Finite Group Databasebeta

← C₁₃×ΓL(1,𝔽₈)×C₁₁₃ #44524747

Subgroups — C₁₃×ΓL(1,𝔽₈)×C₁₁₃ #44524747

Order 30849.

16 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₃×C₁₁₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₁₃×C₁₁₃	—	7	1	no	maximal, Hall
3	ΓL(1,𝔽₈)×C₁₁₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₁₃×C₁₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
5	C₇×C₁₁₃	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₁₃	—	7	1	no	Hall
7	C₁₃×ΓL(1,𝔽₈)	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₁₁₃	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
9	C₇×C₁₃	C₃×C₁₁₃	1	1	yes	characteristic, complemented, Hall
10	C₃×C₁₃	—	7	1	no	Hall
11	ΓL(1,𝔽₈)	C₁₃×C₁₁₃	1	1	yes	characteristic, direct product factor, Hall
12	C₁₃	ΓL(1,𝔽₈)×C₁₁₃	1	1	yes	characteristic, direct product factor, Sylow
13	C₇	C₃×C₁₃×C₁₁₃	1	1	yes	characteristic, complemented, Sylow
14	C₃	—	7	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃×C₁₁₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₁₃×C₁₁₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₁₁₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₁₃×C₁₁₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₃×C₁₁₃	C₃	semidirect
3. ΓL(1,𝔽₈)×C₁₁₃	C₁₃	direct (N × H)
4. C₁₃×C₁₁₃	ΓL(1,𝔽₈)	direct (N × H)
5. C₇×C₁₁₃	C₃×C₁₃	semidirect
7. C₁₃×ΓL(1,𝔽₈)	C₁₁₃	direct (N × H)
8. C₁₁₃	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
9. C₇×C₁₃	C₃×C₁₁₃	semidirect
11. ΓL(1,𝔽₈)	C₁₃×C₁₁₃	direct (N × H)
12. C₁₃	ΓL(1,𝔽₈)×C₁₁₃	direct (N × H)
13. C₇	C₃×C₁₃×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₁₃×C₁₁₃	3
C₃×C₁₃×C₁₁₃	7
ΓL(1,𝔽₈)×C₁₁₃	13
C₁₃×ΓL(1,𝔽₈)	113