Subgroups — C₉×#676 #10623565

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	C₅×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#676	C₃	1	1	yes	maximal, characteristic
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	C₉×M₁₆	—	5	1	no	maximal, Hall
9	C₃×C₄×D₅	C₂×C₃	1	1	yes	characteristic
10	C₃×C₅×C₈	C₂×C₃	1	1	yes	characteristic
11	C₃×#168	C₂×C₃	1	1	yes	characteristic
12	C₂×C₅×C₉	C₂×C₄	1	1	yes	characteristic
13	C₉×D₅	—	2	1	no	—
14	#676	C₉	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₄×C₉	—	5	1	no	—
16	C₈×C₉	—	5	1	no	—
17	C₈×C₉	D₅	1	1	yes	characteristic, complemented
18	C₂×C₃×D₅	C₃×C₄	1	1	yes	characteristic
19	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
20	C₃×Dic₅	C₃×C₄	1	1	yes	characteristic
21	C₃×M₁₆	—	5	1	no	—
22	C₅×C₉	M₁₆	1	1	yes	characteristic, complemented, Hall
23	C₄×D₅	C₂×C₉	1	1	yes	characteristic
24	C₅×C₈	C₂×C₉	1	1	yes	characteristic, complemented
25	#168	C₂×C₉	1	1	yes	characteristic, complemented
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×C₃×C₄	1	1	yes	characteristic
30	C₃×D₅	—	2	1	no	—
31	C₂×C₃×C₄	—	5	1	no	—
32	C₃×C₈	—	5	1	no	—
33	C₃×C₈	C₃×D₅	1	1	yes	characteristic
34	C₂×D₅	C₄×C₉	1	1	yes	characteristic
35	C₄×C₅	C₂²×C₉	1	1	yes	characteristic
36	Dic₅	C₄×C₉	1	1	yes	characteristic
37	C₂×C₉	—	10	1	no	—
38	C₂×C₉	C₄×D₅	1	1	yes	characteristic
39	M₁₆	—	5	1	no	Sylow
40	C₃×C₅	C₃×M₁₆	1	1	yes	characteristic
41	C₂²×C₃	—	5	1	no	—
42	C₃×C₄	—	5	1	no	—
43	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
44	C₂×C₅	C₂×C₄×C₉	1	1	yes	characteristic
45	D₅	—	2	1	no	—
46	C₉	#676	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×C₄	—	5	1	no	—
48	C₈	—	5	1	no	—
49	C₈	C₉×D₅	1	1	yes	characteristic, complemented
50	C₂×C₃	—	10	1	no	—
51	C₂×C₃	C₃×C₄×D₅	1	1	yes	characteristic
52	C₅	C₉×M₁₆	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	5	1	no	—
54	C₄	—	5	1	no	—
55	C₄	C₂×C₉×D₅	1	1	yes	characteristic
56	C₃	C₃×#676	1	1	yes	characteristic
57	C₂	—	10	1	no	—
58	C₂	C₄×C₉×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	C₂×D₅
commutator	C₂×C₅	C₂×C₄×C₉
Frattini	C₃×C₄	C₂×C₃×D₅
Fitting	C₅×C₈×C₉	C₂
socle	C₂×C₃×C₅	C₂×C₃×C₄
prime radical	C₂×C₃×D₅	C₃×C₄
cosocle	C₃×C₄×C₅	C₂²×C₃
hypercenter	C₈×C₉	D₅
hypocenter	C₅	C₉×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₉ ◁ C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×C₉	C₂	semidirect
3. C₉×#168	C₂	semidirect
14. #676	C₉	direct (N × H)
17. C₈×C₉	D₅	semidirect
22. C₅×C₉	M₁₆	semidirect
24. C₅×C₈	C₂×C₉	semidirect
25. #168	C₂×C₉	semidirect
46. C₉	#676	direct (N × H)
49. C₈	C₉×D₅	semidirect
52. C₅	C₉×M₁₆	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
C₅×C₈×C₉	2
C₉×#168	2
C₃×#676	3
C₉×M₁₆	5

Subgroups — C9×#676 #10623565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#676 #10623565