Subgroups — #13092572

Order 1472.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#67275	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#10624992	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₃	—	2	1	no	—
9	C₂×Dic₄₆	—	2	1	no	—
10	C₂×Dic₄₆	—	2	4	no	—
11	C₂×Dic₄₆	C₂²	1	2	yes	—
12	C₄⋊C₄×C₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₃	—	2	1	no	—
15	#67275	—	2	4	no	—
16	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
17	C₂³×C₂₃	D₄	1	1	yes	characteristic
18	Q₈×C₂₃	—	2	2	no	—
19	C₂×C₄×C₂₃	—	2	2	no	—
20	C₂×C₄×C₂₃	—	2	1	no	—
21	C₂×C₄×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	4	no	—
25	C₂×Dic₂₃	—	2	2	no	—
26	Dic₄₆	—	2	4	no	—
27	Dic₄₆	—	2	2	no	—
28	Dic₄₆	D₄	1	4	yes	—
29	C₂₃⋊C₈	—	2	2	no	—
30	C₂²×C₂₃	—	2	1	no	—
31	C₂²×C₂₃	—	2	1	no	—
32	C₂²×C₂₃	Q₁₆	1	2	yes	complemented
33	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₃	—	2	1	no	—
35	C₄×C₂₃	—	4	1	no	—
36	C₄×C₂₃	—	4	1	no	—
37	C₄×C₂₃	C₂×D₄	1	2	yes	—
38	Dic₂₃	—	2	4	no	—
39	#451	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	2	2	no	—
41	C₂×C₂₃	#120	1	1	yes	characteristic
42	C₂×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
43	C₂×C₂₃	#137	1	1	yes	characteristic
44	C₂²×Q₈	—	23	1	no	—
45	C₂×Q₁₆	—	23	2	no	—
46	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic
47	#103	—	23	2	no	—
48	C₂²⋊C₈	—	23	1	no	—
49	C₂₃	#451	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	—	23	2	no	—
51	C₂×Q₈	—	46	4	no	—
52	C₂×Q₈	—	46	1	no	—
53	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
54	C₂²×C₄	—	46	1	no	—
55	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
56	Q₁₆	—	46	4	no	—
57	C₂×C₈	—	23	2	no	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
60	C₂²⋊C₄	—	2	1	no	—
61	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	Q₈	—	23	4	no	—
64	Q₈	—	46	4	no	—
65	Q₈	—	46	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	46	4	no	—
70	C₂×C₄	—	46	2	no	—
71	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₈	—	46	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#67275	1	2	yes	complemented
77	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	46	4	no	—
82	C₄	D₄×D₂₃	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#10625108	1	1	yes	characteristic
85	C₂	C₂×#67275	1	1	yes	characteristic
86	C₂	#10625122	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×C₂²⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂³×C₂₃	D₄
hypercenter	C₂²⋊Q₈	D₂₃
hypocenter	C₂₃	#451
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×#67275	C₂	semidirect
5. #10624992	C₂	semidirect
32. C₂²×C₂₃	Q₁₆	semidirect
49. C₂₃	#451	semidirect
76. C₂²	#67275	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₆	2
C₂₃×C₂²⋊Q₈	2
C₂×#67275	2
C₂²⋊(C₂₃⋊C₈)	2
#10624992	2
#451	23