Subgroups — #13092702

Order 1472.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625040	C₂	1	1	yes	maximal, characteristic
3	#10625017	C₂	1	2	yes	maximal
4	#10624990	C₂	1	2	yes	maximal
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₄₆	—	2	1	no	—
8	C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₄²×C₂₃	—	2	1	no	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₃	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	Q₈×C₂₃	—	2	1	no	—
16	Q₈×C₂₃	Q₈	1	2	yes	complemented
17	C₂×C₄×C₂₃	—	2	1	no	—
18	C₂×C₄×C₂₃	—	2	1	no	—
19	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₃	—	2	2	no	—
23	Dic₄₆	—	4	2	no	—
24	C₂₃⋊C₈	—	2	2	no	—
25	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₃	—	2	2	no	—
27	C₄×C₂₃	—	2	1	no	—
28	C₄×C₂₃	—	4	1	no	—
29	C₄×C₂₃	SD₁₆	1	2	yes	—
30	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₂₃	#41	1	1	yes	characteristic
32	Dic₂₃	—	4	2	no	—
33	#475	—	23	1	no	maximal, Sylow
34	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
35	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
36	C₂×C₂₃	#137	1	1	yes	characteristic
37	C₄⋊Q₈	—	23	1	no	—
38	C₄×Q₈	D₂₃	1	1	yes	characteristic
39	#106	—	23	2	no	—
40	C₄⋊C₈	—	23	1	no	—
41	#103	—	23	2	no	—
42	C₂₃	#475	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	—	46	1	no	—
44	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
45	C₂×C₈	—	23	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	23	2	no	—
48	C₄⋊C₄	—	46	1	no	—
49	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂×D₂₃	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	92	2	no	—
54	Q₈	Dic₄₆	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	46	2	no	—
58	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
61	C₈	—	46	2	no	—
62	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	1	no	—
66	C₄	—	92	2	no	—
67	C₄	#67274	1	2	yes	—
68	C₄	C₂×Dic₄₆	1	1	yes	characteristic
69	C₄	#67288	1	1	yes	characteristic
70	C₂	#10625095	1	1	yes	characteristic
71	C₂	C₂×#67274	1	1	yes	characteristic
72	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	C₄×Q₈×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#475
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. Q₈×C₂₃	Q₈	semidirect
42. C₂₃	#475	semidirect
54. Q₈	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₃	2
#10625040	2
#10625017	2
#10624990	2
C₄⋊(C₂₃⋊C₈)	2
#475	23