semigroup of operator under the norm $L(U,E)$

Question

Let $E, U$ be two Banach spaces. Let $(S(t),t\in[0,T]) : U\to E$ be a semigroup of bounded linear operator. What does \begin{align*} |S(t)|_{L(U,E)} \end{align*} mean explicitly? I think, the above is by definition (of operator norm), namely \begin{align*} |S_t|_{L(U,E)} :=\sup_{|x|_U\le 1}|S_t(x)|_E\le C|x|_U\quad C\,\,\text{depends on $S_t$} \end{align*} Could anyone correct me above? How can one see $C$ depends on $S(t)$, can someone show me?