Subgroups — C₉×#688 #10623577

Order 720.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₉×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#688	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×D₅	—	2	1	no	—
6	C₄×C₅×C₉	—	2	1	no	—
7	C₄×C₅×C₉	C₂²	1	1	yes	characteristic
8	C₉×SD₁₆	—	5	1	no	maximal, Hall
9	C₃×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×D₂₀	C₂×C₃	1	1	yes	characteristic
11	C₃×#168	C₂×C₃	1	1	yes	characteristic
12	C₂×C₅×C₉	D₄	1	1	yes	characteristic
13	C₉×D₅	—	4	1	no	—
14	#688	C₉	1	1	yes	characteristic, direct product factor, Hall
15	Q₈×C₉	D₅	1	1	yes	characteristic, complemented
16	D₄×C₉	—	5	1	no	—
17	C₈×C₉	—	5	1	no	—
18	C₂×C₃×D₅	—	2	1	no	—
19	C₃×C₄×C₅	—	2	1	no	—
20	C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
21	C₃×SD₁₆	—	5	1	no	—
22	C₅×C₉	SD₁₆	1	1	yes	characteristic, complemented, Hall
23	C₅×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
24	D₂₀	C₂×C₉	1	1	yes	characteristic
25	#168	C₂×C₉	1	1	yes	characteristic, complemented
26	C₂²×C₉	—	10	1	no	—
27	C₄×C₉	—	2	1	no	—
28	C₄×C₉	C₂×D₅	1	1	yes	characteristic
29	C₂×C₃×C₅	C₃×D₄	1	1	yes	characteristic
30	C₃×D₅	—	4	1	no	—
31	C₃×Q₈	C₃×D₅	1	1	yes	characteristic
32	C₃×D₄	—	5	1	no	—
33	C₃×C₈	—	5	1	no	—
34	C₂×D₅	—	2	1	no	—
35	C₄×C₅	—	2	1	no	—
36	C₄×C₅	C₂²×C₉	1	1	yes	characteristic
37	C₂×C₉	—	20	1	no	—
38	C₂×C₉	C₂²⋊D₅	1	1	yes	characteristic
39	SD₁₆	—	5	1	no	Sylow
40	C₃×C₅	C₃×SD₁₆	1	1	yes	characteristic
41	C₂²×C₃	—	10	1	no	—
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
44	C₂×C₅	D₄×C₉	1	1	yes	characteristic
45	D₅	—	4	1	no	—
46	C₉	#688	1	1	yes	characteristic, direct product factor, Sylow
47	Q₈	C₉×D₅	1	1	yes	characteristic, complemented
48	D₄	—	5	1	no	—
49	C₈	—	5	1	no	—
50	C₂×C₃	—	20	1	no	—
51	C₂×C₃	C₃×C₂²⋊D₅	1	1	yes	characteristic
52	C₅	C₉×SD₁₆	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	10	1	no	—
54	C₄	—	2	1	no	—
55	C₄	C₂×C₉×D₅	1	1	yes	characteristic
56	C₃	C₃×#688	1	1	yes	characteristic
57	C₂	—	20	1	no	—
58	C₂	C₉×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×D₅
Fitting	C₅×Q₈×C₉	C₂
socle	C₂×C₃×C₅	C₃×D₄
prime radical	C₃×D₂₀	C₂×C₃
cosocle	C₃×C₄×C₅	C₂²×C₃
hypercenter	Q₈×C₉	D₅
hypocenter	C₅	C₉×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×C₉	C₂	semidirect
3. C₉×#168	C₂	semidirect
14. #688	C₉	direct (N × H)
15. Q₈×C₉	D₅	semidirect
22. C₅×C₉	SD₁₆	semidirect
23. C₅×Q₈	C₂×C₉	semidirect
25. #168	C₂×C₉	semidirect
46. C₉	#688	direct (N × H)
47. Q₈	C₉×D₅	semidirect
52. C₅	C₉×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×C₉	2
C₉×D₂₀	2
C₉×#168	2
C₃×#688	3
C₉×SD₁₆	5

Subgroups — C9×#688 #10623577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#688 #10623577