Subgroups — #10623713

Order 720.

120 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	3	yes	maximal, complemented
3	C₄×D₄₅	C₂	1	3	yes	maximal, complemented
4	#6821	—	3	1	no	maximal
5	C₂×D₄₅	C₂²	1	3	yes	—
6	C₄×C₅×C₉	C₂²	1	3	yes	—
7	Dic₄₅	C₂²	1	1	yes	characteristic
8	#3701	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₈	D₃	1	1	yes	characteristic
10	D₆₀	—	3	3	no	—
11	C₄×D₁₅	—	3	3	no	—
12	C₂×C₅×C₉	C₂³	1	1	yes	characteristic
13	D₄₅	—	2	3	no	—
14	#713	—	9	1	no	Hall
15	Q₈×C₉	D₅	1	1	yes	characteristic, complemented
16	D₃₆	—	5	3	no	—
17	C₄×D₉	—	5	3	no	—
18	C₂×D₁₅	—	3	3	no	—
19	C₃×C₄×C₅	C₂×D₃	1	3	yes	—
20	Dic₁₅	—	3	1	no	—
21	#239	—	15	1	no	—
22	C₅×C₉	#41	1	1	yes	characteristic, complemented, Hall
23	C₅×Q₈	D₉	1	1	yes	characteristic, complemented
24	D₂₀	—	9	3	no	—
25	C₄×D₅	—	9	3	no	—
26	C₂×D₉	—	5	3	no	—
27	C₄×C₉	C₂×D₅	1	3	yes	—
28	Dic₉	—	5	1	no	—
29	C₂×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
30	D₁₅	—	6	3	no	—
31	C₃×Q₈	D₁₅	1	1	yes	characteristic
32	D₁₂	—	15	3	no	—
33	C₄×D₃	—	15	3	no	—
34	C₂×D₅	—	9	3	no	—
35	C₄×C₅	C₂×D₉	1	3	yes	—
36	Dic₅	—	9	1	no	—
37	C₂×C₉	C₂²×D₅	1	1	yes	characteristic
38	D₉	—	10	3	no	—
39	#41	—	45	1	no	Sylow
40	C₃×C₅	#239	1	1	yes	characteristic
41	C₂×D₃	—	15	3	no	—
42	C₃×C₄	C₂×D₁₅	1	3	yes	—
43	Dic₃	—	15	1	no	—
44	C₂×C₅	C₂²×D₉	1	1	yes	characteristic
45	D₅	—	18	3	no	—
46	C₉	#713	1	1	yes	characteristic, complemented, Sylow
47	Q₈	D₄₅	1	1	yes	characteristic, complemented
48	D₄	—	45	3	no	—
49	C₂×C₄	—	45	3	no	—
50	C₂×C₃	C₂²×D₁₅	1	1	yes	characteristic
51	D₃	—	30	3	no	—
52	C₅	#3701	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	45	3	no	—
54	C₄	—	45	1	no	—
55	C₄	C₂×D₄₅	1	3	yes	—
56	C₃	#6821	1	1	yes	characteristic
57	C₂	—	90	3	no	—
58	C₂	C₂²×D₄₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₅
commutator, cosocle	C₂×C₅×C₉	C₂³
Frattini	C₂×C₃	C₂²×D₁₅
Fitting	C₅×Q₈×C₉	C₂
socle	C₂×C₃×C₅	C₂²×D₃
hypercenter	Q₈	D₄₅
hypocenter	C₅×C₉	#41
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×C₉	C₂	semidirect
2. D₁₈₀	C₂	semidirect
3. C₄×D₄₅	C₂	semidirect
15. Q₈×C₉	D₅	semidirect
22. C₅×C₉	#41	semidirect
23. C₅×Q₈	D₉	semidirect
46. C₉	#713	semidirect
47. Q₈	D₄₅	semidirect
52. C₅	#3701	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
C₄×D₄₅	2
#6821	3
#3701	5