Subgroups — #15671449

Order 2944.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#13092168	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#499	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#125	—	2	1	no	—
5	C₂₃×#107	C₄	1	1	yes	characteristic
6	C₂₃×#107	C₄	1	1	yes	characteristic
7	C₂₃×#106	—	2	1	no	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
13	C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₄⋊C₄×C₂₃	—	4	1	no	—
16	C₄⋊C₄×C₂₃	—	4	1	no	—
17	C₄²×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	—	4	1	no	—
19	C₂₃⋊C₁₆	—	4	1	no	—
20	C₂×C₄×C₂₃	—	4	1	no	—
21	C₂×C₄×C₂₃	—	4	1	no	—
22	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
25	C₈×C₂₃	—	2	1	no	—
26	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
27	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
28	C₂₃⋊C₈	—	8	1	no	—
29	#1363	—	23	1	no	maximal, Sylow
30	C₂²×C₂₃	#102	1	1	yes	characteristic
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	—	2	1	no	—
33	C₄×C₂₃	—	8	1	no	—
34	C₄×C₂₃	—	8	1	no	—
35	C₄×C₂₃	#101	1	1	yes	characteristic
36	C₄×C₂₃	#103	1	1	yes	characteristic
37	#499	D₂₃	1	1	yes	characteristic
38	C₄⋊C₁₆	—	23	1	no	—
39	#334	—	23	1	no	—
40	C₂×C₂₃	#332	1	1	yes	characteristic
41	C₂×C₂₃	#359	1	1	yes	characteristic
42	C₂×C₂₃	#360	1	1	yes	characteristic
43	#125	—	2	1	no	—
44	C₂×C₁₆	—	46	1	no	—
45	#107	Dic₂₃	1	1	yes	characteristic
46	#107	Dic₂₃	1	1	yes	characteristic
47	#106	—	2	1	no	—
48	C₄⋊C₈	—	46	1	no	—
49	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
50	C₂₃	#1363	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₈	—	92	1	no	—
52	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	—	4	1	no	—
58	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₁₆	—	92	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	#67272	1	1	yes	characteristic
63	C₂×C₄	#67273	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	184	1	no	—
67	C₈	#67274	1	1	yes	characteristic
68	C₈	#67275	1	1	yes	characteristic
69	C₂²	#10625014	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	8	1	no	—
73	C₄	—	8	1	no	—
74	C₄	#10625017	1	1	yes	characteristic
75	C₄	#10625018	1	1	yes	characteristic
76	C₂	#13092163	1	1	yes	characteristic
77	C₂	#13092224	1	1	yes	characteristic
78	C₂	#13092227	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625014
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#499	C₂
socle, prime radical	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#499	D₂₃
hypocenter	C₂₃	#1363
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
50. C₂₃	#1363	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₃⋊C₁₆)	2
#13092168	2
C₂₃×#499	2
#1363	23