Subgroups — #10623543

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₈₀	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6660	—	3	1	no	maximal
5	C₂×D₄₅	—	2	1	no	—
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₅×Dic₉	—	2	1	no	—
8	#3664	—	5	1	no	maximal, Hall
9	D₆₀	—	3	1	no	—
10	C₅×Dic₆	—	3	1	no	—
11	C₃×#168	D₃	1	1	yes	characteristic
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	D₄₅	—	4	1	no	—
14	#688	—	9	1	no	Hall
15	D₃₆	—	5	1	no	—
16	Dic₁₈	D₅	1	1	yes	characteristic, complemented
17	C₈×C₉	—	5	1	no	—
18	C₂×D₁₅	—	6	1	no	—
19	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	C₅×Dic₃	—	6	1	no	—
21	#204	—	15	1	no	—
22	C₅×C₉	SD₁₆	1	1	yes	characteristic, complemented, Hall
23	C₅×Q₈	—	9	1	no	—
24	D₂₀	—	9	1	no	—
25	#168	D₉	1	1	yes	characteristic, complemented
26	C₂×D₉	—	10	1	no	—
27	C₄×C₉	C₂×D₅	1	1	yes	characteristic
28	Dic₉	—	2	1	no	—
29	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
30	D₁₅	—	12	1	no	—
31	D₁₂	—	15	1	no	—
32	Dic₆	—	3	1	no	—
33	C₃×C₈	—	5	1	no	—
34	C₂×D₅	—	18	1	no	—
35	C₄×C₅	—	18	1	no	—
36	C₄×C₅	C₂×D₉	1	1	yes	characteristic
37	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
38	D₉	—	20	1	no	—
39	SD₁₆	—	45	1	no	Sylow
40	C₃×C₅	#204	1	1	yes	characteristic
41	C₂×D₃	—	30	1	no	—
42	C₃×C₄	D₃×D₅	1	1	yes	characteristic
43	Dic₃	—	6	1	no	—
44	C₂×C₅	D₃₆	1	1	yes	characteristic
45	D₅	—	36	1	no	—
46	C₉	#688	1	1	yes	characteristic, complemented, Sylow
47	Q₈	—	9	1	no	—
48	D₄	—	45	1	no	—
49	C₈	—	5	1	no	—
50	C₂×C₃	#1203	1	1	yes	characteristic
51	D₃	—	60	1	no	—
52	C₅	#3664	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	90	1	no	—
54	C₄	—	18	1	no	—
55	C₄	D₅×D₉	1	1	yes	characteristic
56	C₃	#6660	1	1	yes	characteristic
57	C₂	—	180	1	no	—
58	C₂	#67080	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67080
commutator, Fitting, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	D₃×D₅
socle	C₂×C₃×C₅	D₁₂
prime radical	D₁₈₀	C₂
hypercenter	C₄	D₅×D₉
hypocenter	C₅×C₉	SD₁₆
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×Dic₁₈	C₂	semidirect
3. C₉×#168	C₂	semidirect
16. Dic₁₈	D₅	semidirect
22. C₅×C₉	SD₁₆	semidirect
25. #168	D₉	semidirect
46. C₉	#688	semidirect
52. C₅	#3664	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×Dic₁₈	2
C₉×#168	2
#6660	3
#3664	5