Subgroups — #10623610

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₈₀	C₂	1	1	yes	maximal, characteristic
3	C₉⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6717	—	3	1	no	maximal
5	C₂×D₄₅	—	2	1	no	—
6	C₄×C₅×C₉	—	2	1	no	—
7	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
8	#3675	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₈	D₃	1	1	yes	characteristic
10	D₆₀	—	3	1	no	—
11	C₃⋊#168	—	3	1	no	—
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	D₄₅	—	4	1	no	—
14	#688	—	9	1	no	Hall
15	Q₈×C₉	D₅	1	1	yes	characteristic, complemented
16	D₃₆	—	5	1	no	—
17	C₉⋊C₈	—	5	1	no	—
18	C₂×D₁₅	—	6	1	no	—
19	C₃×C₄×C₅	—	2	1	no	—
20	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
21	#215	—	15	1	no	—
22	C₅×C₉	SD₁₆	1	1	yes	characteristic, complemented, Hall
23	C₅×Q₈	D₉	1	1	yes	characteristic, complemented
24	D₂₀	—	9	1	no	—
25	#168	—	9	1	no	—
26	C₂×D₉	—	10	1	no	—
27	C₄×C₉	—	2	1	no	—
28	C₄×C₉	C₂×D₅	1	1	yes	characteristic
29	C₂×C₃×C₅	#67	1	1	yes	characteristic
30	D₁₅	—	12	1	no	—
31	C₃×Q₈	D₁₅	1	1	yes	characteristic
32	D₁₂	—	15	1	no	—
33	C₃⋊C₈	—	15	1	no	—
34	C₂×D₅	—	18	1	no	—
35	C₄×C₅	—	2	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	#215	1	1	yes	characteristic
41	C₂×D₃	—	30	1	no	—
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
44	C₂×C₅	#611	1	1	yes	characteristic
45	D₅	—	36	1	no	—
46	C₉	#688	1	1	yes	characteristic, complemented, Sylow
47	Q₈	D₄₅	1	1	yes	characteristic, complemented
48	D₄	—	45	1	no	—
49	C₈	—	45	1	no	—
50	C₂×C₃	#1220	1	1	yes	characteristic
51	D₃	—	60	1	no	—
52	C₅	#3675	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	90	1	no	—
54	C₄	—	2	1	no	—
55	C₄	C₂×D₄₅	1	1	yes	characteristic
56	C₃	#6717	1	1	yes	characteristic
57	C₂	—	180	1	no	—
58	C₂	#67099	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67099
commutator, cosocle	C₄×C₅×C₉	C₂²
Frattini	C₃×C₄	C₂×D₁₅
Fitting	C₅×Q₈×C₉	C₂
socle	C₂×C₃×C₅	#67
prime radical	D₁₈₀	C₂
hypercenter	Q₈	D₄₅
hypocenter	C₅×C₉	SD₁₆
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×C₉	C₂	semidirect
3. C₉⋊#168	C₂	semidirect
15. Q₈×C₉	D₅	semidirect
22. C₅×C₉	SD₁₆	semidirect
23. C₅×Q₈	D₉	semidirect
46. C₉	#688	semidirect
47. Q₈	D₄₅	semidirect
52. C₅	#3675	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×C₉	2
D₁₈₀	2
C₉⋊#168	2
#6717	3
#3675	5