Subgroups — C₁₇×C₃₁×ΓL(1,𝔽₁₆) #44539362

Order 31620.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₁₇×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₁₇×AGL(1,𝔽₅)×C₃₁	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₇×C₃₁	C₄	1	1	yes	characteristic, complemented, Hall
4	Dic₃×C₁₇×C₃₁	—	5	1	no	maximal, Hall
5	D₅×C₁₇×C₃₁	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₇×C₃₁	—	5	1	no	—
7	C₅×C₁₇×C₃₁	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₄×C₁₇×C₃₁	—	15	1	no	Hall
9	C₃₁×ΓL(1,𝔽₁₆)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₁₇×C₃₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₇×C₃₁	—	5	1	no	—
12	C₁₇×ΓL(1,𝔽₁₆)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×D₅×C₃₁	C₂×C₁₇	1	1	yes	characteristic
14	AGL(1,𝔽₅)×C₃₁	—	3	1	no	Hall
15	C₁₇×C₃₁	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
16	C₃×D₅×C₁₇	C₂×C₃₁	1	1	yes	characteristic
17	C₃×C₅×C₃₁	C₄×C₁₇	1	1	yes	characteristic, complemented, Hall
18	Dic₃×C₃₁	—	5	1	no	Hall
19	C₁₇×AGL(1,𝔽₅)	—	3	1	no	Hall
20	D₅×C₃₁	D₃×C₁₇	1	1	yes	characteristic
21	C₃×C₅×C₁₇	C₄×C₃₁	1	1	yes	characteristic, complemented, Hall
22	Dic₃×C₁₇	—	5	1	no	Hall
23	C₂×C₃×C₃₁	—	5	1	no	—
24	D₅×C₁₇	D₃×C₃₁	1	1	yes	characteristic
25	C₅×C₃₁	Dic₃×C₁₇	1	1	yes	characteristic, complemented, Hall
26	C₄×C₃₁	—	15	1	no	Hall
27	C₂×C₃×C₁₇	—	5	1	no	—
28	C₃×C₃₁	C₁₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
29	C₅×C₁₇	Dic₃×C₃₁	1	1	yes	characteristic, complemented, Hall
30	C₄×C₁₇	—	15	1	no	Hall
31	C₂×C₃₁	—	5	1	no	—
32	ΓL(1,𝔽₁₆)	C₁₇×C₃₁	1	1	yes	characteristic, direct product factor, Hall
33	C₃×C₁₇	AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic, complemented, Hall
34	C₂×C₁₇	—	5	1	no	—
35	C₃₁	C₁₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
36	C₃×D₅	C₂×C₁₇×C₃₁	1	1	yes	characteristic
37	AGL(1,𝔽₅)	—	3	1	no	Hall
38	C₁₇	C₃₁×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
39	C₃×C₅	C₄×C₁₇×C₃₁	1	1	yes	characteristic, complemented, Hall
40	Dic₃	—	5	1	no	Hall
41	D₅	D₃×C₁₇×C₃₁	1	1	yes	characteristic
42	C₂×C₃	—	5	1	no	—
43	C₅	Dic₃×C₁₇×C₃₁	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	15	1	no	Sylow
45	C₃	C₁₇×AGL(1,𝔽₅)×C₃₁	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₇×C₃₁	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₁₇×C₃₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₇×C₃₁	C₄
prime radical	C₃×D₅×C₁₇×C₃₁	C₂
cosocle	C₃×D₅	C₂×C₁₇×C₃₁
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₁₇×C₃₁

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₇×C₃₁	C₄	semidirect
7. C₅×C₁₇×C₃₁	Dic₃	semidirect
9. C₃₁×ΓL(1,𝔽₁₆)	C₁₇	direct (N × H)
10. C₃×C₁₇×C₃₁	AGL(1,𝔽₅)	semidirect
12. C₁₇×ΓL(1,𝔽₁₆)	C₃₁	direct (N × H)
15. C₁₇×C₃₁	ΓL(1,𝔽₁₆)	direct (N × H)
17. C₃×C₅×C₃₁	C₄×C₁₇	semidirect
21. C₃×C₅×C₁₇	C₄×C₃₁	semidirect
25. C₅×C₃₁	Dic₃×C₁₇	semidirect
28. C₃×C₃₁	C₁₇×AGL(1,𝔽₅)	semidirect
29. C₅×C₁₇	Dic₃×C₃₁	semidirect
32. ΓL(1,𝔽₁₆)	C₁₇×C₃₁	direct (N × H)
33. C₃×C₁₇	AGL(1,𝔽₅)×C₃₁	semidirect
35. C₃₁	C₁₇×ΓL(1,𝔽₁₆)	direct (N × H)
38. C₁₇	C₃₁×ΓL(1,𝔽₁₆)	direct (N × H)
39. C₃×C₅	C₄×C₁₇×C₃₁	semidirect
43. C₅	Dic₃×C₁₇×C₃₁	semidirect
45. C₃	C₁₇×AGL(1,𝔽₅)×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₁₇×C₃₁	2
C₁₇×AGL(1,𝔽₅)×C₃₁	3
Dic₃×C₁₇×C₃₁	5
C₃₁×ΓL(1,𝔽₁₆)	17
C₁₇×ΓL(1,𝔽₁₆)	31

Subgroups — C17×C31×ΓL(1,𝔽16) #44539362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×C₃₁×ΓL(1,𝔽₁₆) #44539362