Finite Group Databasebeta

← C₃×ΓL(1,𝔽₈)×C₂₅₇ #30633716

Subgroups — C₃×ΓL(1,𝔽₈)×C₂₅₇ #30633716

Order 16191.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₂₅₇	C₃	1	1	yes	maximal, characteristic, complemented
2	ΓL(1,𝔽₈)×C₂₅₇	C₃	1	3	yes	maximal, direct product factor
3	C₃²×C₂₅₇	—	7	1	no	maximal, Hall
4	C₇×C₂₅₇	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₂₅₇	—	7	3	no	—
6	C₃×C₂₅₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
7	C₂₅₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×ΓL(1,𝔽₈)	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₇	C₃×C₂₅₇	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)	C₃×C₂₅₇	1	3	yes	direct product factor
11	C₃²	—	7	1	no	Sylow
12	C₇	C₃²×C₂₅₇	1	1	yes	characteristic, complemented, Sylow
13	C₃	—	7	3	no	—
14	C₃	ΓL(1,𝔽₈)×C₂₅₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₂₅₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃²×C₂₅₇
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₂₅₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₂₅₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×C₂₅₇	C₃	semidirect
2. ΓL(1,𝔽₈)×C₂₅₇	C₃	direct (N × H)
4. C₇×C₂₅₇	C₃²	semidirect
6. C₃×C₂₅₇	ΓL(1,𝔽₈)	direct (N × H)
7. C₂₅₇	C₃×ΓL(1,𝔽₈)	direct (N × H)
8. C₃×ΓL(1,𝔽₈)	C₂₅₇	direct (N × H)
9. C₃×C₇	C₃×C₂₅₇	semidirect
10. ΓL(1,𝔽₈)	C₃×C₂₅₇	direct (N × H)
12. C₇	C₃²×C₂₅₇	semidirect
14. C₃	ΓL(1,𝔽₈)×C₂₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×C₂₅₇	3
ΓL(1,𝔽₈)×C₂₅₇	3
C₃²×C₂₅₇	7
C₃×ΓL(1,𝔽₈)	257