Subgroups — #13092755

Order 1472.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	1	yes	maximal, characteristic
3	#10625017	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625017	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10624990	C₂	1	1	yes	maximal, characteristic, complemented
7	#10624989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²⋊C₄×C₂₃	—	2	1	no	—
13	C₂²⋊Dic₂₃	—	2	1	no	—
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₃	—	2	1	no	—
17	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂³×C₂₃	D₄	1	1	yes	characteristic
20	Q₈×C₂₃	—	2	1	no	—
21	Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	—	2	1	no	—
25	C₂×C₄×C₂₃	—	2	1	no	—
26	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₃	—	2	1	no	—
29	C₂×Dic₂₃	—	2	1	no	—
30	C₂₃⋊C₈	—	2	1	no	—
31	C₂₃⋊C₈	—	2	1	no	—
32	C₂²×C₂₃	—	2	1	no	—
33	C₂²×C₂₃	—	2	1	no	—
34	C₂²×C₂₃	—	2	1	no	—
35	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₃	—	2	1	no	—
37	C₄×C₂₃	—	4	1	no	—
38	C₄×C₂₃	—	4	1	no	—
39	C₄×C₂₃	#41	1	1	yes	characteristic
40	C₄×C₂₃	#41	1	1	yes	characteristic
41	Dic₂₃	—	4	1	no	—
42	Dic₂₃	—	4	1	no	—
43	#485	—	23	1	no	maximal, Sylow
44	C₂×C₂₃	—	4	1	no	—
45	C₂×C₂₃	#123	1	1	yes	characteristic
46	C₂×C₂₃	#135	1	1	yes	characteristic
47	C₂×C₂₃	#137	1	1	yes	characteristic
48	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic
49	#117	—	23	1	no	—
50	#107	—	23	1	no	—
51	#106	—	23	1	no	—
52	#103	—	23	1	no	—
53	#103	—	23	1	no	—
54	C₂²⋊C₈	—	23	1	no	—
55	C₂₃	#485	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
58	C₂×C₈	—	23	1	no	—
59	C₂×C₈	—	23	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	23	1	no	—
62	C₄⋊C₄	—	23	1	no	—
63	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	46	1	no	—
66	C₄²	—	46	1	no	—
67	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	—	46	1	no	—
76	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	C₈	—	46	1	no	—
79	C₈	—	46	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	92	1	no	—
88	C₄	—	92	1	no	—
89	C₄	#67290	1	1	yes	characteristic
90	C₄	#67290	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#10625106	1	1	yes	characteristic
93	C₂	#10625113	1	1	yes	characteristic
94	C₂	#10625121	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	#485
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. #10625017	C₂	semidirect
4. #10625017	C₂	semidirect
6. #10624990	C₂	semidirect
7. #10624989	C₂	semidirect
55. C₂₃	#485	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₂²⋊Q₈	2
#10625097	2
#10625017	2
C₂²⋊(C₂₃⋊C₈)	2
#10624990	2
#10624989	2
#485	23