Subgroups — C₁₇×AGL(1,𝔽₅)×C₁₃⋊C₃ #27210495

Order 13260.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₁₇×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₁₇×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₇×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₄×C₁₇×C₁₃⋊C₃	—	5	1	no	maximal, Hall
5	D₅×C₁₃×C₁₇	C₂×C₃	1	1	yes	characteristic
6	C₂×C₁₇×C₁₃⋊C₃	—	5	1	no	—
7	C₅×C₁₃×C₁₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₇×AGL(1,𝔽₅)	—	13	1	no	maximal, Hall
9	C₄×C₁₃×C₁₇	—	5	1	no	Hall
10	AGL(1,𝔽₅)×C₁₃⋊C₃	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₁₇×C₁₃⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
12	C₃×D₅×C₁₇	—	13	1	no	—
13	C₂×C₁₃×C₁₇	—	5	1	no	—
14	D₅×C₁₃⋊C₃	C₂×C₁₇	1	1	yes	characteristic
15	C₁₇×AGL(1,𝔽₅)	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
16	C₁₃×AGL(1,𝔽₅)	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
17	C₃×C₅×C₁₇	—	13	1	no	Hall
18	C₁₃×C₁₇	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
19	C₃×C₄×C₁₇	—	65	1	no	Hall
20	C₅×C₁₃⋊C₃	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
21	D₅×C₁₇	C₂×C₁₃⋊C₃	1	1	yes	characteristic
22	C₄×C₁₃⋊C₃	—	5	1	no	Hall
23	D₅×C₁₃	C₂×C₃×C₁₇	1	1	yes	characteristic
24	C₂×C₃×C₁₇	—	65	1	no	—
25	C₅×C₁₇	C₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₁₃⋊C₃	—	5	1	no	—
27	C₄×C₁₇	—	5	1	no	Hall
28	C₅×C₁₃	C₃×C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
29	C₃×AGL(1,𝔽₅)	—	13	1	no	Hall
30	C₄×C₁₃	—	5	1	no	Hall
31	C₃×C₁₇	—	13	1	no	Hall
32	C₁₃⋊C₃	C₁₇×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₁₇	—	5	1	no	—
34	C₃×D₅	—	13	1	no	—
35	C₂×C₁₃	—	5	1	no	—
36	AGL(1,𝔽₅)	C₁₇×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
37	C₁₇	AGL(1,𝔽₅)×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
38	C₃×C₅	—	13	1	no	Hall
39	C₁₃	C₃×C₁₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₄	—	65	1	no	Hall
41	D₅	C₂×C₁₇×C₁₃⋊C₃	1	1	yes	characteristic
42	C₂×C₃	—	65	1	no	—
43	C₅	C₄×C₁₇×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	5	1	no	Sylow
45	C₃	—	13	1	no	Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇	AGL(1,𝔽₅)×C₁₃⋊C₃
commutator, hypocenter	C₅×C₁₃	C₃×C₄×C₁₇
Frattini	C₁	G
Fitting, socle	C₅×C₁₃×C₁₇	C₃×C₄
prime radical	D₅×C₁₇×C₁₃⋊C₃	C₂
cosocle	D₅×C₁₃	C₂×C₃×C₁₇
LCS	G ▷ C₅×C₁₃
UCS	C₁ ◁ C₁₇

Complements

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

N	H ≅ G/N	type
2. C₁₃×C₁₇×AGL(1,𝔽₅)	C₃	semidirect
3. C₅×C₁₇×C₁₃⋊C₃	C₄	semidirect
7. C₅×C₁₃×C₁₇	C₃×C₄	semidirect
10. AGL(1,𝔽₅)×C₁₃⋊C₃	C₁₇	direct (N × H)
11. C₁₇×C₁₃⋊C₃	AGL(1,𝔽₅)	direct (N × H)
15. C₁₇×AGL(1,𝔽₅)	C₁₃⋊C₃	direct (N × H)
16. C₁₃×AGL(1,𝔽₅)	C₃×C₁₇	semidirect
18. C₁₃×C₁₇	C₃×AGL(1,𝔽₅)	semidirect
20. C₅×C₁₃⋊C₃	C₄×C₁₇	semidirect
25. C₅×C₁₇	C₄×C₁₃⋊C₃	semidirect
28. C₅×C₁₃	C₃×C₄×C₁₇	semidirect
32. C₁₃⋊C₃	C₁₇×AGL(1,𝔽₅)	direct (N × H)
36. AGL(1,𝔽₅)	C₁₇×C₁₃⋊C₃	direct (N × H)
37. C₁₇	AGL(1,𝔽₅)×C₁₃⋊C₃	direct (N × H)
39. C₁₃	C₃×C₁₇×AGL(1,𝔽₅)	semidirect
43. C₅	C₄×C₁₇×C₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₁₇×C₁₃⋊C₃	2
C₁₃×C₁₇×AGL(1,𝔽₅)	3
C₄×C₁₇×C₁₃⋊C₃	5
C₃×C₁₇×AGL(1,𝔽₅)	13
AGL(1,𝔽₅)×C₁₃⋊C₃	17

Subgroups — C17×AGL(1,𝔽5)×C13⋊C3 #27210495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×AGL(1,𝔽₅)×C₁₃⋊C₃ #27210495