Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
420 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C5×#390 | C2 |
| prime radical | C22×C5×D4 | C22 |
| hypercenter | #390 | D5 |
| hypocenter | C5 | #2644 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #65274 | C2 | semidirect |
| 5. #65262 | C2 | semidirect |
| 6. #65242 | C2 | semidirect |
| 7. #65241 | C2 | semidirect |
| 8. #65241 | C2 | semidirect |
| 9. #65237 | C2 | semidirect |
| 10. #65221 | C2 | semidirect |
| 11. #65217 | C2 | semidirect |
| 238. C5 | #2644 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#390 | 2 |
| #65510 | 2 |
| #65274 | 2 |
| #65262 | 2 |
| #65242 | 2 |
| #65241 | 2 |
| #65237 | 2 |
| #65221 | 2 |
| #65217 | 2 |
| #2644 | 5 |