CGM
You have the order right. There are two extra sub units below Company in an infantry Battalion they are Platoon and Section. The numbers and structures changed during the war but as a guide in 1917:
A Rifle Section would have 10 men in it commanded by a corporal or sergeant. There were other specialist sections for weapons such as Lewis Guns etc but Ill keep it simple.
A Platoon comprised 3 Rifle sections I Lewis Gun section and a HQ group of I Officer (normally a 2nd Lieutenant) and 4 other ranks. Total 46 men.
A Company comprised 4 Platoons and an HQ group of 2 Officers and 57 other ranks. Generally commanded by a Major.
A battalion comprised an HQ unit and 4 Companies with a total of 29 Officers and 1007 other ranks. Commanded by a Lieutenant Colonel
A Brigade comprised 4 Battalions and an HQ unit. Commanded by a Brigadier General
A Division comprised 3 Brigades plus Artillery Engineers etc. Commanded by a Major General
You can see the composition of the 2nd Division as an example here:
http://www.unsw.adfa.edu.au/~rmallett/index.htmlAbove this was a Corps and an Army.
Artillery was arranged in 1917 as follows:
A Detachment of 10 other ranks and an 18 Pounder gun or 4.5 inch Howitzer commanded by a Sergeant
A Section for an 18 pounder this comprised 2 detachments, 1 officer (a Lieutenant), 16 drivers and two other ranks (Howitzers had 2extra drivers)
A Battery comprising an HQ unit and 2 Sections and commanded by a Major
A Brigade which comprised 3 18-poubnder batteries, 1 4.5 inch Howitzer Battery and a HQ unit to a total of 20 Officers and 579 Other ranks. Commanded by a Lieutenant Colonel.
Hope that helps.
Tim B