Finite Group Databasebeta

← C2×C11×ΓL(1,𝔽8)×D13 #26046521

Subgroups — C2×C11×ΓL(1,𝔽8)×D13 #26046521

Order 12012.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C11×ΓL(1,𝔽8)×D13C212yesmaximal, direct product factor
2C2×C11×C13×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×C7×C11×D13C311yesmaximal, characteristic, complemented, Hall
4C11×C13×ΓL(1,𝔽8)C2211yescharacteristic, complemented, Hall
5C7×C11×D13C2×C312yescomplemented
6C2×C7×C11×C13C2×C311yescharacteristic, complemented
7C2×C3×C11×D13—71nomaximal, Hall
8C2×ΓL(1,𝔽8)×D13C1111yesmaximal, characteristic, direct product factor, Hall
9C7×C11×C13C22×C311yescharacteristic, complemented, Hall
10C22×C11×ΓL(1,𝔽8)—131nomaximal, Hall
11C2×C3×C11×C13—71no—
12C3×C11×D13—72no—
13C2×C11×D13ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
14C2×C13×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented
15ΓL(1,𝔽8)×D13C2×C1112yesdirect product factor
16C2×C11×ΓL(1,𝔽8)—132no—
17C2×C11×ΓL(1,𝔽8)D1311yescharacteristic, direct product factor
18C3×C11×C13—71noHall
19C2×C7×D13C3×C1111yescharacteristic, complemented, Hall
20C22×C7×C11—131noHall
21C2×C11×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
22C11×D13C2×ΓL(1,𝔽8)12yesdirect product factor
23C13×ΓL(1,𝔽8)C22×C1111yescharacteristic, complemented, Hall
24C11×ΓL(1,𝔽8)C2×D1311yescharacteristic, direct product factor, Hall
25C2×C7×C13C2×C3×C1111yescharacteristic, complemented
26C7×D13C2×C3×C1112yescomplemented
27C2×C3×D13—71noHall
28C2×C7×C11—132no—
29C2×C7×C11C3×D1311yescharacteristic, complemented
30C11×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
31C22×C3×C11—911noHall
32C7×C13C22×C3×C1111yescharacteristic, complemented, Hall
33C22×ΓL(1,𝔽8)—131noHall
34C2×C3×C13—71no—
35C3×D13—72no—
36C7×C11C2×C3×D1311yescharacteristic, complemented, Hall
37C2×C3×C11—71no—
38C2×C3×C11—912no—
39C2×D13C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
40C22×C11—131noHall
41C2×ΓL(1,𝔽8)—132no—
42C2×ΓL(1,𝔽8)C11×D1311yescharacteristic, direct product factor
43C3×C13—71noHall
44C3×C11—71noHall
45C22×C7—131noHall
46C2×C13C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
47D13C2×C11×ΓL(1,𝔽8)12yesdirect product factor
48C2×C11—132no—
49C2×C11ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
50ΓL(1,𝔽8)C2×C11×D1311yescharacteristic, direct product factor, Hall
51C2×C7—132no—
52C2×C7C3×C11×D1311yescharacteristic, complemented
53C13C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
54C22×C3—911noHall
55C11C2×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor, Sylow
56C7C2×C3×C11×D1311yescharacteristic, complemented, Sylow
57C2×C3—71no—
58C2×C3—912no—
59C22—131noSylow
60C3—71noSylow
61C2—132no—
62C2C11×ΓL(1,𝔽8)×D1311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)×D13
commutator, cosocle, hypocenterC7×C13C22×C3×C11
FrattiniC1G
Fitting, socleC2×C7×C11×C13C2×C3
LCSG ▷ C7×C13
UCSC1 ◁ C2×C11

Complements

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

NH ≅ G/Ntype
1. C11×ΓL(1,𝔽8)×D13C2direct (N × H)
2. C2×C11×C13×ΓL(1,𝔽8)C2semidirect
3. C2×C7×C11×D13C3semidirect
4. C11×C13×ΓL(1,𝔽8)C22semidirect
5. C7×C11×D13C2×C3semidirect
6. C2×C7×C11×C13C2×C3semidirect
8. C2×ΓL(1,𝔽8)×D13C11direct (N × H)
9. C7×C11×C13C22×C3semidirect
13. C2×C11×D13ΓL(1,𝔽8)direct (N × H)
14. C2×C13×ΓL(1,𝔽8)C2×C11semidirect
15. ΓL(1,𝔽8)×D13C2×C11direct (N × H)
17. C2×C11×ΓL(1,𝔽8)D13direct (N × H)
19. C2×C7×D13C3×C11semidirect
21. C2×C11×C13C2×ΓL(1,𝔽8)semidirect
22. C11×D13C2×ΓL(1,𝔽8)direct (N × H)
23. C13×ΓL(1,𝔽8)C22×C11semidirect
24. C11×ΓL(1,𝔽8)C2×D13direct (N × H)
25. C2×C7×C13C2×C3×C11semidirect
26. C7×D13C2×C3×C11semidirect
29. C2×C7×C11C3×D13semidirect
30. C11×C13C22×ΓL(1,𝔽8)semidirect
32. C7×C13C22×C3×C11semidirect
36. C7×C11C2×C3×D13semidirect
39. C2×D13C11×ΓL(1,𝔽8)direct (N × H)
42. C2×ΓL(1,𝔽8)C11×D13direct (N × H)
46. C2×C13C2×C11×ΓL(1,𝔽8)semidirect
47. D13C2×C11×ΓL(1,𝔽8)direct (N × H)
49. C2×C11ΓL(1,𝔽8)×D13direct (N × H)
50. ΓL(1,𝔽8)C2×C11×D13direct (N × H)
52. C2×C7C3×C11×D13semidirect
53. C13C22×C11×ΓL(1,𝔽8)semidirect
55. C11C2×ΓL(1,𝔽8)×D13direct (N × H)
56. C7C2×C3×C11×D13semidirect
62. C2C11×ΓL(1,𝔽8)×D13direct (N × H)

Maximal subgroups

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

subgroupindex
C11×ΓL(1,𝔽8)×D132
C2×C11×C13×ΓL(1,𝔽8)2
C2×C7×C11×D133
C2×C3×C11×D137
C2×ΓL(1,𝔽8)×D1311
C22×C11×ΓL(1,𝔽8)13