consecutive 連続した
 It has been raining for five consecutive days. (キクタン990)
consecutively 連続して、連続的に、継続的な
consequent ~の結果として起こる
 the oil embargo and consequent fuel shortages
  石油輸出停止とその結果として起きた燃料不足 (ムズイ!) weblio
consequently その結果
 It started to rain heavily and consequently the baseball game was called off. (キクタン800)
subsequent (~の)後[次]の、後に起こる(to~)≒following
 Subsequent to Mr. Brown's resignation, Mr.Harrison was appointed to the position of CEO.  (キクタン990)
subsequently 後で

sequ=続く
 sequel to ~ ~の続編
 sequence 連続 (シーケンサー=音楽の打ち込み自動演奏機器)
Subは「下」なので、下に続く=「後」でよいでしょう。

consequent と consecutive は単純に暗記します。(cとqの違いが腹立つ!)