  We consider normalizers of an irreducible inclusion $N\subseteq M$ of
$\mathrm{II}_1$ factors. In the infinite index setting an inclusion
$uNu^*\subseteq N$ can be strict, forcing us to also investigate the semigroup
of one-sided normalizers. We relate these normalizers of $N$ in $M$ to
projections in the basic construction and show that every trace one projection
in the relative commutant $N'\cap < M,e_N>$ is of the form $u^*e_Nu$ for some
unitary $u\in M$ with $uNu^*\subseteq N$. This enables us to identify the
normalizers and the algebras they generate in several situations. In particular
each normalizer of a tensor product of irreducible subfactors is a tensor
product of normalizers modulo a unitary. We also examine normalizers of
irreducible subfactors arising from subgroup--group inclusions $H\subseteq G$.
Here the normalizers are the normalizing group elements modulo a unitary from
$L(H)$. We are also able to identify the finite trace $L(H)$-bimodules in
$\ell^2(G)$ as double cosets which are also finite unions of left cosets.
