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×#391 | C2 |
| socle, cosocle | C23×C5 | C24 |
| prime radical | C22×D20 | C22 |
| hypercenter | #391 | D5 |
| hypocenter | C5 | #2326 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #391 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q8×Dic5 | C2 | semidirect |
| 4. C5×#391 | C2 | semidirect |
| 5. #65265 | C2 | semidirect |
| 7. #65221 | C2 | semidirect |
| 8. #64958 | C2 | semidirect |
| 9. #64957 | C2 | semidirect |
| 15. #4093 | C4 | semidirect |
| 34. C2×C5×Q8 | C2×C4 | semidirect |
| 52. #685 | C2×C4 | semidirect |
| 57. C4×Dic5 | C2×C4 | semidirect |
| 62. #391 | D5 | semidirect |
| 138. C2×Q8 | C4×D5 | 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×#4093 | 2 |
| C2×Q8×Dic5 | 2 |
| C2×#4039 | 2 |
| C5×#391 | 2 |
| #65265 | 2 |
| #65228 | 2 |
| #65221 | 2 |
| #64958 | 2 |
| #64957 | 2 |
| #2326 | 5 |