[Grammar] Syntactic analysis of complex sentences

Hello,

I have to do a syntactic analysis in terms of SVOAC (subject, verb, object, adverbial, complement) of this sentence (it is a second part of a compound sentence):
"...and then had supper, after which we started for the hills armed with three bags, which, by a happy chance, were in the house."

then had supper, after which - adverbial (time clause?)
we - subject
started for - verb
the hills - adverbial
armed with three bags, which, by a happy chance, were in the house - postponed attribute of subject?

Could you please help me with the analysis? Thanks very much.

[susiedqq]

after which we started for the hills armed with three bags, which, by a happy chance, were in the house."

we / started
for / hills
armed with three bags describes how we "started"

which, by a happy . . . house" decribes the bags.