Subgroups — C₁₉×C₂₉×AGL(1,𝔽₇) #37529034

Order 23142.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×ΓL(1,𝔽₈)×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₇×C₁₉×C₂₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₉×C₂₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₁₉×C₂₉	—	7	1	no	maximal, Hall
5	C₃×C₁₉×C₂₉	—	7	1	no	Hall
6	C₂₉×AGL(1,𝔽₇)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₁₉×C₂₉	—	7	1	no	Hall
8	C₁₉×AGL(1,𝔽₇)	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
9	ΓL(1,𝔽₈)×C₂₉	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
10	C₁₉×C₂₉	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Hall
11	D₇×C₂₉	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
12	C₁₉×ΓL(1,𝔽₈)	C₂×C₂₉	1	1	yes	characteristic, complemented, Hall
13	D₇×C₁₉	C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
14	C₇×C₂₉	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₂₉	—	7	1	no	Hall
16	C₇×C₁₉	C₂×C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₁₉	—	7	1	no	Hall
18	C₃×C₂₉	—	7	1	no	Hall
19	C₂×C₂₉	—	7	1	no	Hall
20	C₃×C₁₉	—	7	1	no	Hall
21	AGL(1,𝔽₇)	C₁₉×C₂₉	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₁₉	—	7	1	no	Hall
23	C₂₉	C₁₉×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
24	ΓL(1,𝔽₈)	C₂×C₁₉×C₂₉	1	1	yes	characteristic, complemented, Hall
25	C₁₉	C₂₉×AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor, Sylow
26	D₇	C₃×C₁₉×C₂₉	1	1	yes	characteristic, complemented, Hall
27	C₇	C₂×C₃×C₁₉×C₂₉	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	7	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₉×C₂₉	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₉×C₂₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₉×C₂₉	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₉×C₂₉

Complements

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

N	H ≅ G/N	type
1. C₁₉×ΓL(1,𝔽₈)×C₂₉	C₂	semidirect
2. D₇×C₁₉×C₂₉	C₃	semidirect
3. C₇×C₁₉×C₂₉	C₂×C₃	semidirect
6. C₂₉×AGL(1,𝔽₇)	C₁₉	direct (N × H)
8. C₁₉×AGL(1,𝔽₇)	C₂₉	direct (N × H)
9. ΓL(1,𝔽₈)×C₂₉	C₂×C₁₉	semidirect
10. C₁₉×C₂₉	AGL(1,𝔽₇)	direct (N × H)
11. D₇×C₂₉	C₃×C₁₉	semidirect
12. C₁₉×ΓL(1,𝔽₈)	C₂×C₂₉	semidirect
13. D₇×C₁₉	C₃×C₂₉	semidirect
14. C₇×C₂₉	C₂×C₃×C₁₉	semidirect
16. C₇×C₁₉	C₂×C₃×C₂₉	semidirect
21. AGL(1,𝔽₇)	C₁₉×C₂₉	direct (N × H)
23. C₂₉	C₁₉×AGL(1,𝔽₇)	direct (N × H)
24. ΓL(1,𝔽₈)	C₂×C₁₉×C₂₉	semidirect
25. C₁₉	C₂₉×AGL(1,𝔽₇)	direct (N × H)
26. D₇	C₃×C₁₉×C₂₉	semidirect
27. C₇	C₂×C₃×C₁₉×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×ΓL(1,𝔽₈)×C₂₉	2
D₇×C₁₉×C₂₉	3
C₂×C₃×C₁₉×C₂₉	7
C₂₉×AGL(1,𝔽₇)	19
C₁₉×AGL(1,𝔽₇)	29

Subgroups — C19×C29×AGL(1,𝔽7) #37529034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×C₂₉×AGL(1,𝔽₇) #37529034