Finite Group Databasebeta

← C3×ΓL(1,𝔽8)×D187 #37555638

Subgroups — C3×ΓL(1,𝔽8)×D187 #37555638

Order 23562.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C11×C17×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented, Hall
2ΓL(1,𝔽8)×D187C313yesmaximal, direct product factor
3C3×C7×D187C311yesmaximal, characteristic, complemented
4C11×C17×ΓL(1,𝔽8)C2×C313yescomplemented
5C3×C7×C11×C17C2×C311yescharacteristic, complemented
6C32×D187—71nomaximal, Hall
7C7×D187C3211yescharacteristic, complemented, Hall
8C3×ΓL(1,𝔽8)×D17—111nomaximal, Hall
9C32×C11×C17—71noHall
10C3×ΓL(1,𝔽8)×D11—171nomaximal, Hall
11C7×C11×C17C2×C3211yescharacteristic, complemented, Hall
12C3×D187—73no—
13C3×D187ΓL(1,𝔽8)11yescharacteristic, direct product factor
14C3×C17×ΓL(1,𝔽8)D1111yescharacteristic, complemented, Hall
15C3×C7×D17—111no—
16ΓL(1,𝔽8)×D17—113no—
17C3×C11×ΓL(1,𝔽8)D1711yescharacteristic, complemented, Hall
18C3×C11×C17—73no—
19C3×C11×C17C2×ΓL(1,𝔽8)11yescharacteristic, complemented
20C3×C7×D11—171no—
21ΓL(1,𝔽8)×D11—173no—
22D187C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
23C3×C7×C17C3×D1111yescharacteristic, complemented
24C17×ΓL(1,𝔽8)C3×D1113yescomplemented
25C32×D17—771noHall
26C7×D17—111noHall
27C3×C7×C11C3×D1711yescharacteristic, complemented
28C11×ΓL(1,𝔽8)C3×D1713yescomplemented
29C32×D11—1191noHall
30C11×C17C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
31C7×D11—171noHall
32C32×C17—71noHall
33C2×C3×ΓL(1,𝔽8)—1871noHall
34C7×C17C32×D1111yescharacteristic, complemented, Hall
35C3×D17—111no—
36C3×D17—773no—
37C32×C11—71noHall
38C7×C11C32×D1711yescharacteristic, complemented, Hall
39C3×D11—171no—
40C3×D11—1193no—
41C3×ΓL(1,𝔽8)D18711yescharacteristic, direct product factor, Hall
42C3×C17—73no—
43C3×C17ΓL(1,𝔽8)×D1111yescharacteristic, complemented
44C2×C3×C7—1871no—
45C2×ΓL(1,𝔽8)—1873no—
46D17—111noHall
47C3×C11—73no—
48C3×C11ΓL(1,𝔽8)×D1711yescharacteristic, complemented
49D11—171noHall
50C3×C7C3×D18711yescharacteristic, complemented
51ΓL(1,𝔽8)C3×D18713yesdirect product factor
52C2×C32—13091noHall
53C17C3×ΓL(1,𝔽8)×D1111yescharacteristic, complemented, Sylow
54C2×C7—1871noHall
55C11C3×ΓL(1,𝔽8)×D1711yescharacteristic, complemented, Sylow
56C32—71noSylow
57C7C32×D18711yescharacteristic, complemented, Sylow
58C2×C3—13093no—
59C2×C3—1871no—
60C3—73no—
61C3ΓL(1,𝔽8)×D18711yescharacteristic, direct product factor
62C2—1871noSylow
subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D187
commutator, cosocle, hypocenterC7×C11×C17C2×C32
FrattiniC1G
Fitting, socleC3×C7×C11×C17C2×C3
LCSG ▷ C7×C11×C17
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×C11×C17×ΓL(1,𝔽8)C2semidirect
2. ΓL(1,𝔽8)×D187C3direct (N × H)
3. C3×C7×D187C3semidirect
4. C11×C17×ΓL(1,𝔽8)C2×C3semidirect
5. C3×C7×C11×C17C2×C3semidirect
7. C7×D187C32semidirect
11. C7×C11×C17C2×C32semidirect
13. C3×D187ΓL(1,𝔽8)direct (N × H)
14. C3×C17×ΓL(1,𝔽8)D11semidirect
17. C3×C11×ΓL(1,𝔽8)D17semidirect
19. C3×C11×C17C2×ΓL(1,𝔽8)semidirect
22. D187C3×ΓL(1,𝔽8)direct (N × H)
23. C3×C7×C17C3×D11semidirect
24. C17×ΓL(1,𝔽8)C3×D11semidirect
27. C3×C7×C11C3×D17semidirect
28. C11×ΓL(1,𝔽8)C3×D17semidirect
30. C11×C17C2×C3×ΓL(1,𝔽8)semidirect
34. C7×C17C32×D11semidirect
38. C7×C11C32×D17semidirect
41. C3×ΓL(1,𝔽8)D187direct (N × H)
43. C3×C17ΓL(1,𝔽8)×D11semidirect
48. C3×C11ΓL(1,𝔽8)×D17semidirect
50. C3×C7C3×D187semidirect
51. ΓL(1,𝔽8)C3×D187direct (N × H)
53. C17C3×ΓL(1,𝔽8)×D11semidirect
55. C11C3×ΓL(1,𝔽8)×D17semidirect
57. C7C32×D187semidirect
61. C3ΓL(1,𝔽8)×D187direct (N × H)

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
ΓL(1,𝔽8)×D1873
C3×C7×D1873
C32×D1877
C3×ΓL(1,𝔽8)×D1711
C3×ΓL(1,𝔽8)×D1117