Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
996 conjugacy classes of subgroups; only the 509 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 509 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×#543 | C2 |
| prime radical | C24×C5 | C23 |
| hypercenter | #543 | D5 |
| hypocenter | C5 | #3458 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #543 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #65957 | C2 | semidirect |
| 3. #65956 | C2 | semidirect |
| 10. Q8×Dic5 | C22 | semidirect |
| 11. #4090 | C22 | semidirect |
| 15. #4035 | C22 | semidirect |
| 16. #4034 | C22 | semidirect |
| 17. #4033 | C22 | semidirect |
| 25. C22×Dic5 | Q8 | semidirect |
| 111. C5 | #3458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#543 | 2 |
| #65957 | 2 |
| #65956 | 2 |
| #65817 | 2 |
| C2×#4023 | 2 |
| #3458 | 5 |