Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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×#389 | C2 |
| prime radical | C22×D20 | C22 |
| hypercenter | #389 | D5 |
| hypocenter | C5 | #2808 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #389 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#389 | C2 | semidirect |
| 2. #65282 | C2 | semidirect |
| 4. #65279 | C2 | semidirect |
| 5. #65270 | C2 | semidirect |
| 7. #65218 | C2 | semidirect |
| 8. #64962 | C2 | semidirect |
| 9. #64959 | C2 | semidirect |
| 65. #389 | D5 | semidirect |
| 216. C5 | #2808 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#389 | 2 |
| #65282 | 2 |
| #65280 | 2 |
| #65279 | 2 |
| #65270 | 2 |
| #65228 | 2 |
| #65218 | 2 |
| #64962 | 2 |
| #64959 | 2 |
| #2808 | 5 |