Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #11763549 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11762998 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11762997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11762979 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11762601 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11762597 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C17×#352 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #10587691 | C22 | 1 | 1 | yes | characteristic |
| 9 | C17×#123 | C22 | 1 | 1 | yes | characteristic |
| 10 | #10587626 | C22 | 1 | 1 | yes | characteristic |
| 11 | #10587623 | C22 | 1 | 1 | yes | characteristic |
| 12 | #10587616 | — | 2 | 1 | no | — |
| 13 | #10587615 | — | 2 | 1 | no | — |
| 14 | #10587594 | — | 2 | 1 | no | — |
| 15 | #10587584 | — | 2 | 1 | no | — |
| 16 | #10587581 | — | 2 | 1 | no | — |
| 17 | #10587570 | — | 2 | 1 | no | — |
| 18 | #10587569 | — | 2 | 1 | no | — |
| 19 | #10587568 | — | 2 | 1 | no | — |
| 20 | #10587568 | — | 2 | 1 | no | — |
| 21 | C17×#100 | C22 | 1 | 1 | yes | characteristic |
| 22 | C17×#99 | C22 | 1 | 1 | yes | characteristic |
| 23 | #10587505 | C22 | 1 | 1 | yes | characteristic |
| 24 | #10587496 | — | 2 | 1 | no | — |
| 25 | #10587477 | — | 2 | 1 | no | — |
| 26 | #10587477 | — | 2 | 1 | no | — |
| 27 | C23×D17 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×D4×C17 | C23 | 1 | 1 | yes | characteristic |
| 29 | C22×C4×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 31 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 32 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 33 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 34 | C2×C22⋊D17 | — | 2 | 1 | no | — |
| 35 | #63254 | — | 4 | 1 | no | — |
| 36 | D4×D17 | — | 4 | 1 | no | — |
| 37 | #63252 | — | 2 | 1 | no | — |
| 38 | C2×D68 | — | 2 | 1 | no | — |
| 39 | C2×D68 | — | 2 | 1 | no | — |
| 40 | C2×D68 | — | 4 | 1 | no | — |
| 41 | C2×D68 | D4 | 1 | 1 | yes | characteristic, complemented |
| 42 | C2×C4×D17 | — | 2 | 1 | no | — |
| 43 | M16×C17 | — | 2 | 1 | no | — |
| 44 | C4⋊C4×C17 | — | 2 | 1 | no | — |
| 45 | C22⋊C4×C17 | — | 2 | 1 | no | — |
| 46 | C22⋊C4×C17 | — | 2 | 1 | no | — |
| 47 | C22⋊C4×C17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 48 | C22⋊Dic17 | — | 2 | 1 | no | — |
| 49 | C22⋊Dic17 | D4 | 1 | 1 | yes | characteristic |
| 50 | #63227 | — | 2 | 1 | no | — |
| 51 | #63227 | — | 2 | 1 | no | — |
| 52 | #63227 | — | 2 | 1 | no | — |
| 53 | #63227 | — | 2 | 1 | no | — |
| 54 | #63227 | — | 4 | 1 | no | — |
| 55 | #63227 | — | 4 | 1 | no | — |
| 56 | C4⋊Dic17 | — | 2 | 1 | no | — |
| 57 | #63225 | — | 2 | 1 | no | — |
| 58 | #63225 | — | 4 | 1 | no | — |
| 59 | C4×Dic17 | D4 | 1 | 1 | yes | characteristic, complemented |
| 60 | D136 | — | 4 | 1 | no | — |
| 61 | #63219 | — | 4 | 1 | no | — |
| 62 | C23×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C23×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×D17 | — | 2 | 1 | no | — |
| 65 | C22×D17 | — | 2 | 1 | no | — |
| 66 | C22×D17 | — | 2 | 1 | no | — |
| 67 | C22×D17 | — | 2 | 1 | no | — |
| 68 | C22×D17 | — | 4 | 1 | no | — |
| 69 | C22×D17 | — | 8 | 1 | no | — |
| 70 | D4×C17 | — | 4 | 1 | no | — |
| 71 | C2×C4×C17 | — | 2 | 1 | no | — |
| 72 | C2×C4×C17 | — | 2 | 1 | no | — |
| 73 | C2×C4×C17 | — | 4 | 1 | no | — |
| 74 | C2×C4×C17 | — | 4 | 1 | no | — |
| 75 | C2×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C22⋊D17 | — | 4 | 1 | no | — |
| 77 | C22⋊D17 | — | 4 | 1 | no | — |
| 78 | C22⋊D17 | — | 4 | 1 | no | — |
| 79 | C22⋊D17 | — | 8 | 1 | no | — |
| 80 | C22⋊D17 | — | 8 | 1 | no | — |
| 81 | C22⋊D17 | — | 8 | 1 | no | — |
| 82 | C2×Dic17 | — | 2 | 1 | no | — |
| 83 | C2×Dic17 | — | 2 | 1 | no | — |
| 84 | C2×Dic17 | — | 2 | 1 | no | — |
| 85 | C2×Dic17 | — | 2 | 1 | no | — |
| 86 | C2×Dic17 | — | 4 | 1 | no | — |
| 87 | D68 | — | 2 | 1 | no | — |
| 88 | D68 | — | 4 | 1 | no | — |
| 89 | D68 | — | 8 | 1 | no | — |
| 90 | D68 | — | 8 | 1 | no | — |
| 91 | D68 | — | 8 | 2 | no | — |
| 92 | C4×D17 | — | 4 | 1 | no | — |
| 93 | C4×D17 | — | 8 | 1 | no | — |
| 94 | Dic34 | — | 2 | 1 | no | — |
| 95 | C8×C17 | — | 4 | 1 | no | — |
| 96 | #2200 | — | 17 | 1 | no | maximal, Sylow |
| 97 | C22×C17 | — | 2 | 1 | no | — |
| 98 | C22×C17 | — | 2 | 1 | no | — |
| 99 | C22×C17 | — | 4 | 1 | no | — |
| 100 | C22×C17 | — | 4 | 1 | no | — |
| 101 | C22×C17 | #120 | 1 | 1 | yes | characteristic |
| 102 | C2×D17 | — | 4 | 1 | no | — |
| 103 | C2×D17 | — | 4 | 1 | no | — |
| 104 | C2×D17 | — | 4 | 1 | no | — |
| 105 | C2×D17 | — | 4 | 1 | no | — |
| 106 | C2×D17 | — | 4 | 2 | no | — |
| 107 | C2×D17 | — | 4 | 1 | no | — |
| 108 | C2×D17 | — | 4 | 1 | no | — |
| 109 | C2×D17 | — | 4 | 1 | no | — |
| 110 | C2×D17 | — | 8 | 1 | no | — |
| 111 | C2×D17 | — | 8 | 1 | no | — |
| 112 | C2×D17 | — | 8 | 1 | no | — |
| 113 | C4×C17 | — | 2 | 1 | no | — |
| 114 | C4×C17 | — | 4 | 1 | no | — |
| 115 | C4×C17 | — | 4 | 1 | no | — |
| 116 | C4×C17 | — | 8 | 1 | no | — |
| 117 | Dic17 | — | 4 | 1 | no | — |
| 118 | Dic17 | — | 4 | 1 | no | — |
| 119 | Dic17 | — | 4 | 1 | no | — |
| 120 | Dic17 | — | 8 | 1 | no | — |
| 121 | #560 | — | 17 | 1 | no | — |
| 122 | #458 | — | 17 | 1 | no | — |
| 123 | B(4,𝔽2) | — | 17 | 1 | no | — |
| 124 | #453 | — | 17 | 1 | no | — |
| 125 | #353 | — | 17 | 1 | no | — |
| 126 | #352 | D17 | 1 | 1 | yes | characteristic, complemented |
| 127 | AΣL(1,𝔽16) | — | 17 | 1 | no | — |
| 128 | C2×C17 | — | 2 | 1 | no | — |
| 129 | C2×C17 | — | 4 | 1 | no | — |
| 130 | C2×C17 | — | 4 | 1 | no | — |
| 131 | C2×C17 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 132 | D17 | — | 8 | 1 | no | — |
| 133 | D17 | — | 8 | 1 | no | — |
| 134 | D17 | — | 8 | 1 | no | — |
| 135 | H(5,𝔽2) | — | 17 | 1 | no | — |
| 136 | AGL(1,ℤ/8) | — | 34 | 1 | no | — |
| 137 | #127 | — | 17 | 1 | no | — |
| 138 | #126 | — | 34 | 1 | no | — |
| 139 | #123 | — | 34 | 1 | no | — |
| 140 | #123 | — | 34 | 1 | no | — |
| 141 | #123 | C2×D17 | 1 | 1 | yes | characteristic |
| 142 | #121 | — | 34 | 1 | no | — |
| 143 | #121 | — | 34 | 1 | no | — |
| 144 | #121 | — | 34 | 1 | no | — |
| 145 | #120 | — | 17 | 1 | no | — |
| 146 | #120 | — | 34 | 1 | no | — |
| 147 | #120 | — | 34 | 1 | no | — |
| 148 | #104 | — | 34 | 1 | no | — |
| 149 | #100 | C2×D17 | 1 | 1 | yes | characteristic |
| 150 | #99 | — | 17 | 1 | no | — |
| 151 | #99 | — | 34 | 1 | no | — |
| 152 | #99 | — | 34 | 1 | no | — |
| 153 | #99 | C2×D17 | 1 | 1 | yes | characteristic |
| 154 | C17 | #2200 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | C24 | — | 17 | 1 | no | — |
| 156 | #41 | — | 34 | 1 | no | — |
| 157 | #41 | — | 68 | 1 | no | — |
| 158 | C2×D4 | — | 17 | 1 | no | — |
| 159 | C2×D4 | — | 34 | 1 | no | — |
| 160 | C2×D4 | — | 34 | 1 | no | — |
| 161 | C2×D4 | — | 34 | 1 | no | — |
| 162 | C2×D4 | — | 34 | 1 | no | — |
| 163 | C2×D4 | — | 34 | 1 | no | — |
| 164 | C2×D4 | — | 34 | 1 | no | — |
| 165 | C2×D4 | — | 34 | 1 | no | — |
| 166 | C2×D4 | — | 68 | 1 | no | — |
| 167 | C2×D4 | — | 68 | 1 | no | — |
| 168 | C2×D4 | C22×D17 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | — | 34 | 1 | no | — |
| 170 | C22×C4 | D68 | 1 | 1 | yes | characteristic, complemented |
| 171 | SD16 | — | 68 | 1 | no | — |
| 172 | D8 | — | 68 | 1 | no | — |
| 173 | M16 | — | 2 | 1 | no | — |
| 174 | C4⋊C4 | — | 2 | 1 | no | — |
| 175 | C4⋊C4 | — | 34 | 1 | no | — |
| 176 | C4⋊C4 | — | 34 | 1 | no | — |
| 177 | C4⋊C4 | — | 68 | 1 | no | — |
| 178 | C22⋊C4 | — | 2 | 1 | no | — |
| 179 | C22⋊C4 | — | 2 | 1 | no | — |
| 180 | C22⋊C4 | — | 17 | 1 | no | — |
| 181 | C22⋊C4 | — | 34 | 1 | no | — |
| 182 | C22⋊C4 | — | 34 | 1 | no | — |
| 183 | C22⋊C4 | — | 34 | 1 | no | — |
| 184 | C22⋊C4 | — | 34 | 1 | no | — |
| 185 | C22⋊C4 | — | 34 | 1 | no | — |
| 186 | C22⋊C4 | — | 68 | 1 | no | — |
| 187 | C22⋊C4 | — | 68 | 1 | no | — |
| 188 | C22⋊C4 | D68 | 1 | 1 | yes | characteristic, complemented |
| 189 | C42 | — | 17 | 1 | no | — |
| 190 | C23 | — | 34 | 1 | no | — |
| 191 | C23 | — | 34 | 1 | no | — |
| 192 | C23 | — | 34 | 1 | no | — |
| 193 | C23 | — | 34 | 1 | no | — |
| 194 | C23 | — | 68 | 1 | no | — |
| 195 | C23 | — | 136 | 1 | no | — |
| 196 | C23 | C2×D68 | 1 | 1 | yes | characteristic |
| 197 | C23 | D4×D17 | 1 | 1 | yes | characteristic |
| 198 | Q8 | — | 34 | 1 | no | — |
| 199 | D4 | — | 4 | 1 | no | — |
| 200 | D4 | — | 34 | 1 | no | — |
| 201 | D4 | — | 68 | 1 | no | — |
| 202 | D4 | — | 68 | 1 | no | — |
| 203 | D4 | — | 68 | 1 | no | — |
| 204 | D4 | — | 68 | 1 | no | — |
| 205 | D4 | — | 136 | 1 | no | — |
| 206 | D4 | — | 136 | 1 | no | — |
| 207 | D4 | — | 136 | 2 | no | — |
| 208 | D4 | — | 136 | 1 | no | — |
| 209 | D4 | — | 136 | 1 | no | — |
| 210 | D4 | — | 136 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 2 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 4 | 1 | no | — |
| 215 | C2×C4 | — | 34 | 1 | no | — |
| 216 | C2×C4 | — | 34 | 1 | no | — |
| 217 | C2×C4 | — | 34 | 1 | no | — |
| 218 | C2×C4 | — | 34 | 1 | no | — |
| 219 | C2×C4 | — | 68 | 1 | no | — |
| 220 | C2×C4 | — | 68 | 1 | no | — |
| 221 | C2×C4 | — | 136 | 1 | no | — |
| 222 | C2×C4 | D4×D17 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 4 | 1 | no | — |
| 224 | C22 | — | 2 | 1 | no | — |
| 225 | C22 | — | 2 | 1 | no | — |
| 226 | C22 | — | 4 | 1 | no | — |
| 227 | C22 | — | 4 | 1 | no | — |
| 228 | C22 | — | 68 | 1 | no | — |
| 229 | C22 | — | 68 | 1 | no | — |
| 230 | C22 | — | 68 | 1 | no | — |
| 231 | C22 | — | 68 | 1 | no | — |
| 232 | C22 | — | 68 | 2 | no | — |
| 233 | C22 | — | 68 | 1 | no | — |
| 234 | C22 | — | 68 | 1 | no | — |
| 235 | C22 | — | 68 | 1 | no | — |
| 236 | C22 | — | 136 | 1 | no | — |
| 237 | C22 | — | 136 | 1 | no | — |
| 238 | C22 | — | 136 | 1 | no | — |
| 239 | C22 | #10587568 | 1 | 1 | yes | characteristic |
| 240 | C4 | — | 2 | 1 | no | — |
| 241 | C4 | — | 4 | 1 | no | — |
| 242 | C4 | — | 4 | 1 | no | — |
| 243 | C4 | — | 8 | 1 | no | — |
| 244 | C4 | — | 68 | 1 | no | — |
| 245 | C4 | — | 68 | 1 | no | — |
| 246 | C4 | — | 68 | 1 | no | — |
| 247 | C4 | — | 136 | 1 | no | — |
| 248 | C2 | — | 2 | 1 | no | — |
| 249 | C2 | — | 4 | 1 | no | — |
| 250 | C2 | — | 4 | 1 | no | — |
| 251 | C2 | — | 136 | 1 | no | — |
| 252 | C2 | — | 136 | 1 | no | — |
| 253 | C2 | — | 136 | 1 | no | — |
| 254 | C2 | #11762997 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763549 | C2 | semidirect |
| 2. #11762998 | C2 | semidirect |
| 3. #11762997 | C2 | semidirect |
| 4. #11762979 | C2 | semidirect |
| 5. #11762601 | C2 | semidirect |
| 6. #11762597 | C2 | semidirect |
| 7. C17×#352 | C2 | semidirect |
| 29. C22×C4×C17 | D4 | semidirect |
| 41. C2×D68 | D4 | semidirect |
| 47. C22⋊C4×C17 | D4 | semidirect |
| 59. C4×Dic17 | D4 | semidirect |
| 126. #352 | D17 | semidirect |
| 154. C17 | #2200 | semidirect |
| 170. C22×C4 | D68 | semidirect |
| 188. C22⋊C4 | D68 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763549 | 2 |
| #11762998 | 2 |
| #11762997 | 2 |
| #11762979 | 2 |
| #11762601 | 2 |
| #11762597 | 2 |
| C17×#352 | 2 |
| #2200 | 17 |