Subgroups — Dic₆₇₆₀ #72236282

Order 27040.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₃₈₀	C₂	1	2	yes	maximal
2	Dic₁₆₉₀	—	2	2	no	—
3	Dic₁₃₅₂	—	5	1	no	maximal, Hall
4	Dic₆₇₆	—	5	2	no	—
5	Dic₅₂₀	—	13	1	no	maximal
6	C₅×C₁₆×C₁₆₉	C₂	1	1	yes	maximal, characteristic
7	C₅×C₈×C₁₆₉	C₂²	1	1	yes	characteristic
8	C₁₆×C₁₆₉	D₅	1	1	yes	characteristic
9	Dic₈₄₅	—	4	2	no	—
10	C₄×C₅×C₁₆₉	D₄	1	1	yes	characteristic
11	C₂×C₅×C₁₆₉	D₈	1	1	yes	characteristic
12	Dic₃₃₈	—	10	2	no	—
13	C₈×C₁₆₉	C₂×D₅	1	1	yes	characteristic
14	Dic₂₆₀	—	13	2	no	—
15	C₅×C₁₃×C₁₆	D₁₃	1	1	yes	characteristic
16	C₅×C₁₆₉	Q₃₂	1	1	yes	characteristic, complemented, Hall
17	C₄×C₁₆₉	D₂₀	1	1	yes	characteristic
18	Dic₁₆₉	—	20	2	no	—
19	Dic₁₃₀	—	26	2	no	—
20	C₅×C₈×C₁₃	C₂×D₁₃	1	1	yes	characteristic
21	Dic₁₀₄	—	65	1	no	—
22	C₂×C₁₆₉	D₄₀	1	1	yes	characteristic
23	C₄×C₅×C₁₃	D₅₂	1	1	yes	characteristic
24	Dic₆₅	—	52	2	no	—
25	Dic₅₂	—	65	2	no	—
26	C₁₃×C₁₆	D₆₅	1	1	yes	characteristic
27	C₁₆₉	Dic₄₀	1	1	yes	characteristic, complemented, Sylow
28	Dic₄₀	—	169	1	no	Hall
29	C₂×C₅×C₁₃	D₁₀₄	1	1	yes	characteristic
30	Dic₂₆	—	130	2	no	—
31	C₈×C₁₃	C₂×D₆₅	1	1	yes	characteristic
32	Dic₂₀	—	169	2	no	—
33	C₅×C₁₆	D₁₆₉	1	1	yes	characteristic
34	C₅×C₁₃	Dic₁₀₄	1	1	yes	characteristic
35	C₄×C₁₃	D₂₆₀	1	1	yes	characteristic
36	Dic₁₃	—	260	2	no	—
37	Dic₁₀	—	338	2	no	—
38	C₅×C₈	C₂×D₁₆₉	1	1	yes	characteristic
39	Q₃₂	—	845	1	no	Sylow
40	C₂×C₁₃	D₅₂₀	1	1	yes	characteristic
41	C₄×C₅	D₆₇₆	1	1	yes	characteristic
42	Dic₅	—	676	2	no	—
43	Q₁₆	—	845	2	no	—
44	C₁₆	D₈₄₅	1	1	yes	characteristic
45	C₁₃	Dic₅₂₀	1	1	yes	characteristic
46	C₂×C₅	D₁₃₅₂	1	1	yes	characteristic
47	Q₈	—	1690	2	no	—
48	C₈	C₂×D₈₄₅	1	1	yes	characteristic
49	C₅	Dic₁₃₅₂	1	1	yes	characteristic, complemented, Sylow
50	C₄	—	3380	2	no	—
51	C₄	D₃₃₈₀	1	1	yes	characteristic
52	C₂	D₆₇₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₆₇₆₀
commutator, cosocle	C₅×C₈×C₁₆₉	C₂²
Frattini	C₈×C₁₃	C₂×D₆₅
Fitting	C₅×C₁₆×C₁₆₉	C₂
socle, prime radical	C₂×C₅×C₁₃	D₁₀₄
hypercenter	C₁₆	D₈₄₅
hypocenter	C₅×C₁₆₉	Q₃₂
LCS	G ▷ C₅×C₈×C₁₆₉ ▷ C₄×C₅×C₁₆₉ ▷ C₂×C₅×C₁₆₉ ▷ C₅×C₁₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
16. C₅×C₁₆₉	Q₃₂	semidirect
27. C₁₆₉	Dic₄₀	semidirect
49. C₅	Dic₁₃₅₂	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₃₈₀	2
Dic₁₃₅₂	5
Dic₅₂₀	13
C₅×C₁₆×C₁₆₉	2

Subgroups — Dic6760 #72236282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₆₇₆₀ #72236282