Finite Group Databasebeta

← C11×ΓL(1,𝔽8)×D51 #37555644

Subgroups — C11×ΓL(1,𝔽8)×D51 #37555644

Order 23562.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C11×C17×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented, Hall
2C11×ΓL(1,𝔽8)×D17—31nomaximal
3C7×C11×D51C311yesmaximal, characteristic, complemented
4C11×C17×ΓL(1,𝔽8)—21no—
5C11×C17×ΓL(1,𝔽8)D311yescharacteristic, complemented
6C3×C7×C11×C17C2×C311yescharacteristic, complemented
7C3×C11×D51—71nomaximal, Hall
8C7×C11×D17—31noHall
9ΓL(1,𝔽8)×D51C1111yesmaximal, characteristic, direct product factor, Hall
10C32×C11×C17—71noHall
11D3×C11×ΓL(1,𝔽8)—171nomaximal, Hall
12C7×C11×C17C3×D311yescharacteristic, complemented, Hall
13C11×D51ΓL(1,𝔽8)11yescharacteristic, direct product factor
14C3×C11×D17—211no—
15C3×C17×ΓL(1,𝔽8)C2×C1111yescharacteristic, complemented, Hall
16C7×D51C3×C1111yescharacteristic, complemented
17ΓL(1,𝔽8)×D17—31no—
18C3×C11×ΓL(1,𝔽8)D1711yescharacteristic, complemented, Hall
19C3×C11×C17—71no—
20C3×C11×C17—141no—
21C3×C11×C17C2×ΓL(1,𝔽8)11yescharacteristic, complemented
22D3×C7×C11—171no—
23C2×C11×ΓL(1,𝔽8)—511no—
24C11×D17—31noHall
25C3×C7×C17C2×C3×C1111yescharacteristic, complemented
26C17×ΓL(1,𝔽8)—21no—
27C17×ΓL(1,𝔽8)D3×C1111yescharacteristic, complemented
28C3×D51—71noHall
29C7×D17—31noHall
30C3×C7×C11C3×D1711yescharacteristic, complemented
31C11×ΓL(1,𝔽8)—21no—
32C11×ΓL(1,𝔽8)D5111yescharacteristic, direct product factor
33C3×D3×C11—1191noHall
34C11×C17D3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
35C2×C7×C11—511noHall
36C32×C17—71noHall
37D3×ΓL(1,𝔽8)—171noHall
38C7×C17C3×D3×C1111yescharacteristic, complemented, Hall
39D51C11×ΓL(1,𝔽8)11yescharacteristic, direct product factor
40C3×D17—211no—
41C32×C11—71noHall
42C7×C11C3×D5111yescharacteristic, complemented, Hall
43C2×C3×C11—3571no—
44D3×C11—171no—
45C3×ΓL(1,𝔽8)C11×D1711yescharacteristic, complemented, Hall
46C3×C17—71no—
47C3×C17—141no—
48C3×C17C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
49D3×C7—171no—
50C2×ΓL(1,𝔽8)—511no—
51D17—31noHall
52C3×C11—71no—
53C3×C11—141no—
54C3×C11ΓL(1,𝔽8)×D1711yescharacteristic, complemented
55C2×C11—511noHall
56C3×C7C3×C11×D1711yescharacteristic, complemented
57ΓL(1,𝔽8)—21no—
58ΓL(1,𝔽8)C11×D5111yescharacteristic, direct product factor
59C3×D3—1191noHall
60C17D3×C11×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
61C2×C7—511noHall
62C11ΓL(1,𝔽8)×D5111yescharacteristic, direct product factor, Sylow
63C32—71noSylow
64C7C3×C11×D5111yescharacteristic, complemented, Sylow
65C2×C3—3571no—
66D3—171no—
67C3—71no—
68C3—141no—
69C3C11×ΓL(1,𝔽8)×D1711yescharacteristic, complemented
70C2—511noSylow
subgroupG/H
center, hypercenterC11ΓL(1,𝔽8)×D51
commutator, cosocle, hypocenterC3×C7×C17C2×C3×C11
FrattiniC1G
Fitting, socleC3×C7×C11×C17C2×C3
LCSG ▷ C3×C7×C17
UCSC1 ◁ C11

Complements

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

NH ≅ G/Ntype
1. C3×C11×C17×ΓL(1,𝔽8)C2semidirect
3. C7×C11×D51C3semidirect
5. C11×C17×ΓL(1,𝔽8)D3semidirect
6. C3×C7×C11×C17C2×C3semidirect
9. ΓL(1,𝔽8)×D51C11direct (N × H)
12. C7×C11×C17C3×D3semidirect
13. C11×D51ΓL(1,𝔽8)direct (N × H)
15. C3×C17×ΓL(1,𝔽8)C2×C11semidirect
16. C7×D51C3×C11semidirect
18. C3×C11×ΓL(1,𝔽8)D17semidirect
21. C3×C11×C17C2×ΓL(1,𝔽8)semidirect
25. C3×C7×C17C2×C3×C11semidirect
27. C17×ΓL(1,𝔽8)D3×C11semidirect
30. C3×C7×C11C3×D17semidirect
32. C11×ΓL(1,𝔽8)D51direct (N × H)
34. C11×C17D3×ΓL(1,𝔽8)semidirect
38. C7×C17C3×D3×C11semidirect
39. D51C11×ΓL(1,𝔽8)direct (N × H)
42. C7×C11C3×D51semidirect
45. C3×ΓL(1,𝔽8)C11×D17semidirect
48. C3×C17C2×C11×ΓL(1,𝔽8)semidirect
54. C3×C11ΓL(1,𝔽8)×D17semidirect
56. C3×C7C3×C11×D17semidirect
58. ΓL(1,𝔽8)C11×D51direct (N × H)
60. C17D3×C11×ΓL(1,𝔽8)semidirect
62. C11ΓL(1,𝔽8)×D51direct (N × H)
64. C7C3×C11×D51semidirect
69. C3C11×ΓL(1,𝔽8)×D17semidirect

Maximal subgroups

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

subgroupindex
C3×C11×C17×ΓL(1,𝔽8)2
C11×ΓL(1,𝔽8)×D173
C7×C11×D513
C3×C11×D517
ΓL(1,𝔽8)×D5111
D3×C11×ΓL(1,𝔽8)17