Subgroups — #10623549

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₈₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6653	—	3	1	no	maximal
5	C₂×D₄₅	—	2	1	no	—
6	C₂×C₉×D₅	—	2	1	no	—
7	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
8	#3673	—	5	1	no	maximal, Hall
9	D₆₀	—	3	1	no	—
10	C₃×D₂₀	D₃	1	1	yes	characteristic
11	C₅×C₃⋊C₈	—	3	1	no	—
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	D₄₅	—	4	1	no	—
14	C₉×D₅	—	4	1	no	—
15	D₄₀	—	9	1	no	Hall
16	D₄×C₉	—	5	1	no	—
17	D₃₆	—	5	1	no	—
18	C₉⋊C₈	D₅	1	1	yes	characteristic, complemented
19	C₂×D₁₅	—	6	1	no	—
20	C₂×C₃×D₅	—	2	1	no	—
21	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
22	#213	—	15	1	no	—
23	C₅×C₉	D₈	1	1	yes	characteristic, complemented, Hall
24	D₂₀	—	9	1	no	—
25	D₂₀	D₉	1	1	yes	characteristic, complemented
26	C₅×C₈	—	9	1	no	—
27	C₂²×C₉	—	10	1	no	—
28	C₂×D₉	—	10	1	no	—
29	C₄×C₉	C₂×D₅	1	1	yes	characteristic
30	C₂×C₃×C₅	#67	1	1	yes	characteristic
31	D₁₅	—	12	1	no	—
32	C₃×D₅	—	4	1	no	—
33	C₃×D₄	—	5	1	no	—
34	D₁₂	—	15	1	no	—
35	C₃⋊C₈	—	3	1	no	—
36	C₂×D₅	—	2	1	no	—
37	C₂×D₅	—	18	1	no	—
38	C₄×C₅	C₂×D₉	1	1	yes	characteristic
39	C₂×C₉	—	20	1	no	—
40	C₂×C₉	D₂₀	1	1	yes	characteristic
41	D₉	—	20	1	no	—
42	D₈	—	45	1	no	Sylow
43	C₃×C₅	#213	1	1	yes	characteristic
44	C₂²×C₃	—	10	1	no	—
45	C₂×D₃	—	30	1	no	—
46	C₃×C₄	D₃×D₅	1	1	yes	characteristic
47	C₂×C₅	#611	1	1	yes	characteristic
48	D₅	—	4	1	no	—
49	D₅	—	36	1	no	—
50	C₉	D₄₀	1	1	yes	characteristic, complemented, Sylow
51	D₄	—	5	1	no	—
52	D₄	—	45	1	no	—
53	C₈	—	9	1	no	—
54	C₂×C₃	—	20	1	no	—
55	C₂×C₃	#1202	1	1	yes	characteristic
56	D₃	—	60	1	no	—
57	C₅	#3673	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	10	1	no	—
59	C₂²	—	90	1	no	—
60	C₄	D₅×D₉	1	1	yes	characteristic
61	C₃	#6653	1	1	yes	characteristic
62	C₂	—	20	1	no	—
63	C₂	—	180	1	no	—
64	C₂	#67083	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67083
commutator, Fitting, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	D₃×D₅
socle	C₂×C₃×C₅	#67
hypercenter	C₄	D₅×D₉
hypocenter	C₅×C₉	D₈
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
1. D₁₈₀	C₂	semidirect
2. C₉×D₂₀	C₂	semidirect
3. C₅×C₉⋊C₈	C₂	semidirect
18. C₉⋊C₈	D₅	semidirect
23. C₅×C₉	D₈	semidirect
25. D₂₀	D₉	semidirect
50. C₉	D₄₀	semidirect
57. C₅	#3673	semidirect

Maximal subgroups

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

subgroup	index
D₁₈₀	2
C₉×D₂₀	2
C₅×C₉⋊C₈	2
#6653	3
#3673	5