Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
488 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 165 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 | C2×C5×#117 | C2 |
| prime radical | C24×C5 | C23 |
| hypercenter | C2×#117 | D5 |
| hypocenter | C5 | #2628 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4015 | C2 | semidirect |
| 4. #65275 | C2 | semidirect |
| 5. #65268 | C2 | semidirect |
| 8. #65225 | C2 | semidirect |
| 9. #65217 | C2 | semidirect |
| 33. C22×C4×C5 | Q8 | semidirect |
| 136. C22×C4 | Dic10 | semidirect |
| 188. C5 | #2628 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#117 | 2 |
| C2×#4015 | 2 |
| #65501 | 2 |
| #65275 | 2 |
| #65268 | 2 |
| #65238 | 2 |
| #65236 | 2 |
| #65225 | 2 |
| #65217 | 2 |
| #2628 | 5 |