Subgroups — C₃×AΓL(1,𝔽₉) #89090

Order 432.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×GO⁺(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₃×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic, complemented
4	AΓL(1,𝔽₉)	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×D₃²	—	2	1	no	—
6	C₃×#157	—	2	1	no	—
7	C₃×#157	C₂²	1	1	yes	characteristic
8	PSU(3,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
9	GO⁺(4,𝔽₂)	C₂×C₃	1	1	yes	characteristic
10	AGL(1,𝔽₉)	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
12	C₃²×D₃	—	4	1	no	—
13	C₃×SD₁₆	—	9	1	no	maximal
14	C₂×C₃×D₃	—	12	1	no	—
15	D₃²	—	2	1	no	—
16	#157	—	2	1	no	—
17	#157	C₂²×C₃	1	1	yes	characteristic
18	C₃³	SD₁₆	1	1	yes	characteristic, complemented, Sylow
19	C₃×Q₈	—	9	1	no	—
20	C₃×D₄	—	9	1	no	—
21	C₃×C₈	—	9	1	no	—
22	C₂×C₃²	—	12	1	no	—
23	C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
24	C₃×D₃	—	4	1	no	—
25	C₃×D₃	—	4	1	no	—
26	C₃×D₃	—	8	1	no	—
27	C₃×D₃	—	12	1	no	—
28	SD₁₆	—	9	1	no	Sylow
29	C₂²×C₃	—	18	1	no	—
30	C₂×D₃	—	12	1	no	—
31	C₃×C₄	—	9	1	no	—
32	C₃×C₄	—	18	1	no	—
33	C₃²	—	4	1	no	—
34	C₃²	—	8	1	no	—
35	C₃²	C₃×SD₁₆	1	1	yes	characteristic, complemented
36	Q₈	—	9	1	no	—
37	D₄	—	9	1	no	—
38	C₈	—	9	1	no	—
39	C₂×C₃	—	9	1	no	—
40	C₂×C₃	—	12	1	no	—
41	C₂×C₃	—	12	1	no	—
42	C₂×C₃	—	24	1	no	—
43	D₃	—	4	1	no	—
44	D₃	—	12	1	no	—
45	C₂²	—	18	1	no	—
46	C₄	—	9	1	no	—
47	C₄	—	18	1	no	—
48	C₃	—	4	1	no	—
49	C₃	—	8	1	no	—
50	C₃	AΓL(1,𝔽₉)	1	1	yes	characteristic, direct product factor
51	C₂	—	9	1	no	—
52	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₃	AΓL(1,𝔽₉)
commutator, cosocle	#157	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃³	SD₁₆
prime radical	C₃×GO⁺(4,𝔽₂)	C₂
hypocenter	C₃²	C₃×SD₁₆
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×PSU(3,𝔽₂)	C₂	semidirect
3. C₃×AGL(1,𝔽₉)	C₂	semidirect
4. AΓL(1,𝔽₉)	C₃	direct (N × H)
8. PSU(3,𝔽₂)	C₂×C₃	semidirect
10. AGL(1,𝔽₉)	C₂×C₃	semidirect
18. C₃³	SD₁₆	semidirect
35. C₃²	C₃×SD₁₆	semidirect
50. C₃	AΓL(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
C₃×PSU(3,𝔽₂)	2
C₃×GO⁺(4,𝔽₂)	2
C₃×AGL(1,𝔽₉)	2
AΓL(1,𝔽₉)	3
C₃×SD₁₆	9

Subgroups — C3×AΓL(1,𝔽9) #89090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×AΓL(1,𝔽₉) #89090