Subgroups — C₄×C₅⋊AGL(1,ℤ/9) #11761867

Order 1080.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₁₅	—	3	1	no	maximal
5	C₄×D₄₅	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₅⋊AGL(1,ℤ/9)	C₄	1	2	yes	direct product factor
8	C₄×AGL(1,ℤ/9)	—	5	1	no	maximal, Hall
9	C₂×C₃×D₁₅	—	3	1	no	—
10	C₃²×C₄×C₅	D₃	1	1	yes	characteristic
11	C₃×Dic₁₅	—	3	1	no	—
12	C₂×D₄₅	C₂×C₃	1	1	yes	characteristic
13	C₄×C₅×C₉	—	2	1	no	—
14	C₄×C₅×C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	Dic₄₅	C₂×C₃	1	1	yes	characteristic, complemented
16	C₅×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented, Hall
17	C₄×D₁₅	—	3	1	no	—
18	C₃×C₄×D₅	—	9	1	no	—
19	C₂×AGL(1,ℤ/9)	—	5	1	no	—
20	C₄×3₋¹⁺²	D₅	1	1	yes	characteristic, complemented
21	3₋¹⁺²⋊C₄	—	5	1	no	—
22	C₂×C₃²×C₅	C₂×D₃	1	1	yes	characteristic
23	C₃×D₁₅	—	3	2	no	—
24	C₂×C₅×C₉	—	2	1	no	—
25	C₂×C₅×C₉	C₂²×C₃	1	1	yes	characteristic
26	D₄₅	C₃×C₄	1	2	yes	complemented
27	C₃×C₄×D₃	—	15	1	no	—
28	C₄×D₉	—	5	1	no	—
29	C₂×D₁₅	—	3	1	no	—
30	C₂×C₃×D₅	—	9	1	no	—
31	C₃×C₄×C₅	—	3	1	no	—
32	C₃×C₄×C₅	C₃×D₃	1	1	yes	characteristic
33	Dic₁₅	—	3	1	no	—
34	C₃×Dic₅	—	9	1	no	—
35	C₂×3₋¹⁺²	C₂×D₅	1	1	yes	characteristic
36	AGL(1,ℤ/9)	—	5	2	no	—
37	C₃²×C₅	C₄×D₃	1	1	yes	characteristic
38	C₅×C₉	—	2	1	no	—
39	C₅×C₉	C₂×C₃×C₄	1	1	yes	characteristic, complemented
40	C₄×D₅	—	9	1	no	Hall
41	C₂×C₃×D₃	—	15	1	no	—
42	C₃²×C₄	D₁₅	1	1	yes	characteristic
43	C₃×Dic₃	—	15	1	no	—
44	C₂×D₉	—	5	1	no	—
45	C₄×C₉	—	2	1	no	—
46	C₄×C₉	C₃×D₅	1	1	yes	characteristic, complemented
47	Dic₉	—	5	1	no	—
48	C₂×C₃×C₅	—	3	1	no	—
49	C₂×C₃×C₅	C₂×C₃×D₃	1	1	yes	characteristic
50	D₁₅	—	3	2	no	—
51	C₃×D₅	—	9	2	no	—
52	3₋¹⁺²	C₄×D₅	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₃×C₄	—	45	1	no	—
54	C₄×D₃	—	15	1	no	—
55	C₂×D₅	—	9	1	no	—
56	C₄×C₅	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
57	Dic₅	—	9	1	no	—
58	C₂×C₃²	C₂×D₁₅	1	1	yes	characteristic
59	C₃×D₃	—	15	2	no	—
60	C₂×C₉	—	2	1	no	—
61	C₂×C₉	C₂×C₃×D₅	1	1	yes	characteristic
62	D₉	—	5	2	no	—
63	C₃×C₅	—	3	1	no	—
64	C₃×C₅	C₃×C₄×D₃	1	1	yes	characteristic
65	C₂²×C₃	—	45	1	no	—
66	C₂×D₃	—	15	1	no	—
67	C₃×C₄	—	3	1	no	—
68	C₃×C₄	—	45	1	no	—
69	C₃×C₄	C₃×D₁₅	1	1	yes	characteristic
70	Dic₃	—	15	1	no	—
71	C₂×C₅	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
72	D₅	—	9	2	no	—
73	C₃²	C₄×D₁₅	1	1	yes	characteristic
74	C₉	—	2	1	no	—
75	C₉	C₃×C₄×D₅	1	1	yes	characteristic, complemented
76	C₂×C₄	—	45	1	no	Sylow
77	C₂×C₃	—	3	1	no	—
78	C₂×C₃	—	45	2	no	—
79	C₂×C₃	C₂×C₃×D₁₅	1	1	yes	characteristic
80	D₃	—	15	2	no	—
81	C₅	C₄×AGL(1,ℤ/9)	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	45	1	no	—
83	C₄	—	45	1	no	—
84	C₄	C₅⋊AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
85	C₃	—	3	1	no	—
86	C₃	C₃×C₄×D₁₅	1	1	yes	characteristic
87	C₂	—	45	2	no	—
88	C₂	C₂×C₅⋊AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₅⋊AGL(1,ℤ/9)
commutator, hypocenter	C₅×C₉	C₂×C₃×C₄
Frattini	C₂×C₃	C₂×C₃×D₁₅
Fitting	C₄×C₅×3₋¹⁺²	C₂
socle	C₂×C₃×C₅	C₂×C₃×D₃
prime radical	C₂×C₅⋊AGL(1,ℤ/9)	C₂
cosocle	C₂×C₅×C₉	C₂²×C₃
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²⋊Dic₅	C₂	semidirect
5. C₄×D₄₅	C₃	semidirect
7. C₅⋊AGL(1,ℤ/9)	C₄	direct (N × H)
14. C₄×C₅×C₉	C₂×C₃	semidirect
15. Dic₄₅	C₂×C₃	semidirect
16. C₅×3₋¹⁺²	C₂×C₄	semidirect
20. C₄×3₋¹⁺²	D₅	semidirect
26. D₄₅	C₃×C₄	semidirect
39. C₅×C₉	C₂×C₃×C₄	semidirect
46. C₄×C₉	C₃×D₅	semidirect
52. 3₋¹⁺²	C₄×D₅	semidirect
56. C₄×C₅	AGL(1,ℤ/9)	semidirect
75. C₉	C₃×C₄×D₅	semidirect
81. C₅	C₄×AGL(1,ℤ/9)	semidirect
84. C₄	C₅⋊AGL(1,ℤ/9)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅⋊AGL(1,ℤ/9)	2
C₄×C₅×3₋¹⁺²	2
3₋¹⁺²⋊Dic₅	2
C₃×C₄×D₁₅	3
C₄×D₄₅	3
C₄×AGL(1,ℤ/9)	5

Subgroups — C4×C5⋊AGL(1,ℤ/9) #11761867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅⋊AGL(1,ℤ/9) #11761867