Subgroups — #10623627

Order 720.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#67079	C₂	1	1	yes	maximal, characteristic
2	C₉⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	#6738	—	3	1	no	maximal
5	C₂×D₄₅	C₄	1	1	yes	characteristic
6	C₉×Dic₅	C₂²	1	1	yes	characteristic
7	C₅×Dic₉	C₄	1	1	yes	characteristic
8	C₈×D₉	—	5	1	no	maximal, Hall
9	#1200	—	3	1	no	—
10	C₃⋊#170	—	3	1	no	—
11	C₃×#170	D₃	1	1	yes	characteristic
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	D₄₅	C₈	1	2	yes	complemented
14	D₅⋊C₈	—	9	1	no	Hall
15	C₄×D₉	—	5	1	no	—
16	C₈×C₉	—	5	1	no	—
17	C₉⋊C₈	—	5	1	no	—
18	C₂×D₁₅	—	3	1	no	—
19	C₃×Dic₅	C₂×D₃	1	1	yes	characteristic
20	C₅×Dic₃	—	3	1	no	—
21	D₃×C₈	—	15	1	no	—
22	C₅×C₉	C₂×C₈	1	1	yes	characteristic, complemented, Hall
23	C₄×D₅	—	9	1	no	—
24	#170	—	9	1	no	—
25	#170	D₉	1	1	yes	characteristic, complemented
26	C₂×D₉	—	5	1	no	—
27	C₄×C₉	—	5	1	no	—
28	Dic₉	AGL(1,𝔽₅)	1	1	yes	characteristic
29	C₂×C₃×C₅	C₄×D₃	1	1	yes	characteristic
30	D₁₅	—	3	2	no	—
31	C₄×D₃	—	15	1	no	—
32	C₃×C₈	—	5	1	no	—
33	C₃⋊C₈	—	15	1	no	—
34	C₂×D₅	—	9	1	no	—
35	C₄×C₅	—	9	1	no	—
36	Dic₅	C₂×D₉	1	1	yes	characteristic
37	C₂×C₉	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
38	D₉	—	5	2	no	—
39	C₂×C₈	—	45	1	no	Sylow
40	C₃×C₅	D₃×C₈	1	1	yes	characteristic
41	C₂×D₃	—	15	1	no	—
42	C₃×C₄	—	5	1	no	—
43	Dic₃	—	3	1	no	—
44	C₂×C₅	C₄×D₉	1	1	yes	characteristic
45	D₅	—	9	2	no	—
46	C₉	D₅⋊C₈	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄	—	45	1	no	—
48	C₈	—	5	1	no	—
49	C₈	—	45	1	no	—
50	C₂×C₃	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
51	D₃	—	15	2	no	—
52	C₅	C₈×D₉	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	45	1	no	—
54	C₄	—	5	1	no	—
55	C₄	—	9	1	no	—
56	C₃	#6738	1	1	yes	characteristic
57	C₂	—	45	2	no	—
58	C₂	D₉×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	D₉×AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₉	C₂×C₈
Frattini	C₂×C₃	D₃×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₉	C₂×C₄
socle	C₂×C₃×C₅	C₄×D₃
prime radical	C₂×D₄₅	C₄
cosocle	C₉×Dic₅	C₂²
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₉⋊#170	C₂	semidirect
3. C₉×#170	C₂	semidirect
13. D₄₅	C₈	semidirect
22. C₅×C₉	C₂×C₈	semidirect
25. #170	D₉	semidirect
46. C₉	D₅⋊C₈	semidirect
52. C₅	C₈×D₉	semidirect

Maximal subgroups

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

subgroup	index
#67079	2
C₉⋊#170	2
C₉×#170	2
#6738	3
C₈×D₉	5