  We consider the Andr\'e/Bruck-Bose representation of the projective plane
$\mathrm{PG}(2,q^n)$ in $\mathrm{PG}(2n,q)$. We investigate the representation
of $\mathbb{F}_{q^k}$-sublines and $\mathbb{F}_{q^k}$-subplanes of
$\mathrm{PG}(2,q^n)$, extending the results for $n=3$ of \cite{BarJack2} and
correcting the general result of \cite{BarJack1}. We characterise the
representation of $\mathbb{F}_{q^k}$-sublines tangent to or contained in the
line at infinity, $\mathbb{F}_q$-sublines external to the line at infinity,
$\mathbb{F}_q$-subplanes tangent to and $\mathbb{F}_{q^k}$-subplanes secant to
the line at infinity.
