Finite Group Databasebeta

← ΓL(1,𝔽₈)×C₃₁×C₄₃ #43241981

Subgroups — ΓL(1,𝔽₈)×C₃₁×C₄₃ #43241981

Order 27993.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃₁×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₃₁×C₄₃	—	7	1	no	maximal, Hall
3	C₃₁×C₄₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
4	ΓL(1,𝔽₈)×C₄₃	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
5	ΓL(1,𝔽₈)×C₃₁	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₇×C₄₃	C₃×C₃₁	1	1	yes	characteristic, complemented, Hall
7	C₇×C₃₁	C₃×C₄₃	1	1	yes	characteristic, complemented, Hall
8	C₃×C₄₃	—	7	1	no	Hall
9	C₃×C₃₁	—	7	1	no	Hall
10	C₄₃	ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
11	C₃₁	ΓL(1,𝔽₈)×C₄₃	1	1	yes	characteristic, direct product factor, Sylow
12	ΓL(1,𝔽₈)	C₃₁×C₄₃	1	1	yes	characteristic, direct product factor, Hall
13	C₇	C₃×C₃₁×C₄₃	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃₁×C₄₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₃₁×C₄₃
Frattini	C₁	G
Fitting, socle	C₇×C₃₁×C₄₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃₁×C₄₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₃₁×C₄₃	C₃	semidirect
3. C₃₁×C₄₃	ΓL(1,𝔽₈)	direct (N × H)
4. ΓL(1,𝔽₈)×C₄₃	C₃₁	direct (N × H)
5. ΓL(1,𝔽₈)×C₃₁	C₄₃	direct (N × H)
6. C₇×C₄₃	C₃×C₃₁	semidirect
7. C₇×C₃₁	C₃×C₄₃	semidirect
10. C₄₃	ΓL(1,𝔽₈)×C₃₁	direct (N × H)
11. C₃₁	ΓL(1,𝔽₈)×C₄₃	direct (N × H)
12. ΓL(1,𝔽₈)	C₃₁×C₄₃	direct (N × H)
13. C₇	C₃×C₃₁×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃₁×C₄₃	3
C₃×C₃₁×C₄₃	7
ΓL(1,𝔽₈)×C₄₃	31
ΓL(1,𝔽₈)×C₃₁	43