Finite Group Databasebeta

← C2×C7×ΓL(1,𝔽8)×C79 #37551163

Subgroups — C2×C7×ΓL(1,𝔽8)×C79 #37551163

Order 23226.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×ΓL(1,𝔽8)×C79C211yesmaximal, characteristic, direct product factor, Hall
2C2×C72×C79C311yesmaximal, characteristic, complemented, Hall
3C72×C79C2×C311yescharacteristic, complemented, Hall
4C2×ΓL(1,𝔽8)×C79C711yesmaximal, characteristic, direct product factor
5C2×C3×C7×C79—71nomaximal
6C3×C7×C79—71no—
7ΓL(1,𝔽8)×C79C2×C711yescharacteristic, direct product factor
8C2×C7×C79—32no—
9C2×C7×C79ΓL(1,𝔽8)11yescharacteristic, direct product factor
10C2×C7×C79C3×C711yescharacteristic, complemented
11C7×C79—32no—
12C7×C79C2×ΓL(1,𝔽8)11yescharacteristic, direct product factor
13C7×C79C2×C3×C711yescharacteristic, complemented
14C2×C3×C79—71noHall
15C2×C7×ΓL(1,𝔽8)C7911yesmaximal, characteristic, direct product factor, Hall
16C3×C79—71noHall
17C2×C79C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
18C7×ΓL(1,𝔽8)C2×C7911yescharacteristic, direct product factor, Hall
19C2×C72C3×C7911yescharacteristic, complemented, Hall
20C79C2×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
21C72C2×C3×C7911yescharacteristic, complemented, Sylow
22C2×C3×C7—71no—
23C2×ΓL(1,𝔽8)C7×C7911yescharacteristic, direct product factor
24C3×C7—71no—
25ΓL(1,𝔽8)C2×C7×C7911yescharacteristic, direct product factor
26C2×C7—32no—
27C2×C7ΓL(1,𝔽8)×C7911yescharacteristic, direct product factor
28C2×C7C3×C7×C7911yescharacteristic, complemented
29C7—32no—
30C7C2×C3×C7×C7911yescharacteristic, complemented
31C7C2×ΓL(1,𝔽8)×C7911yescharacteristic, direct product factor
32C2×C3—71noHall
33C3—71noSylow
34C2C7×ΓL(1,𝔽8)×C7911yescharacteristic, direct product factor, Sylow
subgroupG/H
center, hypercenterC2×C7×C79ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C7×C79
FrattiniC1G
Fitting, socleC2×C72×C79C3
LCSG ▷ C7
UCSC1 ◁ C2×C7×C79

Complements

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

NH ≅ G/Ntype
1. C7×ΓL(1,𝔽8)×C79C2direct (N × H)
2. C2×C72×C79C3semidirect
3. C72×C79C2×C3semidirect
4. C2×ΓL(1,𝔽8)×C79C7direct (N × H)
7. ΓL(1,𝔽8)×C79C2×C7direct (N × H)
9. C2×C7×C79ΓL(1,𝔽8)direct (N × H)
10. C2×C7×C79C3×C7semidirect
12. C7×C79C2×ΓL(1,𝔽8)direct (N × H)
13. C7×C79C2×C3×C7semidirect
15. C2×C7×ΓL(1,𝔽8)C79direct (N × H)
17. C2×C79C7×ΓL(1,𝔽8)direct (N × H)
18. C7×ΓL(1,𝔽8)C2×C79direct (N × H)
19. C2×C72C3×C79semidirect
20. C79C2×C7×ΓL(1,𝔽8)direct (N × H)
21. C72C2×C3×C79semidirect
23. C2×ΓL(1,𝔽8)C7×C79direct (N × H)
25. ΓL(1,𝔽8)C2×C7×C79direct (N × H)
27. C2×C7ΓL(1,𝔽8)×C79direct (N × H)
28. C2×C7C3×C7×C79semidirect
30. C7C2×C3×C7×C79semidirect
31. C7C2×ΓL(1,𝔽8)×C79direct (N × H)
34. C2C7×ΓL(1,𝔽8)×C79direct (N × H)

Maximal subgroups

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

subgroupindex
C7×ΓL(1,𝔽8)×C792
C2×C72×C793
C2×ΓL(1,𝔽8)×C797
C2×C3×C7×C797
C2×C7×ΓL(1,𝔽8)79