Why does Ubuntu not see disk partitions when reinstalling, but only the physical disk itself?

Question

linux

This problem arose after updating Ubuntu from version 14.04 to 16.04
On a computer win7x64 (works like a clock) + Ubuntu 16.04 (before the update it worked like a clock, now it does not work)
During normal startup, Ubuntu does not display an image, but a startup sound is heard, and when starting in recovery mode does not skip past the password, so I decided to reinstall, but the partitions are not visible.
I found out that the matter is most likely in the partition table, but there is no experience in fixing such problems.

sudo fdisk -l /dev/sda
Диск /dev/sda: 698,7 GiB, 750156374016 байтов, 1465149168 секторов
Единицы измерения: секторов из 1 * 512 = 512 байтов
Размер сектора (логический/физический): 512 байт / 4096 байт
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Тип метки диска: dos
Идентификатор диска: 0xc9c82ac5

Устр-во    Загрузочный     Start Конец Секторы   Size Id Тип
/dev/sda1                   2048   31459327   31457280    15G 27 Hidden NTFS WinRE
/dev/sda2  *            31459328   31664127     204800   100M  7 HPFS/NTFS/exFAT
/dev/sda3               31666174 1465145343 1433479170 683,5G  f W95 расшир. (LBA)
/dev/sda4              226975744  584341503  357365760 170,4G  7 HPFS/NTFS/exFAT
/dev/sda5               31666176   66820095   35153920  16,8G 83 Linux
/dev/sda6               66822144  223070207  156248064  74,5G 83 Linux
/dev/sda7              223072256  226975743    3903488   1,9G 82 Linux своп / Solaris
/dev/sda8              584343552 1465145343  880801792   420G  7 HPFS/NTFS/exFAT

Partition 3 does not start on physical sector boundary.
Записи таблицы разделов не соответствуют порядку разделов на диске.

boot rescue disk info
Advise what to do, demolish everything and reinstall it

Answer 1

you have
/dev/sda3 31666174 1465145343 1433479170 683.5G f W95 ext. (LBA)
overlaps
/dev/sda4 226975744 584341503 357365760 170.4G 7 HPFS/NTFS/exFAT
were there exactly 4 primary partitions?
Or did the window recovery section pop up out of nowhere?
or someone in the middle of the extended section poked the main one?
save markup
boot from Gparted live
move