Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
612 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 137 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×#123 | C2 |
| prime radical | C2×#4086 | C2 |
| hypercenter | C2×#123 | D5 |
| hypocenter | C5 | #2709 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×#123 | C2 | semidirect |
| 3. #65509 | C2 | semidirect |
| 4. #65502 | C2 | semidirect |
| 5. #65280 | C2 | semidirect |
| 6. #65267 | C2 | semidirect |
| 7. #65248 | C2 | semidirect |
| 8. #65248 | C2 | semidirect |
| 9. #65247 | C2 | semidirect |
| 72. C2×#123 | D5 | semidirect |
| 260. C5 | #2709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×#123 | 2 |
| C2×#4086 | 2 |
| #65509 | 2 |
| #65502 | 2 |
| #65280 | 2 |
| #65267 | 2 |
| #65248 | 2 |
| #65247 | 2 |
| #2709 | 5 |