Finite Group DatabasebetaFinite Group DatabasebetaOrder 640.
680 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 680 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×#4083 | C2 |
| hypercenter | C2×#123 | D5 |
| hypocenter | C5 | #2697 |
| 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. #65510 | C2 | semidirect |
| 4. #65502 | C2 | semidirect |
| 5. #65283 | C2 | semidirect |
| 6. #65279 | C2 | semidirect |
| 7. #65248 | C2 | semidirect |
| 8. #65244 | C2 | semidirect |
| 9. #65241 | C2 | semidirect |
| 12. C23×C4×C5 | C22 | semidirect |
| 15. C2×C5×C4⋊C4 | C22 | semidirect |
| 17. C2×C5×C22⋊C4 | C22 | semidirect |
| 20. C2×C22⋊Dic5 | C22 | semidirect |
| 27. C2×C4⋊Dic5 | C22 | semidirect |
| 28. C2×#683 | C22 | semidirect |
| 29. C2×C4×Dic5 | C22 | semidirect |
| 30. #3963 | C22 | semidirect |
| 31. #3963 | C22 | semidirect |
| 32. #3963 | C22 | semidirect |
| 71. C2×#123 | D5 | semidirect |
| 127. C23×C4 | C2×D5 | semidirect |
| 132. C2×C4⋊C4 | C2×D5 | semidirect |
| 142. C2×C22⋊C4 | C2×D5 | semidirect |
| 265. C5 | #2697 | 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×#4083 | 2 |
| #65510 | 2 |
| #65502 | 2 |
| #65283 | 2 |
| #65279 | 2 |
| #65248 | 2 |
| #65244 | 2 |
| #65241 | 2 |
| #2697 | 5 |