Subgroups — #10623541

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₅	C₂	1	1	yes	maximal, characteristic
2	C₉×#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6651	—	3	1	no	maximal
5	C₂×D₄₅	C₄	1	1	yes	characteristic
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	Dic₄₅	C₄	1	1	yes	characteristic
8	#3663	—	5	1	no	maximal, Hall
9	C₄×D₁₅	—	3	1	no	—
10	C₃×#168	D₃	1	1	yes	characteristic
11	C₅×C₃⋊C₈	—	3	1	no	—
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	D₄₅	—	2	1	no	—
14	#676	—	9	1	no	Hall
15	C₄×D₉	—	5	1	no	—
16	C₈×C₉	—	5	1	no	—
17	C₉⋊C₈	D₅	1	1	yes	characteristic, complemented
18	C₂×D₁₅	—	3	1	no	—
19	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	Dic₁₅	—	3	1	no	—
21	ΓL(1,𝔽₂₅)	—	15	1	no	—
22	C₅×C₉	M₁₆	1	1	yes	characteristic, complemented, Hall
23	C₄×D₅	—	9	1	no	—
24	C₅×C₈	—	9	1	no	—
25	#168	D₉	1	1	yes	characteristic, complemented
26	C₂×D₉	—	5	1	no	—
27	C₄×C₉	C₂×D₅	1	1	yes	characteristic
28	Dic₉	—	5	1	no	—
29	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
30	D₁₅	—	6	1	no	—
31	C₄×D₃	—	15	1	no	—
32	C₃×C₈	—	5	1	no	—
33	C₃⋊C₈	—	3	1	no	—
34	C₂×D₅	—	9	1	no	—
35	C₄×C₅	C₂×D₉	1	1	yes	characteristic
36	Dic₅	—	9	1	no	—
37	C₂×C₉	C₄×D₅	1	1	yes	characteristic
38	D₉	—	10	1	no	—
39	M₁₆	—	45	1	no	Sylow
40	C₃×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
41	C₂×D₃	—	15	1	no	—
42	C₃×C₄	D₃×D₅	1	1	yes	characteristic
43	Dic₃	—	15	1	no	—
44	C₂×C₅	C₄×D₉	1	1	yes	characteristic
45	D₅	—	18	1	no	—
46	C₉	#676	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	45	1	no	—
48	C₈	—	5	1	no	—
49	C₈	—	9	1	no	—
50	C₂×C₃	#1200	1	1	yes	characteristic
51	D₃	—	30	1	no	—
52	C₅	#3663	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	45	1	no	—
54	C₄	—	45	1	no	—
55	C₄	D₅×D₉	1	1	yes	characteristic
56	C₃	#6651	1	1	yes	characteristic
57	C₂	—	90	1	no	—
58	C₂	#67079	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	D₅×D₉
commutator	C₂×C₅×C₉	C₂×C₄
Frattini	C₃×C₄	D₃×D₅
Fitting, cosocle	C₄×C₅×C₉	C₂²
socle	C₂×C₃×C₅	C₄×D₃
prime radical	C₂×D₄₅	C₄
hypocenter	C₅×C₉	M₁₆
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₉×#168	C₂	semidirect
3. C₅×C₉⋊C₈	C₂	semidirect
17. C₉⋊C₈	D₅	semidirect
22. C₅×C₉	M₁₆	semidirect
25. #168	D₉	semidirect
46. C₉	#676	semidirect
52. C₅	#3663	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄₅	2
C₉×#168	2
C₅×C₉⋊C₈	2
#6651	3
#3663	5