Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
424 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22571074 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22562157 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22561313 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22558201 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C67×#1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16864196 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #16864194 | — | 2 | 1 | no | — |
| 8 | #16863960 | — | 2 | 1 | no | — |
| 9 | #16863834 | — | 2 | 2 | no | — |
| 10 | #16863721 | — | 2 | 2 | no | — |
| 11 | #16863649 | C22 | 1 | 2 | yes | — |
| 12 | #16863632 | — | 2 | 1 | no | — |
| 13 | #16863532 | — | 2 | 2 | no | — |
| 14 | #16863362 | — | 2 | 1 | no | — |
| 15 | #16863207 | — | 2 | 2 | no | — |
| 16 | C67⋊#338 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #408×C67 | C22 | 1 | 1 | yes | characteristic |
| 18 | #349×C67 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×#11761628 | — | 2 | 1 | no | — |
| 20 | C22×D268 | — | 2 | 1 | no | — |
| 21 | #14524076 | — | 4 | 1 | no | — |
| 22 | #14524074 | — | 4 | 1 | no | — |
| 23 | #14524073 | — | 4 | 1 | no | — |
| 24 | #14524071 | — | 4 | 1 | no | — |
| 25 | #14524069 | — | 4 | 1 | no | — |
| 26 | #14524068 | — | 4 | 1 | no | — |
| 27 | #14524065 | — | 2 | 2 | no | — |
| 28 | #14524063 | — | 4 | 2 | no | — |
| 29 | C2×D536 | — | 2 | 1 | no | — |
| 30 | C2×D536 | D4 | 1 | 2 | yes | — |
| 31 | C2×#11761612 | — | 2 | 1 | no | — |
| 32 | #14524039 | — | 4 | 1 | no | — |
| 33 | #14524032 | — | 4 | 1 | no | — |
| 34 | #14524017 | — | 2 | 2 | no | — |
| 35 | #14524017 | — | 2 | 2 | no | — |
| 36 | #14524011 | — | 4 | 1 | no | — |
| 37 | #14524011 | — | 4 | 1 | no | — |
| 38 | #14524002 | — | 2 | 1 | no | — |
| 39 | #14523988 | — | 4 | 2 | no | — |
| 40 | D1072 | — | 4 | 2 | no | — |
| 41 | #14523977 | — | 2 | 1 | no | — |
| 42 | #14523975 | D4 | 1 | 2 | yes | — |
| 43 | #14523965 | — | 2 | 2 | no | — |
| 44 | #14523965 | — | 2 | 2 | no | — |
| 45 | (C2×D67)⋊C8 | — | 4 | 2 | no | — |
| 46 | C2×#41×C67 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×M16×C67 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C67 | D4 | 1 | 1 | yes | characteristic |
| 49 | M32×C67 | — | 2 | 2 | no | — |
| 50 | C22⋊C8×C67 | — | 2 | 2 | no | — |
| 51 | C23×D67 | — | 4 | 1 | no | — |
| 52 | #11761628 | — | 4 | 1 | no | — |
| 53 | #11761628 | — | 4 | 1 | no | — |
| 54 | #11761628 | — | 4 | 1 | no | — |
| 55 | #11761628 | — | 4 | 1 | no | — |
| 56 | C2×Dic134 | — | 2 | 1 | no | — |
| 57 | C2×D268 | — | 2 | 1 | no | — |
| 58 | C2×D268 | — | 2 | 2 | no | — |
| 59 | C2×D268 | — | 4 | 1 | no | — |
| 60 | C2×D268 | — | 8 | 1 | no | — |
| 61 | C2×D268 | — | 8 | 1 | no | — |
| 62 | C2×C22⋊D67 | — | 4 | 1 | no | — |
| 63 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 64 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 65 | C2×C4×D67 | — | 4 | 1 | no | — |
| 66 | Dic268 | — | 2 | 2 | no | — |
| 67 | #11761618 | — | 4 | 2 | no | — |
| 68 | #11761617 | — | 4 | 2 | no | — |
| 69 | #11761616 | — | 4 | 2 | no | — |
| 70 | #11761615 | — | 4 | 2 | no | — |
| 71 | D536 | — | 2 | 2 | no | — |
| 72 | D536 | — | 4 | 2 | no | — |
| 73 | D536 | — | 8 | 1 | no | — |
| 74 | #11761613 | — | 4 | 2 | no | — |
| 75 | #11761612 | — | 4 | 1 | no | — |
| 76 | #11761612 | — | 4 | 1 | no | — |
| 77 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 78 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 79 | C22⋊Dic67 | — | 4 | 1 | no | — |
| 80 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 81 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 82 | C4×Dic67 | — | 4 | 1 | no | — |
| 83 | #41×C67 | — | 4 | 1 | no | — |
| 84 | #41×C67 | — | 4 | 1 | no | — |
| 85 | C2×Q8×C67 | D8 | 1 | 1 | yes | characteristic |
| 86 | C2×D4×C67 | — | 2 | 1 | no | — |
| 87 | C2×D4×C67 | D8 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C67 | — | 2 | 1 | no | — |
| 89 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | M16×C67 | — | 4 | 1 | no | — |
| 91 | C2×C8×C67 | — | 2 | 1 | no | — |
| 92 | C2×C8×C67 | — | 4 | 1 | no | — |
| 93 | C2×C8×C67 | C2×D4 | 1 | 2 | yes | — |
| 94 | C16×C67 | — | 4 | 2 | no | — |
| 95 | C23×C67 | — | 2 | 1 | no | — |
| 96 | C23×C67 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 97 | C22×D67 | — | 4 | 1 | no | — |
| 98 | C22×D67 | — | 4 | 2 | no | — |
| 99 | C22×D67 | — | 8 | 2 | no | — |
| 100 | C22×D67 | — | 16 | 2 | no | — |
| 101 | Q8×C67 | — | 4 | 1 | no | — |
| 102 | D4×C67 | — | 4 | 1 | no | — |
| 103 | D4×C67 | — | 8 | 1 | no | — |
| 104 | C2×C4×C67 | — | 2 | 1 | no | — |
| 105 | C2×C4×C67 | — | 2 | 1 | no | — |
| 106 | C2×C4×C67 | — | 2 | 1 | no | — |
| 107 | C2×C4×C67 | — | 4 | 1 | no | — |
| 108 | C2×C4×C67 | — | 4 | 1 | no | — |
| 109 | C2×C4×C67 | #120 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C67 | C2×D8 | 1 | 1 | yes | characteristic |
| 111 | C22⋊D67 | — | 8 | 1 | no | — |
| 112 | C22⋊D67 | — | 8 | 1 | no | — |
| 113 | C22⋊D67 | — | 16 | 2 | no | — |
| 114 | C22⋊D67 | — | 16 | 2 | no | — |
| 115 | C2×Dic67 | — | 4 | 1 | no | — |
| 116 | C2×Dic67 | — | 4 | 2 | no | — |
| 117 | D268 | — | 4 | 1 | no | — |
| 118 | D268 | — | 4 | 2 | no | — |
| 119 | D268 | — | 4 | 1 | no | — |
| 120 | D268 | — | 4 | 2 | no | — |
| 121 | D268 | — | 8 | 2 | no | — |
| 122 | C4×D67 | — | 8 | 1 | no | — |
| 123 | C4×D67 | — | 8 | 1 | no | — |
| 124 | Dic134 | — | 4 | 1 | no | — |
| 125 | Dic134 | — | 4 | 1 | no | — |
| 126 | C8×C67 | — | 2 | 2 | no | — |
| 127 | C8×C67 | — | 4 | 1 | no | — |
| 128 | C67⋊C8 | — | 4 | 2 | no | — |
| 129 | C22×C67 | — | 2 | 1 | no | — |
| 130 | C22×C67 | — | 4 | 1 | no | — |
| 131 | C22×C67 | — | 4 | 2 | no | — |
| 132 | C22×C67 | — | 4 | 1 | no | — |
| 133 | C22×C67 | #447 | 1 | 1 | yes | characteristic |
| 134 | C2×D67 | — | 8 | 2 | no | — |
| 135 | C2×D67 | — | 8 | 2 | no | — |
| 136 | C2×D67 | — | 8 | 2 | no | — |
| 137 | C2×D67 | — | 8 | 1 | no | — |
| 138 | C2×D67 | — | 8 | 2 | no | — |
| 139 | C2×D67 | — | 16 | 4 | no | — |
| 140 | C4×C67 | — | 2 | 1 | no | — |
| 141 | C4×C67 | — | 4 | 1 | no | — |
| 142 | C4×C67 | #453 | 1 | 1 | yes | characteristic |
| 143 | C4×C67 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 144 | Dic67 | — | 8 | 2 | no | — |
| 145 | Dic67 | — | 8 | 1 | no | — |
| 146 | #12117 | — | 67 | 1 | no | maximal, Sylow |
| 147 | C2×C67 | — | 2 | 1 | no | — |
| 148 | C2×C67 | — | 4 | 1 | no | — |
| 149 | C2×C67 | — | 8 | 1 | no | — |
| 150 | C2×C67 | #1595 | 1 | 1 | yes | characteristic |
| 151 | D67 | — | 16 | 1 | no | — |
| 152 | D67 | — | 16 | 2 | no | — |
| 153 | #3199 | — | 67 | 1 | no | — |
| 154 | #2190 | — | 67 | 2 | no | — |
| 155 | #2017 | — | 67 | 1 | no | — |
| 156 | #1341 | — | 67 | 2 | no | — |
| 157 | #1321 | D67 | 1 | 1 | yes | characteristic, complemented |
| 158 | C67 | #12117 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | #575 | — | 67 | 1 | no | — |
| 160 | #575 | — | 134 | 1 | no | — |
| 161 | #535 | — | 134 | 1 | no | — |
| 162 | #509 | — | 134 | 2 | no | — |
| 163 | #482 | — | 134 | 2 | no | — |
| 164 | #469 | — | 67 | 2 | no | — |
| 165 | #466 | — | 134 | 1 | no | — |
| 166 | #447 | — | 134 | 2 | no | — |
| 167 | #408 | — | 134 | 1 | no | — |
| 168 | #408 | C2×D67 | 1 | 1 | yes | characteristic |
| 169 | #360 | — | 134 | 2 | no | — |
| 170 | #349 | C2×D67 | 1 | 2 | yes | complemented |
| 171 | #338 | — | 67 | 1 | no | — |
| 172 | C2×#41 | — | 134 | 1 | no | — |
| 173 | C2×#41 | D268 | 1 | 1 | yes | characteristic |
| 174 | C22×D4 | — | 134 | 1 | no | — |
| 175 | #137 | — | 268 | 1 | no | — |
| 176 | #137 | — | 268 | 1 | no | — |
| 177 | #137 | — | 268 | 1 | no | — |
| 178 | AGL(1,ℤ/8) | — | 268 | 1 | no | — |
| 179 | AGL(1,ℤ/8) | — | 268 | 1 | no | — |
| 180 | AGL(1,ℤ/8) | — | 268 | 1 | no | — |
| 181 | #135 | — | 268 | 2 | no | — |
| 182 | #135 | — | 134 | 2 | no | — |
| 183 | C2×D8 | — | 67 | 2 | no | — |
| 184 | C2×D8 | — | 134 | 1 | no | — |
| 185 | C2×M16 | — | 134 | 1 | no | — |
| 186 | C2×M16 | C22×D67 | 1 | 1 | yes | characteristic |
| 187 | C22×C8 | D268 | 1 | 1 | yes | characteristic |
| 188 | #127 | — | 268 | 1 | no | — |
| 189 | #124 | — | 268 | 1 | no | — |
| 190 | #121 | — | 134 | 2 | no | — |
| 191 | #121 | — | 134 | 2 | no | — |
| 192 | #120 | — | 268 | 1 | no | — |
| 193 | #120 | — | 268 | 1 | no | — |
| 194 | #117 | — | 134 | 1 | no | — |
| 195 | SD32 | — | 268 | 2 | no | — |
| 196 | D16 | — | 268 | 2 | no | — |
| 197 | M32 | — | 2 | 2 | no | — |
| 198 | #107 | — | 134 | 1 | no | — |
| 199 | #106 | — | 67 | 2 | no | — |
| 200 | #102 | — | 134 | 2 | no | — |
| 201 | #102 | — | 134 | 2 | no | — |
| 202 | C22⋊C8 | — | 2 | 2 | no | — |
| 203 | C22⋊C8 | — | 268 | 2 | no | — |
| 204 | C24 | — | 268 | 1 | no | — |
| 205 | #41 | — | 4 | 1 | no | — |
| 206 | #41 | — | 4 | 1 | no | — |
| 207 | #41 | — | 268 | 1 | no | — |
| 208 | #41 | — | 268 | 1 | no | — |
| 209 | #41 | — | 268 | 1 | no | — |
| 210 | #41 | — | 268 | 1 | no | — |
| 211 | C2×Q8 | — | 134 | 1 | no | — |
| 212 | C2×Q8 | D536 | 1 | 1 | yes | characteristic |
| 213 | C2×D4 | — | 2 | 1 | no | — |
| 214 | C2×D4 | — | 268 | 1 | no | — |
| 215 | C2×D4 | — | 268 | 1 | no | — |
| 216 | C2×D4 | — | 268 | 2 | no | — |
| 217 | C2×D4 | — | 268 | 2 | no | — |
| 218 | C2×D4 | — | 536 | 1 | no | — |
| 219 | C2×D4 | — | 536 | 1 | no | — |
| 220 | C2×D4 | — | 134 | 1 | no | — |
| 221 | C2×D4 | — | 134 | 2 | no | — |
| 222 | C2×D4 | D536 | 1 | 1 | yes | characteristic |
| 223 | C22×C4 | — | 2 | 1 | no | — |
| 224 | C22×C4 | — | 268 | 1 | no | — |
| 225 | C22×C4 | C2×D268 | 1 | 1 | yes | characteristic |
| 226 | Q16 | — | 268 | 2 | no | — |
| 227 | Q16 | — | 134 | 2 | no | — |
| 228 | SD16 | — | 268 | 2 | no | — |
| 229 | SD16 | — | 268 | 2 | no | — |
| 230 | SD16 | — | 268 | 2 | no | — |
| 231 | D8 | — | 268 | 2 | no | — |
| 232 | D8 | — | 268 | 2 | no | — |
| 233 | D8 | — | 536 | 1 | no | — |
| 234 | D8 | — | 134 | 2 | no | — |
| 235 | M16 | — | 4 | 1 | no | — |
| 236 | M16 | — | 268 | 1 | no | — |
| 237 | M16 | — | 268 | 1 | no | — |
| 238 | C2×C8 | — | 2 | 1 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 134 | 2 | no | — |
| 241 | C2×C8 | D4×D67 | 1 | 2 | yes | — |
| 242 | C4⋊C4 | — | 134 | 2 | no | — |
| 243 | C22⋊C4 | — | 268 | 2 | no | — |
| 244 | C22⋊C4 | — | 268 | 1 | no | — |
| 245 | C22⋊C4 | — | 268 | 2 | no | — |
| 246 | C42 | — | 268 | 1 | no | — |
| 247 | C16 | — | 4 | 2 | no | — |
| 248 | C23 | — | 2 | 1 | no | — |
| 249 | C23 | — | 268 | 1 | no | — |
| 250 | C23 | — | 268 | 2 | no | — |
| 251 | C23 | — | 536 | 2 | no | — |
| 252 | C23 | — | 1072 | 2 | no | — |
| 253 | C23 | #14524071 | 1 | 1 | yes | characteristic |
| 254 | Q8 | — | 4 | 1 | no | — |
| 255 | Q8 | — | 268 | 1 | no | — |
| 256 | Q8 | — | 268 | 1 | no | — |
| 257 | D4 | — | 4 | 1 | no | — |
| 258 | D4 | — | 8 | 1 | no | — |
| 259 | D4 | — | 268 | 1 | no | — |
| 260 | D4 | — | 268 | 2 | no | — |
| 261 | D4 | — | 268 | 2 | no | — |
| 262 | D4 | — | 268 | 1 | no | — |
| 263 | D4 | — | 536 | 2 | no | — |
| 264 | D4 | — | 536 | 1 | no | — |
| 265 | D4 | — | 536 | 1 | no | — |
| 266 | D4 | — | 1072 | 2 | no | — |
| 267 | D4 | — | 1072 | 2 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 1 | no | — |
| 272 | C2×C4 | — | 4 | 1 | no | — |
| 273 | C2×C4 | — | 268 | 1 | no | — |
| 274 | C2×C4 | — | 268 | 2 | no | — |
| 275 | C2×C4 | — | 536 | 1 | no | — |
| 276 | C2×C4 | — | 536 | 1 | no | — |
| 277 | C2×C4 | #14524013 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | C2×D536 | 1 | 1 | yes | characteristic |
| 279 | C8 | — | 2 | 2 | no | — |
| 280 | C8 | — | 4 | 1 | no | — |
| 281 | C8 | — | 268 | 2 | no | — |
| 282 | C22 | — | 2 | 1 | no | — |
| 283 | C22 | — | 4 | 1 | no | — |
| 284 | C22 | — | 4 | 2 | no | — |
| 285 | C22 | — | 4 | 1 | no | — |
| 286 | C22 | — | 536 | 2 | no | — |
| 287 | C22 | — | 536 | 2 | no | — |
| 288 | C22 | — | 536 | 2 | no | — |
| 289 | C22 | — | 536 | 1 | no | — |
| 290 | C22 | — | 536 | 2 | no | — |
| 291 | C22 | — | 1072 | 4 | no | — |
| 292 | C22 | #16863532 | 1 | 1 | yes | characteristic |
| 293 | C4 | — | 2 | 1 | no | — |
| 294 | C4 | — | 4 | 1 | no | — |
| 295 | C4 | — | 536 | 2 | no | — |
| 296 | C4 | — | 536 | 1 | no | — |
| 297 | C4 | #16863566 | 1 | 1 | yes | characteristic |
| 298 | C4 | #16863584 | 1 | 1 | yes | characteristic |
| 299 | C2 | — | 2 | 1 | no | — |
| 300 | C2 | — | 4 | 1 | no | — |
| 301 | C2 | — | 8 | 1 | no | — |
| 302 | C2 | — | 1072 | 2 | no | — |
| 303 | C2 | — | 1072 | 1 | no | — |
| 304 | C2 | #22559129 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22571074 | C2 | semidirect |
| 2. #22562157 | C2 | semidirect |
| 3. #22561313 | C2 | semidirect |
| 4. #22558201 | C2 | semidirect |
| 5. C67×#1321 | C2 | semidirect |
| 6. #16864196 | C22 | semidirect |
| 16. C67⋊#338 | C22 | semidirect |
| 18. #349×C67 | C22 | semidirect |
| 157. #1321 | D67 | semidirect |
| 158. C67 | #12117 | semidirect |
| 170. #349 | C2×D67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22571074 | 2 |
| #22562157 | 2 |
| #22561313 | 2 |
| #22558201 | 2 |
| C67×#1321 | 2 |
| #12117 | 67 |