I'm not sure if the comment on your question was a comment from Random or your deleted post but from your second screen-cap you do in fact have an extended partition at the start of your drive that is housing the three Linux partitions, for a total of 3 actual primary partitions.
This is corroborated by the following, though it appears that Linux is enumerating the primary partitions first even though they are located towards the end of the disk.
Device Boot Start End Blocks Id System
/dev/sda1 6536 28220 174183530+ 7 HPFS/NTFS
/dev/sda2 28221 38913 85891522+ c W95 FAT32 (LBA)
/dev/sda3 1 6535 52492356 5 Extended
/dev/sda5 * 1 3246 26073432 83 Linux
/dev/sda6 6015 6535 4184901 82 Linux swap / Solaris
/dev/sda7 3247 6014 22233928+ 83 Linux
Specifically note that the extended partition is at the start of the disk and encloses the Linux partitions (the second and third numbers in the columns are block start and stop numbers).
Windows does not appear to handle this case very well and while it is correctly reporting the positions of the partitions it is not actually showing the extended partition.
Your partition scheme is technically legal and I doubt it will cause any problems, but it is a bit unusual to say the least.