= Solution
The <stabilizer> of the indicated flag consists of block upper triangular matrices with diagonal block sizes $1,2,2$. The diagonal blocks can be any elements of $\operatorname{GL}(1,q)$, $\operatorname{GL}(2,q)$ and $\operatorname{GL}(2,q)$. The upper off-diagonal blocks have $2+2+4=8$ arbitrary entries. Thus the <stabilizer> order is $(q-1)|\operatorname{GL}(2,q)|^2q^8$. The <orbit-stabilizer theorem> gives
$$
\boxed{|\Omega|=\frac{|\operatorname{GL}(5,q)|}{(q-1)|\operatorname{GL}(2,q)|^2q^8}
=\frac{(q^5-1)(q^4-1)(q^3-1)}{(q-1)^2(q^2-1)}}.
$$
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