Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
556 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 137 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×#385 | C2 |
| prime radical | #65249 | C2 |
| hypercenter | #385 | D5 |
| hypocenter | C5 | #2712 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #385 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4031 | C2 | semidirect |
| 2. C5×#385 | C2 | semidirect |
| 3. #65502 | C2 | semidirect |
| 4. #65280 | C2 | semidirect |
| 5. #65263 | C2 | semidirect |
| 7. #65242 | C2 | semidirect |
| 8. #64962 | C2 | semidirect |
| 9. #64959 | C2 | semidirect |
| 54. C5×C22⋊C4 | D4 | semidirect |
| 71. #385 | D5 | semidirect |
| 182. C22⋊C4 | D20 | semidirect |
| 227. C5 | #2712 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4031 | 2 |
| C5×#385 | 2 |
| #65502 | 2 |
| #65280 | 2 |
| #65263 | 2 |
| #65249 | 2 |
| #65242 | 2 |
| #64962 | 2 |
| #64959 | 2 |
| #2712 | 5 |