= Solution
Relative to the three successive blocks of sizes $1,2,2$, the stabilizer consists of the block upper triangular <matrices>
$$
\begin{pmatrix}a&B&C\\0&D&E\\0&0&F\end{pmatrix},\qquad a\in K^\times,\quad D,F\in\operatorname{GL}(2,K).
$$
The off-diagonal blocks contain $2+2+4=8$ arbitrary <field> entries. Thus its order is $(q-1)|\operatorname{GL}(2,q)|^2q^8$. By <orbit-stabilizer theorem>,
$$
\boxed{|\Omega|=\frac{\prod_{j=0}^4(q^5-q^j)}{q^8(q-1)(q^2-1)^2(q^2-q)^2}.}
$$
Equivalently, choose the line first and then a two-dimensional subspace of the four-dimensional quotient:
$$
|\Omega|=\frac{q^5-1}{q-1}\frac{(q^4-1)(q^4-q)}{(q^2-1)(q^2-q)}.
$$
This also checks the stabilizer count independently.
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