Subgroups — C₁₁×C₃₇×ΓL(1,𝔽₁₆) #37567598

Order 24420.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×C₁₁×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₁₁×AGL(1,𝔽₅)×C₃₇	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₁×C₃₇	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₁×Dic₃×C₃₇	—	5	1	no	maximal, Hall
5	D₅×C₁₁×C₃₇	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₁×C₃₇	—	5	1	no	—
7	C₃₇×ΓL(1,𝔽₁₆)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₅×C₁₁×C₃₇	Dic₃	1	1	yes	characteristic, complemented, Hall
9	C₄×C₁₁×C₃₇	—	15	1	no	Hall
10	C₃×C₁₁×C₃₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
11	C₃×D₅×C₃₇	C₂×C₁₁	1	1	yes	characteristic
12	C₂×C₁₁×C₃₇	—	5	1	no	—
13	AGL(1,𝔽₅)×C₃₇	—	3	1	no	Hall
14	C₁₁×ΓL(1,𝔽₁₆)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₃×C₅×C₃₇	C₄×C₁₁	1	1	yes	characteristic, complemented, Hall
16	Dic₃×C₃₇	—	5	1	no	Hall
17	C₁₁×C₃₇	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Hall
18	D₅×C₃₇	D₃×C₁₁	1	1	yes	characteristic
19	C₃×D₅×C₁₁	C₂×C₃₇	1	1	yes	characteristic
20	C₂×C₃×C₃₇	—	5	1	no	—
21	C₁₁×AGL(1,𝔽₅)	—	3	1	no	Hall
22	C₅×C₃₇	C₁₁×Dic₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅×C₁₁	C₄×C₃₇	1	1	yes	characteristic, complemented, Hall
24	C₄×C₃₇	—	15	1	no	Hall
25	C₁₁×Dic₃	—	5	1	no	Hall
26	C₃×C₃₇	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
27	D₅×C₁₁	D₃×C₃₇	1	1	yes	characteristic
28	C₂×C₃₇	—	5	1	no	—
29	C₂×C₃×C₁₁	—	5	1	no	—
30	ΓL(1,𝔽₁₆)	C₁₁×C₃₇	1	1	yes	characteristic, direct product factor, Hall
31	C₅×C₁₁	Dic₃×C₃₇	1	1	yes	characteristic, complemented, Hall
32	C₄×C₁₁	—	15	1	no	Hall
33	C₃₇	C₁₁×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
34	C₃×C₁₁	AGL(1,𝔽₅)×C₃₇	1	1	yes	characteristic, complemented, Hall
35	C₃×D₅	C₂×C₁₁×C₃₇	1	1	yes	characteristic
36	C₂×C₁₁	—	5	1	no	—
37	AGL(1,𝔽₅)	—	3	1	no	Hall
38	C₃×C₅	C₄×C₁₁×C₃₇	1	1	yes	characteristic, complemented, Hall
39	Dic₃	—	5	1	no	Hall
40	C₁₁	C₃₇×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
41	D₅	D₃×C₁₁×C₃₇	1	1	yes	characteristic
42	C₂×C₃	—	5	1	no	—
43	C₅	C₁₁×Dic₃×C₃₇	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	15	1	no	Sylow
45	C₃	C₁₁×AGL(1,𝔽₅)×C₃₇	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁×C₃₇	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₁₁×C₃₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₁×C₃₇	C₄
prime radical	C₃×D₅×C₁₁×C₃₇	C₂
cosocle	C₃×D₅	C₂×C₁₁×C₃₇
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₁₁×C₃₇

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×C₁₁×C₃₇	C₄	semidirect
7. C₃₇×ΓL(1,𝔽₁₆)	C₁₁	direct (N × H)
8. C₅×C₁₁×C₃₇	Dic₃	semidirect
10. C₃×C₁₁×C₃₇	AGL(1,𝔽₅)	semidirect
14. C₁₁×ΓL(1,𝔽₁₆)	C₃₇	direct (N × H)
15. C₃×C₅×C₃₇	C₄×C₁₁	semidirect
17. C₁₁×C₃₇	ΓL(1,𝔽₁₆)	direct (N × H)
22. C₅×C₃₇	C₁₁×Dic₃	semidirect
23. C₃×C₅×C₁₁	C₄×C₃₇	semidirect
26. C₃×C₃₇	C₁₁×AGL(1,𝔽₅)	semidirect
30. ΓL(1,𝔽₁₆)	C₁₁×C₃₇	direct (N × H)
31. C₅×C₁₁	Dic₃×C₃₇	semidirect
33. C₃₇	C₁₁×ΓL(1,𝔽₁₆)	direct (N × H)
34. C₃×C₁₁	AGL(1,𝔽₅)×C₃₇	semidirect
38. C₃×C₅	C₄×C₁₁×C₃₇	semidirect
40. C₁₁	C₃₇×ΓL(1,𝔽₁₆)	direct (N × H)
43. C₅	C₁₁×Dic₃×C₃₇	semidirect
45. C₃	C₁₁×AGL(1,𝔽₅)×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×C₁₁×C₃₇	2
C₁₁×AGL(1,𝔽₅)×C₃₇	3
C₁₁×Dic₃×C₃₇	5
C₃₇×ΓL(1,𝔽₁₆)	11
C₁₁×ΓL(1,𝔽₁₆)	37

Subgroups — C11×C37×ΓL(1,𝔽16) #37567598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×C₃₇×ΓL(1,𝔽₁₆) #37567598