Finite Group Databasebeta

← C₂×ΓL(1,𝔽₈)×C₆₄₃ #41008866

Subgroups — C₂×ΓL(1,𝔽₈)×C₆₄₃ #41008866

Order 27006.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₆₄₃	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇×C₆₄₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₆₄₃	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×C₃×C₆₄₃	—	7	1	no	maximal, Hall
5	C₃×C₆₄₃	—	7	1	no	Hall
6	C₂×C₆₄₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
7	C₆₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
8	C₂×ΓL(1,𝔽₈)	C₆₄₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	ΓL(1,𝔽₈)	C₂×C₆₄₃	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₇	C₃×C₆₄₃	1	1	yes	characteristic, complemented, Hall
11	C₇	C₂×C₃×C₆₄₃	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃	—	7	1	no	Hall
13	C₃	—	7	1	no	Sylow
14	C₂	ΓL(1,𝔽₈)×C₆₄₃	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₆₄₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₆₄₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₆₄₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₆₄₃

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₆₄₃	C₂	direct (N × H)
2. C₂×C₇×C₆₄₃	C₃	semidirect
3. C₇×C₆₄₃	C₂×C₃	semidirect
6. C₂×C₆₄₃	ΓL(1,𝔽₈)	direct (N × H)
7. C₆₄₃	C₂×ΓL(1,𝔽₈)	direct (N × H)
8. C₂×ΓL(1,𝔽₈)	C₆₄₃	direct (N × H)
9. ΓL(1,𝔽₈)	C₂×C₆₄₃	direct (N × H)
10. C₂×C₇	C₃×C₆₄₃	semidirect
11. C₇	C₂×C₃×C₆₄₃	semidirect
14. C₂	ΓL(1,𝔽₈)×C₆₄₃	direct (N × H)

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₆₄₃	2
C₂×C₇×C₆₄₃	3
C₂×C₃×C₆₄₃	7
C₂×ΓL(1,𝔽₈)	643