Subgroups — #13092624

Order 1472.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#67275	C₂	1	2	yes	maximal
3	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
4	#10625017	C₂	1	2	yes	maximal
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	#67275	—	2	4	no	—
12	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
13	Dic₂₃⋊C₄	—	2	1	no	—
14	C₄×Dic₂₃	—	2	1	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	Q₈×C₂₃	—	2	2	no	—
17	Q₈×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	—	2	2	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	1	no	—
23	C₂×Dic₂₃	—	2	1	no	—
24	Dic₄₆	—	2	1	no	—
25	Dic₄₆	D₄	1	2	yes	—
26	C₂₃⋊C₈	—	2	2	no	—
27	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₃	—	2	1	no	—
29	C₄×C₂₃	—	4	2	no	—
30	C₄×C₂₃	Q₁₆	1	2	yes	complemented
31	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₃	#41	1	1	yes	characteristic
33	Dic₂₃	—	2	2	no	—
34	Dic₂₃	—	4	1	no	—
35	#462	—	23	1	no	maximal, Sylow
36	C₂×C₂₃	#121	1	1	yes	characteristic
37	C₂×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₂₃	#137	1	1	yes	characteristic
39	C₂×Q₁₆	—	23	2	no	—
40	C₄⋊Q₈	D₂₃	1	1	yes	characteristic
41	C₄×Q₈	—	23	1	no	—
42	C₄⋊C₈	—	23	1	no	—
43	#103	—	23	2	no	—
44	C₂₃	#462	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₈	—	23	1	no	—
46	C₂×Q₈	C₂×D₂₃	1	2	yes	—
47	Q₁₆	—	46	4	no	—
48	C₂×C₈	—	23	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	23	1	no	—
52	C₄⋊C₄	—	46	1	no	—
53	C₄²	—	46	1	no	—
54	C₄²	C₂×D₂₃	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	23	2	no	—
58	Q₈	—	46	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	46	1	no	—
61	C₂×C₄	—	46	1	no	—
62	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
65	C₈	—	46	2	no	—
66	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	2	no	—
69	C₄	—	46	2	no	—
70	C₄	—	92	1	no	—
71	C₄	#67275	1	2	yes	complemented
72	C₄	D₄×D₂₃	1	1	yes	characteristic
73	C₄	#67290	1	1	yes	characteristic
74	C₂	#10625109	1	1	yes	characteristic
75	C₂	#10625113	1	1	yes	characteristic
76	C₂	C₂×#67275	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×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#462
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
30. C₄×C₂₃	Q₁₆	semidirect
44. C₂₃	#462	semidirect
71. C₄	#67275	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
C₂×#67275	2
C₄×Dic₄₆	2
#10625017	2
C₄⋊(C₂₃⋊C₈)	2
#462	23