conditional for sub category on archive page

You are looking for the cat_is_ancestor_of() function. https://codex.wordpress.org/Function_Reference/cat_is_ancestor_of

The example in the WP codex is very close to what you are attempting for your archive template.

<?php 
  // if the category is music or a music SUBcategory, 
   if (cat_is_ancestor_of( 4, $cat ) or is_category( 4 ) ):  ?>
   <div id="music_subnav_menu" class="subnav_menu">
    <?php wp_nav_menu( array( 'menu' => 'Music' ) ); ?>
  </div>
 <?php endif; ?>