1. self orthogonality with codes

Hello,
I'm at a loss when it comes to the next step here
I've got the question:
Let C be the [6; 3] binary code with generator matrix
G =
1 1 0 0 0 0
0 1 1 0 0 0
1 1 1 1 1 1

Is C self-orthogonal? Do the codewords whose weights are divisible by

the generator for the dual code, is the parity matrix for the code C
and a code C is self-orthgonal if $\displaystyle C \subseteq C^\perp$
and if $\displaystyle G=[I_k|A]$ is a generator matrix for the [n,k] code C in standard form then $\displaystyle H=[-A^T|I_{n-k}]$ is a parity check matrix for C

from this I took the matrix and got
G=
1 0 0 1 1 1
0 1 0 1 1 1
0 0 1 1 1 1
by row operations
then I get
$\displaystyle H=C^\perp =$
1 1 1 1 0 0
1 1 1 0 1 0
1 1 1 0 0 1

I don't know what to do from here to show $\displaystyle C \subseteq C^\perp$ or that it's not the case

I'm also unsure of what to do about the second part of the question, where it asks if the codewords of weight divisible by 4 form a subcode

edit:
ok so I know it's not the case
C is not self orthogonal, and the codewords of weight divisible by 4 are not a subcode of C
how do I prove this?