Torsion on elliptic curves in isogeny classes

Let $E$ be an elliptic curve over a number field $K$ and $\mathcal C$ its $K$-isogeny class. We are interested in determining the orders and the types of torsion groups $E(K)_{\textrm{tors}}$ in $\mathcal C$. For a prime $l$, we give the range of possible types of $l$-primary parts $E(K)_{(l)}$ of $E(K)_{\textrm{tors}}$ when $E$ runs over $\mathcal C$. One of our results immediately gives a simple proof of a theorem of Katz on the order $\sup_{E \in \mathcal C}\vert E(K)_{(l)}\vert$ of maximal $l$-primary torsion in $\mathcal C$.