How do I find out if my wireless card supports 5 GHz?

Find out the interface name, by running iwconfig

$ iwconfig
eth0      no wireless extensions.

lo        no wireless extensions.

wlan0     IEEE 802.11bgn  ESSID:"EvanCarroll"  
          Mode:Managed  Frequency:2.437 GHz  Access Point: D8:50:E6:44:B2:C8   
          Bit Rate=19.5 Mb/s   Tx-Power=15 dBm   
          Retry  long limit:7   RTS thr:off   Fragment thr:off
          Power Management:off
          Link Quality=61/70  Signal level=-49 dBm  
          Rx invalid nwid:0  Rx invalid crypt:0  Rx invalid frag:0
          Tx excessive retries:1  Invalid misc:80   Missed beacon:0

In this case it is wlan0, then run iwlist <interface> freq,

$ iwlist wlan0 freq
wlan0     13 channels in total; available frequencies :
          Channel 01 : 2.412 GHz
          Channel 02 : 2.417 GHz
          Channel 03 : 2.422 GHz
          Channel 04 : 2.427 GHz
          Channel 05 : 2.432 GHz
          Channel 06 : 2.437 GHz
          Channel 07 : 2.442 GHz
          Channel 08 : 2.447 GHz
          Channel 09 : 2.452 GHz
          Channel 10 : 2.457 GHz
          Channel 11 : 2.462 GHz
          Channel 12 : 2.467 GHz
          Channel 13 : 2.472 GHz
          Current Frequency:2.437 GHz (Channel 6)

None of these channels are outside of 2.4 GHz. It does not support 5 GHz.

If you're trying to discover what your card supports, iw phy is a nice solution with a lot more information (including supported bands).

iwlist is showing more what's available and/or allowed in your locale, what was disabled due to DFS channels, etc., not what your device supports. From the iwlist man page:

   freq[uency]/channel
          Give  the  list of available frequencies in the device and the number of defined channels. Please note that usually the
          driver returns the total number of channels and only the frequencies available in the present locale, so  there  is  no
          one-to-one mapping between frequencies displayed and channel numbers.