Finite Group Databasebeta

← C2×C7×ΓL(1,𝔽8)×C109 #44545255

Subgroups — C2×C7×ΓL(1,𝔽8)×C109 #44545255

Order 32046.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×ΓL(1,𝔽8)×C109C211yesmaximal, characteristic, direct product factor, Hall
2C2×C72×C109C311yesmaximal, characteristic, complemented, Hall
3C72×C109C2×C311yescharacteristic, complemented, Hall
4C2×ΓL(1,𝔽8)×C109C711yesmaximal, characteristic, direct product factor
5C2×C3×C7×C109—71nomaximal
6C3×C7×C109—71no—
7ΓL(1,𝔽8)×C109C2×C711yescharacteristic, direct product factor
8C2×C7×C109—32no—
9C2×C7×C109ΓL(1,𝔽8)11yescharacteristic, direct product factor
10C2×C7×C109C3×C711yescharacteristic, complemented
11C7×C109—32no—
12C7×C109C2×ΓL(1,𝔽8)11yescharacteristic, direct product factor
13C7×C109C2×C3×C711yescharacteristic, complemented
14C2×C3×C109—71noHall
15C3×C109—71noHall
16C2×C7×ΓL(1,𝔽8)C10911yesmaximal, characteristic, direct product factor, Hall
17C2×C109C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
18C7×ΓL(1,𝔽8)C2×C10911yescharacteristic, direct product factor, Hall
19C109C2×C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Sylow
20C2×C72C3×C10911yescharacteristic, complemented, Hall
21C72C2×C3×C10911yescharacteristic, complemented, Sylow
22C2×C3×C7—71no—
23C2×ΓL(1,𝔽8)C7×C10911yescharacteristic, direct product factor
24C3×C7—71no—
25ΓL(1,𝔽8)C2×C7×C10911yescharacteristic, direct product factor
26C2×C7—32no—
27C2×C7ΓL(1,𝔽8)×C10911yescharacteristic, direct product factor
28C2×C7C3×C7×C10911yescharacteristic, complemented
29C7—32no—
30C7C2×C3×C7×C10911yescharacteristic, complemented
31C7C2×ΓL(1,𝔽8)×C10911yescharacteristic, direct product factor
32C2×C3—71noHall
33C3—71noSylow
34C2C7×ΓL(1,𝔽8)×C10911yescharacteristic, direct product factor, Sylow
subgroupG/H
center, hypercenterC2×C7×C109ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C7×C109
FrattiniC1G
Fitting, socleC2×C72×C109C3
LCSG ▷ C7
UCSC1 ◁ C2×C7×C109

Complements

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

NH ≅ G/Ntype
1. C7×ΓL(1,𝔽8)×C109C2direct (N × H)
2. C2×C72×C109C3semidirect
3. C72×C109C2×C3semidirect
4. C2×ΓL(1,𝔽8)×C109C7direct (N × H)
7. ΓL(1,𝔽8)×C109C2×C7direct (N × H)
9. C2×C7×C109ΓL(1,𝔽8)direct (N × H)
10. C2×C7×C109C3×C7semidirect
12. C7×C109C2×ΓL(1,𝔽8)direct (N × H)
13. C7×C109C2×C3×C7semidirect
16. C2×C7×ΓL(1,𝔽8)C109direct (N × H)
17. C2×C109C7×ΓL(1,𝔽8)direct (N × H)
18. C7×ΓL(1,𝔽8)C2×C109direct (N × H)
19. C109C2×C7×ΓL(1,𝔽8)direct (N × H)
20. C2×C72C3×C109semidirect
21. C72C2×C3×C109semidirect
23. C2×ΓL(1,𝔽8)C7×C109direct (N × H)
25. ΓL(1,𝔽8)C2×C7×C109direct (N × H)
27. C2×C7ΓL(1,𝔽8)×C109direct (N × H)
28. C2×C7C3×C7×C109semidirect
30. C7C2×C3×C7×C109semidirect
31. C7C2×ΓL(1,𝔽8)×C109direct (N × H)
34. C2C7×ΓL(1,𝔽8)×C109direct (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)×C1092
C2×C72×C1093
C2×ΓL(1,𝔽8)×C1097
C2×C3×C7×C1097
C2×C7×ΓL(1,𝔽8)109