Let $G$ be a finite group (non $p$-group) with the following properties:

a: For every prime divisor $p$ of $\vert G\vert$, there exists only one minimal subgroup of order $p$.

b: For every pair of distinct prime divisors of $\vert G\vert$ say $p$ and $q$, If $P_{1}, P_{2},..., P_{m}$ are all of the $p$-subgroups of $G$ and similarly $Q_{1}, Q_{2},..., Q_{n}$ are all of the $q$-subgroups of $G$, then $G$ has $mn$ subgroups of order $p^{\alpha}q^{\beta}$($\alpha,\beta\geq 1$). ( If $\vert G\vert=p^{s}q^{t}$ for some naturals $s,t$, then we count it in this computation )

Can we say that $G$ is a Dedekind group.( i.e. is every subgroup of $G$, a normal subgroup? )