/************************************************************\
* LAST MODIFIED COLUMN IN /ADMIN
\************************************************************/
function add_last_modified_column($columns) {
    return array_merge( $columns,
              array("last_modified" => __("Last Modified")) );
}
add_filter("manage_posts_columns" , "add_last_modified_column");

function last_modified_column( $column, $post_id ){
  if( ! "last_modified" == $column ) {
  the_modified_time('F j, Y');
  }
}

add_action( "manage_posts_custom_column" , "last_modified_column", 10, 2 );

function last_modified_column_register_sortable( $columns ) {
    $columns["last_modified"] = "last_modified";

    return $columns;
}

add_filter( "manage_edit-post_sortable_columns", "last_modified_column_register_sortable" ); // replace "post" with the slug of your custom post type

function sort_column_by_modified( $vars ){
    if ( isset( $vars["orderby"] ) && "last_modified" == $vars["orderby"] ) {
        $vars = array_merge( $vars, array(
            "orderby" => "modified"
        ) );
    }
    return $vars;
}
add_filter( "request", "sort_column_by_modified" );