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<ul>
  <?php
      $categoryID = get_category_by_slug( 'blog' ); /*Retrieve category object by category slug*/
      $categoryID = $categoryID ->term_id; /*Get the term id*/
      $args = array(
        'child_of' => $categoryID,
        'title_li' => __( '' )
      ); 
      wp_list_categories( $args ); /*Display or retrieve the HTML list of categories.*/
  ?>
</ul>