If the head of cockroach is removed, it may live for few days because:

the supra-oesophageal ganglia of the cockroach are situated in ventral part of abdomen.

the cockroach does not have nervous system.

the head holds a small proportion of a nervous system while the rest is situated along the ventral part of its body.

the head holds a $\displaystyle 1 / 3^{\mathrm{rd}}$ of a nervous system while the rest is situated along the dorsal part of its body.