Subgroups — #10623544

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	C₅×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6654	—	3	1	no	maximal
5	C₂×D₄₅	—	2	1	no	—
6	C₂×C₅×D₉	—	2	1	no	—
7	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
8	D₇₂	—	5	1	no	maximal, Hall
9	D₆₀	—	3	1	no	—
10	C₅×D₁₂	—	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	C₅×D₉	—	4	1	no	—
15	#686	—	9	1	no	Hall
16	D₃₆	—	5	1	no	—
17	D₃₆	D₅	1	1	yes	characteristic, complemented
18	C₈×C₉	—	5	1	no	—
19	C₂×D₁₅	—	6	1	no	—
20	C₂×C₅×D₃	—	6	1	no	—
21	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
22	D₂₄	—	15	1	no	—
23	C₅×C₉	D₈	1	1	yes	characteristic, complemented, Hall
24	C₅×D₄	—	9	1	no	—
25	D₂₀	—	9	1	no	—
26	#168	D₉	1	1	yes	characteristic, complemented
27	C₂×D₉	—	2	1	no	—
28	C₂×D₉	—	10	1	no	—
29	C₄×C₉	C₂×D₅	1	1	yes	characteristic
30	C₂×C₃×C₅	D₁₂	1	1	yes	characteristic
31	D₁₅	—	12	1	no	—
32	C₅×D₃	—	12	1	no	—
33	D₁₂	—	3	1	no	—
34	D₁₂	—	15	1	no	—
35	C₃×C₈	—	5	1	no	—
36	C₂²×C₅	—	18	1	no	—
37	C₂×D₅	—	18	1	no	—
38	C₄×C₅	C₂×D₉	1	1	yes	characteristic
39	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
40	D₉	—	4	1	no	—
41	D₉	—	20	1	no	—
42	D₈	—	45	1	no	Sylow
43	C₃×C₅	D₂₄	1	1	yes	characteristic
44	C₂×D₃	—	6	1	no	—
45	C₂×D₃	—	30	1	no	—
46	C₃×C₄	D₃×D₅	1	1	yes	characteristic
47	C₂×C₅	—	36	1	no	—
48	C₂×C₅	D₃₆	1	1	yes	characteristic
49	D₅	—	36	1	no	—
50	C₉	#686	1	1	yes	characteristic, complemented, Sylow
51	D₄	—	9	1	no	—
52	D₄	—	45	1	no	—
53	C₈	—	5	1	no	—
54	C₂×C₃	#1203	1	1	yes	characteristic
55	D₃	—	12	1	no	—
56	D₃	—	60	1	no	—
57	C₅	D₇₂	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	18	1	no	—
59	C₂²	—	90	1	no	—
60	C₄	D₅×D₉	1	1	yes	characteristic
61	C₃	#6654	1	1	yes	characteristic
62	C₂	—	36	1	no	—
63	C₂	—	180	1	no	—
64	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
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. C₅×D₃₆	C₂	semidirect
3. C₉×#168	C₂	semidirect
17. D₃₆	D₅	semidirect
23. C₅×C₉	D₈	semidirect
26. #168	D₉	semidirect
50. C₉	#686	semidirect
57. 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×D₃₆	2
C₉×#168	2
#6654	3
D₇₂	5