Finite Group Databasebeta

← C2×C3×ΓL(1,𝔽8)⋊D127 #44544860

Subgroups — C2×C3×ΓL(1,𝔽8)⋊D127 #44544860

Order 32004.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×ΓL(1,𝔽8)×C127C211yesmaximal, characteristic, complemented
2C3×ΓL(1,𝔽8)⋊D127C212yesmaximal, direct product factor
3C2×C3×D889C311yesmaximal, characteristic, complemented
4C2×ΓL(1,𝔽8)⋊D127C313yesmaximal, direct product factor
5C3×ΓL(1,𝔽8)×C127C2211yescharacteristic, complemented, Hall
6ΓL(1,𝔽8)⋊D127C2×C316yesdirect product factor
7C3×D889C2×C312yescomplemented
8C2×ΓL(1,𝔽8)×C127C2×C313yescomplemented
9C2×C3×C7×C127C2×C311yescharacteristic, complemented
10C2×C32×D127—71nomaximal, Hall
11C2×D889C3211yescharacteristic, complemented, Hall
12C3×C7×C127C22×C311yescharacteristic, complemented
13ΓL(1,𝔽8)×C127C22×C313yescomplemented
14C32×D127—72no—
15C2×C32×C127—71no—
16C2×C7×C127C2×C3211yescharacteristic, complemented
17D889C2×C3212yescomplemented
18C2×C3×D127—73no—
19C2×C3×D127—71no—
20C32×C127—71noHall
21C7×C127C22×C3211yescharacteristic, complemented, Hall
22C2×C3×C127—73no—
23C2×C3×C127AGL(1,𝔽7)11yescharacteristic, complemented
24C3×D127—76no—
25C3×D127—72no—
26C2×D127—71noHall
27C3×C127—73no—
28C3×C127C2×AGL(1,𝔽7)11yescharacteristic, complemented
29C2×C127C3×AGL(1,𝔽7)11yescharacteristic, complemented
30D127—72no—
31C2×C3×AGL(1,𝔽7)—1271nomaximal, Hall
32C127C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
33C2×C3×ΓL(1,𝔽8)D12711yescharacteristic, complemented
34C3×AGL(1,𝔽7)—1272no—
35C2×C3×D7—1271no—
36C2×AGL(1,𝔽7)—1273no—
37C3×ΓL(1,𝔽8)C2×D12711yescharacteristic, complemented, Hall
38C2×C3×C7C3×D12711yescharacteristic, complemented
39C3×D7—1272no—
40C2×ΓL(1,𝔽8)C3×D12713yescomplemented
41AGL(1,𝔽7)—1276no—
42C22×C32—8891noHall
43C2×D7—1271noHall
44C3×C7C2×C3×D12711yescharacteristic, complemented
45ΓL(1,𝔽8)C2×C3×D12713yescomplemented
46C2×C32—71no—
47C2×C32—8892no—
48C2×C7C32×D12711yescharacteristic, complemented
49D7—1272no—
50C22×C3—8893no—
51C22×C3—8891no—
52C32—71noSylow
53C7C2×C32×D12711yescharacteristic, complemented, Sylow
54C2×C3—73no—
55C2×C3—8896no—
56C2×C3—8892no—
57C2×C3ΓL(1,𝔽8)⋊D12711yescharacteristic, direct product factor
58C22—8891noSylow
59C3—73no—
60C3C2×ΓL(1,𝔽8)⋊D12711yescharacteristic, direct product factor
61C2—8892no—
62C2C3×ΓL(1,𝔽8)⋊D12711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)⋊D127
commutator, cosocle, hypocenterC7×C127C22×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C127C2×C3
LCSG ▷ C7×C127
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×ΓL(1,𝔽8)×C127C2semidirect
2. C3×ΓL(1,𝔽8)⋊D127C2direct (N × H)
3. C2×C3×D889C3semidirect
4. C2×ΓL(1,𝔽8)⋊D127C3direct (N × H)
5. C3×ΓL(1,𝔽8)×C127C22semidirect
6. ΓL(1,𝔽8)⋊D127C2×C3direct (N × H)
7. C3×D889C2×C3semidirect
8. C2×ΓL(1,𝔽8)×C127C2×C3semidirect
9. C2×C3×C7×C127C2×C3semidirect
11. C2×D889C32semidirect
12. C3×C7×C127C22×C3semidirect
13. ΓL(1,𝔽8)×C127C22×C3semidirect
16. C2×C7×C127C2×C32semidirect
17. D889C2×C32semidirect
21. C7×C127C22×C32semidirect
23. C2×C3×C127AGL(1,𝔽7)semidirect
28. C3×C127C2×AGL(1,𝔽7)semidirect
29. C2×C127C3×AGL(1,𝔽7)semidirect
32. C127C2×C3×AGL(1,𝔽7)semidirect
33. C2×C3×ΓL(1,𝔽8)D127semidirect
37. C3×ΓL(1,𝔽8)C2×D127semidirect
38. C2×C3×C7C3×D127semidirect
40. C2×ΓL(1,𝔽8)C3×D127semidirect
44. C3×C7C2×C3×D127semidirect
45. ΓL(1,𝔽8)C2×C3×D127semidirect
48. C2×C7C32×D127semidirect
53. C7C2×C32×D127semidirect
57. C2×C3ΓL(1,𝔽8)⋊D127direct (N × H)
60. C3C2×ΓL(1,𝔽8)⋊D127direct (N × H)
62. C2C3×ΓL(1,𝔽8)⋊D127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×ΓL(1,𝔽8)×C1272
C3×ΓL(1,𝔽8)⋊D1272
C2×C3×D8893
C2×ΓL(1,𝔽8)⋊D1273
C2×C32×D1277
C2×C3×AGL(1,𝔽7)127