Subgroups — ΓL(1,𝔽₈)×C₂₉×D₁₉ #37529026

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	C₇×C₂₉×D₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₉×C₂₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×C₂₉×D₁₉	—	7	1	no	maximal, Hall
5	C₃×C₁₉×C₂₉	—	7	1	no	Hall
6	C₂×ΓL(1,𝔽₈)×C₂₉	—	19	1	no	maximal, Hall
7	C₂₉×D₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	ΓL(1,𝔽₈)×D₁₉	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
9	ΓL(1,𝔽₈)×C₂₉	D₁₉	1	1	yes	characteristic, direct product factor, Hall
10	C₁₉×C₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
11	C₂×C₇×C₂₉	—	19	1	no	Hall
12	C₁₉×ΓL(1,𝔽₈)	C₂×C₂₉	1	1	yes	characteristic, complemented, Hall
13	C₇×D₁₉	C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
14	C₇×C₂₉	C₃×D₁₉	1	1	yes	characteristic, complemented, Hall
15	C₂×C₃×C₂₉	—	133	1	no	Hall
16	C₇×C₁₉	C₂×C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
17	C₃×D₁₉	—	7	1	no	Hall
18	C₃×C₂₉	—	7	1	no	Hall
19	C₂×C₂₉	—	19	1	no	Hall
20	C₃×C₁₉	—	7	1	no	Hall
21	C₂×ΓL(1,𝔽₈)	—	19	1	no	Hall
22	D₁₉	ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, direct product factor, Hall
23	C₂₉	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, direct product factor, Sylow
24	ΓL(1,𝔽₈)	C₂₉×D₁₉	1	1	yes	characteristic, direct product factor, Hall
25	C₁₉	C₂×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₇	—	19	1	no	Hall
27	C₇	C₃×C₂₉×D₁₉	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	133	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	—	19	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₂₉	ΓL(1,𝔽₈)×D₁₉
commutator, cosocle, hypocenter	C₇×C₁₉	C₂×C₃×C₂₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₉×C₂₉	C₂×C₃
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ 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. C₇×C₂₉×D₁₉	C₃	semidirect
3. C₇×C₁₉×C₂₉	C₂×C₃	semidirect
7. C₂₉×D₁₉	ΓL(1,𝔽₈)	direct (N × H)
8. ΓL(1,𝔽₈)×D₁₉	C₂₉	direct (N × H)
9. ΓL(1,𝔽₈)×C₂₉	D₁₉	direct (N × H)
10. C₁₉×C₂₉	C₂×ΓL(1,𝔽₈)	semidirect
12. C₁₉×ΓL(1,𝔽₈)	C₂×C₂₉	semidirect
13. C₇×D₁₉	C₃×C₂₉	semidirect
14. C₇×C₂₉	C₃×D₁₉	semidirect
16. C₇×C₁₉	C₂×C₃×C₂₉	semidirect
22. D₁₉	ΓL(1,𝔽₈)×C₂₉	direct (N × H)
23. C₂₉	ΓL(1,𝔽₈)×D₁₉	direct (N × H)
24. ΓL(1,𝔽₈)	C₂₉×D₁₉	direct (N × H)
25. C₁₉	C₂×ΓL(1,𝔽₈)×C₂₉	semidirect
27. C₇	C₃×C₂₉×D₁₉	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
C₇×C₂₉×D₁₉	3
C₃×C₂₉×D₁₉	7
C₂×ΓL(1,𝔽₈)×C₂₉	19
ΓL(1,𝔽₈)×D₁₉	29

Subgroups — ΓL(1,𝔽8)×C29×D19 #37529026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₉×D₁₉ #37529026