Subgroups — C₉×AGL(1,ℤ/9) #92255

Order 486.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×AGL(1,ℤ/9)	C₃	1	1	yes	maximal, characteristic
3	C₃×D₃×C₉	—	3	1	no	maximal
4	D₉⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₉×D₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
7	C₃²×C₉	D₃	1	1	yes	characteristic
8	C₉⋊C₉	—	2	2	no	—
9	C₉⋊C₉	C₂×C₃	1	2	yes	complemented
10	C₃²⋊C₉	—	2	1	no	—
11	C₉²	—	2	1	no	—
12	C₉²	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃²×D₃	—	3	1	no	—
14	C₂×C₃×C₉	—	9	1	no	—
15	AGL(1,ℤ/9)	C₉	1	3	yes	direct product factor
16	D₃×C₉	—	3	1	no	—
17	D₃×C₉	—	3	2	no	—
18	C₃×D₉	C₃²	1	1	yes	characteristic
19	C₃³	C₃×D₃	1	1	yes	characteristic
20	3₋¹⁺²	—	2	3	no	—
21	3₋¹⁺²	C₂×C₉	1	3	yes	complemented
22	C₃×C₉	—	2	1	no	—
23	C₃×C₉	—	2	2	no	—
24	C₃×C₉	—	2	1	no	—
25	C₃×C₉	—	3	1	no	—
26	C₃×C₉	—	6	1	no	—
27	C₃×C₉	C₃×D₃	1	1	yes	characteristic
28	C₃×C₉	C₃×D₃	1	2	yes	—
29	C₃×C₉	C₂×C₃²	1	1	yes	characteristic
30	C₂×C₃²	—	9	1	no	—
31	C₃×D₃	—	3	3	no	—
32	C₃×D₃	—	3	1	no	—
33	C₂×C₉	—	9	2	no	—
34	C₂×C₉	—	9	1	no	—
35	D₉	C₃×C₉	1	1	yes	characteristic, complemented
36	C₃²	—	3	1	no	—
37	C₃²	—	6	1	no	—
38	C₃²	D₃×C₉	1	3	yes	—
39	C₃²	C₃²×D₃	1	1	yes	characteristic
40	C₉	—	2	3	no	—
41	C₉	—	2	1	no	—
42	C₉	—	2	1	no	—
43	C₉	—	3	2	no	—
44	C₉	—	6	2	no	—
45	C₉	—	6	1	no	—
46	C₉	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
47	C₉	C₂×C₃×C₉	1	1	yes	characteristic, complemented
48	C₂×C₃	—	9	3	no	—
49	C₂×C₃	—	9	1	no	—
50	D₃	—	3	1	no	—
51	C₃	—	2	1	no	—
52	C₃	—	3	3	no	—
53	C₃	C₃×D₃×C₉	1	1	yes	characteristic
54	C₃	C₃×AGL(1,ℤ/9)	1	1	yes	characteristic
55	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₉	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂×C₃×C₉
Frattini, socle	C₃²	C₃²×D₃
Fitting	C₉×3₋¹⁺²	C₂
prime radical	C₃×AGL(1,ℤ/9)	C₃
cosocle	C₃×C₉	C₂×C₃²
LCS	G ▷ C₉
UCS	C₁ ◁ C₉

Complements

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

N	H ≅ G/N	type
1. C₉×3₋¹⁺²	C₂	semidirect
4. D₉⋊C₉	C₃	semidirect
5. C₉×D₉	C₃	semidirect
9. C₉⋊C₉	C₂×C₃	semidirect
12. C₉²	C₂×C₃	semidirect
15. AGL(1,ℤ/9)	C₉	direct (N × H)
21. 3₋¹⁺²	C₂×C₉	semidirect
35. D₉	C₃×C₉	semidirect
46. C₉	AGL(1,ℤ/9)	direct (N × H)
47. C₉	C₂×C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×3₋¹⁺²	2
C₃×AGL(1,ℤ/9)	3
C₃×D₃×C₉	3
D₉⋊C₉	3
C₉×D₉	3

Subgroups — C9×AGL(1,ℤ/9) #92255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×AGL(1,ℤ/9) #92255