CREATE TABLE `chains` (
  `chain_id` int(10) unsigned NOT NULL AUTO_INCREMENT,
...
  PRIMARY KEY (`chain_id`),
...
) ENGINE=InnoDB

CREATE TABLE `docs` (
  `doc_id` int(10) unsigned NOT NULL AUTO_INCREMENT,
  `chain_id` int(10) unsigned NOT NULL,
  `id_status` tinyint(3) unsigned NOT NULL COMMENT 'open,sent,inbox,read,etc',
  `date_created` int(10) unsigned NOT NULL DEFAULT '0',
...
  PRIMARY KEY (`doc_id`),
  KEY `id_status` (`id_status`),
  KEY `family` (`family`),
  KEY `chain_id` (`chain_id`),
  KEY `date_created` (`date_created`),
) ENGINE=InnoDB

SELECT `chains`.`chain_id`
FROM `chains`
JOIN `docs` USING (`chain_id`)
WHERE `docs`.`family`=1
AND `docs`.`id_status` IN (4,5)
AND `chains`.`is_archive`=0
GROUP BY `chain_id`
ORDER BY `date_created` DESC
LIMIT 0,200

+----+-------------+--------+------+---------------------------------+------------+---------+--------------------------+---------+----------------------------------------------+
| id | select_type | table  | type | possible_keys                   | key        | key_len | ref                      | rows    | Extra                                        |
+----+-------------+--------+------+---------------------------------+------------+---------+--------------------------+---------+----------------------------------------------+
|  1 | SIMPLE      | chains | ref  | PRIMARY,is_archive              | is_archive | 1       | const                    | 2789080 | Using index; Using temporary; Using filesort |
|  1 | SIMPLE      | docs   | ref  | id_status,family,chain_id,test2 | chain_id   | 4       | for_test.chains.chain_id |       1 | Using where                                  |
+----+-------------+--------+------+---------------------------------+------------+---------+--------------------------+---------+----------------------------------------------+

SELECT d.chain_id
FROM docs AS d
WHERE d.family = 1
  AND d.id_status IN (4,5)
  AND EXISTS
      ( SELECT *
        FROM chains AS c
        WHERE c.is_archive = 0
          AND c.chain_id = d.chain_id
      )
GROUP BY d.chain_id
ORDER BY MAX(d.date_created) DESC
LIMIT 200 OFFSET 0 ;

SELECT d.chain_id
FROM docs AS d
  JOIN ( SELECT c.chain_id
         FROM chains AS c
         WHERE c.is_archive = 0
       ) AS c
    ON c.chain_id = d.chain_id
WHERE d.family = 1
  AND d.id_status IN (4,5)
GROUP BY d.chain_id
ORDER BY MAX(d.date_created) DESC
LIMIT 200 OFFSET 0 ;

docs
(id_status, family, chain_id, date_created)

chains
(id_archive, chain_id)