Subgroups — #15671452

Order 2944.

86 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	#13092170	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#500	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊Q₈	—	2	1	no	—
5	C₂₃×#107	—	2	1	no	—
6	C₂₃×#107	C₄	1	2	yes	—
7	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃⋊C₁₆	—	2	1	no	—
9	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
10	C₂×Q₈×C₂₃	—	4	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	2	no	—
14	C₄⋊C₄×C₂₃	—	4	1	no	—
15	C₄²×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₂₃⋊C₈	—	4	1	no	—
17	C₂₃⋊C₁₆	—	4	1	no	—
18	Q₈×C₂₃	—	8	1	no	—
19	Q₈×C₂₃	—	8	1	no	—
20	C₂×C₄×C₂₃	—	4	2	no	—
21	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
24	C₈×C₂₃	—	2	1	no	—
25	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
26	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
27	C₂₃⋊C₈	—	8	1	no	—
28	#1364	—	23	1	no	maximal, Sylow
29	C₂²×C₂₃	#102	1	1	yes	characteristic
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	—	8	2	no	—
32	C₄×C₂₃	#101	1	1	yes	characteristic
33	C₄×C₂₃	#103	1	1	yes	characteristic
34	C₄×C₂₃	D₁₆	1	1	yes	characteristic
35	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
36	#500	D₂₃	1	1	yes	characteristic
37	C₄⋊C₁₆	—	23	1	no	—
38	#335	—	23	1	no	—
39	C₂×C₂₃	#332	1	1	yes	characteristic
40	C₂×C₂₃	#357	1	1	yes	characteristic
41	C₂×C₂₃	#362	1	1	yes	characteristic
42	C₄⋊Q₈	—	2	1	no	—
43	C₂×C₁₆	—	46	1	no	—
44	#107	—	2	1	no	—
45	#107	Dic₂₃	1	2	yes	—
46	C₄⋊C₈	—	46	1	no	—
47	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
48	C₂₃	#1364	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	92	1	no	—
51	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
56	C₁₆	—	92	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#67272	1	1	yes	characteristic
61	C₂×C₄	#67273	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	184	1	no	—
65	C₈	#67274	1	1	yes	characteristic
66	C₈	#67275	1	1	yes	characteristic
67	C₂²	#10625014	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#10625008	1	1	yes	characteristic
71	C₄	#10625009	1	1	yes	characteristic
72	C₄	#10625017	1	1	yes	characteristic
73	C₄	#10625018	1	1	yes	characteristic
74	C₂	#13092163	1	1	yes	characteristic
75	C₂	#13092218	1	1	yes	characteristic
76	C₂	#13092233	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625014
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#500	C₂
socle, prime radical	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#500	D₂₃
hypocenter	C₂₃	#1364
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
48. C₂₃	#1364	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
#13092170	2
C₂₃×#500	2
#1364	23