Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
476 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 141 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 | #2727 |
| 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. C2×#4044 | C2 | semidirect |
| 2. C5×#389 | C2 | semidirect |
| 3. #65283 | C2 | semidirect |
| 5. #65274 | C2 | semidirect |
| 7. #65217 | C2 | semidirect |
| 8. #64966 | C2 | semidirect |
| 9. #64959 | C2 | semidirect |
| 47. C5×C4⋊C4 | D4 | semidirect |
| 69. #389 | D5 | semidirect |
| 168. C4⋊C4 | D20 | semidirect |
| 218. C5 | #2727 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4044 | 2 |
| C5×#389 | 2 |
| #65283 | 2 |
| #65280 | 2 |
| #65274 | 2 |
| #65228 | 2 |
| #65217 | 2 |
| #64966 | 2 |
| #64959 | 2 |
| #2727 | 5 |