How to display posts in hyperlink with nested slugs?

This can be achieved by adding rewrite rules.

First, add the query vars that will store the custom values in our rules:

function wpd_family_query_var( $query_vars ) {
    $query_vars[] = 'wpd_family';
    $query_vars[] = 'wpd_members';
    return $query_vars;
}
add_filter( 'query_vars', 'wpd_family_query_var' );

Next, add the rules to capture both displaypage/family/ and displaypage/family/members/ requests:

function wpd_add_family_rewrite_rules() {
    add_rewrite_rule(
        'displaypage/([^/]+)/?$',
        'index.php?pagename=displaypage&wpd_family=$matches[1]',
        'top'
    );
    add_rewrite_rule(
        'displaypage/([^/]+)/members/?$',
        'index.php?pagename=displaypage&wpd_family=$matches[1]&wpd_members=true',
        'top'
    );
}
add_action( 'init', 'wpd_add_family_rewrite_rules' );

Don’t forget, you must flush rewrite rules after changing them.

You will then be able to access the values of your query vars anywhere after the parse_query action with get_query_var():

if( $family = get_query_var( 'wpd_family' ) ){
    echo 'family: ' . $family;
}

if( $members = get_query_var( 'wpd_members' ) ){
    echo 'members: ' . $members;
}