Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of Galois modules. Let us assume that this sequence splits, i.e. $E[p]\simeq\mathbb{Z}/p\times\mu_p$ as Galois modules. Set $E':=E/\mu_p$. Now, my questions is:

Is the set of bad primes for the elliptic curve $E'$ the same as the set of bad primes for $E$?

I am also wondering whether there is a general theory in this line, for example, how to compute the set of bad rimes for the quotient elliptic curve $E/H$ for "Galois stable" subgroup $H$ of $E$.