Why is a cyclic subgroup chosen in ECDSA? One thing which I am wondering for a long time and to which I did not find an answer after doing a web search and hope to find an answer here.

When we construct the elliptic curve over a prime field why do we actually select a cyclic subgroup instead of taking the entire group of the elliptic curve?

On a side note the thing that confuses me most about this choice: We know that the cyclic subgroup of prime order p is isomorphic to Z/pZ and finding the isomorphism would mean solving the discrete log. 

Switching to a cyclic group seems actually rather like making the problem easier in comparison to staying with the full elliptic curve.
The security of the discrete logarithm problem in a group is only as hard as that of the largest prime subgroup.

Because of this, there is no security gain from working in the larger group. However; it's worse. If you're working in the larger group you must make sure to not accidentally ending up in a (much smaller) subgroup when multiplying.