Finite Group Databasebeta

← C₇×ΓL(1,𝔽₈)×C₃₁ #17986949

Subgroups — C₇×ΓL(1,𝔽₈)×C₃₁ #17986949

Order 4557.

20 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇×C₃₁	—	7	1	no	maximal
3	ΓL(1,𝔽₈)×C₃₁	C₇	1	1	yes	maximal, characteristic, direct product factor
4	C₇×C₃₁	—	3	2	no	—
5	C₇×C₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
6	C₇×C₃₁	C₃×C₇	1	1	yes	characteristic, complemented
7	C₇×ΓL(1,𝔽₈)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₃₁	—	7	1	no	Hall
9	C₇²	C₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
10	C₃₁	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
11	C₃×C₇	—	7	1	no	—
12	ΓL(1,𝔽₈)	C₇×C₃₁	1	1	yes	characteristic, direct product factor
13	C₇	—	3	2	no	—
14	C₇	ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, direct product factor
15	C₇	C₃×C₇×C₃₁	1	1	yes	characteristic, complemented
16	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇×C₃₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₇×C₃₁
Frattini	C₁	G
Fitting, socle	C₇²×C₃₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₇×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₇²×C₃₁	C₃	semidirect
3. ΓL(1,𝔽₈)×C₃₁	C₇	direct (N × H)
5. C₇×C₃₁	ΓL(1,𝔽₈)	direct (N × H)
6. C₇×C₃₁	C₃×C₇	semidirect
7. C₇×ΓL(1,𝔽₈)	C₃₁	direct (N × H)
9. C₇²	C₃×C₃₁	semidirect
10. C₃₁	C₇×ΓL(1,𝔽₈)	direct (N × H)
12. ΓL(1,𝔽₈)	C₇×C₃₁	direct (N × H)
14. C₇	ΓL(1,𝔽₈)×C₃₁	direct (N × H)
15. C₇	C₃×C₇×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₇²×C₃₁	3
C₃×C₇×C₃₁	7
ΓL(1,𝔽₈)×C₃₁	7
C₇×ΓL(1,𝔽₈)	31