In eukaryotic cells, the stages of the cell cycle are divided into two major phases: interphase and the mitotic (M) phase.
During the interphase,
the cell grows physically larger, copies organelles, and makes the molecular building blocks (G1 phase);
the cell synthesizes a complete copy of the DNA in its nucleus. It also duplicates a microtubule-organizing structure called the centrosome (S phase); and
the cell grows still more bigger, makes proteins and organelles, and begins to reorganize its contents in preparation for mitosis (G2 phase).
During the mitotic (M) phase, the cell divides its copied DNA and cytoplasm to make two new cells. M phase involves two distinct division-related processes: mitosis and cytokinesis.
So, if I have got your question right, it is during the interphase (consisting of G1, S and G2) that the cell does not split, but rather prepares itself for the M phase during which the cell divides its DNA and cytoplasm to make two new cells.