  Given any irreducible inclusion $\mB \subset \mA$ of unital $C^*$-algebras
with a finite-index conditional expectation $E: \mA \to \mB$, we show that the
set of $E$-compatible intermediate $C^*$-subalgebras is finite, thereby
generalizing a finiteness result of Ino and Watatani (from \cite{IW}). A
finiteness result for a certain collection of intermediate $C^*$-subalgebras of
a non-irreducible inclusion of simple unital $C^*$-algebras is also obtained,
which provides a $C^*$-version of a finiteness result of Khoshkam and Mashood
(from \cite{KM}).
  Apart from these finiteness results, comparisons between various notions of
distance between subalgebras of operator algebras by Kadison-Kastler,
Christensen and Mashood-Taylor are made. Further, these comparisons are used
satisfactorily to provide some concrete calculations of distance between
operator algebras associated to two distinct subgroups of a given discrete
group.
