I've been using an online wallet and I don't have a problem.
These kinds of sites have their own security, you can always connect your email, some have their 2FA, some even require a phone number and I am wondering how can someone access your wallet without you being notified of that. Two points to consider, it is either your fault or the sites fault but I don't think the site will just let someone use your balance without logging to your account.
This seems to be a personal error, it is not possible that it was the fault of Blockchain. I also use Blockchain and there are no problems while using this wallet. Bitcoin will not move on its own if no one controls the address and most likely it is because of hackers. The money moves 2 times in two different addresses and stops at this address 1L5tLc6ayc3RXAyicHRv2c2uSNVJzoP4Yp
https://www.walletexplorer.com/wallet/Huobi.com-2?from_address=1L5tLc6ayc3RXAyicHRv2c2uSNVJzoP4YpThat's a wallet belonging to Huobi... I'd contact their support department and ask them to look into this transaction... However, since the funds moved 2 times, it's perfectly possible the thief used a mixer...
Just to summarise:
1FgpbfQ8nVMS523Btnyv3rxhF7YxtiEdr7 belonged to you...
The thief robbed you in tx 4d412c58ab3dba8a138557e41e9ace1c8b36ee3d44107a930c8f31f0198d2b93 and funded what is most likely either his own address or an address belonging to a mixer => 1PdXxtu8Laq773RXT5q5pp6qtaG3HvSw2r
The unspent output created by the thief was combined with unspent outputs funding the following addresses:
1MXVwghCaz7yPvuyJU35yQdQHnpA9ea8PA (0.06180723 BTC - Output)
1H6htL41r9v8nAAZdpmaxw35kyjCewS9um (0.04020064 BTC - Output)
1H6htL41r9v8nAAZdpmaxw35kyjCewS9um (0.35708617 BTC - Output)
1JHLWuWsDqYwxzrjYcxZ6K7Feiv5R9SyvF (0.03159732 BTC - Output)
1PdXxtu8Laq773RXT5q5pp6qtaG3HvSw2r (0.03435393 BTC - Output)
1DbovmqydEvgFjLPfA6brASV98cmEy8gy2 (0.13459853 BTC - Output)
1MXVwghCaz7yPvuyJU35yQdQHnpA9ea8PA (0.01590557 BTC - Output)
1Q2qYzdeqd1KJdRJwMBqjft9mnMzc98fdp (0.01895248 BTC - Output)
1M7T8LZeJopJMffYRcpvviFvb11eKMLb1j (0.03038733 BTC - Output)
1Q2qYzdeqd1KJdRJwMBqjft9mnMzc98fdp (0.00673544 BTC - Output)
1FWSCxbScSfwrdetudCqArD5VND71JfPiy (0.0296867 BTC - Output)
1Q7SmKLD5Vs597oBZBAaviSzTfLwkZbXLd (0.11753431 BTC - Output)
to fund 1L5tLc6ayc3RXAyicHRv2c2uSNVJzoP4Yp
1L5tLc6ayc3RXAyicHRv2c2uSNVJzoP4Yp belongs to huobi... So somebody is exchanging those funds, either to other altcoins or to fiat... If huobi requires KYC, the police should be able to either find the thief, or the person who was unlucky enough to use a mixer and end up with the thief's funds after mixing...