Subgroups — ΓL(1,𝔽₈)×D₁₉×C₄₇ #49175809

Order 37506.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×ΓL(1,𝔽₈)×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×D₁₉×C₄₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₉×C₄₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×D₁₉×C₄₇	—	7	1	no	maximal, Hall
5	C₃×C₁₉×C₄₇	—	7	1	no	Hall
6	C₂×ΓL(1,𝔽₈)×C₄₇	—	19	1	no	maximal, Hall
7	D₁₉×C₄₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	ΓL(1,𝔽₈)×C₄₇	D₁₉	1	1	yes	characteristic, direct product factor, Hall
9	C₁₉×C₄₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)×D₁₉	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₇×C₄₇	—	19	1	no	Hall
12	C₁₉×ΓL(1,𝔽₈)	C₂×C₄₇	1	1	yes	characteristic, complemented, Hall
13	C₇×C₄₇	C₃×D₁₉	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₄₇	—	133	1	no	Hall
15	C₇×D₁₉	C₃×C₄₇	1	1	yes	characteristic, complemented, Hall
16	C₃×C₄₇	—	7	1	no	Hall
17	C₇×C₁₉	C₂×C₃×C₄₇	1	1	yes	characteristic, complemented, Hall
18	C₃×D₁₉	—	7	1	no	Hall
19	C₂×C₄₇	—	19	1	no	Hall
20	C₃×C₁₉	—	7	1	no	Hall
21	C₄₇	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×ΓL(1,𝔽₈)	—	19	1	no	Hall
23	D₁₉	ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, direct product factor, Hall
24	ΓL(1,𝔽₈)	D₁₉×C₄₇	1	1	yes	characteristic, direct product factor, Hall
25	C₁₉	C₂×ΓL(1,𝔽₈)×C₄₇	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₇	—	19	1	no	Hall
27	C₇	C₃×D₁₉×C₄₇	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	C₄₇	ΓL(1,𝔽₈)×D₁₉
commutator, cosocle, hypocenter	C₇×C₁₉	C₂×C₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₉×C₄₇	C₂×C₃
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₄₇

Complements

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

N	H ≅ G/N	type
1. C₁₉×ΓL(1,𝔽₈)×C₄₇	C₂	semidirect
2. C₇×D₁₉×C₄₇	C₃	semidirect
3. C₇×C₁₉×C₄₇	C₂×C₃	semidirect
7. D₁₉×C₄₇	ΓL(1,𝔽₈)	direct (N × H)
8. ΓL(1,𝔽₈)×C₄₇	D₁₉	direct (N × H)
9. C₁₉×C₄₇	C₂×ΓL(1,𝔽₈)	semidirect
10. ΓL(1,𝔽₈)×D₁₉	C₄₇	direct (N × H)
12. C₁₉×ΓL(1,𝔽₈)	C₂×C₄₇	semidirect
13. C₇×C₄₇	C₃×D₁₉	semidirect
15. C₇×D₁₉	C₃×C₄₇	semidirect
17. C₇×C₁₉	C₂×C₃×C₄₇	semidirect
21. C₄₇	ΓL(1,𝔽₈)×D₁₉	direct (N × H)
23. D₁₉	ΓL(1,𝔽₈)×C₄₇	direct (N × H)
24. ΓL(1,𝔽₈)	D₁₉×C₄₇	direct (N × H)
25. C₁₉	C₂×ΓL(1,𝔽₈)×C₄₇	semidirect
27. C₇	C₃×D₁₉×C₄₇	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)×C₄₇	2
C₇×D₁₉×C₄₇	3
C₃×D₁₉×C₄₇	7
C₂×ΓL(1,𝔽₈)×C₄₇	19
ΓL(1,𝔽₈)×D₁₉	47

Subgroups — ΓL(1,𝔽8)×D19×C47 #49175809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₁₉×C₄₇ #49175809