  We show that for any positive integer $N$, there are only finitely many
holomorphic eta quotients of level $N$, none of which is a product of two
holomorphic eta quotients other than 1 and itself. This result is an analog of
Zagier's conjecture/ Mersmann's theorem which states that: Of any given weight,
there are only finitely many irreducible holomorphic eta quotients, none of
which is an integral rescaling of another eta quotient. We construct such eta
quotients for all cubefree levels. In particular, our construction demonstrates
the existence of irreducible holomorphic eta quotients of arbitrarily large
weights.
