Yes, it is possible.
You are doing something wrong, most likely tagging frames on the router but the trunk has VLAN 1 as native.
To confirm what your trunking settings are use:
show interfaces trunk
I like to keep the subinterface IDs matching the VLAN number although it is not required. Use a configuration like:
! Switch
! Port to Laptop
interface Fax/x
switchport mode access
switchport access vlan 1
spanning-tree portfast
description Port to PC
! Port to Netgear
interface Fax/x
switchport mode access
switchport access vlan 1
spanning-tree portfast
description Port to Netgear
! Port to Cisco router
interface Fax/x
switchport trunk encapsulation dot1q
switchport mode trunk
spanning-tree portfast trunk
description port to Cisco router
For the router:
! Router
! Port to switch
interface E0/0.1
encapsulation dot1q 1 native
ip add 192.168.0.70 255.255.255.0
description VLAN1 to Netgear
interface E0/0.10
encapsulation dot1q 10
ip add 192.168.10.70 255.255.255.0
description VLAN10
interface E0/0.20
encapsulation dot1q 20
ip add 192.168.20.70 255.255.255.0
description VLAN20
ip routing
ip route 0.0.0.0 0.0.0.0 192.168.0.1
Obviously replace IP with your desired IP in the subnet.