Finite Group DatabasebetaFinite Group DatabasebetaOrder 17457.
104 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C11×C232 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 2 | C3×C232 | C11 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C3×C11×C23 | C23 | 1 | 24 | yes | maximal, direct product factor |
| 4 | C232 | C3×C11 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 5 | C11×C23 | C3×C23 | 1 | 24 | yes | direct product factor |
| 6 | C3×C23 | C11×C23 | 1 | 24 | yes | direct product factor |
| 7 | C3×C11 | C232 | 1 | 1 | yes | characteristic, direct product factor, Hall |
| 8 | C23 | C3×C11×C23 | 1 | 24 | yes | direct product factor |
| 9 | C11 | C3×C232 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 10 | C3 | C11×C232 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C11×C232 | C3 | direct (N × H) |
| 2. C3×C232 | C11 | direct (N × H) |
| 3. C3×C11×C23 | C23 | direct (N × H) |
| 4. C232 | C3×C11 | direct (N × H) |
| 5. C11×C23 | C3×C23 | direct (N × H) |
| 6. C3×C23 | C11×C23 | direct (N × H) |
| 7. C3×C11 | C232 | direct (N × H) |
| 8. C23 | C3×C11×C23 | direct (N × H) |
| 9. C11 | C3×C232 | direct (N × H) |
| 10. C3 | C11×C232 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C11×C232 | 3 |
| C3×C232 | 11 |
| C3×C11×C23 | 23 |