Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
4638 conjugacy classes of subgroups; only the 3000 normal subgroups are shown.
This group has 4638 conjugacy classes of subgroups — too many to draw. Only its 3000 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C4 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C23×C4×C8 |
| Frattini | C22 | C24×AGL(1,𝔽5) |
| Fitting | C24×C4×C5 | C4 |
| socle, prime radical | C25×C5 | C2×C4 |
| cosocle | C2×Dic5 | C25 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4×#170 | C2 | direct (N × H) |
| 7. C23×#170 | C4 | direct (N × H) |
| 11. C2×C4×#170 | C22 | direct (N × H) |
| 14. C23×C4×C5 | C8 | semidirect |
| 16. C23×Dic5 | C8 | semidirect |
| 21. C22×#170 | C2×C4 | direct (N × H) |
| 25. C4×#170 | C23 | direct (N × H) |
| 33. C22×C4×C5 | C2×C8 | semidirect |
| 36. C22×Dic5 | C2×C8 | semidirect |
| 41. C2×#170 | C22×C4 | direct (N × H) |
| 53. C23×C5 | C4×C8 | semidirect |
| 58. C2×C4×C5 | C22×C8 | semidirect |
| 61. C2×Dic5 | C22×C8 | semidirect |
| 66. #170 | C23×C4 | direct (N × H) |
| 73. C23×C4 | #170 | direct (N × H) |
| 80. C22×C5 | C2×C4×C8 | semidirect |
| 85. C4×C5 | C23×C8 | semidirect |
| 87. Dic5 | C23×C8 | semidirect |
| 99. C22×C4 | C2×#170 | direct (N × H) |
| 105. C2×C5 | C22×C4×C8 | semidirect |
| 109. C23 | C4×#170 | direct (N × H) |
| 120. C2×C4 | C22×#170 | direct (N × H) |
| 123. C5 | C23×C4×C8 | semidirect |
| 124. C22 | C2×C4×#170 | direct (N × H) |
| 132. C4 | C23×#170 | direct (N × H) |
| 133. C2 | C22×C4×#170 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#170 | 2 |
| C23×C4×Dic5 | 2 |
| C22×C4×#170 | 2 |
| C23×C4×C8 | 5 |