Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
544 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 185 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 | C4×C5×C22⋊C4 | C2 |
| prime radical | C24×C5 | C23 |
| hypercenter | C4×C22⋊C4 | D5 |
| hypocenter | C5 | #2549 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #65218 | C2 | semidirect |
| 7. #65217 | C2 | semidirect |
| 8. #64950 | C2 | semidirect |
| 9. #64945 | C2 | semidirect |
| 35. C22×C4×C5 | Q8 | semidirect |
| 51. C5×C22⋊C4 | Q8 | semidirect |
| 152. C22×C4 | Dic10 | semidirect |
| 164. C22⋊C4 | Dic10 | semidirect |
| 204. C5 | #2549 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4024 | 2 |
| C4×C5×C22⋊C4 | 2 |
| #65501 | 2 |
| C2×#3963 | 2 |
| #65239 | 2 |
| #65218 | 2 |
| #65217 | 2 |
| #64950 | 2 |
| #64945 | 2 |
| #2549 | 5 |