Subgroups — D₃₆₀ #10623600

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₈₀	C₂	1	2	yes	maximal, complemented
2	C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₂₀	—	3	1	no	maximal
4	C₂×D₄₅	—	2	2	no	—
5	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
6	D₇₂	—	5	1	no	maximal, Hall
7	D₆₀	—	3	2	no	—
8	C₃×C₅×C₈	D₃	1	1	yes	characteristic
9	C₂×C₅×C₉	D₄	1	1	yes	characteristic
10	D₄₅	—	4	2	no	—
11	D₄₀	—	9	1	no	Hall
12	D₃₆	—	5	2	no	—
13	C₈×C₉	D₅	1	1	yes	characteristic, complemented
14	C₂×D₁₅	—	6	2	no	—
15	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
16	D₂₄	—	15	1	no	—
17	C₅×C₉	D₈	1	1	yes	characteristic, complemented, Hall
18	D₂₀	—	9	2	no	—
19	C₅×C₈	D₉	1	1	yes	characteristic, complemented
20	C₂×D₉	—	10	2	no	—
21	C₄×C₉	C₂×D₅	1	1	yes	characteristic
22	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
23	D₁₅	—	12	2	no	—
24	D₁₂	—	15	2	no	—
25	C₃×C₈	D₁₅	1	1	yes	characteristic
26	C₂×D₅	—	18	2	no	—
27	C₄×C₅	C₂×D₉	1	1	yes	characteristic
28	C₂×C₉	D₂₀	1	1	yes	characteristic
29	D₉	—	20	2	no	—
30	D₈	—	45	1	no	Sylow
31	C₃×C₅	D₂₄	1	1	yes	characteristic
32	C₂×D₃	—	30	2	no	—
33	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
34	C₂×C₅	D₃₆	1	1	yes	characteristic
35	D₅	—	36	2	no	—
36	C₉	D₄₀	1	1	yes	characteristic, complemented, Sylow
37	D₄	—	45	2	no	—
38	C₈	D₄₅	1	1	yes	characteristic, complemented
39	C₂×C₃	D₆₀	1	1	yes	characteristic
40	D₃	—	60	2	no	—
41	C₅	D₇₂	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	90	2	no	—
43	C₄	C₂×D₄₅	1	1	yes	characteristic
44	C₃	D₁₂₀	1	1	yes	characteristic
45	C₂	—	180	2	no	—
46	C₂	D₁₈₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₈₀
commutator, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	C₂×D₁₅
Fitting	C₅×C₈×C₉	C₂
socle	C₂×C₃×C₅	D₁₂
hypercenter	C₈	D₄₅
hypocenter	C₅×C₉	D₈
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. D₁₈₀	C₂	semidirect
2. C₅×C₈×C₉	C₂	semidirect
13. C₈×C₉	D₅	semidirect
17. C₅×C₉	D₈	semidirect
19. C₅×C₈	D₉	semidirect
36. C₉	D₄₀	semidirect
38. C₈	D₄₅	semidirect
41. C₅	D₇₂	semidirect

Maximal subgroups

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

subgroup	index
D₁₈₀	2
C₅×C₈×C₉	2
D₁₂₀	3
D₇₂	5

Subgroups — D360 #10623600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₆₀ #10623600