Subgroups — #13092467

Order 1472.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Dic₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₃	C₄	1	1	yes	characteristic
8	C₂²×Dic₂₃	C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂²	1	2	yes	—
10	C₄×Dic₂₃	C₄	1	1	yes	characteristic
11	C₄×Dic₂₃	C₄	1	1	yes	characteristic
12	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	—	2	2	no	—
20	C₂×Dic₂₃	C₈	1	4	yes	complemented
21	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₃	—	2	2	no	—
26	C₂₃⋊C₈	—	2	2	no	—
27	C₂²×C₂₃	—	2	1	no	—
28	C₂²×C₂₃	—	2	1	no	—
29	C₂²×C₂₃	C₂×C₈	1	2	yes	—
30	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	#41	1	2	yes	—
33	Dic₂₃	—	2	1	no	—
34	Dic₂₃	C₂×C₈	1	2	yes	—
35	Dic₂₃	C₂×C₈	1	2	yes	—
36	Dic₂₃	M₁₆	1	2	yes	—
37	#431	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	2	2	no	—
39	C₂×C₂₃	#117	1	1	yes	characteristic
40	C₂×C₂₃	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄²	—	23	1	no	—
43	C₄⋊C₈	—	23	2	no	—
44	C₂²⋊C₈	—	23	1	no	—
45	C₂²⋊C₈	D₂₃	1	1	yes	characteristic
46	C₄×C₈	—	23	2	no	—
47	C₂₃	#431	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₄	—	23	1	no	—
49	C₂²×C₄	—	23	1	no	—
50	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
51	C₂×C₈	—	23	2	no	—
52	C₂×C₈	C₂×D₂₃	1	2	yes	—
53	C₄²	—	23	1	no	—
54	C₄²	—	23	1	no	—
55	C₄²	—	23	1	no	—
56	C₄²	—	23	1	no	—
57	C₂³	C₄×D₂₃	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	23	4	no	—
60	C₂×C₄	—	23	1	no	—
61	C₂×C₄	—	23	1	no	—
62	C₂×C₄	—	23	1	no	—
63	C₂×C₄	—	23	1	no	—
64	C₂×C₄	—	46	2	no	—
65	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	46	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₈×D₂₃	1	2	yes	—
72	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	23	2	no	—
75	C₄	—	23	2	no	—
76	C₄	—	23	2	no	—
77	C₄	—	46	1	no	—
78	C₄	#67290	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#10625048	1	1	yes	characteristic
81	C₂	C₂×C₈×D₂₃	1	1	yes	characteristic
82	C₂	M₁₆×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₈
Fitting	C₂₃×C₂²⋊C₈	C₂
socle	C₂²×C₂₃	C₂²×C₄
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#431
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. Dic₂₃⋊C₈	C₂	semidirect
5. C₈×Dic₂₃	C₂	semidirect
20. C₂×Dic₂₃	C₈	semidirect
47. C₂₃	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃	2
C₂₃×C₂²⋊C₈	2
C₂²⋊(C₂₃⋊C₈)	2
Dic₂₃⋊C₈	2
C₈×Dic₂₃	2
#431	23