Subgroups — C₁₁×AGL(1,𝔽₅)×ΓL(1,𝔽₃₂) #46802068

Order 34100.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₁₁×ΓL(1,𝔽₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₁×ΓL(1,𝔽₃₂)	C₄	1	1	yes	characteristic, complemented, Hall
3	C₁₁×AGL(1,𝔽₅)×C₃₁	C₅	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₁×ΓL(1,𝔽₃₂)	—	5	1	no	maximal
5	D₅×C₁₁×C₃₁	C₂×C₅	1	1	yes	characteristic
6	C₂×C₁₁×ΓL(1,𝔽₃₂)	—	5	1	no	—
7	AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₅×C₁₁×C₃₁	C₄×C₅	1	1	yes	characteristic, complemented
9	C₁₁×ΓL(1,𝔽₃₂)	—	4	1	no	—
10	C₁₁×ΓL(1,𝔽₃₂)	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
11	D₅×ΓL(1,𝔽₃₂)	C₂×C₁₁	1	1	yes	characteristic
12	C₄×C₁₁×C₃₁	—	5	1	no	Hall
13	C₅×C₁₁×AGL(1,𝔽₅)	—	31	1	no	maximal, Hall
14	C₅×ΓL(1,𝔽₃₂)	C₄×C₁₁	1	1	yes	characteristic, complemented, Hall
15	C₂×C₁₁×C₃₁	—	5	1	no	—
16	AGL(1,𝔽₅)×C₃₁	C₅×C₁₁	1	1	yes	characteristic, complemented
17	C₄×ΓL(1,𝔽₃₂)	—	5	1	no	—
18	C₅×D₅×C₁₁	—	31	1	no	—
19	C₁₁×C₃₁	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
20	D₅×C₃₁	C₂×C₅×C₁₁	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₃₂)	—	5	1	no	—
22	C₅²×C₁₁	—	31	1	no	Hall
23	C₁₁×AGL(1,𝔽₅)	ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
24	C₄×C₅×C₁₁	—	155	1	no	—
25	C₅×C₃₁	C₄×C₅×C₁₁	1	1	yes	characteristic, complemented
26	ΓL(1,𝔽₃₂)	—	4	1	no	—
27	ΓL(1,𝔽₃₂)	C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
28	C₄×C₃₁	—	5	1	no	Hall
29	C₂×C₅×C₁₁	—	155	1	no	—
30	D₅×C₁₁	C₂×ΓL(1,𝔽₃₂)	1	1	yes	characteristic
31	C₅×AGL(1,𝔽₅)	—	31	1	no	Hall
32	C₂×C₃₁	—	5	1	no	—
33	C₅×C₁₁	—	31	1	no	—
34	C₅×C₁₁	—	124	1	no	—
35	C₅×C₁₁	C₄×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented
36	C₅×D₅	—	31	1	no	—
37	C₄×C₁₁	—	5	1	no	Hall
38	C₃₁	C₅×C₁₁×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
39	C₅²	—	31	1	no	Sylow
40	C₂×C₁₁	—	5	1	no	—
41	AGL(1,𝔽₅)	C₁₁×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor
42	C₄×C₅	—	155	1	no	—
43	C₁₁	AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₅	—	155	1	no	—
45	D₅	C₂×C₁₁×ΓL(1,𝔽₃₂)	1	1	yes	characteristic
46	C₅	—	31	1	no	—
47	C₅	—	124	1	no	—
48	C₅	C₄×C₁₁×ΓL(1,𝔽₃₂)	1	1	yes	characteristic, complemented
49	C₄	—	5	1	no	Sylow
50	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁	AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)
commutator, hypocenter	C₅×C₃₁	C₄×C₅×C₁₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₃₁	C₄×C₅
prime radical	D₅×C₁₁×ΓL(1,𝔽₃₂)	C₂
cosocle	D₅×C₃₁	C₂×C₅×C₁₁
LCS	G ▷ C₅×C₃₁
UCS	C₁ ◁ C₁₁

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₁×ΓL(1,𝔽₃₂)	C₄	semidirect
3. C₁₁×AGL(1,𝔽₅)×C₃₁	C₅	semidirect
7. AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)	C₁₁	direct (N × H)
8. C₅×C₁₁×C₃₁	C₄×C₅	semidirect
10. C₁₁×ΓL(1,𝔽₃₂)	AGL(1,𝔽₅)	direct (N × H)
14. C₅×ΓL(1,𝔽₃₂)	C₄×C₁₁	semidirect
16. AGL(1,𝔽₅)×C₃₁	C₅×C₁₁	semidirect
19. C₁₁×C₃₁	C₅×AGL(1,𝔽₅)	semidirect
23. C₁₁×AGL(1,𝔽₅)	ΓL(1,𝔽₃₂)	direct (N × H)
25. C₅×C₃₁	C₄×C₅×C₁₁	semidirect
27. ΓL(1,𝔽₃₂)	C₁₁×AGL(1,𝔽₅)	direct (N × H)
35. C₅×C₁₁	C₄×ΓL(1,𝔽₃₂)	semidirect
38. C₃₁	C₅×C₁₁×AGL(1,𝔽₅)	semidirect
41. AGL(1,𝔽₅)	C₁₁×ΓL(1,𝔽₃₂)	direct (N × H)
43. C₁₁	AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)	direct (N × H)
48. C₅	C₄×C₁₁×ΓL(1,𝔽₃₂)	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₁₁×ΓL(1,𝔽₃₂)	2
C₁₁×AGL(1,𝔽₅)×C₃₁	5
C₄×C₁₁×ΓL(1,𝔽₃₂)	5
AGL(1,𝔽₅)×ΓL(1,𝔽₃₂)	11
C₅×C₁₁×AGL(1,𝔽₅)	31

Subgroups — C11×AGL(1,𝔽5)×ΓL(1,𝔽32) #46802068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×AGL(1,𝔽₅)×ΓL(1,𝔽₃₂) #46802068