Finite Group Databasebeta

← C₁₇×ΓL(1,𝔽₈)×C₈₉ #44540978

Subgroups — C₁₇×ΓL(1,𝔽₈)×C₈₉ #44540978

Order 31773.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×C₈₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₁₇×C₈₉	—	7	1	no	maximal, Hall
3	ΓL(1,𝔽₈)×C₈₉	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₁₇×C₈₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
5	C₇×C₈₉	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
6	C₁₇×ΓL(1,𝔽₈)	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₈₉	—	7	1	no	Hall
8	C₇×C₁₇	C₃×C₈₉	1	1	yes	characteristic, complemented, Hall
9	C₈₉	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
10	C₃×C₁₇	—	7	1	no	Hall
11	ΓL(1,𝔽₈)	C₁₇×C₈₉	1	1	yes	characteristic, direct product factor, Hall
12	C₁₇	ΓL(1,𝔽₈)×C₈₉	1	1	yes	characteristic, direct product factor, Sylow
13	C₇	C₃×C₁₇×C₈₉	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₇×C₈₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₁₇×C₈₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₇×C₈₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₇×C₈₉

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×C₈₉	C₃	semidirect
3. ΓL(1,𝔽₈)×C₈₉	C₁₇	direct (N × H)
4. C₁₇×C₈₉	ΓL(1,𝔽₈)	direct (N × H)
5. C₇×C₈₉	C₃×C₁₇	semidirect
6. C₁₇×ΓL(1,𝔽₈)	C₈₉	direct (N × H)
8. C₇×C₁₇	C₃×C₈₉	semidirect
9. C₈₉	C₁₇×ΓL(1,𝔽₈)	direct (N × H)
11. ΓL(1,𝔽₈)	C₁₇×C₈₉	direct (N × H)
12. C₁₇	ΓL(1,𝔽₈)×C₈₉	direct (N × H)
13. C₇	C₃×C₁₇×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₇×C₈₉	3
C₃×C₁₇×C₈₉	7
ΓL(1,𝔽₈)×C₈₉	17
C₁₇×ΓL(1,𝔽₈)	89