Subgroups — #10623547

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×D₅	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	#6649	—	3	1	no	maximal
5	C₂×C₉×D₅	C₄	1	1	yes	characteristic
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	C₉×Dic₅	C₄	1	1	yes	characteristic
8	#3667	—	5	1	no	maximal, Hall
9	C₃×C₄×D₅	D₃	1	1	yes	characteristic
10	C₃⋊#168	—	3	1	no	—
11	C₅×C₃⋊C₈	—	3	1	no	—
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	C₉×D₅	—	2	1	no	—
14	#676	—	9	1	no	Hall
15	C₂×C₄×C₉	—	5	1	no	—
16	C₉⋊C₈	—	5	1	no	—
17	C₉⋊C₈	D₅	1	1	yes	characteristic, complemented
18	C₂×C₃×D₅	Dic₃	1	1	yes	characteristic
19	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	C₃×Dic₅	Dic₃	1	1	yes	characteristic
21	#208	—	15	1	no	—
22	C₅×C₉	M₁₆	1	1	yes	characteristic, complemented, Hall
23	C₄×D₅	D₉	1	1	yes	characteristic
24	C₅×C₈	—	9	1	no	—
25	#168	—	9	1	no	—
26	C₂²×C₉	—	5	1	no	—
27	C₄×C₉	—	5	1	no	—
28	C₄×C₉	C₂×D₅	1	1	yes	characteristic
29	C₂×C₃×C₅	C₂×Dic₃	1	1	yes	characteristic
30	C₃×D₅	—	2	1	no	—
31	C₂×C₃×C₄	—	5	1	no	—
32	C₃⋊C₈	—	3	1	no	—
33	C₃⋊C₈	—	15	1	no	—
34	C₂×D₅	Dic₉	1	1	yes	characteristic
35	C₄×C₅	C₂×D₉	1	1	yes	characteristic
36	Dic₅	Dic₉	1	1	yes	characteristic
37	C₂×C₉	—	10	1	no	—
38	C₂×C₉	C₄×D₅	1	1	yes	characteristic
39	M₁₆	—	45	1	no	Sylow
40	C₃×C₅	#208	1	1	yes	characteristic
41	C₂²×C₃	—	5	1	no	—
42	C₃×C₄	—	5	1	no	—
43	C₃×C₄	D₃×D₅	1	1	yes	characteristic
44	C₂×C₅	C₂×Dic₉	1	1	yes	characteristic
45	D₅	—	2	1	no	—
46	C₉	#676	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	5	1	no	—
48	C₈	—	9	1	no	—
49	C₈	—	45	1	no	—
50	C₂×C₃	—	10	1	no	—
51	C₂×C₃	D₅×Dic₃	1	1	yes	characteristic
52	C₅	#3667	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	5	1	no	—
54	C₄	—	5	1	no	—
55	C₄	D₅×D₉	1	1	yes	characteristic
56	C₃	#6649	1	1	yes	characteristic
57	C₂	—	10	1	no	—
58	C₂	D₅×Dic₉	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	C₂×Dic₃
prime radical	C₂×C₃×D₅	Dic₃
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
46. C₉	#676	semidirect
52. C₅	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₉×D₅	2
C₉⋊#168	2
C₅×C₉⋊C₈	2
#6649	3
#3667	5