tl; dr: ZigBee isn’t a subset of WiFi (802.11). It is a different protocol family, just as Bluetooth is another different protocol family. ZigBee, BT and WiFi are not directly compatible with each other, but will co-exist in the same band.
What Is ZigBee, And How Is It Different From 802.11, Really?
ZigBee (802.15.4) is a family of protocols used for low- to medium data rates, targeting small devices / Internet of Things. ZigBee is designed to be inexpensive, and lower-power compared to WiFi.
ZigBee top throughput is 250kbps when it operates in the 2.4GHz ISM band it shares with 802.11 and Bluetooth. (ZigBee also supports 868 and 915MHz bands at lower data rates.) ZigBee employs Direct Sequence Spread Spectrum (DSSS) and uses an Offset Quadrature Phase Shift Keying (O-QPSK) to modulate the RF carrier, on one 2MHz channel at a time (no frequency-hopping, e.g., like Bluetooth).
In contrast, the 802.11 family covers a broad range of coding and modulation schemes that range from very simple to quite complex (e.g., MIMO), with data rates starting out at 11Mbit all the way up to multi-Gbit throughput. 802.11 channels are also much fatter: 20, or 40MHz wide in the 2.4GHz band.
Finally, 802.11 RF power starts out at 100mW and goes up from there to as high as 250mW in the 2.4GHz band. ZigBee is between 1mW and 100mW.
ZigBee And WiFi Together?
Not that I was aware of, at least in the same chip, but see below. It is technically possible to build a dual-radio ZigBee + 802.11, just as there are dual-radio 802.11 + BT chips. For a long time there wasn’t a big push for that. This probably has to do with ZigBee’s sharper focus on low power and low-bandwidth data, and not streaming applications like headphone audio that are the realm of Bluetooth.
EDIT: Apparently such a multi-radio beast exists: https://developer.qualcomm.com/blog/need-wi-fi-ble-and-zigbee-same-iot-application-qca4020-qca4024-development-kits-now-available
Can They Get Along?
There are 16 ZigBee channels defined in the 2.4GHz band, spaced 5MHz apart, which are shared with WiFi. The graphic below shows how ZigBee channels overlay 802.11:
From here: https://support.metageek.com/hc/en-us/articles/203845040-ZigBee-and-WiFi-Coexistence
At times WiFi and ZigBee (and Bluetooth) will also overlap each other on application use. For example, newer voice-capable DVR and TV set-tops use either RF4CE (that is, the ZigBee RF4CE profile) or Bluetooth / BLE to support wireless voice remotes, and will also have WiFi to do over-the-top streaming.
Nevertheless, despite using the same band, and sometimes even being present in the same device, ZigBee and WiFi are not directly compatible with each other at the wireless level. This is the exact same situation as for Bluetooth.
ZigBee and 802.11 will see each other as noise, and work to avoid each other through various methods collectively referred to as co-existence, described here.