subsheaf of free sheaves

Let $X$ be an irreducible nodal curve, $E:=oplus_{i=1}^r mathcal{O}_X$ be a free sheaf on $X$ and $F subset E$ a (coherent) subsheaf. Is it possible to write $F$ as a direct sum of subsheaves $F_1,..,F_j$ for $j le r$ such that $F_i subset mathcal{O}_X$? Can we take $F_i$ to be the image of $F$ under the composition $F to E xrightarrow{mathrm{pr}_i} mathcal{O}_X$, where $mathrm{pr}_i$ is the projection onto the $i$-th coordinate.