Subgroups — Dic₆₁₆₀ #72224331

Order 24640.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₀₈₀	C₂	1	2	yes	maximal
2	Dic₁₅₄₀	—	2	2	no	—
3	Dic₁₂₃₂	—	5	1	no	maximal, Hall
4	Dic₈₈₀	—	7	1	no	maximal, Hall
5	Dic₇₇₀	—	4	2	no	—
6	Dic₆₁₆	—	5	2	no	—
7	Dic₅₆₀	—	11	1	no	maximal, Hall
8	C₅×C₇×C₁₁×C₃₂	C₂	1	1	yes	maximal, characteristic
9	C₅×C₇×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
10	C₅×C₇×C₈×C₁₁	D₄	1	1	yes	characteristic
11	C₇×C₁₁×C₃₂	D₅	1	1	yes	characteristic
12	Dic₄₄₀	—	7	2	no	—
13	C₅×C₁₁×C₃₂	D₇	1	1	yes	characteristic
14	C₄×C₅×C₇×C₁₁	D₈	1	1	yes	characteristic
15	Dic₃₈₅	—	8	2	no	—
16	Dic₃₀₈	—	10	2	no	—
17	C₇×C₁₁×C₁₆	C₂×D₅	1	1	yes	characteristic
18	Dic₂₈₀	—	11	2	no	—
19	C₅×C₇×C₃₂	D₁₁	1	1	yes	characteristic
20	Dic₂₂₀	—	14	2	no	—
21	C₅×C₁₁×C₁₆	C₂×D₇	1	1	yes	characteristic
22	C₂×C₅×C₇×C₁₁	D₁₆	1	1	yes	characteristic
23	Dic₁₇₆	—	35	1	no	Hall
24	Dic₁₅₄	—	20	2	no	—
25	C₇×C₈×C₁₁	D₂₀	1	1	yes	characteristic
26	Dic₁₄₀	—	22	2	no	—
27	C₅×C₇×C₁₆	C₂×D₁₁	1	1	yes	characteristic
28	Dic₁₁₂	—	55	1	no	Hall
29	Dic₁₁₀	—	28	2	no	—
30	C₅×C₈×C₁₁	D₂₈	1	1	yes	characteristic
31	C₅×C₇×C₁₁	Q₆₄	1	1	yes	characteristic, complemented, Hall
32	Dic₈₈	—	35	2	no	—
33	C₁₁×C₃₂	D₃₅	1	1	yes	characteristic
34	Dic₈₀	—	77	1	no	Hall
35	C₄×C₇×C₁₁	D₄₀	1	1	yes	characteristic
36	Dic₇₇	—	40	2	no	—
37	Dic₇₀	—	44	2	no	—
38	C₅×C₇×C₈	D₄₄	1	1	yes	characteristic
39	Dic₅₆	—	55	2	no	—
40	C₇×C₃₂	D₅₅	1	1	yes	characteristic
41	C₄×C₅×C₁₁	D₅₆	1	1	yes	characteristic
42	Dic₅₅	—	56	2	no	—
43	Dic₄₄	—	70	2	no	—
44	C₁₁×C₁₆	C₂×D₃₅	1	1	yes	characteristic
45	Dic₄₀	—	77	2	no	—
46	C₅×C₃₂	D₇₇	1	1	yes	characteristic
47	C₂×C₇×C₁₁	D₈₀	1	1	yes	characteristic
48	C₄×C₅×C₇	D₈₈	1	1	yes	characteristic
49	Dic₃₅	—	88	2	no	—
50	Dic₂₈	—	110	2	no	—
51	C₇×C₁₆	C₂×D₅₅	1	1	yes	characteristic
52	C₂×C₅×C₁₁	D₁₁₂	1	1	yes	characteristic
53	Dic₂₂	—	140	2	no	—
54	C₈×C₁₁	D₁₄₀	1	1	yes	characteristic
55	Dic₂₀	—	154	2	no	—
56	C₅×C₁₆	C₂×D₇₇	1	1	yes	characteristic
57	C₇×C₁₁	Dic₈₀	1	1	yes	characteristic, complemented, Hall
58	C₂×C₅×C₇	D₁₇₆	1	1	yes	characteristic
59	Q₆₄	—	385	1	no	Sylow
60	Dic₁₄	—	220	2	no	—
61	C₇×C₈	D₂₂₀	1	1	yes	characteristic
62	C₅×C₁₁	Dic₁₁₂	1	1	yes	characteristic, complemented, Hall
63	C₄×C₁₁	D₂₈₀	1	1	yes	characteristic
64	Dic₁₁	—	280	2	no	—
65	Dic₁₀	—	308	2	no	—
66	C₅×C₈	D₃₀₈	1	1	yes	characteristic
67	C₅×C₇	Dic₁₇₆	1	1	yes	characteristic, complemented, Hall
68	Q₃₂	—	385	2	no	—
69	C₃₂	D₃₈₅	1	1	yes	characteristic
70	C₄×C₇	D₄₄₀	1	1	yes	characteristic
71	Dic₇	—	440	2	no	—
72	C₂×C₁₁	D₅₆₀	1	1	yes	characteristic
73	C₄×C₅	D₆₁₆	1	1	yes	characteristic
74	Dic₅	—	616	2	no	—
75	Q₁₆	—	770	2	no	—
76	C₁₆	C₂×D₃₈₅	1	1	yes	characteristic
77	C₂×C₇	D₈₈₀	1	1	yes	characteristic
78	C₁₁	Dic₅₆₀	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₅	D₁₂₃₂	1	1	yes	characteristic
80	Q₈	—	1540	2	no	—
81	C₈	D₁₅₄₀	1	1	yes	characteristic
82	C₇	Dic₈₈₀	1	1	yes	characteristic, complemented, Sylow
83	C₅	Dic₁₂₃₂	1	1	yes	characteristic, complemented, Sylow
84	C₄	—	3080	2	no	—
85	C₄	D₃₀₈₀	1	1	yes	characteristic
86	C₂	D₆₁₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₆₁₆₀
commutator, cosocle	C₅×C₇×C₁₁×C₁₆	C₂²
Frattini	C₁₆	C₂×D₃₈₅
Fitting	C₅×C₇×C₁₁×C₃₂	C₂
socle, prime radical	C₂×C₅×C₇×C₁₁	D₁₆
hypercenter	C₃₂	D₃₈₅
hypocenter	C₅×C₇×C₁₁	Q₆₄
LCS	G ▷ C₅×C₇×C₁₁×C₁₆ ▷ C₅×C₇×C₈×C₁₁ ▷ C₄×C₅×C₇×C₁₁ ▷ C₂×C₅×C₇×C₁₁ ▷ C₅×C₇×C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
31. C₅×C₇×C₁₁	Q₆₄	semidirect
57. C₇×C₁₁	Dic₈₀	semidirect
62. C₅×C₁₁	Dic₁₁₂	semidirect
67. C₅×C₇	Dic₁₇₆	semidirect
78. C₁₁	Dic₅₆₀	semidirect
82. C₇	Dic₈₈₀	semidirect
83. C₅	Dic₁₂₃₂	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₀₈₀	2
Dic₁₂₃₂	5
Dic₈₈₀	7
Dic₅₆₀	11
C₅×C₇×C₁₁×C₃₂	2

Subgroups — Dic6160 #72224331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆₁₆₀ #72224331