Subgroups — C₅×AΓL(1,𝔽₉) #10624310

Order 720.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×GO⁺(4,𝔽₂)	C₂	1	1	yes	maximal, characteristic
3	C₅×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₃²	—	2	1	no	—
5	C₅×#157	—	2	1	no	—
6	C₅×#157	C₂²	1	1	yes	characteristic
7	AΓL(1,𝔽₉)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₅×C₃⋊D₃	D₄	1	1	yes	characteristic
9	C₃×C₅×D₃	—	4	1	no	—
10	C₅×SD₁₆	—	9	1	no	maximal, Hall
11	PSU(3,𝔽₂)	C₂×C₅	1	1	yes	characteristic, complemented
12	GO⁺(4,𝔽₂)	C₂×C₅	1	1	yes	characteristic
13	AGL(1,𝔽₉)	C₂×C₅	1	1	yes	characteristic, complemented
14	C₂×C₅×D₃	—	12	1	no	—
15	C₃²×C₅	SD₁₆	1	1	yes	characteristic, complemented, Hall
16	C₅×Q₈	—	9	1	no	—
17	C₅×D₄	—	9	1	no	—
18	C₅×C₈	—	9	1	no	—
19	D₃²	—	2	1	no	—
20	#157	—	2	1	no	—
21	#157	C₂²×C₅	1	1	yes	characteristic
22	C₂×C₃×C₅	—	12	1	no	—
23	C₅×D₃	—	4	1	no	—
24	C₅×D₃	—	12	1	no	—
25	C₂²×C₅	—	18	1	no	—
26	C₄×C₅	—	9	1	no	—
27	C₄×C₅	—	18	1	no	—
28	C₃⋊D₃	C₅×D₄	1	1	yes	characteristic
29	C₃×D₃	—	4	1	no	—
30	SD₁₆	—	9	1	no	Sylow
31	C₃×C₅	—	4	1	no	—
32	C₂×D₃	—	12	1	no	—
33	C₂×C₅	—	9	1	no	—
34	C₂×C₅	—	12	1	no	—
35	C₃²	C₅×SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	Q₈	—	9	1	no	—
37	D₄	—	9	1	no	—
38	C₈	—	9	1	no	—
39	C₂×C₃	—	12	1	no	—
40	D₃	—	4	1	no	—
41	D₃	—	12	1	no	—
42	C₅	AΓL(1,𝔽₉)	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²	—	18	1	no	—
44	C₄	—	9	1	no	—
45	C₄	—	18	1	no	—
46	C₃	—	4	1	no	—
47	C₂	—	9	1	no	—
48	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₅	AΓL(1,𝔽₉)
commutator, cosocle	#157	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅	SD₁₆
prime radical	C₅×GO⁺(4,𝔽₂)	C₂
hypocenter	C₃²	C₅×SD₁₆
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₅

Complements

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

N	H ≅ G/N	type
1. C₅×PSU(3,𝔽₂)	C₂	semidirect
3. C₅×AGL(1,𝔽₉)	C₂	semidirect
7. AΓL(1,𝔽₉)	C₅	direct (N × H)
11. PSU(3,𝔽₂)	C₂×C₅	semidirect
13. AGL(1,𝔽₉)	C₂×C₅	semidirect
15. C₃²×C₅	SD₁₆	semidirect
35. C₃²	C₅×SD₁₆	semidirect
42. C₅	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
C₅×PSU(3,𝔽₂)	2
C₅×GO⁺(4,𝔽₂)	2
C₅×AGL(1,𝔽₉)	2
AΓL(1,𝔽₉)	5
C₅×SD₁₆	9

Subgroups — C5×AΓL(1,𝔽9) #10624310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×AΓL(1,𝔽₉) #10624310