Subgroups — C₂×AΓL(1,𝔽₉) #64379

Order 288.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×GO⁺(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₂×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic, complemented
4	AΓL(1,𝔽₉)	C₂	1	4	yes	maximal, direct product factor
5	C₂×D₃²	—	2	1	no	—
6	C₂×#157	—	2	1	no	—
7	C₂×#157	C₂²	1	1	yes	characteristic
8	PSU(3,𝔽₂)	—	2	1	no	—
9	PSU(3,𝔽₂)	C₂²	1	2	yes	complemented
10	GO⁺(4,𝔽₂)	—	2	1	no	—
11	GO⁺(4,𝔽₂)	C₂²	1	2	yes	—
12	AGL(1,𝔽₉)	C₂²	1	2	yes	complemented
13	C₂×C₃⋊D₃	D₄	1	1	yes	characteristic
14	C₂×C₃×D₃	—	4	1	no	—
15	D₃²	—	2	2	no	—
16	D₃²	—	4	1	no	—
17	#157	—	2	2	no	—
18	#157	D₄	1	1	yes	characteristic
19	#157	C₂³	1	1	yes	characteristic
20	C₂×SD₁₆	—	9	1	no	maximal, Sylow
21	C₂²×D₃	—	12	1	no	—
22	C₂×C₃²	SD₁₆	1	1	yes	characteristic, complemented
23	C₃⋊D₃	SD₁₆	1	1	yes	characteristic, complemented
24	C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
25	C₃×D₃	—	4	2	no	—
26	C₂×Q₈	—	9	1	no	—
27	C₂×D₄	—	9	1	no	—
28	SD₁₆	—	9	4	no	—
29	C₂×C₈	—	9	1	no	—
30	C₂²×C₃	—	12	1	no	—
31	C₂×D₃	—	4	1	no	—
32	C₂×D₃	—	12	1	no	—
33	C₂×D₃	—	12	2	no	—
34	C₂×D₃	—	12	2	no	—
35	C₃²	C₂×SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂³	—	18	1	no	—
37	Q₈	—	9	2	no	—
38	Q₈	—	18	1	no	—
39	D₄	—	9	2	no	—
40	D₄	—	18	1	no	—
41	C₂×C₄	—	9	1	no	—
42	C₂×C₄	—	18	1	no	—
43	C₈	—	9	2	no	—
44	C₂×C₃	—	4	1	no	—
45	C₂×C₃	—	12	2	no	—
46	D₃	—	4	2	no	—
47	D₃	—	12	1	no	—
48	D₃	—	12	1	no	—
49	C₂²	—	9	1	no	—
50	C₂²	—	12	1	no	—
51	C₂²	—	18	2	no	—
52	C₂²	—	36	1	no	—
53	C₄	—	9	1	no	—
54	C₄	—	9	1	no	—
55	C₄	—	18	2	no	—
56	C₃	—	4	1	no	—
57	C₂	—	9	1	no	—
58	C₂	—	9	1	no	—
59	C₂	—	12	2	no	—
60	C₂	AΓL(1,𝔽₉)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	AΓL(1,𝔽₉)
commutator, cosocle	#157	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²	SD₁₆
prime radical	C₂×GO⁺(4,𝔽₂)	C₂
hypocenter	C₃²	C₂×SD₁₆
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×PSU(3,𝔽₂)	C₂	semidirect
3. C₂×AGL(1,𝔽₉)	C₂	semidirect
4. AΓL(1,𝔽₉)	C₂	direct (N × H)
9. PSU(3,𝔽₂)	C₂²	semidirect
12. AGL(1,𝔽₉)	C₂²	semidirect
22. C₂×C₃²	SD₁₆	semidirect
23. C₃⋊D₃	SD₁₆	semidirect
35. C₃²	C₂×SD₁₆	semidirect
60. C₂	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
C₂×PSU(3,𝔽₂)	2
C₂×GO⁺(4,𝔽₂)	2
C₂×AGL(1,𝔽₉)	2
AΓL(1,𝔽₉)	2
C₂×SD₁₆	9

Subgroups — C2×AΓL(1,𝔽9) #64379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×AΓL(1,𝔽₉) #64379