So we seem to be looking at ...
Charles de Bourbon, Duke of Vendome (to 1537), b. 1489, d. 1537, m. Francoise d'Alencon
3) Marguerite de Bourbon, b. 1516, d. 1579, m. Francis, Duke of Nevers, has issue
4) Antoine de Bourbon, Duke of Vendome (1537 to 1549), b. 1518, d. 1549, m. Jeanne of Navarre, no issue
8) Charles X of France, prev. Archbishop of Rouen, b. 1523, r. 1589 to 1590, never married
12) Louis de Bourbon, Prince of Conde, b. 1530, d. 1569, m1. Eleanor de Roye, m2. Francois d'Orleans
1a) Henri, Prince of Conde, b. 1552, d. 1588, m1. Marie of Cleeves, m2. Charlotte Catherine de La Tremoile
a1) Catherine, b. 1574, d. 1595
b1) Eleanor de Bourbon, b. 1587, d. 1619, m. William, Prince of Orange, no issue
b2) Henri IV of France, b. 1588, r. 1590 to 1646, m. Anne of Austria (1601 to 1666)
1) Anne I of France, b. 1638, r. 1646 to 1715, m. Henri V of France (1628 to 1709)
a) Henri, Dauphin of France, b. 1661
2) Charlotte of France, b. 1640, m. Louis, Count of Armagnac (descendant of Antoinette de Bourbon, sister of Charles, Duke of Vendome)
a) several children
1d) Francois, Prince of Conti, b. 1558, d. 1614, married, only one child who died in infancy
1e) Charles, Archbishop of Rouen, b. 1562, d. 1594, Regent of France 1590 to 1594
2a) Charles, Count of Soissons, b. 1566, d. 1612, Regent of France, 1594 to 1606, m. Anne de Montafie
a) Louis, Count of Soissons, b. 1604, d. 1641, First Prince of the Blood, never married
1) Louis Henri, Batard de Soissons, b. 1640, nominal claimant to French throne as Louis XIII
b) Louise de Bourbon, Countess of Longueville, b. 1603, d. 1637, m. Henri d'Orleans, Duke of Longueville
1) Marie d'Orleans Longueville, b. 1625, d. 1707, m. Henri of Savoy, Duke of Nemours, no issue
c) Marie de Bourbon, Princess of Carignano, b. 1606, d. 1692, m. Thomas Francis, Prince of Carignano
2) Louise of Savoy, b. 1627
3) Henri V of France, (Emmanuel Philibert of Savoy), b. 1628, r. 1646 to 1709, (e). Anne I of France
x) for issue, see line of Anne I of France
5) Joseph Emmanuel, Duke of Maine, b. 1631
6) Eugene Maurice, Duke of Normandy, b. 1633
The Pragmatic Concordat of 1644 states that the French crown may be inherited through, but not directly by, a female claimant.
The crown may be transmitted across only a single generation - thus the closest claimants after the House of Savoy are the House of Lorraine as they descend from the sons of Antoinette de Bourbon, thus why Charlotte of France is married to the Count of Armagnac (the Duke of Joyuese is proposed first, but he dies in 1654)