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No. This follows from a beautiful theorem of B.H. Neumann:

Let $G$ be a group. If $\{x_iH_i\}_{i=1}^n$ is a covering of $G$ by cosets of proper subgroups, then $n \geq \min_{i} [G:H_i]$.

Explicitly, this is Lemma 4.1 in

http://www.math.uga.edu/~pete/Neumann54.pdf

As Neumann remarks, the identity $gH = (g H g^{-1}) g$ shows that it is no loss of generality to restrict to coverings by left cosets.

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No. This follows from a beautiful theorem of B.H. Neumann:

Let $G$ be a group. If $\{x_iH_i\}_{i=1}^n$ is a covering of $G$ by cosets of proper subgroups, then $n \geq \min_{i} [G:H_i]$.