The order that kata is taught in is dependent on the school and the instructor. Some even tailor it to the individual.

In my style Kusanku Dai is the kata taught after Bassai Dai.

There are many more than 15 karate katas. Each style has their own set of katas and some do not overlap.

In my style there are 20:

5 Kihon Kata

5 Pinan Kata

3 Bassai Kata

3 Naihanchi Kata

2 Kusanku Kata

Chinto

Gojushiho

Depends upon what you consider complicated kicks. The Kusanku katas in Shorin Ryu, as well as Chinto have some tricky kicks in them, but not really complicated.

karate katas:

basic stationary exercize

basic moving exercize

naihanchi shodan kata

basic kicking exercize

naihanchi nidan kata

naihanchi sandan kata

tomari seisan kata

pinan shodan kata

pinan nidan kata

pinan sandan kata

pinan yondan kata

pinan godan kata

passia shodan kata

passia nidan kata

passia sandan kata

passia yondan kata

passia godan kata

heion shodan kata

heion nidan kata

heion sandan kata

heion yondan kata

heion godan kata

heion dai

lot of katas huh

in karate i like doing katas.

the karate kata are called

basic stationary exercise

basic moving exercise

naihanchi shodan kata

basic kicking exercise

naihanchi nidan kata

naihanchi sandan kata

tomari seisan kata

pinan shodan kata

pinan nidan kata

pinan sandan kata

pinan seisan kata

pinan yondan kata

pinan godan kata

lot of katas huh