EMB-145 Captain reported an engine failure occurred during cruise flight and diverted to an alternate airport.

Narrative

We had leveled at FL290 when there was a noise and the aircraft did a 10-degree to the left and the autopilot brought [it] right back and on the EICAS we got the Engine 1 out message; SHED BUS OFF; and RUD HDOV PROTFAIL. I kept on flying and started to do the radios and the navigation. The First Officer went to the QRH and did the Engine Out Checklist. Also; we decided to do a restart with the APU because it was already on per engine out procedure without being able to get a restart. During this time I sent a message to Dispatch; informed ATC; and requested if ATC could get me for ZZZ1 that was 80 miles away; but Dispatch informed us that ZZZ2 was good; the visibility was 10 miles; and the ceiling at 1600; so informed ATC that we would be going to ZZZ2. Also we requested lower right away for 2 reasons; one was because of the single pack and the other because the speed was degrading. Also informed the Flight Attendant a few times what was going on [and] told her that it was going to be a normal landing. Also kept the passengers informed by doing a few PAs. We also did QRH for RUD HDOV PROTFAIL which in the air nothing to be done. At no time did we deviated from altitude or course. All altitude and heading changes were done by request with ATC or ATC given. Also all the checklists were done from the QRH.