Finite Group Databasebeta

← C23×C5×AGL(1,𝔽7) #13132952

Subgroups — C23×C5×AGL(1,𝔽7) #13132952

Order 1680.

536 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C22×C5×AGL(1,𝔽7)C2114yesmaximal, direct product factor
3C23×C5×D7C311yesmaximal, characteristic, complemented, Hall
4C22×C5×ΓL(1,𝔽8)C2217yescomplemented
5C2×C5×AGL(1,𝔽7)C22128yesdirect product factor
6C23×AGL(1,𝔽7)C511yesmaximal, characteristic, direct product factor, Hall
7C23×C5×C7C2×C311yescharacteristic, complemented
8C22×C5×D7C2×C3114yescomplemented
9C24×C3×C5—71nomaximal, Hall
10C2×C5×ΓL(1,𝔽8)C2317yescomplemented
11C5×AGL(1,𝔽7)C2318yesdirect product factor
12C23×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
13C22×AGL(1,𝔽7)C2×C5114yesdirect product factor
14C22×C5×C7C22×C317yescomplemented
15C2×C5×D7C22×C3128yescomplemented
16C23×C3×C5—714no—
17C23×C3×C5—71no—
18C23×D7C3×C511yescharacteristic, complemented, Hall
19C5×ΓL(1,𝔽8)C2411yescharacteristic, complemented, Hall
20C22×ΓL(1,𝔽8)C22×C517yescomplemented
21C2×AGL(1,𝔽7)C22×C5128yesdirect product factor
22C24×C5—71noHall
23C2×C5×C7C23×C317yescomplemented
24C5×D7C23×C318yescomplemented
25C22×C3×C5—728no—
26C22×C3×C5—77no—
27C23×C7C2×C3×C511yescharacteristic, complemented
28C22×D7C2×C3×C5114yescomplemented
29C24×C3—71noHall
30C2×ΓL(1,𝔽8)C23×C517yescomplemented
31AGL(1,𝔽7)C23×C518yesdirect product factor
32C23×C5—714no—
33C23×C5AGL(1,𝔽7)11yescharacteristic, direct product factor
34C5×C7C24×C311yescharacteristic, complemented, Hall
35C2×C3×C5—78no—
36C2×C3×C5—77no—
37C22×C7C22×C3×C517yescomplemented
38C2×D7C22×C3×C5128yescomplemented
39C23×C3—714no—
40C23×C3—71no—
41ΓL(1,𝔽8)C24×C511yescharacteristic, complemented, Hall
42C22×C5—728no—
43C22×C5C2×AGL(1,𝔽7)17yesdirect product factor
44C24—71noSylow
45C3×C5—71noHall
46C2×C7C23×C3×C517yescomplemented
47D7C23×C3×C518yescomplemented
48C22×C3—728no—
49C22×C3—77no—
50C2×C5—78no—
51C2×C5C22×AGL(1,𝔽7)17yesdirect product factor
52C23—714no—
53C23C5×AGL(1,𝔽7)11yescharacteristic, direct product factor
54C7C24×C3×C511yescharacteristic, complemented, Sylow
55C2×C3—78no—
56C2×C3—77no—
57C5C23×AGL(1,𝔽7)11yescharacteristic, direct product factor, Sylow
58C22—728no—
59C22C2×C5×AGL(1,𝔽7)17yesdirect product factor
60C3—71noSylow
61C2—78no—
62C2C22×C5×AGL(1,𝔽7)17yesdirect product factor
subgroupG/H
center, hypercenterC23×C5AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C24×C3×C5
FrattiniC1G
Fitting, socleC23×C5×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C23×C5

Complements

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

NH ≅ G/Ntype
1. C23×C5×ΓL(1,𝔽8)C2semidirect
2. C22×C5×AGL(1,𝔽7)C2direct (N × H)
3. C23×C5×D7C3semidirect
4. C22×C5×ΓL(1,𝔽8)C22semidirect
5. C2×C5×AGL(1,𝔽7)C22direct (N × H)
6. C23×AGL(1,𝔽7)C5direct (N × H)
7. C23×C5×C7C2×C3semidirect
8. C22×C5×D7C2×C3semidirect
10. C2×C5×ΓL(1,𝔽8)C23semidirect
11. C5×AGL(1,𝔽7)C23direct (N × H)
12. C23×ΓL(1,𝔽8)C2×C5semidirect
13. C22×AGL(1,𝔽7)C2×C5direct (N × H)
14. C22×C5×C7C22×C3semidirect
15. C2×C5×D7C22×C3semidirect
18. C23×D7C3×C5semidirect
19. C5×ΓL(1,𝔽8)C24semidirect
20. C22×ΓL(1,𝔽8)C22×C5semidirect
21. C2×AGL(1,𝔽7)C22×C5direct (N × H)
23. C2×C5×C7C23×C3semidirect
24. C5×D7C23×C3semidirect
27. C23×C7C2×C3×C5semidirect
28. C22×D7C2×C3×C5semidirect
30. C2×ΓL(1,𝔽8)C23×C5semidirect
31. AGL(1,𝔽7)C23×C5direct (N × H)
33. C23×C5AGL(1,𝔽7)direct (N × H)
34. C5×C7C24×C3semidirect
37. C22×C7C22×C3×C5semidirect
38. C2×D7C22×C3×C5semidirect
41. ΓL(1,𝔽8)C24×C5semidirect
43. C22×C5C2×AGL(1,𝔽7)direct (N × H)
46. C2×C7C23×C3×C5semidirect
47. D7C23×C3×C5semidirect
51. C2×C5C22×AGL(1,𝔽7)direct (N × H)
53. C23C5×AGL(1,𝔽7)direct (N × H)
54. C7C24×C3×C5semidirect
57. C5C23×AGL(1,𝔽7)direct (N × H)
59. C22C2×C5×AGL(1,𝔽7)direct (N × H)
62. C2C22×C5×AGL(1,𝔽7)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×ΓL(1,𝔽8)2
C22×C5×AGL(1,𝔽7)2
C23×C5×D73
C23×AGL(1,𝔽7)5
C24×C3×C57