How does the order of Q affect the time it takes to solve ECDLP? I use Sagemath's built-in function discrete_log() to solve ECDLP and according to the documentation it uses Pohling-Hellman algorithm to solve an ECDLP. 

This is the case for my EC:

P-order =&gt; 176648753626148987385153422443570230330
Q-order() =&gt; 2


The P-order has these prime factors: 2 * 3 * 5 * 5888291787538299579505114081452341011

Even though the highest prime factor is super-large, the ECDLP through discrete_log() is solved within a second. 

I Read about the Pohling-hellman attack and it only mentions that P should have small prime-factors in order for the attack to be successful. It never mentions the order of Q. 

How does the order of Q affect the time it takes for the attack to be successful?
Here is what I believe Sagemath is doing:


Given the problem of finding $x$ s.t. $xP = Q$, one step is computing $x' = x \bmod 5888291787538299579505114081452341011$; it does this by computing $P' = (2\cdot 3 \cdot 5)P$, and $Q' = (2 \cdot 3 \cdot 5)Q$, and trying to solve $x'P' = Q'$
As the order of $P'$ is circa $2^{122}$, we would normally expect this computation to take circa $2^{61}$ time (quite a while).  However, in this case, $Q'$ is the neutral element.  I suspect that Sagemath notices this, and so immediately concludes that $x' = 0$
Every other computation involved in Pohlig-Hellman is relatively quick, and so it reports the answer ($3 \cdot 5 \cdot 5888291787538299579505114081452341011$) quickly.


The summary is: if your implementation special cases this $Q' = 0$ case, then we can skip any prime that does not appear in the order of $Q$; the time taken by Polhig-Hellman is dominated by the largest prime that appears both in the order of $P$ and the order of $Q$.