Subgroups — #13092700

Order 1472.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625059	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#103	C₂	1	1	yes	maximal, characteristic
4	#10625017	C₂	1	1	yes	maximal, characteristic
5	#10624999	C₂	1	1	yes	maximal, characteristic
6	Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#10624990	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₆	—	2	1	no	—
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	Dic₂₃⋊C₄	—	2	1	no	—
16	C₄×Dic₂₃	—	2	1	no	—
17	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×C₂₃	—	2	1	no	—
20	Q₈×C₂₃	Q₈	1	2	yes	complemented
21	C₂×C₄×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	1	no	—
25	C₂×Dic₂₃	—	2	1	no	—
26	C₂×Dic₂₃	D₄	1	1	yes	characteristic
27	C₂×Dic₂₃	D₄	1	1	yes	characteristic
28	Dic₄₆	—	4	2	no	—
29	C₈×C₂₃	—	2	1	no	—
30	C₂₃⋊C₈	—	2	1	no	—
31	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₃	—	2	2	no	—
33	C₄×C₂₃	—	4	1	no	—
34	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₃	#41	1	1	yes	characteristic
36	Dic₂₃	—	2	1	no	—
37	Dic₂₃	—	4	1	no	—
38	Dic₂₃	—	4	1	no	—
39	Dic₂₃	SD₁₆	1	2	yes	—
40	#475	—	23	1	no	maximal, Sylow
41	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
43	C₂×C₂₃	#137	1	1	yes	characteristic
44	C₄⋊Q₈	—	23	1	no	—
45	C₄×Q₈	—	23	1	no	—
46	#106	—	23	1	no	—
47	#106	—	23	1	no	—
48	C₄⋊C₈	—	23	1	no	—
49	#103	—	23	1	no	—
50	#103	D₂₃	1	1	yes	characteristic
51	C₂₃	#475	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	46	1	no	—
53	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
54	C₂×C₈	—	23	1	no	—
55	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
56	C₄⋊C₄	—	23	1	no	—
57	C₄⋊C₄	—	23	1	no	—
58	C₄⋊C₄	—	46	1	no	—
59	C₄⋊C₄	—	46	1	no	—
60	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
61	C₄²	—	23	1	no	—
62	C₄²	—	46	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	92	2	no	—
65	Q₈	Dic₄₆	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	23	1	no	—
69	C₂×C₄	—	23	1	no	—
70	C₂×C₄	—	46	1	no	—
71	C₂×C₄	—	46	1	no	—
72	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	46	1	no	—
75	C₂²	D₄×D₂₃	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	23	2	no	—
79	C₄	—	46	1	no	—
80	C₄	—	92	1	no	—
81	C₄	—	92	1	no	—
82	C₄	C₂×Dic₄₆	1	1	yes	characteristic
83	C₄	#67290	1	1	yes	characteristic
84	C₂	#10625049	1	1	yes	characteristic
85	C₂	SD₁₆×D₂₃	1	1	yes	characteristic
86	C₂	#10625089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#103	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	#103	D₂₃
hypocenter	C₂₃	#475
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
20. Q₈×C₂₃	Q₈	semidirect
51. C₂₃	#475	semidirect
65. Q₈	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₃	2
#10625059	2
C₂₃×#103	2
#10625017	2
#10624999	2
Dic₂₃⋊C₈	2
#10624990	2
#475	23