Subgroups — #10623602

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉⋊#168	C₂	1	2	yes	maximal, complemented
3	#6710	—	3	1	no	maximal
4	C₂²×C₅×C₉	C₄	1	1	yes	characteristic
5	C₄×C₅×C₉	C₄	1	1	yes	characteristic
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	#3667	—	5	1	no	maximal, Hall
8	C₂×C₃×C₄×C₅	D₃	1	1	yes	characteristic
9	C₃⋊#168	—	3	2	no	—
10	C₂×C₅×C₉	—	2	1	no	—
11	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
12	#681	—	9	1	no	Hall
13	C₂×C₄×C₉	D₅	1	1	yes	characteristic
14	C₉⋊C₈	—	5	2	no	—
15	C₂²×C₃×C₅	Dic₃	1	1	yes	characteristic
16	C₃×C₄×C₅	Dic₃	1	1	yes	characteristic
17	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
18	#208	—	15	1	no	—
19	C₅×C₉	M₁₆	1	1	yes	characteristic, complemented, Hall
20	C₂×C₄×C₅	D₉	1	1	yes	characteristic
21	#168	—	9	2	no	—
22	C₂²×C₉	Dic₅	1	1	yes	characteristic
23	C₄×C₉	Dic₅	1	1	yes	characteristic
24	C₄×C₉	C₂×D₅	1	1	yes	characteristic
25	C₂×C₃×C₅	—	2	1	no	—
26	C₂×C₃×C₅	C₂×Dic₃	1	1	yes	characteristic
27	C₂×C₃×C₄	D₁₅	1	1	yes	characteristic
28	C₃⋊C₈	—	15	2	no	—
29	C₂²×C₅	Dic₉	1	1	yes	characteristic
30	C₄×C₅	Dic₉	1	1	yes	characteristic
31	C₄×C₅	C₂×D₉	1	1	yes	characteristic
32	C₂×C₉	—	2	1	no	—
33	C₂×C₉	C₂×Dic₅	1	1	yes	characteristic
34	M₁₆	—	45	1	no	Sylow
35	C₃×C₅	#208	1	1	yes	characteristic
36	C₂²×C₃	Dic₁₅	1	1	yes	characteristic
37	C₃×C₄	Dic₁₅	1	1	yes	characteristic
38	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
39	C₂×C₅	—	2	1	no	—
40	C₂×C₅	C₂×Dic₉	1	1	yes	characteristic
41	C₉	#681	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄	D₄₅	1	1	yes	characteristic
43	C₈	—	45	2	no	—
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	C₂×Dic₁₅	1	1	yes	characteristic
46	C₅	#3667	1	1	yes	characteristic, complemented, Sylow
47	C₂²	Dic₄₅	1	1	yes	characteristic
48	C₄	Dic₄₅	1	1	yes	characteristic
49	C₄	C₂×D₄₅	1	1	yes	characteristic
50	C₃	#6710	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	C₂×Dic₄₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄₅
commutator	C₂×C₅×C₉	C₂×C₄
Frattini	C₃×C₄	C₂×D₁₅
Fitting	C₂×C₄×C₅×C₉	C₂
socle	C₂×C₃×C₅	C₂×Dic₃
prime radical	C₂²×C₃×C₅	Dic₃
cosocle	C₄×C₅×C₉	C₂²
hypercenter	C₂×C₄	D₄₅
hypocenter	C₅×C₉	M₁₆
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₉⋊#168	C₂	semidirect
19. C₅×C₉	M₁₆	semidirect
41. C₉	#681	semidirect
46. C₅	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₉	2
C₉⋊#168	2
#6710	3
#3667	5