Subgroups — #13092856

Order 1472.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625083	C₂	1	1	yes	maximal, characteristic
3	C₂₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625011	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625009	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#10624980	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×D₂₃	—	2	1	no	—
9	#67290	—	2	1	no	—
10	Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	D₈×C₂₃	C₂²	1	1	yes	characteristic
12	#67275	—	2	1	no	—
13	#67273	—	2	1	no	—
14	Dic₉₂	C₂²	1	1	yes	characteristic
15	Dic₉₂	C₂²	1	1	yes	characteristic
16	C₈×D₂₃	C₂²	1	1	yes	characteristic
17	C₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	Q₈×C₂₃	—	2	1	no	—
20	D₄×C₂₃	—	2	1	no	—
21	C₂²⋊D₂₃	—	4	1	no	—
22	C₂×Dic₂₃	—	4	1	no	—
23	C₄×D₂₃	—	4	1	no	—
24	C₄×D₂₃	D₄	1	1	yes	characteristic
25	Dic₄₆	—	2	1	no	—
26	Dic₄₆	—	2	1	no	—
27	Dic₄₆	—	4	1	no	—
28	C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃⋊C₈	D₄	1	1	yes	characteristic
30	C₂²×C₂₃	—	4	1	no	—
31	C₂×D₂₃	D₈	1	1	yes	characteristic
32	C₄×C₂₃	—	4	1	no	—
33	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	Dic₂₃	—	4	1	no	—
35	Dic₂₃	—	4	1	no	—
36	Dic₂₃	D₈	1	1	yes	characteristic
37	#510	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	8	1	no	—
39	C₂×C₂₃	C₂×D₈	1	1	yes	characteristic
40	D₂₃	—	2	1	no	—
41	#135	—	23	1	no	—
42	C₂×Q₁₆	—	23	1	no	—
43	Q₃₂	—	23	1	no	—
44	Q₃₂	—	23	1	no	—
45	SD₃₂	—	23	1	no	—
46	SD₃₂	D₂₃	1	1	yes	characteristic, complemented
47	M₃₂	—	23	1	no	—
48	C₂₃	#510	1	1	yes	characteristic, complemented, Sylow
49	#41	—	46	1	no	—
50	C₂×Q₈	—	46	1	no	—
51	Q₁₆	—	23	1	no	—
52	Q₁₆	—	23	1	no	—
53	Q₁₆	—	46	1	no	—
54	Q₁₆	C₂×D₂₃	1	1	yes	characteristic
55	SD₁₆	—	46	1	no	—
56	D₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂×C₈	—	23	1	no	—
58	C₁₆	—	23	1	no	—
59	C₁₆	C₂×D₂₃	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	46	1	no	—
62	Q₈	—	46	1	no	—
63	Q₈	—	92	1	no	—
64	D₄	—	2	1	no	—
65	D₄	—	92	1	no	—
66	C₂×C₄	—	23	1	no	—
67	C₂×C₄	—	92	1	no	—
68	C₂×C₄	—	92	1	no	—
69	C₈	—	23	1	no	—
70	C₈	C₂²×D₂₃	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	—	23	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	23	1	no	—
75	C₄	—	92	1	no	—
76	C₄	—	92	1	no	—
77	C₄	D₄×D₂₃	1	1	yes	characteristic
78	C₂	—	8	1	no	—
79	C₂	—	46	1	no	—
80	C₂	D₈×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₃
commutator, cosocle	C₈×C₂₃	C₂³
Frattini	C₈	C₂²×D₂₃
Fitting	C₂₃×SD₃₂	C₂
socle	C₂×C₂₃	C₂×D₈
prime radical	#10625083	C₂
hypercenter	SD₃₂	D₂₃
hypocenter	C₂₃	#510
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. Q₁₆×D₂₃	C₂	semidirect
3. C₂₃×SD₃₂	C₂	semidirect
4. #10625011	C₂	semidirect
5. #10625009	C₂	semidirect
6. Dic₁₈₄	C₂	semidirect
7. #10624980	C₂	semidirect
46. SD₃₂	D₂₃	semidirect
48. C₂₃	#510	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×D₂₃	2
#10625083	2
C₂₃×SD₃₂	2
#10625011	2
#10625009	2
Dic₁₈₄	2
#10624980	2
#510	23