Grouping the sports:
To determine which sport does not belong to the group, let's analyze the similarities and differences among the given options.

Badminton:
- Badminton is a racket sport played with a shuttlecock.
- It is usually played indoors.
- The objective is to hit the shuttlecock over the net and into the opponent's side of the court.

Table Tennis:
- Table tennis, also known as ping pong, is a racket sport played with a small ball on a table divided by a net.
- It is also played indoors.
- The objective is to hit the ball over the net and onto the opponent's side of the table.

Lawn Tennis:
- Lawn tennis, commonly referred to as just tennis, is a racket sport played on a rectangular court divided by a net.
- It is played outdoors on grass, clay, or hard surfaces.
- The objective is to hit the ball over the net and within the boundaries of the court.

Hockey:
- Hockey is a team sport played on a field with curved sticks and a ball.
- It can be played on both grass and synthetic surfaces.
- The objective is to score goals by hitting the ball into the opponent's net.

Rugby:
- Rugby is a contact team sport played with an oval-shaped ball.
- It is typically played on grass fields.
- The objective is to score points by carrying, passing, or kicking the ball into the opponent's goal area.

Identifying the odd one out:
From the analysis above, we can observe that badminton, table tennis, lawn tennis, and hockey are all racket or stick sports played with a ball or shuttlecock. These sports also involve hitting the ball or shuttlecock over a net or towards a goal. Therefore, they form a group based on their similar characteristics.

On the other hand, rugby stands out as the odd one out. It is a different type of sport that involves carrying and passing an oval-shaped ball, rather than using a racket or stick. Additionally, rugby has different rules and objectives compared to the other sports mentioned. Hence, option E, Rugby, is the one that does not belong to the group.