Finite Group DatabasebetaFinite Group DatabasebetaOrder 17152.
438 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 69 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 |
| 2 | #22561858 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22560734 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22560428 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C67×#1391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16864163 | C22 | 1 | 1 | yes | characteristic |
| 7 | #16863960 | C22 | 1 | 2 | yes | — |
| 8 | #16863721 | — | 2 | 2 | no | — |
| 9 | #16863649 | — | 2 | 1 | no | — |
| 10 | #16863632 | — | 2 | 1 | no | — |
| 11 | #16863532 | — | 2 | 2 | no | — |
| 12 | #16863444 | — | 2 | 1 | no | — |
| 13 | #16863400 | — | 2 | 2 | no | — |
| 14 | #16863380 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16863182 | — | 2 | 4 | no | — |
| 16 | #16863161 | — | 2 | 1 | no | — |
| 17 | #16863152 | — | 2 | 2 | no | — |
| 18 | #410×C67 | C22 | 1 | 2 | yes | — |
| 19 | #408×C67 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×D268 | D4 | 1 | 2 | yes | — |
| 21 | C2×D536 | — | 2 | 1 | no | — |
| 22 | C2×D536 | — | 2 | 1 | no | — |
| 23 | #14524050 | — | 4 | 2 | no | — |
| 24 | C2×#11761612 | — | 2 | 2 | no | — |
| 25 | C2×#11761612 | — | 2 | 1 | no | — |
| 26 | C2×#11761612 | — | 2 | 1 | no | — |
| 27 | #14524039 | — | 2 | 2 | no | — |
| 28 | #14524032 | — | 2 | 2 | no | — |
| 29 | #14524017 | — | 2 | 2 | no | — |
| 30 | #14524017 | — | 2 | 2 | no | — |
| 31 | #14524011 | — | 2 | 2 | no | — |
| 32 | #14524011 | — | 2 | 2 | no | — |
| 33 | #14524002 | D4 | 1 | 2 | yes | — |
| 34 | #14523978 | — | 4 | 2 | no | — |
| 35 | #14523977 | — | 2 | 1 | no | — |
| 36 | #14523975 | — | 2 | 1 | no | — |
| 37 | #14523965 | — | 2 | 2 | no | — |
| 38 | #14523965 | — | 2 | 2 | no | — |
| 39 | #14523963 | — | 4 | 4 | no | — |
| 40 | #14523962 | — | 2 | 1 | no | — |
| 41 | #14523960 | D4 | 1 | 2 | yes | — |
| 42 | C2×#41×C67 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×M16×C67 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C8×C67 | D4 | 1 | 1 | yes | characteristic |
| 45 | #117×C67 | — | 2 | 2 | no | — |
| 46 | #99×C67 | — | 2 | 2 | no | — |
| 47 | #99×C67 | — | 2 | 2 | no | — |
| 48 | C22⋊C8×C67 | — | 2 | 2 | no | — |
| 49 | C23×D67 | — | 2 | 2 | no | — |
| 50 | C2×D268 | — | 2 | 2 | no | — |
| 51 | C2×D268 | — | 2 | 2 | no | — |
| 52 | C2×D268 | — | 4 | 2 | no | — |
| 53 | C2×D268 | — | 4 | 2 | no | — |
| 54 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 55 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 56 | D536 | — | 8 | 1 | no | — |
| 57 | D536 | — | 8 | 1 | no | — |
| 58 | #11761612 | — | 2 | 2 | no | — |
| 59 | #11761612 | — | 2 | 2 | no | — |
| 60 | #11761612 | — | 4 | 1 | no | — |
| 61 | #11761612 | — | 4 | 2 | no | — |
| 62 | #11761612 | — | 4 | 1 | no | — |
| 63 | C2×C67⋊C8 | — | 2 | 4 | no | — |
| 64 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 65 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 66 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 67 | C22⋊Dic67 | — | 2 | 2 | no | — |
| 68 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 69 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 70 | C4×Dic67 | — | 2 | 2 | no | — |
| 71 | #41×C67 | — | 4 | 1 | no | — |
| 72 | #41×C67 | — | 4 | 1 | no | — |
| 73 | C2×Q8×C67 | #41 | 1 | 1 | yes | characteristic |
| 74 | C2×D4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×D4×C67 | #41 | 1 | 2 | yes | — |
| 76 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 77 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | M16×C67 | — | 4 | 1 | no | — |
| 79 | C2×C8×C67 | — | 2 | 1 | no | — |
| 80 | C2×C8×C67 | — | 2 | 1 | no | — |
| 81 | C2×C8×C67 | — | 4 | 1 | no | — |
| 82 | C4⋊C4×C67 | — | 4 | 2 | no | — |
| 83 | C22⋊C4×C67 | — | 4 | 2 | no | — |
| 84 | C42×C67 | — | 4 | 2 | no | — |
| 85 | C23×C67 | #121 | 1 | 2 | yes | — |
| 86 | C23×C67 | #124 | 1 | 1 | yes | characteristic |
| 87 | C22×D67 | — | 4 | 2 | no | — |
| 88 | C22×D67 | — | 8 | 2 | no | — |
| 89 | C22×D67 | — | 16 | 2 | no | — |
| 90 | Q8×C67 | — | 4 | 1 | no | — |
| 91 | D4×C67 | — | 4 | 1 | no | — |
| 92 | D4×C67 | — | 4 | 2 | no | — |
| 93 | C2×C4×C67 | — | 2 | 2 | no | — |
| 94 | C2×C4×C67 | — | 2 | 1 | no | — |
| 95 | C2×C4×C67 | — | 2 | 2 | no | — |
| 96 | C2×C4×C67 | — | 2 | 1 | no | — |
| 97 | C2×C4×C67 | — | 8 | 2 | no | — |
| 98 | C2×C4×C67 | #120 | 1 | 1 | yes | characteristic |
| 99 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 100 | C2×C4×C67 | #123 | 1 | 2 | yes | — |
| 101 | C22⋊D67 | — | 16 | 2 | no | — |
| 102 | C22⋊D67 | — | 16 | 2 | no | — |
| 103 | C2×Dic67 | — | 4 | 2 | no | — |
| 104 | D268 | — | 4 | 4 | no | — |
| 105 | D268 | — | 4 | 2 | no | — |
| 106 | D268 | — | 4 | 2 | no | — |
| 107 | C8×C67 | — | 4 | 1 | no | — |
| 108 | C8×C67 | — | 4 | 1 | no | — |
| 109 | C67⋊C8 | — | 4 | 2 | no | — |
| 110 | C67⋊C8 | — | 4 | 2 | no | — |
| 111 | C22×C67 | — | 2 | 2 | no | — |
| 112 | C22×C67 | — | 2 | 1 | no | — |
| 113 | C22×C67 | — | 4 | 1 | no | — |
| 114 | C22×C67 | — | 4 | 2 | no | — |
| 115 | C22×C67 | #394 | 1 | 1 | yes | characteristic |
| 116 | C2×D67 | — | 8 | 2 | no | — |
| 117 | C2×D67 | — | 8 | 2 | no | — |
| 118 | C2×D67 | — | 8 | 2 | no | — |
| 119 | C2×D67 | — | 16 | 4 | no | — |
| 120 | C4×C67 | — | 2 | 1 | no | — |
| 121 | C4×C67 | — | 2 | 2 | no | — |
| 122 | C4×C67 | — | 16 | 2 | no | — |
| 123 | C4×C67 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 124 | C4×C67 | #458 | 1 | 1 | yes | characteristic |
| 125 | Dic67 | — | 8 | 2 | no | — |
| 126 | #13436 | — | 67 | 1 | no | maximal, Sylow |
| 127 | C2×C67 | — | 2 | 1 | no | — |
| 128 | C2×C67 | — | 4 | 2 | no | — |
| 129 | C2×C67 | — | 4 | 1 | no | — |
| 130 | C2×C67 | #2044 | 1 | 1 | yes | characteristic |
| 131 | D67 | — | 16 | 2 | no | — |
| 132 | #3199 | — | 67 | 1 | no | — |
| 133 | #2122 | — | 67 | 2 | no | — |
| 134 | #1910 | — | 67 | 2 | no | — |
| 135 | #1858 | — | 67 | 1 | no | — |
| 136 | #1391 | D67 | 1 | 1 | yes | characteristic, complemented |
| 137 | C67 | #13436 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 138 | #568 | — | 67 | 1 | no | — |
| 139 | #535 | — | 67 | 2 | no | — |
| 140 | #482 | — | 134 | 2 | no | — |
| 141 | #469 | — | 134 | 1 | no | — |
| 142 | #466 | — | 134 | 1 | no | — |
| 143 | #447 | — | 134 | 2 | no | — |
| 144 | #430 | — | 134 | 1 | no | — |
| 145 | #418 | — | 134 | 2 | no | — |
| 146 | #413 | — | 67 | 1 | no | — |
| 147 | #410 | C2×D67 | 1 | 2 | yes | — |
| 148 | #408 | C2×D67 | 1 | 1 | yes | characteristic |
| 149 | AΣL(1,𝔽16) | — | 134 | 4 | no | — |
| 150 | #341 | — | 134 | 1 | no | — |
| 151 | H(3,ℤ/4) | — | 134 | 2 | no | — |
| 152 | C2×#41 | C22×D67 | 1 | 1 | yes | characteristic |
| 153 | C22×D4 | — | 67 | 2 | no | — |
| 154 | C2×D8 | — | 134 | 1 | no | — |
| 155 | C2×D8 | — | 134 | 1 | no | — |
| 156 | #131 | — | 268 | 2 | no | — |
| 157 | C2×M16 | — | 134 | 1 | no | — |
| 158 | C2×M16 | — | 134 | 1 | no | — |
| 159 | C2×M16 | — | 134 | 2 | no | — |
| 160 | C2×M16 | D268 | 1 | 1 | yes | characteristic |
| 161 | C22×C8 | D268 | 1 | 1 | yes | characteristic |
| 162 | #127 | — | 134 | 2 | no | — |
| 163 | #124 | — | 134 | 2 | no | — |
| 164 | #121 | — | 134 | 2 | no | — |
| 165 | #121 | — | 134 | 2 | no | — |
| 166 | #120 | — | 134 | 2 | no | — |
| 167 | #120 | — | 134 | 2 | no | — |
| 168 | #117 | — | 2 | 2 | no | — |
| 169 | #117 | — | 67 | 2 | no | — |
| 170 | #108 | — | 268 | 2 | no | — |
| 171 | #107 | — | 134 | 1 | no | — |
| 172 | #106 | — | 134 | 1 | no | — |
| 173 | #102 | — | 268 | 4 | no | — |
| 174 | #102 | — | 134 | 2 | no | — |
| 175 | #102 | — | 134 | 2 | no | — |
| 176 | #101 | — | 134 | 1 | no | — |
| 177 | #100 | — | 67 | 2 | no | — |
| 178 | #99 | — | 2 | 2 | no | — |
| 179 | #99 | — | 2 | 2 | no | — |
| 180 | C22⋊C8 | — | 2 | 2 | no | — |
| 181 | C24 | — | 134 | 2 | no | — |
| 182 | #41 | — | 4 | 1 | no | — |
| 183 | #41 | — | 4 | 1 | no | — |
| 184 | C2×Q8 | #11761629 | 1 | 1 | yes | characteristic |
| 185 | C2×D4 | — | 268 | 2 | no | — |
| 186 | C2×D4 | — | 268 | 2 | no | — |
| 187 | C2×D4 | — | 268 | 2 | no | — |
| 188 | C2×D4 | — | 268 | 2 | no | — |
| 189 | C2×D4 | — | 134 | 2 | no | — |
| 190 | C2×D4 | — | 134 | 2 | no | — |
| 191 | C2×D4 | D4×D67 | 1 | 1 | yes | characteristic |
| 192 | C2×D4 | #11761628 | 1 | 2 | yes | — |
| 193 | C22×C4 | D4×D67 | 1 | 2 | yes | — |
| 194 | C22×C4 | C2×D268 | 1 | 1 | yes | characteristic |
| 195 | D8 | — | 536 | 1 | no | — |
| 196 | D8 | — | 536 | 1 | no | — |
| 197 | M16 | — | 4 | 1 | no | — |
| 198 | M16 | — | 268 | 2 | no | — |
| 199 | M16 | — | 268 | 1 | no | — |
| 200 | M16 | — | 268 | 1 | no | — |
| 201 | M16 | — | 134 | 2 | no | — |
| 202 | M16 | — | 134 | 2 | no | — |
| 203 | C2×C8 | — | 2 | 1 | no | — |
| 204 | C2×C8 | — | 2 | 1 | no | — |
| 205 | C2×C8 | — | 4 | 1 | no | — |
| 206 | C2×C8 | — | 268 | 1 | no | — |
| 207 | C2×C8 | — | 268 | 1 | no | — |
| 208 | C2×C8 | — | 134 | 4 | no | — |
| 209 | C4⋊C4 | — | 4 | 2 | no | — |
| 210 | C4⋊C4 | — | 134 | 2 | no | — |
| 211 | C22⋊C4 | — | 4 | 2 | no | — |
| 212 | C22⋊C4 | — | 268 | 2 | no | — |
| 213 | C22⋊C4 | — | 268 | 2 | no | — |
| 214 | C22⋊C4 | — | 134 | 2 | no | — |
| 215 | C42 | — | 4 | 2 | no | — |
| 216 | C42 | — | 134 | 2 | no | — |
| 217 | C23 | — | 268 | 2 | no | — |
| 218 | C23 | — | 536 | 2 | no | — |
| 219 | C23 | — | 1072 | 2 | no | — |
| 220 | C23 | #14524016 | 1 | 2 | yes | — |
| 221 | C23 | #14524031 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 4 | 1 | no | — |
| 223 | D4 | — | 4 | 1 | no | — |
| 224 | D4 | — | 4 | 2 | no | — |
| 225 | D4 | — | 268 | 4 | no | — |
| 226 | D4 | — | 268 | 2 | no | — |
| 227 | D4 | — | 268 | 2 | no | — |
| 228 | D4 | — | 1072 | 2 | no | — |
| 229 | D4 | — | 1072 | 2 | no | — |
| 230 | C2×C4 | — | 2 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 2 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 8 | 2 | no | — |
| 235 | C2×C4 | — | 268 | 2 | no | — |
| 236 | C2×C4 | #14524013 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #14524018 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #14524028 | 1 | 2 | yes | — |
| 239 | C8 | — | 4 | 1 | no | — |
| 240 | C8 | — | 4 | 1 | no | — |
| 241 | C8 | — | 268 | 2 | no | — |
| 242 | C8 | — | 268 | 2 | no | — |
| 243 | C22 | — | 2 | 2 | no | — |
| 244 | C22 | — | 2 | 1 | no | — |
| 245 | C22 | — | 4 | 1 | no | — |
| 246 | C22 | — | 4 | 2 | no | — |
| 247 | C22 | — | 536 | 2 | no | — |
| 248 | C22 | — | 536 | 2 | no | — |
| 249 | C22 | — | 536 | 2 | no | — |
| 250 | C22 | — | 1072 | 4 | no | — |
| 251 | C22 | #16863314 | 1 | 1 | yes | characteristic |
| 252 | C4 | — | 2 | 1 | no | — |
| 253 | C4 | — | 2 | 2 | no | — |
| 254 | C4 | — | 16 | 2 | no | — |
| 255 | C4 | — | 536 | 2 | no | — |
| 256 | C4 | #16863584 | 1 | 1 | yes | characteristic |
| 257 | C4 | #16863587 | 1 | 1 | yes | characteristic |
| 258 | C2 | — | 2 | 1 | no | — |
| 259 | C2 | — | 4 | 2 | no | — |
| 260 | C2 | — | 4 | 1 | no | — |
| 261 | C2 | — | 1072 | 2 | no | — |
| 262 | C2 | #22561461 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22561858 | C2 | semidirect |
| 3. #22560734 | C2 | semidirect |
| 4. #22560428 | C2 | semidirect |
| 5. C67×#1391 | C2 | semidirect |
| 136. #1391 | D67 | semidirect |
| 137. C67 | #13436 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22571074 | 2 |
| #22561858 | 2 |
| #22560734 | 2 |
| #22560428 | 2 |
| C67×#1391 | 2 |
| #13436 | 67 |