Subgroups — #19212018

Order 5888.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15675690	C₂	1	1	yes	maximal, characteristic
2	#15671608	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092833	C₄	1	1	yes	characteristic
5	#13092251	C₄	1	1	yes	characteristic
6	#13092250	—	2	1	no	—
7	C₃₂×D₂₃	—	2	1	no	—
8	C₂₃⋊C₆₄	—	2	1	no	—
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂₃×C₆₄	—	2	1	no	—
11	#10625072	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
13	#10624994	C₈	1	1	yes	characteristic
14	#10624980	—	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	#67288	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
21	#67267	C₁₆	1	1	yes	characteristic
22	#67262	—	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	#7514	—	23	1	no	maximal, Sylow
28	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
29	C₂²⋊D₂₃	—	2	1	no	—
30	D₉₂	C₃₂	1	1	yes	characteristic
31	C₄×D₂₃	—	2	1	no	—
32	Dic₄₆	C₃₂	1	1	yes	characteristic
33	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₂₃	M₃₂	1	1	yes	characteristic
35	C₂₃⋊C₈	—	2	1	no	—
36	#2258	—	23	1	no	—
37	M₁₂₈	—	23	1	no	—
38	C₂×C₆₄	D₂₃	1	1	yes	characteristic, complemented
39	C₂²×C₂₃	M₆₄	1	1	yes	characteristic
40	C₂×D₂₃	—	2	1	no	—
41	C₄×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₄×C₂₃	C₂×C₃₂	1	1	yes	characteristic
43	Dic₂₃	—	2	1	no	—
44	#504	—	23	1	no	—
45	M₆₄	—	23	1	no	—
46	M₆₄	—	46	1	no	—
47	C₂×C₃₂	—	46	1	no	—
48	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
49	C₆₄	—	2	1	no	—
50	C₆₄	—	46	1	no	—
51	C₂×C₂₃	—	2	1	no	—
52	C₂×C₂₃	C₂²⋊C₃₂	1	1	yes	characteristic
53	D₂₃	—	4	1	no	—
54	#131	—	23	1	no	—
55	M₃₂	—	23	1	no	—
56	M₃₂	—	46	1	no	—
57	C₂×C₁₆	—	46	1	no	—
58	C₂×C₁₆	C₄×D₂₃	1	1	yes	characteristic
59	C₃₂	—	46	1	no	—
60	C₃₂	D₉₂	1	1	yes	characteristic
61	C₃₂	C₂²⋊D₂₃	1	1	yes	characteristic
62	C₂₃	#7514	1	1	yes	characteristic, complemented, Sylow
63	#41	—	23	1	no	—
64	M₁₆	—	23	1	no	—
65	M₁₆	—	46	1	no	—
66	C₂×C₈	—	46	1	no	—
67	C₂×C₈	C₈×D₂₃	1	1	yes	characteristic
68	C₁₆	—	46	1	no	—
69	C₁₆	#67262	1	1	yes	characteristic
70	C₁₆	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
71	Q₈	—	23	1	no	—
72	D₄	—	23	1	no	—
73	D₄	—	46	1	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	C₁₆×D₂₃	1	1	yes	characteristic
76	C₈	—	46	1	no	—
77	C₈	#10624980	1	1	yes	characteristic
78	C₈	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
79	C₂²	—	46	1	no	—
80	C₂²	#13092250	1	1	yes	characteristic
81	C₄	—	46	1	no	—
82	C₄	(C₂×D₂₃)⋊C₁₆	1	1	yes	characteristic
83	C₄	C₃₂×D₂₃	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	92	1	no	—
86	C₂	(C₂×D₂₃)⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₉₂
commutator	C₄×C₂₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂×C₂₃×C₆₄	C₂
socle	C₂×C₂₃	C₂²⋊C₃₂
hypercenter	C₂×C₆₄	D₂₃
hypocenter	C₂₃	#7514
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #15671608	C₂	semidirect
3. C₂×C₂₃×C₆₄	C₂	semidirect
38. C₂×C₆₄	D₂₃	semidirect
62. C₂₃	#7514	semidirect