Subgroups — C₃×C₅×AGL(1,𝔽₅)×C₅₉ #31799201

Order 17700.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×D₅×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₅×AGL(1,𝔽₅)×C₅₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅²×C₅₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×AGL(1,𝔽₅)×C₅₉	C₅	1	1	yes	maximal, characteristic, direct product factor
5	C₃×C₄×C₅×C₅₉	—	5	1	no	maximal
6	C₅×D₅×C₅₉	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×C₅×C₅₉	—	5	1	no	—
8	C₃×D₅×C₅₉	C₂×C₅	1	1	yes	characteristic
9	C₅²×C₅₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	AGL(1,𝔽₅)×C₅₉	C₃×C₅	1	1	yes	characteristic, direct product factor
11	C₄×C₅×C₅₉	—	5	1	no	—
12	C₃×C₅×C₅₉	—	4	1	no	—
13	C₃×C₅×C₅₉	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃×C₅×C₅₉	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
15	C₃×C₄×C₅₉	—	5	1	no	Hall
16	C₂×C₅×C₅₉	—	5	1	no	—
17	D₅×C₅₉	C₂×C₃×C₅	1	1	yes	characteristic
18	C₂×C₃×C₅₉	—	5	1	no	—
19	C₃×C₅×AGL(1,𝔽₅)	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×C₅₉	—	4	1	no	—
21	C₅×C₅₉	C₃×C₄×C₅	1	1	yes	characteristic, complemented
22	C₅×C₅₉	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
23	C₄×C₅₉	—	5	1	no	Hall
24	C₃×C₅₉	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₅×D₅	C₂×C₅₉	1	1	yes	characteristic
26	C₂×C₅₉	—	5	1	no	—
27	C₅×AGL(1,𝔽₅)	C₃×C₅₉	1	1	yes	characteristic, direct product factor, Hall
28	C₃×C₅²	C₄×C₅₉	1	1	yes	characteristic, complemented, Hall
29	C₃×AGL(1,𝔽₅)	C₅×C₅₉	1	1	yes	characteristic, direct product factor
30	C₃×C₄×C₅	—	5	1	no	—
31	C₅₉	C₃×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
32	C₅×D₅	C₂×C₃×C₅₉	1	1	yes	characteristic
33	C₂×C₃×C₅	—	5	1	no	—
34	C₃×D₅	C₂×C₅×C₅₉	1	1	yes	characteristic
35	C₅²	C₃×C₄×C₅₉	1	1	yes	characteristic, complemented, Sylow
36	AGL(1,𝔽₅)	C₃×C₅×C₅₉	1	1	yes	characteristic, direct product factor
37	C₄×C₅	—	5	1	no	—
38	C₃×C₅	—	4	1	no	—
39	C₃×C₅	C₄×C₅×C₅₉	1	1	yes	characteristic, complemented
40	C₃×C₅	AGL(1,𝔽₅)×C₅₉	1	1	yes	characteristic, direct product factor
41	C₃×C₄	—	5	1	no	Hall
42	C₂×C₅	—	5	1	no	—
43	D₅	C₂×C₃×C₅×C₅₉	1	1	yes	characteristic
44	C₂×C₃	—	5	1	no	—
45	C₅	—	4	1	no	—
46	C₅	C₃×C₄×C₅×C₅₉	1	1	yes	characteristic, complemented
47	C₅	C₃×AGL(1,𝔽₅)×C₅₉	1	1	yes	characteristic, direct product factor
48	C₄	—	5	1	no	Sylow
49	C₃	C₅×AGL(1,𝔽₅)×C₅₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₅×C₅₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃×C₄×C₅×C₅₉
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₅₉	C₄
prime radical	C₃×C₅×D₅×C₅₉	C₂
cosocle	D₅	C₂×C₃×C₅×C₅₉
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₅×C₅₉

Complements

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

N	H ≅ G/N	type
2. C₅×AGL(1,𝔽₅)×C₅₉	C₃	direct (N × H)
3. C₃×C₅²×C₅₉	C₄	semidirect
4. C₃×AGL(1,𝔽₅)×C₅₉	C₅	direct (N × H)
9. C₅²×C₅₉	C₃×C₄	semidirect
10. AGL(1,𝔽₅)×C₅₉	C₃×C₅	direct (N × H)
13. C₃×C₅×C₅₉	C₄×C₅	semidirect
14. C₃×C₅×C₅₉	AGL(1,𝔽₅)	direct (N × H)
19. C₃×C₅×AGL(1,𝔽₅)	C₅₉	direct (N × H)
21. C₅×C₅₉	C₃×C₄×C₅	semidirect
22. C₅×C₅₉	C₃×AGL(1,𝔽₅)	direct (N × H)
24. C₃×C₅₉	C₅×AGL(1,𝔽₅)	direct (N × H)
27. C₅×AGL(1,𝔽₅)	C₃×C₅₉	direct (N × H)
28. C₃×C₅²	C₄×C₅₉	semidirect
29. C₃×AGL(1,𝔽₅)	C₅×C₅₉	direct (N × H)
31. C₅₉	C₃×C₅×AGL(1,𝔽₅)	direct (N × H)
35. C₅²	C₃×C₄×C₅₉	semidirect
36. AGL(1,𝔽₅)	C₃×C₅×C₅₉	direct (N × H)
39. C₃×C₅	C₄×C₅×C₅₉	semidirect
40. C₃×C₅	AGL(1,𝔽₅)×C₅₉	direct (N × H)
46. C₅	C₃×C₄×C₅×C₅₉	semidirect
47. C₅	C₃×AGL(1,𝔽₅)×C₅₉	direct (N × H)
49. C₃	C₅×AGL(1,𝔽₅)×C₅₉	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×C₅×D₅×C₅₉	2
C₅×AGL(1,𝔽₅)×C₅₉	3
C₃×AGL(1,𝔽₅)×C₅₉	5
C₃×C₄×C₅×C₅₉	5
C₃×C₅×AGL(1,𝔽₅)	59

Subgroups — C3×C5×AGL(1,𝔽5)×C59 #31799201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×AGL(1,𝔽₅)×C₅₉ #31799201