Subgroups — Dic₁₆₀₁₆ #72298103

Order 64064.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₈₀₀₈	C₂	1	2	yes	maximal
2	Dic₄₀₀₄	—	2	2	no	—
3	Dic₂₂₈₈	—	7	1	no	maximal, Hall
4	Dic₂₀₀₂	—	4	2	no	—
5	Dic₁₄₅₆	—	11	1	no	maximal, Hall
6	Dic₁₂₃₂	—	13	1	no	maximal, Hall
7	Dic₁₁₄₄	—	7	2	no	—
8	Dic₇₂₈	—	11	2	no	—
9	Dic₆₁₆	—	13	2	no	—
10	Dic₅₇₂	—	14	2	no	—
11	C₇×C₁₁×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic
12	C₇×C₁₁×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₇×C₈×C₁₁×C₁₃	D₄	1	1	yes	characteristic
14	C₁₁×C₁₃×C₃₂	D₇	1	1	yes	characteristic
15	C₇×C₁₃×C₃₂	D₁₁	1	1	yes	characteristic
16	C₇×C₁₁×C₃₂	D₁₃	1	1	yes	characteristic
17	C₁₁×C₁₃×C₁₆	C₂×D₇	1	1	yes	characteristic
18	Dic₁₀₀₁	—	8	2	no	—
19	C₄×C₇×C₁₁×C₁₃	D₈	1	1	yes	characteristic
20	C₂×C₇×C₁₁×C₁₃	D₁₆	1	1	yes	characteristic
21	Dic₃₆₄	—	22	2	no	—
22	C₇×C₁₃×C₁₆	C₂×D₁₁	1	1	yes	characteristic
23	Dic₃₀₈	—	26	2	no	—
24	C₇×C₁₁×C₁₆	C₂×D₁₃	1	1	yes	characteristic
25	Dic₂₈₆	—	28	2	no	—
26	C₈×C₁₁×C₁₃	D₂₈	1	1	yes	characteristic
27	C₇×C₁₁×C₁₃	Q₆₄	1	1	yes	characteristic, complemented, Hall
28	Dic₂₀₈	—	77	1	no	Hall
29	Dic₁₈₂	—	44	2	no	—
30	C₇×C₈×C₁₃	D₄₄	1	1	yes	characteristic
31	Dic₁₇₆	—	91	1	no	Hall
32	Dic₁₅₄	—	52	2	no	—
33	C₇×C₈×C₁₁	D₅₂	1	1	yes	characteristic
34	C₄×C₁₁×C₁₃	D₅₆	1	1	yes	characteristic
35	Dic₁₄₃	—	56	2	no	—
36	Dic₁₁₂	—	143	1	no	Hall
37	Dic₁₀₄	—	77	2	no	—
38	C₁₃×C₃₂	D₇₇	1	1	yes	characteristic
39	C₄×C₇×C₁₃	D₈₈	1	1	yes	characteristic
40	Dic₉₁	—	88	2	no	—
41	Dic₈₈	—	91	2	no	—
42	C₁₁×C₃₂	D₉₁	1	1	yes	characteristic
43	C₄×C₇×C₁₁	D₁₀₄	1	1	yes	characteristic
44	Dic₇₇	—	104	2	no	—
45	C₂×C₁₁×C₁₃	D₁₁₂	1	1	yes	characteristic
46	Dic₅₆	—	143	2	no	—
47	C₇×C₃₂	D₁₄₃	1	1	yes	characteristic
48	Dic₅₂	—	154	2	no	—
49	C₁₃×C₁₆	C₂×D₇₇	1	1	yes	characteristic
50	C₂×C₇×C₁₃	D₁₇₆	1	1	yes	characteristic
51	Dic₄₄	—	182	2	no	—
52	C₁₁×C₁₆	C₂×D₉₁	1	1	yes	characteristic
53	C₂×C₇×C₁₁	D₂₀₈	1	1	yes	characteristic
54	C₁₁×C₁₃	Dic₁₁₂	1	1	yes	characteristic, complemented, Hall
55	Dic₂₈	—	286	2	no	—
56	C₇×C₁₆	C₂×D₁₄₃	1	1	yes	characteristic
57	Dic₂₆	—	308	2	no	—
58	C₈×C₁₃	D₃₀₈	1	1	yes	characteristic
59	C₇×C₁₃	Dic₁₇₆	1	1	yes	characteristic, complemented, Hall
60	Dic₂₂	—	364	2	no	—
61	C₈×C₁₁	D₃₆₄	1	1	yes	characteristic
62	C₇×C₁₁	Dic₂₀₈	1	1	yes	characteristic, complemented, Hall
63	Q₆₄	—	1001	1	no	Sylow
64	Dic₁₄	—	572	2	no	—
65	C₇×C₈	D₅₇₂	1	1	yes	characteristic
66	C₄×C₁₃	D₆₁₆	1	1	yes	characteristic
67	Dic₁₃	—	616	2	no	—
68	C₄×C₁₁	D₇₂₈	1	1	yes	characteristic
69	Dic₁₁	—	728	2	no	—
70	Q₃₂	—	1001	2	no	—
71	C₃₂	D₁₀₀₁	1	1	yes	characteristic
72	C₄×C₇	D₁₁₄₄	1	1	yes	characteristic
73	Dic₇	—	1144	2	no	—
74	C₂×C₁₃	D₁₂₃₂	1	1	yes	characteristic
75	C₂×C₁₁	D₁₄₅₆	1	1	yes	characteristic
76	Q₁₆	—	2002	2	no	—
77	C₁₆	C₂×D₁₀₀₁	1	1	yes	characteristic
78	C₂×C₇	D₂₂₈₈	1	1	yes	characteristic
79	C₁₃	Dic₁₂₃₂	1	1	yes	characteristic, complemented, Sylow
80	C₁₁	Dic₁₄₅₆	1	1	yes	characteristic, complemented, Sylow
81	Q₈	—	4004	2	no	—
82	C₈	D₄₀₀₄	1	1	yes	characteristic
83	C₇	Dic₂₂₈₈	1	1	yes	characteristic, complemented, Sylow
84	C₄	—	8008	2	no	—
85	C₄	D₈₀₀₈	1	1	yes	characteristic
86	C₂	D₁₆₀₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆₀₁₆
commutator, cosocle	C₇×C₁₁×C₁₃×C₁₆	C₂²
Frattini	C₁₆	C₂×D₁₀₀₁
Fitting	C₇×C₁₁×C₁₃×C₃₂	C₂
socle, prime radical	C₂×C₇×C₁₁×C₁₃	D₁₆
hypercenter	C₃₂	D₁₀₀₁
hypocenter	C₇×C₁₁×C₁₃	Q₆₄
LCS	G ▷ C₇×C₁₁×C₁₃×C₁₆ ▷ C₇×C₈×C₁₁×C₁₃ ▷ C₄×C₇×C₁₁×C₁₃ ▷ C₂×C₇×C₁₁×C₁₃ ▷ C₇×C₁₁×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
27. C₇×C₁₁×C₁₃	Q₆₄	semidirect
54. C₁₁×C₁₃	Dic₁₁₂	semidirect
59. C₇×C₁₃	Dic₁₇₆	semidirect
62. C₇×C₁₁	Dic₂₀₈	semidirect
79. C₁₃	Dic₁₂₃₂	semidirect
80. C₁₁	Dic₁₄₅₆	semidirect
83. C₇	Dic₂₂₈₈	semidirect

Maximal subgroups

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

subgroup	index
Dic₈₀₀₈	2
Dic₂₂₈₈	7
Dic₁₄₅₆	11
Dic₁₂₃₂	13
C₇×C₁₁×C₁₃×C₃₂	2

Subgroups — Dic16016 #72298103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₆₀₁₆ #72298103