Subgroups — C₂×C₃²×AGL(1,𝔽₉) #13049840

Order 1296.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#157	C₂	1	1	yes	maximal, characteristic
2	C₃²×AGL(1,𝔽₉)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×AGL(1,𝔽₉)	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₃⋊D₃	C₄	1	1	yes	characteristic
5	C₃²×#157	C₄	1	1	yes	characteristic
6	C₃²×#157	C₂²	1	1	yes	characteristic
7	C₂×C₃×#157	C₂×C₃	1	4	yes	—
8	C₃×AGL(1,𝔽₉)	C₂×C₃	1	8	yes	direct product factor
9	C₂×C₃⁴	C₈	1	1	yes	characteristic, complemented
10	C₃²×C₃⋊D₃	C₈	1	1	yes	characteristic, complemented
11	C₃²×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	C₂×AGL(1,𝔽₉)	C₃²	1	1	yes	characteristic, direct product factor
13	C₂×C₃²×C₈	—	9	1	no	maximal
14	C₂×C₃×C₃⋊D₃	C₃×C₄	1	4	yes	—
15	C₂×C₃²×D₃	—	12	1	no	—
16	C₃×#157	C₃×C₄	1	4	yes	—
17	C₃×#157	C₂²×C₃	1	4	yes	—
18	C₃⁴	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
19	C₂×#157	C₂×C₃²	1	1	yes	characteristic
20	AGL(1,𝔽₉)	C₂×C₃²	1	2	yes	direct product factor
21	C₂×C₃²×C₄	—	9	1	no	—
22	C₃²×C₈	—	9	2	no	—
23	C₂×C₃³	—	4	1	no	—
24	C₂×C₃³	—	8	4	no	—
25	C₂×C₃³	C₃×C₈	1	4	yes	complemented
26	C₃×C₃⋊D₃	C₃×C₈	1	4	yes	complemented
27	C₃×C₃⋊D₃	C₂×C₃×C₄	1	4	yes	—
28	C₃²×D₃	—	12	1	no	—
29	C₃²×D₃	—	12	1	no	—
30	C₂×C₃×C₈	—	9	4	no	—
31	C₂²×C₃²	—	9	1	no	—
32	C₂×C₃⋊D₃	C₃²×C₄	1	1	yes	characteristic
33	C₂×C₃×D₃	—	12	4	no	—
34	#157	C₃²×C₄	1	1	yes	characteristic
35	#157	C₂²×C₃²	1	1	yes	characteristic
36	C₃²×C₄	—	9	1	no	—
37	C₃²×C₄	—	9	1	no	—
38	C₃³	—	4	1	no	—
39	C₃³	—	8	4	no	—
40	C₃³	C₂×C₃×C₈	1	4	yes	complemented
41	C₂×C₃×C₄	—	9	4	no	—
42	C₃×C₈	—	9	8	no	—
43	C₂×C₃²	—	4	4	no	—
44	C₂×C₃²	—	8	4	no	—
45	C₂×C₃²	—	8	6	no	—
46	C₂×C₃²	—	8	4	no	—
47	C₂×C₃²	—	9	1	no	—
48	C₂×C₃²	—	9	1	no	—
49	C₂×C₃²	C₃²×C₈	1	1	yes	characteristic, complemented
50	C₂×C₃²	AGL(1,𝔽₉)	1	1	yes	characteristic, direct product factor
51	C₃⋊D₃	C₃²×C₈	1	1	yes	characteristic, complemented
52	C₃⋊D₃	C₂×C₃²×C₄	1	1	yes	characteristic
53	C₃×D₃	—	12	4	no	—
54	C₃×D₃	—	12	4	no	—
55	C₂×C₈	—	9	1	no	Sylow
56	C₂²×C₃	—	9	4	no	—
57	C₂×D₃	—	12	1	no	—
58	C₃×C₄	—	9	4	no	—
59	C₃×C₄	—	9	4	no	—
60	C₃²	—	4	4	no	—
61	C₃²	—	8	4	no	—
62	C₃²	—	8	6	no	—
63	C₃²	—	8	4	no	—
64	C₃²	C₂×C₃²×C₈	1	1	yes	characteristic, complemented
65	C₃²	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic, direct product factor
66	C₂×C₄	—	9	1	no	—
67	C₈	—	9	2	no	—
68	C₂×C₃	—	4	1	no	—
69	C₂×C₃	—	8	4	no	—
70	C₂×C₃	—	9	4	no	—
71	C₂×C₃	—	9	4	no	—
72	C₂×C₃	C₃×AGL(1,𝔽₉)	1	4	yes	direct product factor
73	D₃	—	12	1	no	—
74	D₃	—	12	1	no	—
75	C₂²	—	9	1	no	—
76	C₄	—	9	1	no	—
77	C₄	—	9	1	no	—
78	C₃	—	4	1	no	—
79	C₃	—	8	4	no	—
80	C₃	C₂×C₃×AGL(1,𝔽₉)	1	4	yes	direct product factor
81	C₂	—	9	1	no	—
82	C₂	—	9	1	no	—
83	C₂	C₃²×AGL(1,𝔽₉)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²	AGL(1,𝔽₉)
commutator, hypocenter	C₃²	C₂×C₃²×C₈
Frattini	C₁	G
Fitting, socle	C₂×C₃⁴	C₈
prime radical	C₂×C₃²×C₃⋊D₃	C₄
cosocle	#157	C₂²×C₃²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
2. C₃²×AGL(1,𝔽₉)	C₂	direct (N × H)
3. C₂×C₃×AGL(1,𝔽₉)	C₃	direct (N × H)
8. C₃×AGL(1,𝔽₉)	C₂×C₃	direct (N × H)
9. C₂×C₃⁴	C₈	semidirect
10. C₃²×C₃⋊D₃	C₈	semidirect
12. C₂×AGL(1,𝔽₉)	C₃²	direct (N × H)
18. C₃⁴	C₂×C₈	semidirect
20. AGL(1,𝔽₉)	C₂×C₃²	direct (N × H)
25. C₂×C₃³	C₃×C₈	semidirect
26. C₃×C₃⋊D₃	C₃×C₈	semidirect
40. C₃³	C₂×C₃×C₈	semidirect
49. C₂×C₃²	C₃²×C₈	semidirect
50. C₂×C₃²	AGL(1,𝔽₉)	direct (N × H)
51. C₃⋊D₃	C₃²×C₈	semidirect
64. C₃²	C₂×C₃²×C₈	semidirect
65. C₃²	C₂×AGL(1,𝔽₉)	direct (N × H)
72. C₂×C₃	C₃×AGL(1,𝔽₉)	direct (N × H)
80. C₃	C₂×C₃×AGL(1,𝔽₉)	direct (N × H)
83. C₂	C₃²×AGL(1,𝔽₉)	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×C₃²×#157	2
C₃²×AGL(1,𝔽₉)	2
C₂×C₃×AGL(1,𝔽₉)	3
C₂×C₃²×C₈	9

Subgroups — C2×C32×AGL(1,𝔽9) #13049840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×AGL(1,𝔽₉) #13049840