Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
478 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #20291006 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20290203 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20289836 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #20284064 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16811644 | — | 2 | 1 | no | — |
| 6 | #16811609 | — | 2 | 1 | no | — |
| 7 | #16811533 | — | 2 | 1 | no | — |
| 8 | #16811531 | C22 | 1 | 2 | yes | complemented |
| 9 | #16811479 | — | 2 | 1 | no | — |
| 10 | #16811478 | C22 | 1 | 2 | yes | complemented |
| 11 | #16811421 | — | 2 | 1 | no | — |
| 12 | #16811169 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16810806 | — | 2 | 1 | no | — |
| 14 | #16810801 | C22 | 1 | 2 | yes | — |
| 15 | #16810796 | — | 2 | 1 | no | — |
| 16 | #16810791 | C4 | 1 | 2 | yes | — |
| 17 | C2×#13133144 | — | 2 | 1 | no | — |
| 18 | #16810607 | C4 | 1 | 1 | yes | characteristic |
| 19 | C53×#585 | C4 | 1 | 1 | yes | characteristic |
| 20 | D53⋊M16 | — | 4 | 1 | no | — |
| 21 | C4⋊#88324 | — | 4 | 1 | no | — |
| 22 | #13133278 | — | 2 | 2 | no | — |
| 23 | #13133277 | — | 4 | 1 | no | — |
| 24 | C22⋊#88324 | — | 4 | 1 | no | — |
| 25 | #13133272 | — | 2 | 2 | no | — |
| 26 | C2×#11719945 | D4 | 1 | 1 | yes | characteristic |
| 27 | #13133248 | — | 4 | 1 | no | — |
| 28 | #13133247 | — | 2 | 2 | no | — |
| 29 | #13133243 | — | 4 | 1 | no | — |
| 30 | #13133242 | — | 2 | 2 | no | — |
| 31 | #13133237 | — | 4 | 2 | no | — |
| 32 | C2×#11719917 | — | 2 | 1 | no | — |
| 33 | #13133215 | D4 | 1 | 1 | yes | characteristic |
| 34 | #13133213 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #13133206 | D4 | 1 | 1 | yes | characteristic |
| 36 | #13133204 | C23 | 1 | 1 | yes | characteristic |
| 37 | #13133189 | — | 4 | 1 | no | — |
| 38 | C4×C22⋊D53 | — | 4 | 1 | no | — |
| 39 | C2×C4×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 40 | #13133144 | — | 2 | 2 | no | — |
| 41 | #13133144 | — | 2 | 2 | no | — |
| 42 | #13133136 | — | 2 | 1 | no | — |
| 43 | #13133134 | C2×C4 | 1 | 2 | yes | — |
| 44 | 2−1+4×C53 | — | 2 | 1 | no | — |
| 45 | 2−1+4×C53 | — | 4 | 1 | no | — |
| 46 | H(5,𝔽2)×C53 | — | 4 | 1 | no | — |
| 47 | H(5,𝔽2)×C53 | — | 4 | 1 | no | — |
| 48 | H(5,𝔽2)×C53 | C2×C4 | 1 | 2 | yes | — |
| 49 | C2×#41×C53 | — | 2 | 1 | no | — |
| 50 | C2×#41×C53 | — | 4 | 1 | no | — |
| 51 | C2×#41×C53 | — | 4 | 2 | no | — |
| 52 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | #41×C53 | — | 2 | 2 | no | — |
| 54 | #41×C53 | — | 2 | 2 | no | — |
| 55 | #41×C53 | — | 4 | 1 | no | — |
| 56 | #41×C53 | — | 4 | 2 | no | — |
| 57 | #41×C53 | — | 4 | 1 | no | — |
| 58 | #41×C53 | — | 4 | 1 | no | — |
| 59 | #41×C53 | — | 4 | 2 | no | — |
| 60 | #41×C53 | — | 4 | 2 | no | — |
| 61 | #41×C53 | — | 4 | 4 | no | — |
| 62 | #41×C53 | — | 4 | 2 | no | — |
| 63 | #41×C53 | — | 4 | 1 | no | — |
| 64 | #41×C53 | — | 4 | 2 | no | — |
| 65 | C2×Q8×C53 | — | 2 | 1 | no | — |
| 66 | C2×Q8×C53 | — | 4 | 2 | no | — |
| 67 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 68 | C2×Q8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C2×D4×C53 | — | 2 | 1 | no | — |
| 70 | C2×D4×C53 | — | 2 | 1 | no | — |
| 71 | C2×D4×C53 | — | 4 | 2 | no | — |
| 72 | C2×D4×C53 | — | 4 | 4 | no | — |
| 73 | C2×D4×C53 | — | 4 | 1 | no | — |
| 74 | C2×D4×C53 | — | 4 | 2 | no | — |
| 75 | C2×D4×C53 | — | 4 | 1 | no | — |
| 76 | C2×D4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C22×C4×C53 | — | 2 | 1 | no | — |
| 78 | C22×C4×C53 | — | 4 | 1 | no | — |
| 79 | C22×C4×C53 | — | 4 | 2 | no | — |
| 80 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 81 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 82 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 83 | C22×Dic53 | — | 2 | 1 | no | — |
| 84 | #11719945 | — | 4 | 1 | no | — |
| 85 | #11719945 | — | 4 | 1 | no | — |
| 86 | C2×D212 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×D53 | — | 2 | 1 | no | — |
| 88 | C2×Dic106 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | #11719940 | — | 4 | 2 | no | — |
| 90 | C2×#88324 | — | 4 | 2 | no | — |
| 91 | #11719936 | — | 8 | 1 | no | — |
| 92 | D53⋊C8 | — | 8 | 1 | no | — |
| 93 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 94 | #11719925 | — | 4 | 2 | no | — |
| 95 | #11719924 | — | 4 | 2 | no | — |
| 96 | #11719923 | — | 4 | 2 | no | — |
| 97 | #11719922 | — | 4 | 2 | no | — |
| 98 | #11719921 | — | 4 | 1 | no | — |
| 99 | #11719919 | — | 4 | 1 | no | — |
| 100 | C4×Dic53 | — | 2 | 1 | no | — |
| 101 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C4×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | #11719917 | — | 2 | 2 | no | — |
| 104 | #11719917 | — | 4 | 1 | no | — |
| 105 | C2×#88322 | — | 4 | 1 | no | — |
| 106 | C23×C53 | — | 2 | 1 | no | — |
| 107 | C23×C53 | — | 4 | 1 | no | — |
| 108 | C23×C53 | — | 4 | 2 | no | — |
| 109 | C23×C53 | #99 | 1 | 1 | yes | characteristic |
| 110 | C22×D53 | — | 2 | 1 | no | — |
| 111 | Q8×C53 | — | 2 | 2 | no | — |
| 112 | Q8×C53 | — | 4 | 1 | no | — |
| 113 | Q8×C53 | — | 4 | 2 | no | — |
| 114 | Q8×C53 | — | 4 | 1 | no | — |
| 115 | D4×C53 | — | 2 | 2 | no | — |
| 116 | D4×C53 | — | 4 | 1 | no | — |
| 117 | D4×C53 | — | 4 | 2 | no | — |
| 118 | D4×C53 | — | 4 | 2 | no | — |
| 119 | D4×C53 | — | 4 | 2 | no | — |
| 120 | D4×C53 | — | 4 | 4 | no | — |
| 121 | D4×C53 | — | 4 | 2 | no | — |
| 122 | D4×C53 | — | 4 | 1 | no | — |
| 123 | C2×C4×C53 | — | 2 | 1 | no | — |
| 124 | C2×C4×C53 | — | 2 | 1 | no | — |
| 125 | C2×C4×C53 | — | 2 | 1 | no | — |
| 126 | C2×C4×C53 | — | 4 | 2 | no | — |
| 127 | C2×C4×C53 | — | 4 | 1 | no | — |
| 128 | C2×C4×C53 | — | 4 | 4 | no | — |
| 129 | C2×C4×C53 | — | 4 | 1 | no | — |
| 130 | C2×C4×C53 | — | 4 | 1 | no | — |
| 131 | C2×C4×C53 | — | 4 | 2 | no | — |
| 132 | C2×C4×C53 | — | 4 | 2 | no | — |
| 133 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 134 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C22⋊D53 | — | 8 | 1 | no | — |
| 136 | C22⋊D53 | — | 8 | 2 | no | — |
| 137 | C2×Dic53 | — | 2 | 1 | no | — |
| 138 | C2×Dic53 | — | 2 | 1 | no | — |
| 139 | C2×Dic53 | — | 2 | 1 | no | — |
| 140 | C2×Dic53 | — | 8 | 1 | no | — |
| 141 | D212 | — | 4 | 1 | no | — |
| 142 | C4×D53 | — | 4 | 1 | no | — |
| 143 | C4×D53 | — | 4 | 1 | no | — |
| 144 | Dic106 | — | 4 | 1 | no | — |
| 145 | #88324 | — | 8 | 2 | no | — |
| 146 | #88324 | — | 8 | 2 | no | — |
| 147 | #88322 | — | 4 | 2 | no | — |
| 148 | #13566 | — | 53 | 1 | no | maximal, Sylow |
| 149 | C22×C53 | — | 2 | 1 | no | — |
| 150 | C22×C53 | — | 4 | 1 | no | — |
| 151 | C22×C53 | — | 4 | 2 | no | — |
| 152 | C22×C53 | — | 4 | 1 | no | — |
| 153 | C22×C53 | — | 8 | 1 | no | — |
| 154 | C22×C53 | — | 16 | 1 | no | — |
| 155 | C22×C53 | — | 16 | 2 | no | — |
| 156 | C22×C53 | C2×#99 | 1 | 1 | yes | characteristic |
| 157 | C2×D53 | — | 4 | 1 | no | — |
| 158 | C2×D53 | — | 8 | 1 | no | — |
| 159 | C4×C53 | — | 2 | 1 | no | — |
| 160 | C4×C53 | — | 4 | 1 | no | — |
| 161 | C4×C53 | — | 4 | 2 | no | — |
| 162 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 163 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 164 | Dic53 | — | 4 | 1 | no | — |
| 165 | Dic53 | — | 4 | 1 | no | — |
| 166 | Dic53 | — | 4 | 1 | no | — |
| 167 | #3018 | — | 53 | 1 | no | — |
| 168 | #2132 | — | 53 | 1 | no | — |
| 169 | #1536 | — | 53 | 1 | no | — |
| 170 | #1403 | — | 53 | 4 | no | — |
| 171 | C2×C53 | — | 2 | 1 | no | — |
| 172 | C2×C53 | — | 4 | 1 | no | — |
| 173 | C2×C53 | — | 8 | 1 | no | — |
| 174 | C2×C53 | — | 8 | 2 | no | — |
| 175 | C2×C53 | C2×AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 176 | D53 | — | 8 | 1 | no | — |
| 177 | #585 | #6157 | 1 | 1 | yes | characteristic |
| 178 | #575 | — | 106 | 1 | no | — |
| 179 | #532 | — | 53 | 1 | no | — |
| 180 | #456 | — | 106 | 1 | no | — |
| 181 | #455 | — | 53 | 2 | no | — |
| 182 | #454 | — | 106 | 1 | no | — |
| 183 | #453 | — | 53 | 2 | no | — |
| 184 | #432 | — | 106 | 1 | no | — |
| 185 | C2×#104 | — | 106 | 1 | no | — |
| 186 | #413 | — | 53 | 1 | no | — |
| 187 | #413 | — | 106 | 1 | no | — |
| 188 | #356 | — | 106 | 1 | no | — |
| 189 | #355 | — | 53 | 2 | no | — |
| 190 | #329 | — | 53 | 2 | no | — |
| 191 | #329 | — | 106 | 1 | no | — |
| 192 | C53 | #13566 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 193 | 2−1+4 | — | 2 | 1 | no | — |
| 194 | 2−1+4 | — | 4 | 1 | no | — |
| 195 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 196 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 197 | H(5,𝔽2) | C2×#6157 | 1 | 2 | yes | — |
| 198 | C2×#41 | — | 2 | 1 | no | — |
| 199 | C2×#41 | — | 4 | 1 | no | — |
| 200 | C2×#41 | — | 4 | 2 | no | — |
| 201 | C2×#41 | — | 53 | 1 | no | — |
| 202 | C2×#41 | C2×#6157 | 1 | 1 | yes | characteristic |
| 203 | #137 | — | 106 | 2 | no | — |
| 204 | #137 | — | 212 | 1 | no | — |
| 205 | AGL(1,ℤ/8) | — | 106 | 2 | no | — |
| 206 | AGL(1,ℤ/8) | — | 212 | 1 | no | — |
| 207 | #135 | — | 212 | 2 | no | — |
| 208 | C2×M16 | — | 106 | 1 | no | — |
| 209 | C2×M16 | — | 212 | 1 | no | — |
| 210 | C4⋊Q8 | — | 53 | 1 | no | — |
| 211 | #127 | — | 53 | 1 | no | — |
| 212 | #124 | — | 53 | 1 | no | — |
| 213 | #124 | — | 53 | 1 | no | — |
| 214 | #121 | — | 212 | 1 | no | — |
| 215 | C4×D4 | — | 212 | 1 | no | — |
| 216 | C2×C42 | — | 53 | 1 | no | — |
| 217 | C4⋊C8 | — | 212 | 1 | no | — |
| 218 | #104 | — | 106 | 2 | no | — |
| 219 | #104 | — | 106 | 2 | no | — |
| 220 | #101 | — | 106 | 1 | no | — |
| 221 | #101 | — | 106 | 2 | no | — |
| 222 | #100 | — | 53 | 2 | no | — |
| 223 | #100 | — | 212 | 1 | no | — |
| 224 | C22⋊C8 | — | 212 | 1 | no | — |
| 225 | #97 | — | 106 | 2 | no | — |
| 226 | #41 | — | 2 | 2 | no | — |
| 227 | #41 | — | 2 | 2 | no | — |
| 228 | #41 | — | 4 | 1 | no | — |
| 229 | #41 | — | 4 | 2 | no | — |
| 230 | #41 | — | 4 | 1 | no | — |
| 231 | #41 | — | 4 | 1 | no | — |
| 232 | #41 | — | 4 | 2 | no | — |
| 233 | #41 | — | 4 | 2 | no | — |
| 234 | #41 | — | 4 | 4 | no | — |
| 235 | #41 | — | 4 | 2 | no | — |
| 236 | #41 | — | 4 | 1 | no | — |
| 237 | #41 | — | 4 | 2 | no | — |
| 238 | #41 | — | 212 | 1 | no | — |
| 239 | #41 | — | 212 | 1 | no | — |
| 240 | C2×Q8 | — | 2 | 1 | no | — |
| 241 | C2×Q8 | — | 4 | 2 | no | — |
| 242 | C2×Q8 | — | 4 | 1 | no | — |
| 243 | C2×Q8 | — | 53 | 1 | no | — |
| 244 | C2×Q8 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 245 | C2×D4 | — | 2 | 1 | no | — |
| 246 | C2×D4 | — | 2 | 1 | no | — |
| 247 | C2×D4 | — | 4 | 2 | no | — |
| 248 | C2×D4 | — | 4 | 4 | no | — |
| 249 | C2×D4 | — | 4 | 1 | no | — |
| 250 | C2×D4 | — | 4 | 2 | no | — |
| 251 | C2×D4 | — | 4 | 1 | no | — |
| 252 | C2×D4 | — | 53 | 1 | no | — |
| 253 | C2×D4 | — | 106 | 1 | no | — |
| 254 | C2×D4 | — | 212 | 1 | no | — |
| 255 | C2×D4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 256 | C22×C4 | — | 2 | 1 | no | — |
| 257 | C22×C4 | — | 4 | 1 | no | — |
| 258 | C22×C4 | — | 4 | 2 | no | — |
| 259 | C22×C4 | — | 106 | 1 | no | — |
| 260 | C22×C4 | — | 106 | 1 | no | — |
| 261 | C22×C4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 262 | Q16 | — | 212 | 2 | no | — |
| 263 | SD16 | — | 212 | 2 | no | — |
| 264 | SD16 | — | 212 | 2 | no | — |
| 265 | D8 | — | 212 | 2 | no | — |
| 266 | M16 | — | 106 | 2 | no | — |
| 267 | M16 | — | 424 | 1 | no | — |
| 268 | M16 | — | 212 | 2 | no | — |
| 269 | M16 | — | 212 | 1 | no | — |
| 270 | C2×C8 | — | 424 | 1 | no | — |
| 271 | C2×C8 | — | 212 | 2 | no | — |
| 272 | C2×C8 | — | 212 | 1 | no | — |
| 273 | C4⋊C4 | — | 212 | 1 | no | — |
| 274 | C22⋊C4 | — | 212 | 1 | no | — |
| 275 | C22⋊C4 | — | 212 | 1 | no | — |
| 276 | C42 | — | 53 | 1 | no | — |
| 277 | C42 | — | 53 | 1 | no | — |
| 278 | C42 | — | 106 | 1 | no | — |
| 279 | C23 | — | 2 | 1 | no | — |
| 280 | C23 | — | 4 | 1 | no | — |
| 281 | C23 | — | 4 | 2 | no | — |
| 282 | C23 | — | 106 | 1 | no | — |
| 283 | C23 | #13133275 | 1 | 1 | yes | characteristic |
| 284 | Q8 | — | 2 | 2 | no | — |
| 285 | Q8 | — | 4 | 1 | no | — |
| 286 | Q8 | — | 4 | 2 | no | — |
| 287 | Q8 | — | 4 | 1 | no | — |
| 288 | Q8 | — | 212 | 1 | no | — |
| 289 | D4 | — | 2 | 2 | no | — |
| 290 | D4 | — | 4 | 1 | no | — |
| 291 | D4 | — | 4 | 2 | no | — |
| 292 | D4 | — | 4 | 2 | no | — |
| 293 | D4 | — | 4 | 2 | no | — |
| 294 | D4 | — | 4 | 4 | no | — |
| 295 | D4 | — | 4 | 2 | no | — |
| 296 | D4 | — | 4 | 1 | no | — |
| 297 | D4 | — | 424 | 1 | no | — |
| 298 | D4 | — | 424 | 2 | no | — |
| 299 | D4 | — | 212 | 1 | no | — |
| 300 | C2×C4 | — | 2 | 1 | no | — |
| 301 | C2×C4 | — | 2 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 4 | 2 | no | — |
| 304 | C2×C4 | — | 4 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 4 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 4 | 1 | no | — |
| 308 | C2×C4 | — | 4 | 2 | no | — |
| 309 | C2×C4 | — | 4 | 2 | no | — |
| 310 | C2×C4 | — | 106 | 1 | no | — |
| 311 | C2×C4 | — | 106 | 1 | no | — |
| 312 | C2×C4 | — | 106 | 1 | no | — |
| 313 | C2×C4 | — | 424 | 1 | no | — |
| 314 | C2×C4 | — | 212 | 1 | no | — |
| 315 | C2×C4 | — | 212 | 1 | no | — |
| 316 | C2×C4 | #13133275 | 1 | 1 | yes | characteristic |
| 317 | C2×C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 318 | C8 | — | 424 | 2 | no | — |
| 319 | C8 | — | 424 | 2 | no | — |
| 320 | C8 | — | 212 | 2 | no | — |
| 321 | C22 | — | 2 | 1 | no | — |
| 322 | C22 | — | 4 | 1 | no | — |
| 323 | C22 | — | 4 | 2 | no | — |
| 324 | C22 | — | 4 | 1 | no | — |
| 325 | C22 | — | 8 | 1 | no | — |
| 326 | C22 | — | 16 | 1 | no | — |
| 327 | C22 | — | 16 | 2 | no | — |
| 328 | C22 | — | 424 | 1 | no | — |
| 329 | C22 | — | 212 | 1 | no | — |
| 330 | C22 | C2×#13133275 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 4 | 1 | no | — |
| 333 | C4 | — | 4 | 2 | no | — |
| 334 | C4 | — | 212 | 1 | no | — |
| 335 | C4 | — | 212 | 1 | no | — |
| 336 | C4 | — | 212 | 1 | no | — |
| 337 | C4 | #16811523 | 1 | 1 | yes | characteristic |
| 338 | C4 | #16811523 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 2 | 1 | no | — |
| 340 | C2 | — | 4 | 1 | no | — |
| 341 | C2 | — | 8 | 1 | no | — |
| 342 | C2 | — | 8 | 2 | no | — |
| 343 | C2 | — | 424 | 1 | no | — |
| 344 | C2 | C2×#16811523 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20291006 | C2 | semidirect |
| 2. #20290203 | C2 | semidirect |
| 3. #20289836 | C2 | semidirect |
| 8. #16811531 | C22 | semidirect |
| 10. #16811478 | C22 | semidirect |
| 192. C53 | #13566 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291006 | 2 |
| #20290203 | 2 |
| #20289836 | 2 |
| #20284064 | 2 |
| #13566 | 53 |