Let $G$ be a finite group and $V$ a finite-dimensional representation of $G$ (or $G$-module) over $\mathbb{C}$. Furthermore, suppose we have a filtration of $G$-modules $$0=V_0\subset V_1\subset\cdots \subset V_r=V,$$ such that each $V_i$ is a $G$-submodule and each quotient $V_i/V_{i-1}$ is isomorphic to one of the simple $G$-modules.
Finally, for some $k\geq 1$ suppose that $v\in V_k\setminus V_{k-1}$. That is, $v$ first appears in the $k^\text{th}$ piece of the filtration. Do we know if $v$ is cyclic in $V_k$? That is, is $\mathbb{C}G\cdot v=V_k$?
Of course, if $v\in V_j$ with $j<k$ then $v$ will not be cyclic in $V_k$ since $V_j$ is a submodule, so we are only looking at elements in the 'top piece'. For $k=1$ the result is obvious, so I figure induction is the way to prove this. That proof would follow from finding an element $g\in\mathbb{C}G$ such that $g\cdot v\in V_{i-1}\setminus V_{i-2}$, but I can't see how to do that.
Any help is appreciated!
David's comment shows a clear counter-example, taking $V_1\subset V_1\oplus V_2$.