1. Identify the group 15 non-metallic elements

   Group 15 elements are: $\mathrm{N,\ P,\ As,\ Sb,\ Bi}$

   Among these, the non-metallic ones are primarily: $\mathrm{N\ and\ P}$

2. Find which one has the weakest $E-E$ bond

   Compare $\mathrm{N-N}$ and $\mathrm{P-P}$ single bonds.

   • In nitrogen, the size is very small, so lone pair-lone pair repulsion is high, making the single bond relatively weak.
   • This is why nitrogen prefers to form a strong multiple bond $\mathrm{N\equiv N}$ rather than stable catenated single bonds.

   Hence, the element $E$ is: $E = \mathrm{N}$

3. Determine the maximum covalency of nitrogen

   Nitrogen has valence shell configuration: $2s^2 2p^3$

   Since nitrogen belongs to the second period, it has no vacant $d$-orbitals and cannot expand its octet.

   Therefore, its maximum covalency is limited to: $4$

   Example: In $\mathrm{NH_4^+}$, nitrogen shows covalency 4.

4. Check the options

   • A: $5$ ❌
   • B: $3$ ❌
   • C: $4$ ✅
   • D: $6$ ❌

Therefore, the correct answer is: $\boxed{4}$