As I understand it, during this time Japan was loosely controlled by Toyotomi Hideyoshi; but his Hashiba clan didn't invade Joseon Korea as he had other clans do that. I check the list of combatants on the Imjin War's Wikipedia page and there are a lot for the Japanese, as I assume most of these were token examples of participation so they could get a slice of the pie.
So which clans were the one or two or three whom provided the majority of the invasion force for Korea (under the direction of the Hashiba)? Was it the Shimazu? The Mori? The Ottomo? The Chosokabe? Etc.
The clan that provided the majority of the forces was in fact the Hashiba clan.
Below is taken from the Mōri clan records for the expedition of 1592. The title of each is a rough translation by me of what they were intended in classical Japan. By the Sengoku (in fact, long before) the titles no longer had any relation to what the title-bearers did. The people that the document labels as of the Hashiba clan or we know from other sources were already going by the Hashiba name at the time are bolded.
| Division | Number | Name and Title | Popularly Known As |
|---|---|---|---|
| First | 5,000 | Hashiba, Lord of Tsushima, Aide-de-camp | Sō Yoshitoshi |
| First | 7,000 | Konishi, Lord of Settsu | Konishi Yukinaga |
| First | 3,000 | Matsuura, Minister of Ceremonial Affairs, Dharma Seal | Matsuura Shigenobu |
| First | 2,000 | Arima, Chief of Palace Repairs | Arima Harunobu |
| First | 1,000 | Ōmura Shinpachi | Ōmura Yoshiaki |
| First | 700 | Gotō, Lord of Yamato | Gotō Sumiharu |
| Second | 10,000 | Katō, Head of Central Accounting | Katō Kiyomasa |
| Second | 12,000 | Nabeshima, Lord of Kaga | Nabeshima Naoshige |
| Second | 800 | Sagara, Junior assistant minister of the Imperial Household | Sagara Yorifusa |
| Third | 5,000 | Kuroda, Lord of Kai | Kuroda Nagamasa |
| Third | 6,000 | Hashiba, Lord of Bungo, Aide-de-camp | Ōtomo Yoshimune |
| Fourth | 2,000 | Mōri, Lord of Iki | Mōri Katsunobu |
| Fourth | 10,000 | Hashiba, Lord of Satsuma, Aide-de-camp | Shimazu Yoshihiro |
| Fourth | Takahashi Kurō | Takahashi Mototane | |
| Fourth | Akizuki Saburō | Akizuki Tanenaga | |
| Fourth | Itō, Senior Assistant Minister of Popular Affairs | Itō Suketaka | |
| Fourth | 2,000* | Shimazu Matashichirō | Shimazu Toyohisa |
| Fifth | 4,800 | Fukushima, Chief of the Imperial Guards of the Left** | Fukushima Masanori |
| Fifth | 3,900 | Toda, Junior Assistant Minister of Popular Affairs | Toda Katsutaka |
| Fifth | 3,000 | Hashiba, Lord of Tosa, Aide-de-camp | Chōsokabe Motochika |
| Fifth | 7,200 | Hachisuka, Lord of Awa | Hachisuka Iemasa |
| Fifth | 5,500 | Ikoma, Head of Department of Songs | Ikoma Chikamasa |
| Fifth | 700 | Kurushima Brothers | Kurushima (Tokui) Michiyuki and Kurushima Michifusa |
| Sixth | 10,000 | Hashiba, Lord of Chikuzen, Aide-de-camp | Kobayakawa Takakage |
| Sixth | 1,500 | Hashiba, Lord of Kurume, Aide-de-camp | Mōri (Kobayakawa) Hidekane |
| Sixth | 2,500 | Hashiba, Lord of Yanagawa, Aide-de-camp | Tachibana Muneshige |
| Sixth | 800 | Takahashi, Imperial Steward | Takahashi (Tachibana) Naotsugu |
| Sixth | 900 | Tsukushi, Deputy of Kōzuke | Tsukushi Hirokado |
| Seventh | 30,000 | Lord Chancellor of Aki | Mōri Terumoto |
| Eighth | 10,000 | Lord Chancellor of Bizen | Ukita Hideie |
| Nighth | 8,000 | Lord Chancellor of Gifu | Hashiba (Toyotomi) Hidekatsu |
| Nighth | 3,500 | Lord of Tango, Major General | Hosokawa Tadaoki |
*Sum of the four.
**The title doesn't exist. Tayū could be a mistake for Daii, or Captain.
Going by the document, of the planned 158,800 invasion force, 89,500 (56.4%) were provided by the Hashiba clan. Every single division had at least one unit provided by the Hashiba clan except for the second division, which had Katō Kiyomasa, who's Hideyoshi's cousin once removed. If we add Katō Kiyomasa and Fukushima Masanori, who's Hideyoshi's cousin and who would later be given the Hashiba name, both of whom were raised as close members of the clan, then the Hashiba clan provided 104,300 (65.7%) of the invasion force. If we include the men who were under Hideyoshi already when Nobunaga was alive (italicized) then the Hashiba clan provided 127,900 men (80.5%)
Of course, going down the list we can see that most of the force was not provided by people who we'd label as members of the Hashiba clan, and who likely did not label themselves, at least in private, as the Hashiba clan. Hideyoshi, like many others, "adopted" important people by giving them his clan name, in an attempt to ensure their loyalty and tie them to his cause. He also awarded them with titles for the same reason. Of course, few people would be loyal to only names and titles, and after Sekigahara the vast majority would abandon the name. Of note that the Tokugawa did the same thing, and many of the clans above officially were members of the Matsudaira clan during the Edo period.
If we don't go by the names on the list, but only those who were already under Hideyoshi when Nobunaga was alive, the Hashiba clan still provided the most men. The men already under Hideyoshi before Nobunaga's death would provide 60,400 (38.0%) of the invasion force. If we include Ukita Hideie, whom Hideyoshi had adopted either before or on his maturity, was close to him, and appointed nominal commander of the invasion force, then the Hashiba clan provided 70,400 (44.3%) of the invasion force. The Mōri (Mōri Terumoto, Kobayakawa Takakage, Mōri Hidekane) came second with 41,500 (26.1%). The Mōri were among Hideyoshi's earliest and most power vassals after Nobunaga's death, who had joined the conquest of Shikoku and Kyūshū and were the only ones to be assigned a division without someone close to the Hashiba, showing Hideyoshi's trust in them.