Subgroups — #13092521

Order 1472.

108 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	C₂×#67275	C₂	1	1	yes	maximal, characteristic
3	#10625058	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#103	C₂	1	1	yes	maximal, characteristic
5	#10624998	C₂	1	1	yes	maximal, characteristic
6	#10624997	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	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	#67275	C₄	1	4	yes	complemented
13	C₄⋊Dic₂₃	—	2	1	no	—
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	Dic₂₃⋊C₄	—	2	1	no	—
16	C₄×Dic₂₃	—	2	1	no	—
17	C₄×Dic₂₃	—	2	1	no	—
18	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	Q₈×C₂₃	—	2	1	no	—
21	Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₃	—	2	1	no	—
26	C₂×Dic₂₃	—	2	1	no	—
27	C₂×Dic₂₃	D₄	1	1	yes	characteristic
28	C₂×Dic₂₃	D₄	1	1	yes	characteristic
29	Dic₄₆	—	2	1	no	—
30	Dic₄₆	C₂×C₄	1	2	yes	—
31	C₈×C₂₃	—	2	1	no	—
32	C₂₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₃	—	2	2	no	—
35	C₄×C₂₃	—	4	1	no	—
36	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₂₃	#41	1	1	yes	characteristic
38	Dic₂₃	—	2	2	no	—
39	Dic₂₃	—	2	1	no	—
40	Dic₂₃	—	2	1	no	—
41	Dic₂₃	—	4	1	no	—
42	#441	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
44	C₂×C₂₃	#137	1	1	yes	characteristic
45	C₂×C₂₃	#137	1	1	yes	characteristic
46	C₂×Q₁₆	—	23	1	no	—
47	C₄×Q₈	—	23	1	no	—
48	C₄×Q₈	—	23	1	no	—
49	#106	—	23	1	no	—
50	#103	—	23	1	no	—
51	#103	D₂₃	1	1	yes	characteristic
52	#97	—	23	1	no	—
53	C₂₃	#441	1	1	yes	characteristic, complemented, Sylow
54	C₂×Q₈	—	23	1	no	—
55	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
56	Q₁₆	—	23	4	no	—
57	C₂×C₈	—	23	1	no	—
58	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	23	1	no	—
60	C₄⋊C₄	—	46	1	no	—
61	C₄⋊C₄	—	46	1	no	—
62	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
63	C₄²	—	23	1	no	—
64	C₄²	—	46	1	no	—
65	C₄²	—	46	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	23	2	no	—
68	Q₈	—	46	1	no	—
69	Q₈	C₄×D₂₃	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	23	1	no	—
73	C₂×C₄	—	23	1	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	—	46	1	no	—
76	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	23	2	no	—
79	C₂²	D₄×D₂₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	46	2	no	—
83	C₄	—	46	1	no	—
84	C₄	—	46	1	no	—
85	C₄	—	92	1	no	—
86	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
87	C₄	#67290	1	1	yes	characteristic
88	C₂	(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
89	C₂	#10625086	1	1	yes	characteristic
90	C₂	#10625089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#103	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#103	D₂₃
hypocenter	C₂₃	#441
LCS	G ▷ 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
12. #67275	C₄	semidirect
53. C₂₃	#441	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
C₂×#67275	2
#10625058	2
C₂₃×#103	2
#10624998	2
#10624997	2
#10624990	2
#441	23