Finite Group Databasebeta

← C₂×C₅×ΓL(1,𝔽₈) #6145

Subgroups — C₂×C₅×ΓL(1,𝔽₈) #6145

Order 210.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₅×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×ΓL(1,𝔽₈)	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented, Hall
5	C₂×C₃×C₅	—	7	1	no	maximal, Hall
6	ΓL(1,𝔽₈)	C₂×C₅	1	1	yes	characteristic, direct product factor, Hall
7	C₃×C₅	—	7	1	no	Hall
8	C₂×C₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
9	C₂×C₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₃	—	7	1	no	Hall
12	C₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
13	C₃	—	7	1	no	Sylow
14	C₂	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₅	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₅×C₇	C₃	semidirect
3. C₂×ΓL(1,𝔽₈)	C₅	direct (N × H)
4. C₅×C₇	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)	C₂×C₅	direct (N × H)
8. C₂×C₇	C₃×C₅	semidirect
9. C₂×C₅	ΓL(1,𝔽₈)	direct (N × H)
10. C₇	C₂×C₃×C₅	semidirect
12. C₅	C₂×ΓL(1,𝔽₈)	direct (N × H)
14. C₂	C₅×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×ΓL(1,𝔽₈)	2
C₂×C₅×C₇	3
C₂×ΓL(1,𝔽₈)	5
C₂×C₃×C₅	7