Subgroups — C₂₇×AGL(1,ℤ/9) #13091023

Order 1458.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₇×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×D₃×C₂₇	—	3	1	no	maximal
3	C₉×AGL(1,ℤ/9)	C₃	1	1	yes	maximal, characteristic
4	D₉⋊C₂₇	C₃	1	2	yes	maximal, complemented
5	D₉×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₂₇	D₃	1	1	yes	characteristic
7	C₉×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
8	C₉⋊C₂₇	—	2	2	no	—
9	C₉⋊C₂₇	C₂×C₃	1	2	yes	complemented
10	C₃²⋊C₂₇	—	2	1	no	—
11	C₉×C₂₇	—	2	1	no	—
12	C₉×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×AGL(1,ℤ/9)	C₉	1	1	yes	characteristic
14	C₃×D₃×C₉	—	3	1	no	—
15	C₂×C₃×C₂₇	—	9	1	no	—
16	D₃×C₂₇	—	3	1	no	—
17	D₃×C₂₇	—	3	2	no	—
18	D₉⋊C₉	C₉	1	2	yes	—
19	C₉×D₉	C₃²	1	1	yes	characteristic
20	C₃×3₋¹⁺²	C₂×C₉	1	1	yes	characteristic
21	C₃²×C₉	C₃×D₃	1	1	yes	characteristic
22	C₃×C₂₇	—	2	2	no	—
23	C₃×C₂₇	—	2	1	no	—
24	C₃×C₂₇	—	3	1	no	—
25	C₃×C₂₇	—	6	1	no	—
26	C₃×C₂₇	C₃×D₃	1	1	yes	characteristic
27	C₃×C₂₇	C₃×D₃	1	2	yes	—
28	C₉⋊C₉	—	2	2	no	—
29	C₉⋊C₉	C₂×C₉	1	2	yes	—
30	C₃²⋊C₉	—	2	1	no	—
31	C₉²	—	2	1	no	—
32	C₉²	C₂×C₃²	1	1	yes	characteristic
33	C₃²×D₃	—	3	1	no	—
34	C₂×C₃×C₉	—	9	1	no	—
35	AGL(1,ℤ/9)	C₂₇	1	3	yes	direct product factor
36	D₃×C₉	—	3	2	no	—
37	D₃×C₉	—	3	1	no	—
38	C₃×D₉	C₃×C₉	1	1	yes	characteristic
39	C₂×C₂₇	—	9	2	no	—
40	C₂×C₂₇	—	9	1	no	—
41	C₃³	D₃×C₉	1	1	yes	characteristic
42	3₋¹⁺²	—	2	3	no	—
43	3₋¹⁺²	C₂×C₂₇	1	3	yes	complemented
44	C₃×C₉	—	2	1	no	—
45	C₃×C₉	—	2	2	no	—
46	C₃×C₉	—	2	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	6	1	no	—
49	C₃×C₉	D₃×C₉	1	2	yes	—
50	C₃×C₉	C₂×C₃×C₉	1	1	yes	characteristic
51	C₃×C₉	C₃²×D₃	1	1	yes	characteristic
52	C₂₇	—	2	1	no	—
53	C₂₇	—	3	2	no	—
54	C₂₇	—	6	2	no	—
55	C₂₇	—	6	1	no	—
56	C₂₇	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
57	C₂×C₃²	—	9	1	no	—
58	C₃×D₃	—	3	3	no	—
59	C₃×D₃	—	3	1	no	—
60	C₂×C₉	—	9	2	no	—
61	C₂×C₉	—	9	1	no	—
62	D₉	C₃×C₂₇	1	1	yes	characteristic, complemented
63	C₃²	—	3	1	no	—
64	C₃²	—	6	1	no	—
65	C₃²	D₃×C₂₇	1	3	yes	—
66	C₃²	C₃×D₃×C₉	1	1	yes	characteristic
67	C₉	—	2	1	no	—
68	C₉	—	2	3	no	—
69	C₉	—	2	1	no	—
70	C₉	—	3	2	no	—
71	C₉	—	6	2	no	—
72	C₉	—	6	1	no	—
73	C₉	C₂×C₃×C₂₇	1	1	yes	characteristic, complemented
74	C₉	C₃×AGL(1,ℤ/9)	1	1	yes	characteristic
75	C₂×C₃	—	9	3	no	—
76	C₂×C₃	—	9	1	no	—
77	D₃	—	3	1	no	—
78	C₃	—	2	1	no	—
79	C₃	—	3	3	no	—
80	C₃	C₉×AGL(1,ℤ/9)	1	1	yes	characteristic
81	C₃	C₃×D₃×C₂₇	1	1	yes	characteristic
82	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₇	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₂×C₃×C₂₇
Frattini	C₃×C₉	C₃²×D₃
Fitting	C₂₇×3₋¹⁺²	C₂
socle	C₃²	C₃×D₃×C₉
prime radical	C₃×AGL(1,ℤ/9)	C₉
cosocle	C₉²	C₂×C₃²
LCS	G ▷ C₉
UCS	C₁ ◁ C₂₇

Complements

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

N	H ≅ G/N	type
1. C₂₇×3₋¹⁺²	C₂	semidirect
4. D₉⋊C₂₇	C₃	semidirect
5. D₉×C₂₇	C₃	semidirect
9. C₉⋊C₂₇	C₂×C₃	semidirect
12. C₉×C₂₇	C₂×C₃	semidirect
35. AGL(1,ℤ/9)	C₂₇	direct (N × H)
43. 3₋¹⁺²	C₂×C₂₇	semidirect
56. C₂₇	AGL(1,ℤ/9)	direct (N × H)
62. D₉	C₃×C₂₇	semidirect
73. C₉	C₂×C₃×C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂₇×3₋¹⁺²	2
C₃×D₃×C₂₇	3
C₉×AGL(1,ℤ/9)	3
D₉⋊C₂₇	3
D₉×C₂₇	3

Subgroups — C27×AGL(1,ℤ/9) #13091023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₇×AGL(1,ℤ/9) #13091023