Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
434 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#459 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #459 | D5 |
| hypocenter | C5 | #3214 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66050 | C2 | semidirect |
| 2. #65882 | C2 | semidirect |
| 3. C5×#459 | C2 | semidirect |
| 4. #65447 | C2 | semidirect |
| 5. #65443 | C2 | semidirect |
| 6. #65361 | C2 | semidirect |
| 7. #65308 | C2 | semidirect |
| 8. #65130 | C2 | semidirect |
| 9. #65124 | C2 | semidirect |
| 10. #65070 | C2 | semidirect |
| 11. #65054 | C2 | semidirect |
| 93. #459 | D5 | semidirect |
| 276. C5 | #3214 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66050 | 2 |
| #65882 | 2 |
| C5×#459 | 2 |
| #65447 | 2 |
| #65443 | 2 |
| #65361 | 2 |
| #65308 | 2 |
| #65130 | 2 |
| #65124 | 2 |
| #65070 | 2 |
| #65054 | 2 |
| #3214 | 5 |