Finite Group DatabasebetaFinite Group DatabasebetaOrder 12890.
8 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C5×C1289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D1289 | — | 5 | 1 | no | maximal, Hall |
| 3 | C1289 | D5 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 4 | D5 | — | 1289 | 1 | no | maximal, Hall |
| 5 | C5 | D1289 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 6 | C2 | — | 6445 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C1289 | C2 |
| LCS | G ▷ C5×C1289 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×C1289 | C2 | semidirect |
| 3. C1289 | D5 | semidirect |
| 5. C5 | D1289 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×C1289 | 2 |
| D1289 | 5 |
| D5 | 1289 |