Subgroups — #13092469

Order 1472.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₃)⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂²	1	2	yes	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₂₃	C₄	1	1	yes	characteristic
12	C₄×Dic₂₃	C₄	1	1	yes	characteristic
13	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₃	—	2	1	no	—
23	C₄×D₂₃	—	2	1	no	—
24	C₄×D₂₃	—	2	2	no	—
25	C₄×D₂₃	C₈	1	4	yes	complemented
26	C₈×C₂₃	—	2	2	no	—
27	C₂₃⋊C₈	—	2	2	no	—
28	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₂₃	—	2	1	no	—
30	C₂×D₂₃	—	2	1	no	—
31	C₂×D₂₃	C₂×C₈	1	2	yes	—
32	C₄×C₂₃	—	2	1	no	—
33	C₄×C₂₃	C₂×C₈	1	2	yes	—
34	C₄×C₂₃	#41	1	1	yes	characteristic
35	C₄×C₂₃	#41	1	1	yes	characteristic
36	Dic₂₃	—	2	1	no	—
37	Dic₂₃	C₂×C₈	1	2	yes	—
38	Dic₂₃	M₁₆	1	2	yes	—
39	#431	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	#117	1	1	yes	characteristic
41	C₂×C₂₃	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
43	D₂₃	—	2	2	no	—
44	C₂×C₄²	—	23	1	no	—
45	C₄⋊C₈	—	23	1	no	—
46	C₄⋊C₈	D₂₃	1	1	yes	characteristic, complemented
47	C₂²⋊C₈	—	23	2	no	—
48	C₄×C₈	—	23	2	no	—
49	C₂₃	#431	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₄	—	23	1	no	—
51	C₂²×C₄	—	23	1	no	—
52	C₂²×C₄	—	23	1	no	—
53	C₂×C₈	—	23	2	no	—
54	C₂×C₈	C₂×D₂₃	1	2	yes	—
55	C₄²	—	23	1	no	—
56	C₄²	—	23	1	no	—
57	C₄²	—	23	1	no	—
58	C₄²	C₂×D₂₃	1	1	yes	characteristic
59	C₂³	—	23	1	no	—
60	C₂×C₄	—	23	1	no	—
61	C₂×C₄	—	23	1	no	—
62	C₂×C₄	—	23	4	no	—
63	C₂×C₄	—	23	1	no	—
64	C₂×C₄	—	46	2	no	—
65	C₂×C₄	—	46	1	no	—
66	C₂×C₄	—	46	1	no	—
67	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
68	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	46	2	no	—
72	C₂²	—	23	2	no	—
73	C₂²	—	46	1	no	—
74	C₂²	—	46	1	no	—
75	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	23	2	no	—
78	C₄	—	23	2	no	—
79	C₄	—	46	1	no	—
80	C₄	C₈×D₂₃	1	2	yes	—
81	C₄	#67290	1	1	yes	characteristic
82	C₄	#67292	1	1	yes	characteristic
83	C₂	—	46	2	no	—
84	C₂	#10625063	1	1	yes	characteristic
85	C₂	C₂×C₈×D₂₃	1	1	yes	characteristic
86	C₂	M₁₆×D₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×C₄⋊C₈	C₂	semidirect
4. C₈×Dic₂₃	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
25. C₄×D₂₃	C₈	semidirect
46. C₄⋊C₈	D₂₃	semidirect
49. C₂₃	#431	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₂₃	2
C₂₃×C₄⋊C₈	2
(C₂×D₂₃)⋊C₈	2
C₈×Dic₂₃	2
C₄⋊(C₂₃⋊C₈)	2
#431	23