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Elements of the group are displayed as matrices in $\GL_{2}(\F_{353})$.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 11P | ||||||
| $C_8.D_{44}$ | 1A | $1$ | $1$ | $C_8.D_{44}$ | 1A | 1A | $ \left(\begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_8.D_{44}$ | 2A | $2$ | $1$ | $C_8.D_{44}$ | 1A | 2A | $ \left(\begin{array}{rr} 352 & 0 \\ 0 & 352 \end{array}\right) $ |
| $C_8.D_{44}$ | 2B | $2$ | $2$ | $C_2\times C_{176}$ | 1A | 2B | $ \left(\begin{array}{rr} 352 & 0 \\ 0 & 1 \end{array}\right) $ |
| $C_8.D_{44}$ | 2C | $2$ | $44$ | $C_2\times C_8$ | 1A | 2C | $ \left(\begin{array}{rr} 0 & 331 \\ 16 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 4A1 | $4$ | $1$ | $C_8.D_{44}$ | 2A | 4A1 | $ \left(\begin{array}{rr} 311 & 0 \\ 0 & 311 \end{array}\right) $ |
| $C_8.D_{44}$ | 4A-1 | $4$ | $1$ | $C_8.D_{44}$ | 2A | 4A-1 | $ \left(\begin{array}{rr} 42 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_8.D_{44}$ | 4B | $4$ | $2$ | $C_2\times C_{176}$ | 2A | 4B | $ \left(\begin{array}{rr} 311 & 0 \\ 0 & 42 \end{array}\right) $ |
| $C_8.D_{44}$ | 4C | $4$ | $44$ | $C_2\times C_8$ | 2A | 4C | $ \left(\begin{array}{rr} 0 & 42 \\ 42 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 8A1 | $8$ | $1$ | $C_8.D_{44}$ | 4A1 | 8A-3 | $ \left(\begin{array}{rr} 283 & 0 \\ 0 & 283 \end{array}\right) $ |
| $C_8.D_{44}$ | 8A-1 | $8$ | $1$ | $C_8.D_{44}$ | 4A-1 | 8A3 | $ \left(\begin{array}{rr} 237 & 0 \\ 0 & 237 \end{array}\right) $ |
| $C_8.D_{44}$ | 8A3 | $8$ | $1$ | $C_8.D_{44}$ | 4A-1 | 8A-1 | $ \left(\begin{array}{rr} 116 & 0 \\ 0 & 116 \end{array}\right) $ |
| $C_8.D_{44}$ | 8A-3 | $8$ | $1$ | $C_8.D_{44}$ | 4A1 | 8A1 | $ \left(\begin{array}{rr} 70 & 0 \\ 0 & 70 \end{array}\right) $ |
| $C_8.D_{44}$ | 8B1 | $8$ | $2$ | $C_2\times C_{176}$ | 4A1 | 8B1 | $ \left(\begin{array}{rr} 283 & 0 \\ 0 & 70 \end{array}\right) $ |
| $C_8.D_{44}$ | 8B-1 | $8$ | $2$ | $C_2\times C_{176}$ | 4A-1 | 8B-1 | $ \left(\begin{array}{rr} 237 & 0 \\ 0 & 116 \end{array}\right) $ |
| $C_8.D_{44}$ | 8C1 | $8$ | $44$ | $C_2\times C_8$ | 4A-1 | 8C1 | $ \left(\begin{array}{rr} 0 & 116 \\ 116 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 8C-1 | $8$ | $44$ | $C_2\times C_8$ | 4A1 | 8C-1 | $ \left(\begin{array}{rr} 0 & 70 \\ 70 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 11A1 | $11$ | $2$ | $C_2\times C_{176}$ | 11A2 | 11A5 | $ \left(\begin{array}{rr} 256 & 0 \\ 0 & 131 \end{array}\right) $ |
| $C_8.D_{44}$ | 11A2 | $11$ | $2$ | $C_2\times C_{176}$ | 11A4 | 11A1 | $ \left(\begin{array}{rr} 231 & 0 \\ 0 & 217 \end{array}\right) $ |
| $C_8.D_{44}$ | 11A3 | $11$ | $2$ | $C_2\times C_{176}$ | 11A5 | 11A4 | $ \left(\begin{array}{rr} 185 & 0 \\ 0 & 187 \end{array}\right) $ |
| $C_8.D_{44}$ | 11A4 | $11$ | $2$ | $C_2\times C_{176}$ | 11A3 | 11A2 | $ \left(\begin{array}{rr} 58 & 0 \\ 0 & 140 \end{array}\right) $ |
| $C_8.D_{44}$ | 11A5 | $11$ | $2$ | $C_2\times C_{176}$ | 11A1 | 11A3 | $ \left(\begin{array}{rr} 22 & 0 \\ 0 & 337 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A1 | $16$ | $2$ | $C_2\times C_{176}$ | 8B1 | 16A5 | $ \left(\begin{array}{rr} 304 & 0 \\ 0 & 60 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A-1 | $16$ | $2$ | $C_2\times C_{176}$ | 8B-1 | 16A-5 | $ \left(\begin{array}{rr} 36 & 0 \\ 0 & 253 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A3 | $16$ | $2$ | $C_2\times C_{176}$ | 8B-1 | 16A-1 | $ \left(\begin{array}{rr} 253 & 0 \\ 0 & 317 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A-3 | $16$ | $2$ | $C_2\times C_{176}$ | 8B1 | 16A1 | $ \left(\begin{array}{rr} 60 & 0 \\ 0 & 49 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A5 | $16$ | $2$ | $C_2\times C_{176}$ | 8B1 | 16A-7 | $ \left(\begin{array}{rr} 293 & 0 \\ 0 & 304 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A-5 | $16$ | $2$ | $C_2\times C_{176}$ | 8B-1 | 16A7 | $ \left(\begin{array}{rr} 100 & 0 \\ 0 & 36 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A7 | $16$ | $2$ | $C_2\times C_{176}$ | 8B-1 | 16A3 | $ \left(\begin{array}{rr} 317 & 0 \\ 0 & 100 \end{array}\right) $ |
| $C_8.D_{44}$ | 16A-7 | $16$ | $2$ | $C_2\times C_{176}$ | 8B1 | 16A-3 | $ \left(\begin{array}{rr} 49 & 0 \\ 0 & 293 \end{array}\right) $ |
| $C_8.D_{44}$ | 16B1 | $16$ | $44$ | $C_{16}$ | 8A3 | 16B-3 | $ \left(\begin{array}{rr} 0 & 78 \\ 92 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 16B-1 | $16$ | $44$ | $C_{16}$ | 8A-3 | 16B3 | $ \left(\begin{array}{rr} 0 & 165 \\ 86 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 16B3 | $16$ | $44$ | $C_{16}$ | 8A1 | 16B-1 | $ \left(\begin{array}{rr} 0 & 223 \\ 82 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 16B-3 | $16$ | $44$ | $C_{16}$ | 8A-1 | 16B1 | $ \left(\begin{array}{rr} 0 & 254 \\ 19 & 0 \end{array}\right) $ |
| $C_8.D_{44}$ | 22A1 | $22$ | $2$ | $C_2\times C_{176}$ | 11A1 | 22A5 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 331 \end{array}\right) $ |
| $C_8.D_{44}$ | 22A3 | $22$ | $2$ | $C_2\times C_{176}$ | 11A3 | 22A7 | $ \left(\begin{array}{rr} 213 & 0 \\ 0 & 295 \end{array}\right) $ |
| $C_8.D_{44}$ | 22A5 | $22$ | $2$ | $C_2\times C_{176}$ | 11A5 | 22A3 | $ \left(\begin{array}{rr} 166 & 0 \\ 0 & 168 \end{array}\right) $ |
| $C_8.D_{44}$ | 22A7 | $22$ | $2$ | $C_2\times C_{176}$ | 11A4 | 22A9 | $ \left(\begin{array}{rr} 136 & 0 \\ 0 & 122 \end{array}\right) $ |
| $C_8.D_{44}$ | 22A9 | $22$ | $2$ | $C_2\times C_{176}$ | 11A2 | 22A1 | $ \left(\begin{array}{rr} 222 & 0 \\ 0 & 97 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B1 | $22$ | $2$ | $C_2\times C_{176}$ | 11A1 | 22B5 | $ \left(\begin{array}{rr} 16 & 0 \\ 0 & 22 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B-1 | $22$ | $2$ | $C_2\times C_{176}$ | 11A1 | 22B-5 | $ \left(\begin{array}{rr} 331 & 0 \\ 0 & 337 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B3 | $22$ | $2$ | $C_2\times C_{176}$ | 11A3 | 22B-7 | $ \left(\begin{array}{rr} 58 & 0 \\ 0 & 213 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B-3 | $22$ | $2$ | $C_2\times C_{176}$ | 11A3 | 22B7 | $ \left(\begin{array}{rr} 140 & 0 \\ 0 & 295 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B5 | $22$ | $2$ | $C_2\times C_{176}$ | 11A5 | 22B3 | $ \left(\begin{array}{rr} 166 & 0 \\ 0 & 185 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B-5 | $22$ | $2$ | $C_2\times C_{176}$ | 11A5 | 22B-3 | $ \left(\begin{array}{rr} 168 & 0 \\ 0 & 187 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B7 | $22$ | $2$ | $C_2\times C_{176}$ | 11A4 | 22B-9 | $ \left(\begin{array}{rr} 136 & 0 \\ 0 & 231 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B-7 | $22$ | $2$ | $C_2\times C_{176}$ | 11A4 | 22B9 | $ \left(\begin{array}{rr} 122 & 0 \\ 0 & 217 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B9 | $22$ | $2$ | $C_2\times C_{176}$ | 11A2 | 22B1 | $ \left(\begin{array}{rr} 222 & 0 \\ 0 & 256 \end{array}\right) $ |
| $C_8.D_{44}$ | 22B-9 | $22$ | $2$ | $C_2\times C_{176}$ | 11A2 | 22B-1 | $ \left(\begin{array}{rr} 97 & 0 \\ 0 & 131 \end{array}\right) $ |
| $C_8.D_{44}$ | 44A1 | $44$ | $2$ | $C_2\times C_{176}$ | 22A1 | 44A5 | $ \left(\begin{array}{rr} 4 & 0 \\ 0 & 265 \end{array}\right) $ |
| $C_8.D_{44}$ | 44A3 | $44$ | $2$ | $C_2\times C_{176}$ | 22A3 | 44A15 | $ \left(\begin{array}{rr} 64 & 0 \\ 0 & 171 \end{array}\right) $ |