Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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 | C22×C5 | C23×C4 |
| Fitting | C5×#390 | C2 |
| socle, cosocle | C23×C5 | C24 |
| prime radical | C22×C5×D4 | C22 |
| hypercenter | #390 | D5 |
| hypocenter | C5 | #2326 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #390 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#4033 | C2 | semidirect |
| 5. #65265 | C2 | semidirect |
| 6. #65235 | C2 | semidirect |
| 7. #65234 | C2 | semidirect |
| 8. #65221 | C2 | semidirect |
| 9. #65218 | C2 | semidirect |
| 18. #4082 | C4 | semidirect |
| 47. C2×Dic10 | C2×C4 | semidirect |
| 52. C22⋊Dic5 | C2×C4 | semidirect |
| 57. C4×Dic5 | C2×C4 | semidirect |
| 198. C5 | #2326 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4082 | 2 |
| C2×D4×Dic5 | 2 |
| C2×#4033 | 2 |
| C5×#390 | 2 |
| #65265 | 2 |
| #65235 | 2 |
| #65234 | 2 |
| #65221 | 2 |
| #65218 | 2 |
| #2326 | 5 |