Subgroups — C₂×C₁₀₉×AGL(1,𝔽₁₁) #37561844

Order 23980.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×AGL(1,𝔽₁₁)	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₁₁⋊C₅×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁⋊C₅×C₁₀₉	C₂²	1	1	yes	characteristic, complemented, Hall
4	C₂×D₁₁×C₁₀₉	C₅	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₁₁×C₁₀₉	C₂×C₅	1	1	yes	characteristic, complemented
6	D₁₁×C₁₀₉	C₂×C₅	1	2	yes	complemented
7	C₂²×C₅×C₁₀₉	—	11	1	no	maximal, Hall
8	C₁₁×C₁₀₉	C₂²×C₅	1	1	yes	characteristic, complemented, Hall
9	C₂×C₅×C₁₀₉	—	11	2	no	—
10	C₂×C₅×C₁₀₉	—	11	1	no	—
11	C₅×C₁₀₉	—	11	1	no	Hall
12	C₂²×C₁₀₉	—	11	1	no	Hall
13	C₂×AGL(1,𝔽₁₁)	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₁₀₉	—	11	2	no	—
15	C₂×C₁₀₉	AGL(1,𝔽₁₁)	1	1	yes	characteristic, direct product factor
16	C₂×C₁₁⋊C₅	C₂×C₁₀₉	1	1	yes	characteristic, complemented
17	AGL(1,𝔽₁₁)	C₂×C₁₀₉	1	2	yes	direct product factor
18	C₁₀₉	C₂×AGL(1,𝔽₁₁)	1	1	yes	characteristic, direct product factor, Sylow
19	C₁₁⋊C₅	C₂²×C₁₀₉	1	1	yes	characteristic, complemented, Hall
20	C₂×D₁₁	C₅×C₁₀₉	1	1	yes	characteristic, complemented, Hall
21	C₂×C₁₁	C₂×C₅×C₁₀₉	1	1	yes	characteristic, complemented
22	D₁₁	C₂×C₅×C₁₀₉	1	2	yes	complemented
23	C₂²×C₅	—	11	1	no	Hall
24	C₁₁	C₂²×C₅×C₁₀₉	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₅	—	11	2	no	—
26	C₂×C₅	—	11	1	no	—
27	C₅	—	11	1	no	Sylow
28	C₂²	—	11	1	no	Sylow
29	C₂	—	11	2	no	—
30	C₂	C₁₀₉×AGL(1,𝔽₁₁)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₁₀₉	AGL(1,𝔽₁₁)
commutator, cosocle, hypocenter	C₁₁	C₂²×C₅×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₁₀₉	C₂×C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂×C₁₀₉

Complements

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

N	H ≅ G/N	type
1. C₁₀₉×AGL(1,𝔽₁₁)	C₂	direct (N × H)
2. C₂×C₁₁⋊C₅×C₁₀₉	C₂	semidirect
3. C₁₁⋊C₅×C₁₀₉	C₂²	semidirect
4. C₂×D₁₁×C₁₀₉	C₅	semidirect
5. C₂×C₁₁×C₁₀₉	C₂×C₅	semidirect
6. D₁₁×C₁₀₉	C₂×C₅	semidirect
8. C₁₁×C₁₀₉	C₂²×C₅	semidirect
13. C₂×AGL(1,𝔽₁₁)	C₁₀₉	direct (N × H)
15. C₂×C₁₀₉	AGL(1,𝔽₁₁)	direct (N × H)
16. C₂×C₁₁⋊C₅	C₂×C₁₀₉	semidirect
17. AGL(1,𝔽₁₁)	C₂×C₁₀₉	direct (N × H)
18. C₁₀₉	C₂×AGL(1,𝔽₁₁)	direct (N × H)
19. C₁₁⋊C₅	C₂²×C₁₀₉	semidirect
20. C₂×D₁₁	C₅×C₁₀₉	semidirect
21. C₂×C₁₁	C₂×C₅×C₁₀₉	semidirect
22. D₁₁	C₂×C₅×C₁₀₉	semidirect
24. C₁₁	C₂²×C₅×C₁₀₉	semidirect
30. C₂	C₁₀₉×AGL(1,𝔽₁₁)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×AGL(1,𝔽₁₁)	2
C₂×C₁₁⋊C₅×C₁₀₉	2
C₂×D₁₁×C₁₀₉	5
C₂²×C₅×C₁₀₉	11
C₂×AGL(1,𝔽₁₁)	109

Subgroups — C2×C109×AGL(1,𝔽11) #37561844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₉×AGL(1,𝔽₁₁) #37561844