Subgroups — C₁₃×C₄₁×ΓL(1,𝔽₁₆) #44544439

Order 31980.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₁₃×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₁₃×AGL(1,𝔽₅)×C₄₁	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₃×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
4	Dic₃×C₁₃×C₄₁	—	5	1	no	maximal, Hall
5	D₅×C₁₃×C₄₁	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₃×C₄₁	—	5	1	no	—
7	C₅×C₁₃×C₄₁	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₄₁×ΓL(1,𝔽₁₆)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₁₃×C₄₁	—	15	1	no	Hall
10	C₃×C₁₃×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
11	C₃×D₅×C₄₁	C₂×C₁₃	1	1	yes	characteristic
12	C₂×C₁₃×C₄₁	—	5	1	no	—
13	AGL(1,𝔽₅)×C₄₁	—	3	1	no	Hall
14	C₁₃×ΓL(1,𝔽₁₆)	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃×C₅×C₄₁	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₁₃×C₄₁	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
17	Dic₃×C₄₁	—	5	1	no	Hall
18	D₅×C₄₁	D₃×C₁₃	1	1	yes	characteristic
19	C₃×D₅×C₁₃	C₂×C₄₁	1	1	yes	characteristic
20	C₁₃×AGL(1,𝔽₅)	—	3	1	no	Hall
21	C₂×C₃×C₄₁	—	5	1	no	—
22	C₅×C₄₁	Dic₃×C₁₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅×C₁₃	C₄×C₄₁	1	1	yes	characteristic, complemented, Hall
24	C₄×C₄₁	—	15	1	no	Hall
25	Dic₃×C₁₃	—	5	1	no	Hall
26	D₅×C₁₃	D₃×C₄₁	1	1	yes	characteristic
27	C₃×C₄₁	C₁₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
28	C₂×C₄₁	—	5	1	no	—
29	C₂×C₃×C₁₃	—	5	1	no	—
30	C₅×C₁₃	Dic₃×C₄₁	1	1	yes	characteristic, complemented, Hall
31	ΓL(1,𝔽₁₆)	C₁₃×C₄₁	1	1	yes	characteristic, direct product factor, Hall
32	C₄×C₁₃	—	15	1	no	Hall
33	C₄₁	C₁₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
34	C₃×C₁₃	AGL(1,𝔽₅)×C₄₁	1	1	yes	characteristic, complemented, Hall
35	C₃×D₅	C₂×C₁₃×C₄₁	1	1	yes	characteristic
36	C₂×C₁₃	—	5	1	no	—
37	AGL(1,𝔽₅)	—	3	1	no	Hall
38	C₃×C₅	C₄×C₁₃×C₄₁	1	1	yes	characteristic, complemented, Hall
39	C₁₃	C₄₁×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
40	Dic₃	—	5	1	no	Hall
41	D₅	D₃×C₁₃×C₄₁	1	1	yes	characteristic
42	C₂×C₃	—	5	1	no	—
43	C₅	Dic₃×C₁₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	15	1	no	Sylow
45	C₃	C₁₃×AGL(1,𝔽₅)×C₄₁	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃×C₄₁	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₁₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃×C₄₁	C₄
prime radical	C₃×D₅×C₁₃×C₄₁	C₂
cosocle	C₃×D₅	C₂×C₁₃×C₄₁
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₁₃×C₄₁

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₃×C₄₁	C₄	semidirect
7. C₅×C₁₃×C₄₁	Dic₃	semidirect
8. C₄₁×ΓL(1,𝔽₁₆)	C₁₃	direct (N × H)
10. C₃×C₁₃×C₄₁	AGL(1,𝔽₅)	semidirect
14. C₁₃×ΓL(1,𝔽₁₆)	C₄₁	direct (N × H)
15. C₃×C₅×C₄₁	C₄×C₁₃	semidirect
16. C₁₃×C₄₁	ΓL(1,𝔽₁₆)	direct (N × H)
22. C₅×C₄₁	Dic₃×C₁₃	semidirect
23. C₃×C₅×C₁₃	C₄×C₄₁	semidirect
27. C₃×C₄₁	C₁₃×AGL(1,𝔽₅)	semidirect
30. C₅×C₁₃	Dic₃×C₄₁	semidirect
31. ΓL(1,𝔽₁₆)	C₁₃×C₄₁	direct (N × H)
33. C₄₁	C₁₃×ΓL(1,𝔽₁₆)	direct (N × H)
34. C₃×C₁₃	AGL(1,𝔽₅)×C₄₁	semidirect
38. C₃×C₅	C₄×C₁₃×C₄₁	semidirect
39. C₁₃	C₄₁×ΓL(1,𝔽₁₆)	direct (N × H)
43. C₅	Dic₃×C₁₃×C₄₁	semidirect
45. C₃	C₁₃×AGL(1,𝔽₅)×C₄₁	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×C₄₁	2
C₁₃×AGL(1,𝔽₅)×C₄₁	3
Dic₃×C₁₃×C₄₁	5
C₄₁×ΓL(1,𝔽₁₆)	13
C₁₃×ΓL(1,𝔽₁₆)	41

Subgroups — C13×C41×ΓL(1,𝔽16) #44544439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₄₁×ΓL(1,𝔽₁₆) #44544439