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Consider a quartic number field $K$ generated by a root of an irreduciblequadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denotethe index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot3^v$ with $u \le 2$ and $v \le 1$. In this paper, we provide sufficientconditions on $a$, $b$ and $c$ for $i(K)$ to be divisible by $2$ or $3$,determining the exact corresponding values of $u$ and $v$ in each case.  In particular, when $i(K) \neq 1$, $K$ cannot be monogenic. We also identifynew infinite parametric families of monogenic quartic number generated by rootsof non-monogenic quadrinomials. We illustrate our results by some computationalexamples. Our method based on a theorem of Ore on the decomposition of primesin number fields \cite{Nar,O}.

On indices and monogenity of quartic number fields defined by quadrinomials