Subgroups — C₁₉×ΓL(1,𝔽₈)×D₁₇ #27214501

Order 13566.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₉×D₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₇×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×C₁₉×D₁₇	—	7	1	no	maximal, Hall
5	C₃×C₁₇×C₁₉	—	7	1	no	Hall
6	C₂×C₁₉×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
7	ΓL(1,𝔽₈)×D₁₇	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₉×D₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
9	C₁₉×ΓL(1,𝔽₈)	D₁₇	1	1	yes	characteristic, direct product factor, Hall
10	C₁₇×ΓL(1,𝔽₈)	C₂×C₁₉	1	1	yes	characteristic, complemented, Hall
11	C₁₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
12	C₂×C₇×C₁₉	—	17	1	no	Hall
13	C₇×D₁₇	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
14	C₇×C₁₉	C₃×D₁₇	1	1	yes	characteristic, complemented, Hall
15	C₇×C₁₇	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
16	C₂×C₃×C₁₉	—	119	1	no	Hall
17	C₃×D₁₇	—	7	1	no	Hall
18	C₃×C₁₉	—	7	1	no	Hall
19	C₃×C₁₇	—	7	1	no	Hall
20	C₂×ΓL(1,𝔽₈)	—	17	1	no	Hall
21	C₂×C₁₉	—	17	1	no	Hall
22	D₁₇	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
23	ΓL(1,𝔽₈)	C₁₉×D₁₇	1	1	yes	characteristic, direct product factor, Hall
24	C₁₉	ΓL(1,𝔽₈)×D₁₇	1	1	yes	characteristic, direct product factor, Sylow
25	C₁₇	C₂×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₇	—	17	1	no	Hall
27	C₇	C₃×C₁₉×D₁₇	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	119	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	—	17	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₉	ΓL(1,𝔽₈)×D₁₇
commutator, cosocle, hypocenter	C₇×C₁₇	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₁₉	C₂×C₃
LCS	G ▷ C₇×C₁₇
UCS	C₁ ◁ C₁₉

Complements

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

N	H ≅ G/N	type
1. C₁₇×C₁₉×ΓL(1,𝔽₈)	C₂	semidirect
2. C₇×C₁₉×D₁₇	C₃	semidirect
3. C₇×C₁₇×C₁₉	C₂×C₃	semidirect
7. ΓL(1,𝔽₈)×D₁₇	C₁₉	direct (N × H)
8. C₁₉×D₁₇	ΓL(1,𝔽₈)	direct (N × H)
9. C₁₉×ΓL(1,𝔽₈)	D₁₇	direct (N × H)
10. C₁₇×ΓL(1,𝔽₈)	C₂×C₁₉	semidirect
11. C₁₇×C₁₉	C₂×ΓL(1,𝔽₈)	semidirect
13. C₇×D₁₇	C₃×C₁₉	semidirect
14. C₇×C₁₉	C₃×D₁₇	semidirect
15. C₇×C₁₇	C₂×C₃×C₁₉	semidirect
22. D₁₇	C₁₉×ΓL(1,𝔽₈)	direct (N × H)
23. ΓL(1,𝔽₈)	C₁₉×D₁₇	direct (N × H)
24. C₁₉	ΓL(1,𝔽₈)×D₁₇	direct (N × H)
25. C₁₇	C₂×C₁₉×ΓL(1,𝔽₈)	semidirect
27. C₇	C₃×C₁₉×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×C₁₉×ΓL(1,𝔽₈)	2
C₇×C₁₉×D₁₇	3
C₃×C₁₉×D₁₇	7
C₂×C₁₉×ΓL(1,𝔽₈)	17
ΓL(1,𝔽₈)×D₁₇	19

Subgroups — C19×ΓL(1,𝔽8)×D17 #27214501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×ΓL(1,𝔽₈)×D₁₇ #27214501