Subgroups — C₁₃×C₁₇×ΓL(1,𝔽₁₆) #27210565

Order 13260.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₁₃×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₁₇×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₃×C₁₇	C₄	1	1	yes	characteristic, complemented, Hall
4	Dic₃×C₁₃×C₁₇	—	5	1	no	maximal, Hall
5	D₅×C₁₃×C₁₇	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₃×C₁₇	—	5	1	no	—
7	C₅×C₁₃×C₁₇	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₁₇×ΓL(1,𝔽₁₆)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₁₃×C₁₇	—	15	1	no	Hall
10	C₁₃×ΓL(1,𝔽₁₆)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₁₃×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₃×D₅×C₁₇	C₂×C₁₃	1	1	yes	characteristic
13	C₂×C₁₃×C₁₇	—	5	1	no	—
14	C₃×D₅×C₁₃	C₂×C₁₇	1	1	yes	characteristic
15	C₁₇×AGL(1,𝔽₅)	—	3	1	no	Hall
16	C₁₃×AGL(1,𝔽₅)	—	3	1	no	Hall
17	C₃×C₅×C₁₇	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
18	C₁₃×C₁₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
19	Dic₃×C₁₇	—	5	1	no	Hall
20	C₃×C₅×C₁₃	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
21	D₅×C₁₇	D₃×C₁₃	1	1	yes	characteristic
22	Dic₃×C₁₃	—	5	1	no	Hall
23	D₅×C₁₃	D₃×C₁₇	1	1	yes	characteristic
24	C₂×C₃×C₁₇	—	5	1	no	—
25	C₅×C₁₇	Dic₃×C₁₃	1	1	yes	characteristic, complemented, Hall
26	C₂×C₃×C₁₃	—	5	1	no	—
27	C₄×C₁₇	—	15	1	no	Hall
28	C₅×C₁₃	Dic₃×C₁₇	1	1	yes	characteristic, complemented, Hall
29	ΓL(1,𝔽₁₆)	C₁₃×C₁₇	1	1	yes	characteristic, direct product factor, Hall
30	C₄×C₁₃	—	15	1	no	Hall
31	C₃×C₁₇	C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
32	C₃×C₁₃	C₁₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
33	C₂×C₁₇	—	5	1	no	—
34	C₃×D₅	C₂×C₁₃×C₁₇	1	1	yes	characteristic
35	C₂×C₁₃	—	5	1	no	—
36	AGL(1,𝔽₅)	—	3	1	no	Hall
37	C₁₇	C₁₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
38	C₃×C₅	C₄×C₁₃×C₁₇	1	1	yes	characteristic, complemented, Hall
39	C₁₃	C₁₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
40	Dic₃	—	5	1	no	Hall
41	D₅	D₃×C₁₃×C₁₇	1	1	yes	characteristic
42	C₂×C₃	—	5	1	no	—
43	C₅	Dic₃×C₁₃×C₁₇	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	15	1	no	Sylow
45	C₃	C₁₃×C₁₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃×C₁₇	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₁₃×C₁₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃×C₁₇	C₄
prime radical	C₃×D₅×C₁₃×C₁₇	C₂
cosocle	C₃×D₅	C₂×C₁₃×C₁₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₁₃×C₁₇

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₃×C₁₇	C₄	semidirect
7. C₅×C₁₃×C₁₇	Dic₃	semidirect
8. C₁₇×ΓL(1,𝔽₁₆)	C₁₃	direct (N × H)
10. C₁₃×ΓL(1,𝔽₁₆)	C₁₇	direct (N × H)
11. C₃×C₁₃×C₁₇	AGL(1,𝔽₅)	semidirect
17. C₃×C₅×C₁₇	C₄×C₁₃	semidirect
18. C₁₃×C₁₇	ΓL(1,𝔽₁₆)	direct (N × H)
20. C₃×C₅×C₁₃	C₄×C₁₇	semidirect
25. C₅×C₁₇	Dic₃×C₁₃	semidirect
28. C₅×C₁₃	Dic₃×C₁₇	semidirect
29. ΓL(1,𝔽₁₆)	C₁₃×C₁₇	direct (N × H)
31. C₃×C₁₇	C₁₃×AGL(1,𝔽₅)	semidirect
32. C₃×C₁₃	C₁₇×AGL(1,𝔽₅)	semidirect
37. C₁₇	C₁₃×ΓL(1,𝔽₁₆)	direct (N × H)
38. C₃×C₅	C₄×C₁₃×C₁₇	semidirect
39. C₁₃	C₁₇×ΓL(1,𝔽₁₆)	direct (N × H)
43. C₅	Dic₃×C₁₃×C₁₇	semidirect
45. C₃	C₁₃×C₁₇×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₁₃×C₁₇	2
C₁₃×C₁₇×AGL(1,𝔽₅)	3
Dic₃×C₁₃×C₁₇	5
C₁₇×ΓL(1,𝔽₁₆)	13
C₁₃×ΓL(1,𝔽₁₆)	17

Subgroups — C13×C17×ΓL(1,𝔽16) #27210565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₁₇×ΓL(1,𝔽₁₆) #27210565