Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
950 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 207 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 | #430×D53 | C2 | 1 | 3 | yes | maximal |
| 2 | C2×#101×D53 | C2 | 1 | 3 | yes | maximal |
| 3 | #20276117 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20276116 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20276115 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #338×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C53×#2070 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×Q8×D53 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×M16×D53 | C22 | 1 | 3 | yes | — |
| 10 | C2×M16×D53 | C22 | 1 | 3 | yes | — |
| 11 | #16810672 | C22 | 1 | 3 | yes | — |
| 12 | #16810671 | C22 | 1 | 3 | yes | — |
| 13 | #16810671 | C22 | 1 | 3 | yes | — |
| 14 | #16810670 | — | 2 | 6 | no | — |
| 15 | #108×D53 | — | 2 | 6 | no | — |
| 16 | C2×#13133136 | C22 | 1 | 3 | yes | — |
| 17 | C2×#13133135 | C22 | 1 | 3 | yes | — |
| 18 | C2×#13133135 | C22 | 1 | 3 | yes | — |
| 19 | #101×D53 | — | 2 | 6 | no | — |
| 20 | #101×D53 | — | 2 | 6 | no | — |
| 21 | #16810381 | C22 | 1 | 3 | yes | — |
| 22 | C53×#430 | C22 | 1 | 3 | yes | — |
| 23 | C2×#101×C53 | C22 | 1 | 3 | yes | — |
| 24 | C53×#338 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22×Dic106 | Q8 | 1 | 2 | yes | — |
| 26 | C22×Dic106 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×Q8×D53 | — | 2 | 3 | no | — |
| 28 | C2×Q8×D53 | — | 2 | 3 | no | — |
| 29 | C2×Q8×D53 | — | 2 | 3 | no | — |
| 30 | C2×Q8×D53 | — | 2 | 3 | no | — |
| 31 | C22×C4×D53 | Q8 | 1 | 2 | yes | — |
| 32 | C22×C4×D53 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×#11719917 | C23 | 1 | 3 | yes | — |
| 34 | C2×#11719917 | C23 | 1 | 3 | yes | — |
| 35 | C2×#11719912 | — | 2 | 3 | no | — |
| 36 | C2×#11719912 | — | 2 | 3 | no | — |
| 37 | M16×D53 | — | 2 | 3 | no | — |
| 38 | M16×D53 | — | 2 | 3 | no | — |
| 39 | M16×D53 | — | 2 | 3 | no | — |
| 40 | M16×D53 | — | 2 | 3 | no | — |
| 41 | M16×D53 | — | 2 | 3 | no | — |
| 42 | M16×D53 | — | 2 | 3 | no | — |
| 43 | M16×D53 | D4 | 1 | 6 | yes | — |
| 44 | M16×D53 | D4 | 1 | 6 | yes | — |
| 45 | C2×C8×D53 | — | 2 | 3 | no | — |
| 46 | C2×C8×D53 | — | 2 | 3 | no | — |
| 47 | #13133153 | — | 2 | 6 | no | — |
| 48 | #13133152 | — | 2 | 6 | no | — |
| 49 | #13133152 | — | 2 | 6 | no | — |
| 50 | #13133136 | — | 2 | 6 | no | — |
| 51 | #13133135 | — | 2 | 6 | no | — |
| 52 | #13133135 | — | 2 | 6 | no | — |
| 53 | C22×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×M16×C53 | C23 | 1 | 3 | yes | — |
| 55 | C2×M16×C53 | C23 | 1 | 3 | yes | — |
| 56 | #108×C53 | — | 2 | 6 | no | — |
| 57 | #101×C53 | — | 2 | 6 | no | — |
| 58 | C23×D53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 59 | C2×Q8×C53 | — | 2 | 3 | no | — |
| 60 | C2×Q8×C53 | — | 2 | 3 | no | — |
| 61 | C22×C4×C53 | C2×Q8 | 1 | 2 | yes | — |
| 62 | C22×C4×C53 | C24 | 1 | 1 | yes | characteristic |
| 63 | C22×Dic53 | C2×Q8 | 1 | 2 | yes | — |
| 64 | C22×Dic53 | C2×Q8 | 1 | 1 | yes | characteristic |
| 65 | Q8×D53 | — | 4 | 3 | no | — |
| 66 | Q8×D53 | — | 4 | 3 | no | — |
| 67 | Q8×D53 | — | 4 | 3 | no | — |
| 68 | Q8×D53 | — | 4 | 3 | no | — |
| 69 | Q8×D53 | — | 4 | 3 | no | — |
| 70 | Q8×D53 | — | 4 | 1 | no | — |
| 71 | C2×C4×D53 | — | 2 | 6 | no | — |
| 72 | C2×C4×D53 | — | 2 | 6 | no | — |
| 73 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 74 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 75 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 76 | C2×C4×D53 | #41 | 1 | 3 | yes | — |
| 77 | C2×Dic106 | — | 2 | 3 | no | — |
| 78 | C2×Dic106 | — | 2 | 3 | no | — |
| 79 | C2×Dic106 | — | 2 | 6 | no | — |
| 80 | C2×Dic106 | — | 2 | 6 | no | — |
| 81 | M16×C53 | — | 2 | 3 | no | — |
| 82 | M16×C53 | — | 2 | 3 | no | — |
| 83 | M16×C53 | — | 2 | 3 | no | — |
| 84 | M16×C53 | C2×D4 | 1 | 6 | yes | — |
| 85 | C2×C8×C53 | — | 2 | 3 | no | — |
| 86 | C2×C8×C53 | — | 2 | 3 | no | — |
| 87 | #11719917 | — | 2 | 3 | no | — |
| 88 | #11719917 | — | 2 | 3 | no | — |
| 89 | #11719917 | — | 2 | 3 | no | — |
| 90 | #11719917 | C2×D4 | 1 | 6 | yes | — |
| 91 | C2×#88322 | — | 2 | 3 | no | — |
| 92 | C2×#88322 | — | 2 | 3 | no | — |
| 93 | #11719912 | — | 2 | 6 | no | — |
| 94 | #11719912 | — | 2 | 6 | no | — |
| 95 | #11719912 | — | 4 | 3 | no | — |
| 96 | #11719912 | — | 4 | 3 | no | — |
| 97 | C8×D53 | — | 2 | 6 | no | — |
| 98 | C8×D53 | — | 2 | 6 | no | — |
| 99 | C8×D53 | — | 4 | 3 | no | — |
| 100 | C8×D53 | — | 4 | 3 | no | — |
| 101 | C23×C53 | C22×Q8 | 1 | 1 | yes | characteristic |
| 102 | C22×D53 | — | 4 | 2 | no | — |
| 103 | C22×D53 | C22⋊Q8 | 1 | 3 | yes | — |
| 104 | C22×D53 | C22⋊Q8 | 1 | 3 | yes | — |
| 105 | Q8×C53 | — | 4 | 3 | no | — |
| 106 | Q8×C53 | — | 4 | 1 | no | — |
| 107 | C2×C4×C53 | — | 2 | 6 | no | — |
| 108 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 109 | C2×C4×C53 | C2×#41 | 1 | 3 | yes | — |
| 110 | C2×Dic53 | — | 2 | 6 | no | — |
| 111 | C2×Dic53 | C22⋊Q8 | 1 | 3 | yes | — |
| 112 | C2×Dic53 | C22⋊Q8 | 1 | 3 | yes | — |
| 113 | C4×D53 | — | 4 | 6 | no | — |
| 114 | C4×D53 | — | 4 | 2 | no | — |
| 115 | C4×D53 | #121 | 1 | 3 | yes | — |
| 116 | C4×D53 | #121 | 1 | 3 | yes | — |
| 117 | C4×D53 | #121 | 1 | 3 | yes | — |
| 118 | C4×D53 | #121 | 1 | 3 | yes | — |
| 119 | C4×D53 | #126 | 1 | 3 | yes | — |
| 120 | C4×D53 | #126 | 1 | 1 | yes | characteristic |
| 121 | Dic106 | — | 4 | 3 | no | — |
| 122 | Dic106 | — | 4 | 1 | no | — |
| 123 | Dic106 | — | 4 | 6 | no | — |
| 124 | Dic106 | — | 4 | 2 | no | — |
| 125 | C8×C53 | — | 2 | 6 | no | — |
| 126 | C8×C53 | — | 4 | 3 | no | — |
| 127 | #88322 | — | 2 | 6 | no | — |
| 128 | #88322 | — | 4 | 3 | no | — |
| 129 | C2×#2070 | — | 53 | 1 | no | maximal, Sylow |
| 130 | C22×C53 | — | 4 | 1 | no | — |
| 131 | C22×C53 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 132 | C2×D53 | — | 2 | 6 | no | — |
| 133 | C2×D53 | — | 4 | 3 | no | — |
| 134 | C2×D53 | #396 | 1 | 3 | yes | — |
| 135 | C2×D53 | #396 | 1 | 1 | yes | characteristic |
| 136 | C4×C53 | — | 4 | 2 | no | — |
| 137 | C4×C53 | C2×#121 | 1 | 3 | yes | — |
| 138 | C4×C53 | C2×#126 | 1 | 1 | yes | characteristic |
| 139 | Dic53 | — | 4 | 2 | no | — |
| 140 | Dic53 | #396 | 1 | 3 | yes | — |
| 141 | Dic53 | #396 | 1 | 1 | yes | characteristic |
| 142 | C2×#430 | — | 53 | 3 | no | — |
| 143 | C22×#101 | — | 53 | 3 | no | — |
| 144 | #2070 | — | 53 | 3 | no | — |
| 145 | #2070 | — | 53 | 3 | no | — |
| 146 | #2070 | — | 53 | 1 | no | — |
| 147 | #2070 | D53 | 1 | 1 | yes | characteristic, direct product factor |
| 148 | C2×#338 | — | 53 | 1 | no | — |
| 149 | C2×C53 | — | 2 | 3 | no | — |
| 150 | C2×C53 | C2×#396 | 1 | 1 | yes | characteristic |
| 151 | D53 | — | 2 | 3 | no | — |
| 152 | D53 | #2070 | 1 | 2 | yes | direct product factor |
| 153 | C23×Q8 | — | 53 | 1 | no | — |
| 154 | C22×M16 | — | 53 | 3 | no | — |
| 155 | C22×M16 | — | 53 | 3 | no | — |
| 156 | #430 | — | 53 | 3 | no | — |
| 157 | #430 | — | 53 | 3 | no | — |
| 158 | #430 | — | 53 | 3 | no | — |
| 159 | #430 | — | 106 | 6 | no | — |
| 160 | #430 | C2×D53 | 1 | 3 | yes | — |
| 161 | C2×#108 | — | 106 | 6 | no | — |
| 162 | C2×#101 | — | 53 | 3 | no | — |
| 163 | C2×#101 | — | 53 | 3 | no | — |
| 164 | C2×#101 | — | 53 | 3 | no | — |
| 165 | C2×#101 | — | 106 | 6 | no | — |
| 166 | C2×#101 | — | 106 | 6 | no | — |
| 167 | C2×#101 | C2×D53 | 1 | 3 | yes | — |
| 168 | #338 | — | 53 | 3 | no | — |
| 169 | #338 | C2×D53 | 1 | 1 | yes | characteristic |
| 170 | C53 | C2×#2070 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C22×Q8 | — | 53 | 2 | no | — |
| 172 | C22×Q8 | — | 53 | 1 | no | — |
| 173 | C22×Q8 | — | 106 | 3 | no | — |
| 174 | C22×Q8 | — | 106 | 3 | no | — |
| 175 | C22×Q8 | — | 106 | 3 | no | — |
| 176 | C22×Q8 | — | 106 | 3 | no | — |
| 177 | C22×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 178 | C23×C4 | — | 53 | 1 | no | — |
| 179 | C23×C4 | — | 53 | 2 | no | — |
| 180 | C2×M16 | — | 53 | 3 | no | — |
| 181 | C2×M16 | — | 53 | 6 | no | — |
| 182 | C2×M16 | — | 53 | 6 | no | — |
| 183 | C2×M16 | — | 53 | 3 | no | — |
| 184 | C2×M16 | — | 106 | 3 | no | — |
| 185 | C2×M16 | — | 106 | 3 | no | — |
| 186 | C2×M16 | — | 106 | 3 | no | — |
| 187 | C2×M16 | — | 106 | 3 | no | — |
| 188 | C2×M16 | — | 106 | 3 | no | — |
| 189 | C2×M16 | — | 106 | 3 | no | — |
| 190 | C2×M16 | — | 106 | 3 | no | — |
| 191 | C2×M16 | — | 106 | 3 | no | — |
| 192 | C2×M16 | C22×D53 | 1 | 3 | yes | — |
| 193 | C2×M16 | C22×D53 | 1 | 3 | yes | — |
| 194 | C22×C8 | — | 106 | 3 | no | — |
| 195 | C22×C8 | — | 106 | 3 | no | — |
| 196 | #108 | — | 2 | 6 | no | — |
| 197 | #108 | — | 106 | 6 | no | — |
| 198 | #108 | — | 106 | 6 | no | — |
| 199 | #108 | — | 106 | 6 | no | — |
| 200 | #101 | — | 2 | 6 | no | — |
| 201 | #101 | — | 106 | 6 | no | — |
| 202 | #101 | — | 106 | 6 | no | — |
| 203 | #101 | — | 106 | 6 | no | — |
| 204 | C24 | — | 53 | 1 | no | — |
| 205 | C2×Q8 | — | 2 | 3 | no | — |
| 206 | C2×Q8 | — | 2 | 3 | no | — |
| 207 | C2×Q8 | — | 106 | 6 | no | — |
| 208 | C2×Q8 | — | 106 | 6 | no | — |
| 209 | C2×Q8 | — | 106 | 3 | no | — |
| 210 | C2×Q8 | — | 106 | 3 | no | — |
| 211 | C2×Q8 | — | 212 | 3 | no | — |
| 212 | C2×Q8 | — | 212 | 3 | no | — |
| 213 | C2×Q8 | — | 212 | 3 | no | — |
| 214 | C2×Q8 | — | 212 | 3 | no | — |
| 215 | C2×Q8 | — | 212 | 1 | no | — |
| 216 | C2×Q8 | — | 212 | 3 | no | — |
| 217 | C22×C4 | — | 53 | 1 | no | — |
| 218 | C22×C4 | — | 53 | 3 | no | — |
| 219 | C22×C4 | — | 53 | 3 | no | — |
| 220 | C22×C4 | — | 53 | 3 | no | — |
| 221 | C22×C4 | — | 53 | 3 | no | — |
| 222 | C22×C4 | — | 53 | 2 | no | — |
| 223 | C22×C4 | — | 106 | 6 | no | — |
| 224 | C22×C4 | — | 106 | 6 | no | — |
| 225 | C22×C4 | Q8×D53 | 1 | 2 | yes | — |
| 226 | C22×C4 | C23×D53 | 1 | 1 | yes | characteristic |
| 227 | M16 | — | 2 | 3 | no | — |
| 228 | M16 | — | 2 | 3 | no | — |
| 229 | M16 | — | 2 | 3 | no | — |
| 230 | M16 | — | 53 | 6 | no | — |
| 231 | M16 | — | 106 | 6 | no | — |
| 232 | M16 | — | 106 | 6 | no | — |
| 233 | M16 | — | 106 | 3 | no | — |
| 234 | M16 | — | 106 | 3 | no | — |
| 235 | M16 | — | 106 | 3 | no | — |
| 236 | M16 | — | 212 | 3 | no | — |
| 237 | M16 | — | 212 | 3 | no | — |
| 238 | M16 | D4×D53 | 1 | 6 | yes | — |
| 239 | C2×C8 | — | 2 | 3 | no | — |
| 240 | C2×C8 | — | 2 | 3 | no | — |
| 241 | C2×C8 | — | 106 | 6 | no | — |
| 242 | C2×C8 | — | 106 | 6 | no | — |
| 243 | C2×C8 | — | 106 | 3 | no | — |
| 244 | C2×C8 | — | 106 | 3 | no | — |
| 245 | C2×C8 | — | 212 | 3 | no | — |
| 246 | C2×C8 | — | 212 | 3 | no | — |
| 247 | C23 | — | 53 | 3 | no | — |
| 248 | C23 | — | 53 | 3 | no | — |
| 249 | C23 | — | 212 | 2 | no | — |
| 250 | C23 | C2×Q8×D53 | 1 | 1 | yes | characteristic |
| 251 | Q8 | — | 4 | 3 | no | — |
| 252 | Q8 | — | 4 | 1 | no | — |
| 253 | Q8 | — | 212 | 6 | no | — |
| 254 | Q8 | — | 212 | 2 | no | — |
| 255 | Q8 | — | 212 | 3 | no | — |
| 256 | Q8 | — | 212 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 6 | no | — |
| 258 | C2×C4 | — | 53 | 3 | no | — |
| 259 | C2×C4 | — | 53 | 3 | no | — |
| 260 | C2×C4 | — | 53 | 3 | no | — |
| 261 | C2×C4 | — | 53 | 3 | no | — |
| 262 | C2×C4 | — | 53 | 3 | no | — |
| 263 | C2×C4 | — | 53 | 3 | no | — |
| 264 | C2×C4 | — | 53 | 1 | no | — |
| 265 | C2×C4 | — | 53 | 3 | no | — |
| 266 | C2×C4 | — | 106 | 6 | no | — |
| 267 | C2×C4 | — | 212 | 2 | no | — |
| 268 | C2×C4 | — | 212 | 6 | no | — |
| 269 | C2×C4 | C2×D4×D53 | 1 | 3 | yes | — |
| 270 | C2×C4 | #41×D53 | 1 | 3 | yes | — |
| 271 | C8 | — | 2 | 6 | no | — |
| 272 | C8 | — | 4 | 3 | no | — |
| 273 | C8 | — | 106 | 6 | no | — |
| 274 | C8 | — | 212 | 3 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 53 | 1 | no | — |
| 277 | C22 | — | 53 | 3 | no | — |
| 278 | C22 | — | 106 | 6 | no | — |
| 279 | C22 | — | 212 | 3 | no | — |
| 280 | C22 | C22⋊Q8×D53 | 1 | 3 | yes | — |
| 281 | C4 | — | 4 | 2 | no | — |
| 282 | C4 | — | 53 | 3 | no | — |
| 283 | C4 | — | 53 | 1 | no | — |
| 284 | C4 | — | 212 | 2 | no | — |
| 285 | C4 | #121×D53 | 1 | 3 | yes | — |
| 286 | C4 | #126×D53 | 1 | 1 | yes | characteristic |
| 287 | C2 | — | 2 | 3 | no | — |
| 288 | C2 | — | 53 | 2 | no | — |
| 289 | C2 | — | 106 | 3 | no | — |
| 290 | C2 | #396×D53 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20276117 | C2 | semidirect |
| 4. #20276116 | C2 | semidirect |
| 5. #20276115 | C2 | semidirect |
| 7. C53×#2070 | C2 | semidirect |
| 147. #2070 | D53 | direct (N × H) |
| 152. D53 | #2070 | direct (N × H) |
| 170. C53 | C2×#2070 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #430×D53 | 2 |
| C2×#101×D53 | 2 |
| #20276117 | 2 |
| #20276116 | 2 |
| #20276115 | 2 |
| #338×D53 | 2 |
| C53×#2070 | 2 |
| C2×#2070 | 53 |