₂	semidirect
38. M₁₂₈	D₂₃	semidirect
63. C₂₃	#7513	semidirect