Subgroups — #10623607

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₄×C₉	C₂	1	1	yes	maximal, characteristic
2	Dic₉₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#6716	—	3	1	no	maximal
5	C₂²×C₅×C₉	—	2	1	no	—
6	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
7	Dic₄₅	—	2	1	no	—
8	#3672	—	5	1	no	maximal, Hall
9	C₃×C₅×D₄	D₃	1	1	yes	characteristic
10	Dic₃₀	—	3	1	no	—
11	C₃⋊#168	—	3	1	no	—
12	C₂×C₅×C₉	—	4	1	no	—
13	C₂×C₅×C₉	D₄	1	1	yes	characteristic
14	#687	—	9	1	no	Hall
15	D₄×C₉	D₅	1	1	yes	characteristic
16	Dic₁₈	—	5	1	no	—
17	C₉⋊C₈	—	5	1	no	—
18	C₂²×C₃×C₅	—	2	1	no	—
19	C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
20	Dic₁₅	—	6	1	no	—
21	#214	—	15	1	no	—
22	C₅×C₉	SD₁₆	1	1	yes	characteristic, complemented, Hall
23	C₅×D₄	D₉	1	1	yes	characteristic
24	Dic₁₀	—	9	1	no	—
25	#168	—	9	1	no	—
26	C₂²×C₉	—	2	1	no	—
27	C₄×C₉	C₂×D₅	1	1	yes	characteristic
28	Dic₉	—	10	1	no	—
29	C₂×C₃×C₅	—	4	1	no	—
30	C₂×C₃×C₅	#67	1	1	yes	characteristic
31	C₃×D₄	D₁₅	1	1	yes	characteristic
32	Dic₆	—	15	1	no	—
33	C₃⋊C₈	—	15	1	no	—
34	C₂²×C₅	—	2	1	no	—
35	C₄×C₅	C₂×D₉	1	1	yes	characteristic
36	Dic₅	—	18	1	no	—
37	C₂×C₉	—	4	1	no	—
38	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
39	SD₁₆	—	45	1	no	Sylow
40	C₃×C₅	#214	1	1	yes	characteristic
41	C₂²×C₃	—	2	1	no	—
42	C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
43	Dic₃	—	30	1	no	—
44	C₂×C₅	—	4	1	no	—
45	C₂×C₅	#611	1	1	yes	characteristic
46	C₉	#687	1	1	yes	characteristic, complemented, Sylow
47	Q₈	—	45	1	no	—
48	D₄	D₄₅	1	1	yes	characteristic
49	C₈	—	45	1	no	—
50	C₂×C₃	—	4	1	no	—
51	C₂×C₃	#1220	1	1	yes	characteristic
52	C₅	#3672	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	2	1	no	—
54	C₄	—	90	1	no	—
55	C₄	C₂×D₄₅	1	1	yes	characteristic
56	C₃	#6716	1	1	yes	characteristic
57	C₂	—	4	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×D₄×C₉	C₂
socle	C₂×C₃×C₅	#67
prime radical	C₃×C₅×D₄	D₃
hypercenter	D₄	D₄₅
hypocenter	C₅×C₉	SD₁₆
LCS	G ▷ C₄×C₅×C₉ ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. Dic₉₀	C₂	semidirect
3. C₉⋊#168	C₂	semidirect
22. C₅×C₉	SD₁₆	semidirect
46. C₉	#687	semidirect
52. C₅	#3672	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₄×C₉	2
Dic₉₀	2
C₉⋊#168	2
#6716	3
#3672	5