Subgroups — #15673011

Order 2944.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092794	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092793	C₂	1	4	yes	maximal, complemented
3	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈×C₂₃	—	2	2	no	—
6	C₂₃×#127	C₂²	1	2	yes	—
7	C₂×#67272	—	2	8	no	—
8	C₂×D₁₈₄	—	2	2	no	—
9	#10625045	C₂²	1	2	yes	—
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂×D₄×C₂₃	—	2	4	no	—
13	C₂×D₄×C₂₃	—	2	2	no	—
14	C₂×D₉₂	—	2	2	no	—
15	C₂×D₉₂	—	2	4	no	—
16	D₈×C₂₃	—	4	4	no	—
17	C₂×C₈×C₂₃	D₄	1	2	yes	—
18	C₄²×C₂₃	C₂³	1	1	yes	characteristic
19	#67272	—	4	16	no	—
20	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
21	D₁₈₄	—	4	4	no	—
22	C₂³×C₂₃	—	4	2	no	—
23	C₂²×D₂₃	—	4	2	no	—
24	D₄×C₂₃	—	4	8	no	—
25	D₄×C₂₃	—	4	4	no	—
26	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
28	D₉₂	—	4	4	no	—
29	D₉₂	—	4	8	no	—
30	C₈×C₂₃	D₈	1	4	yes	complemented
31	C₂₃⋊C₈	D₈	1	8	yes	complemented
32	#1712	—	23	1	no	maximal, Sylow
33	C₂²×C₂₃	—	8	4	no	—
34	C₂²×C₂₃	—	8	2	no	—
35	C₂²×C₂₃	#127	1	1	yes	characteristic
36	C₂×D₂₃	—	8	4	no	—
37	C₂×D₂₃	—	8	2	no	—
38	C₄×C₂₃	C₂×D₈	1	4	yes	—
39	C₄×C₂₃	C₂×D₈	1	2	yes	—
40	#493	—	23	4	no	—
41	#493	—	23	1	no	—
42	#493	D₂₃	1	1	yes	characteristic, complemented
43	C₈²	—	23	1	no	—
44	C₂×C₂₃	—	16	2	no	—
45	C₂×C₂₃	#493	1	2	yes	—
46	C₂×C₂₃	#493	1	1	yes	characteristic
47	D₂₃	—	16	2	no	—
48	C₂×D₈	—	2	2	no	—
49	C₂×D₈	—	46	8	no	—
50	C₂×D₈	—	46	2	no	—
51	#127	—	23	2	no	—
52	#127	C₂×D₂₃	1	2	yes	—
53	C₄×C₈	—	23	2	no	—
54	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₂₃	#1712	1	1	yes	characteristic, complemented, Sylow
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	46	4	no	—
59	C₂×D₄	—	46	2	no	—
60	D₈	—	4	4	no	—
61	D₈	—	92	16	no	—
62	D₈	—	92	4	no	—
63	C₂×C₈	—	23	4	no	—
64	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
65	C₄²	C₂²×D₂₃	1	1	yes	characteristic
66	C₂³	—	4	2	no	—
67	C₂³	—	92	2	no	—
68	D₄	—	4	8	no	—
69	D₄	—	4	4	no	—
70	D₄	—	92	8	no	—
71	D₄	—	92	4	no	—
72	C₂×C₄	D₄×D₂₃	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
74	C₈	—	23	8	no	—
75	C₈	#67272	1	4	yes	complemented
76	C₂²	—	8	4	no	—
77	C₂²	—	8	2	no	—
78	C₂²	—	184	4	no	—
79	C₂²	—	184	2	no	—
80	C₂²	#10625111	1	1	yes	characteristic
81	C₄	D₈×D₂₃	1	4	yes	—
82	C₄	C₂×#67272	1	2	yes	—
83	C₂	—	16	2	no	—
84	C₂	—	368	2	no	—
85	C₂	#13092792	1	2	yes	—
86	C₂	#13092793	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#493	C₂
socle	C₂²×C₂₃	#127
hypercenter	#493	D₂₃
hypocenter	C₂₃	#1712
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #13092794	C₂	semidirect
2. #13092793	C₂	semidirect
3. C₈×C₂₃⋊C₈	C₂	semidirect
4. C₂₃×#493	C₂	semidirect
30. C₈×C₂₃	D₈	semidirect
31. C₂₃⋊C₈	D₈	semidirect
42. #493	D₂₃	semidirect
55. C₂₃	#1712	semidirect
75. C₈	#67272	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092794	2
#13092793	2
C₈×C₂₃⋊C₈	2
C₂₃×#493	2
#1712	23