Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
568 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 149 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×#384 | C2 |
| prime radical | C2×#4041 | C2 |
| hypercenter | #384 | D5 |
| hypocenter | C5 | #2500 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4038 | C2 | semidirect |
| 3. C5×#384 | C2 | semidirect |
| 4. #65280 | C2 | semidirect |
| 5. #65279 | C2 | semidirect |
| 6. #65277 | C2 | semidirect |
| 7. #65266 | C2 | semidirect |
| 8. #65228 | C2 | semidirect |
| 9. C4×#685 | C2 | semidirect |
| 65. #384 | D5 | semidirect |
| 209. C5 | #2500 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4041 | 2 |
| C2×#4038 | 2 |
| C5×#384 | 2 |
| #65280 | 2 |
| #65279 | 2 |
| #65277 | 2 |
| #65266 | 2 |
| #65228 | 2 |
| C4×#685 | 2 |
| #2500 | 5 |