Subgroups — C₃²×AGL(1,𝔽₅)×C₆₁ #24926807

Order 10980.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₅×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₃×AGL(1,𝔽₅)×C₆₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×C₆₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃²×C₄×C₆₁	—	5	1	no	maximal, Hall
5	C₃×D₅×C₆₁	C₂×C₃	1	4	yes	—
6	AGL(1,𝔽₅)×C₆₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
7	C₂×C₃²×C₆₁	—	5	1	no	—
8	C₃×C₅×C₆₁	C₃×C₄	1	4	yes	complemented
9	C₃×C₄×C₆₁	—	5	4	no	—
10	D₅×C₆₁	C₂×C₃²	1	1	yes	characteristic
11	C₃²×C₆₁	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
12	C₂×C₃×C₆₁	—	5	4	no	—
13	C₅×C₆₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
14	C₄×C₆₁	—	5	1	no	Hall
15	C₃×C₆₁	C₃×AGL(1,𝔽₅)	1	4	yes	direct product factor
16	C₃²×AGL(1,𝔽₅)	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₆₁	—	5	1	no	—
18	C₃²×D₅	C₂×C₆₁	1	1	yes	characteristic
19	C₆₁	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×AGL(1,𝔽₅)	C₃×C₆₁	1	4	yes	direct product factor
21	C₃²×C₅	C₄×C₆₁	1	1	yes	characteristic, complemented, Hall
22	C₃²×C₄	—	5	1	no	Hall
23	C₃×D₅	C₂×C₃×C₆₁	1	4	yes	—
24	AGL(1,𝔽₅)	C₃²×C₆₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃²	—	5	1	no	—
26	C₃×C₅	C₃×C₄×C₆₁	1	4	yes	complemented
27	C₃×C₄	—	5	4	no	—
28	D₅	C₂×C₃²×C₆₁	1	1	yes	characteristic
29	C₃²	AGL(1,𝔽₅)×C₆₁	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃	—	5	4	no	—
31	C₅	C₃²×C₄×C₆₁	1	1	yes	characteristic, complemented, Sylow
32	C₄	—	5	1	no	Sylow
33	C₃	C₃×AGL(1,𝔽₅)×C₆₁	1	4	yes	direct product factor
34	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃²×C₆₁	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₄×C₆₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₆₁	C₄
prime radical	C₃²×D₅×C₆₁	C₂
cosocle	D₅	C₂×C₃²×C₆₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₆₁

Complements

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

N	H ≅ G/N	type
2. C₃×AGL(1,𝔽₅)×C₆₁	C₃	direct (N × H)
3. C₃²×C₅×C₆₁	C₄	semidirect
6. AGL(1,𝔽₅)×C₆₁	C₃²	direct (N × H)
8. C₃×C₅×C₆₁	C₃×C₄	semidirect
11. C₃²×C₆₁	AGL(1,𝔽₅)	direct (N × H)
13. C₅×C₆₁	C₃²×C₄	semidirect
15. C₃×C₆₁	C₃×AGL(1,𝔽₅)	direct (N × H)
16. C₃²×AGL(1,𝔽₅)	C₆₁	direct (N × H)
19. C₆₁	C₃²×AGL(1,𝔽₅)	direct (N × H)
20. C₃×AGL(1,𝔽₅)	C₃×C₆₁	direct (N × H)
21. C₃²×C₅	C₄×C₆₁	semidirect
24. AGL(1,𝔽₅)	C₃²×C₆₁	direct (N × H)
26. C₃×C₅	C₃×C₄×C₆₁	semidirect
29. C₃²	AGL(1,𝔽₅)×C₆₁	direct (N × H)
31. C₅	C₃²×C₄×C₆₁	semidirect
33. C₃	C₃×AGL(1,𝔽₅)×C₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₅×C₆₁	2
C₃×AGL(1,𝔽₅)×C₆₁	3
C₃²×C₄×C₆₁	5
C₃²×AGL(1,𝔽₅)	61

Subgroups — C32×AGL(1,𝔽5)×C61 #24926807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×AGL(1,𝔽₅)×C₆₁ #24926807