Finite Group DatabasebetaFinite Group DatabasebetaOrder 510.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×C5×C17 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 2 | D85 | — | 3 | 1 | no | maximal, Hall |
| 3 | D51 | — | 5 | 1 | no | maximal, Hall |
| 4 | C5×C17 | D3 | 1 | 1 | yes | characteristic, complemented, Hall |
| 5 | C3×C17 | D5 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | D17 | — | 15 | 1 | no | Hall |
| 7 | D15 | — | 17 | 1 | no | maximal, Hall |
| 8 | C17 | D15 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C3×C5 | D17 | 1 | 1 | yes | characteristic, complemented, Hall |
| 10 | D5 | — | 51 | 1 | no | Hall |
| 11 | D3 | — | 85 | 1 | no | Hall |
| 12 | C5 | D51 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 13 | C3 | D85 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 14 | C2 | — | 255 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C17 | C2 |
| LCS | G ▷ C3×C5×C17 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C5×C17 | C2 | semidirect |
| 4. C5×C17 | D3 | semidirect |
| 5. C3×C17 | D5 | semidirect |
| 8. C17 | D15 | semidirect |
| 9. C3×C5 | D17 | semidirect |
| 12. C5 | D51 | semidirect |
| 13. C3 | D85 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C5×C17 | 2 |
| D85 | 3 |
| D51 | 5 |
| D15 | 17 |