← #28249215
Subgroups — #28249215
Order 13568.
2144 conjugacy classes of subgroups.
The subgroup lattice has 552 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×#41×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20283847 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20283841 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20283836 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20283830 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #20283825 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | Q8×#11719912 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #20283451 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #20283343 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | #423×D53 | C2 | 1 | 3 | yes | maximal, complemented |
| 11 | C53×#2991 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#41×D53 | C4 | 1 | 1 | yes | characteristic |
| 13 | C2×M16×D53 | C22 | 1 | 6 | yes | — |
| 14 | #41×Dic53 | C4 | 1 | 1 | yes | characteristic |
| 15 | C4×#11719949 | C4 | 1 | 1 | yes | characteristic |
| 16 | C4×#11719949 | C22 | 1 | 1 | yes | characteristic |
| 17 | C4×#11719945 | C22 | 1 | 3 | yes | — |
| 18 | #16811091 | C4 | 1 | 3 | yes | — |
| 19 | C4×#11719947 | C4 | 1 | 3 | yes | — |
| 20 | C4×#11719947 | C22 | 1 | 3 | yes | — |
| 21 | C4×Q8×D53 | C4 | 1 | 1 | yes | characteristic |
| 22 | C4×Q8×D53 | C22 | 1 | 1 | yes | characteristic |
| 23 | C4×D4×D53 | C4 | 1 | 3 | yes | — |
| 24 | C4×D4×D53 | C22 | 1 | 3 | yes | — |
| 25 | #117×D53 | C4 | 1 | 3 | yes | — |
| 26 | C2×C42×D53 | C22 | 1 | 3 | yes | — |
| 27 | #16810753 | C22 | 1 | 6 | yes | complemented |
| 28 | #16810750 | C22 | 1 | 6 | yes | complemented |
| 29 | Q8×#88322 | C22 | 1 | 2 | yes | complemented |
| 30 | #16810701 | C22 | 1 | 6 | yes | complemented |
| 31 | #16810697 | C22 | 1 | 12 | yes | complemented |
| 32 | #16810692 | C22 | 1 | 12 | yes | complemented |
| 33 | C8×D212 | C22 | 1 | 6 | yes | complemented |
| 34 | #16810688 | C22 | 1 | 6 | yes | complemented |
| 35 | #16810679 | C22 | 1 | 3 | yes | complemented |
| 36 | #16810679 | C22 | 1 | 3 | yes | complemented |
| 37 | #16810677 | C22 | 1 | 6 | yes | complemented |
| 38 | #16810676 | C22 | 1 | 3 | yes | complemented |
| 39 | #16810674 | C22 | 1 | 6 | yes | complemented |
| 40 | #16810647 | C22 | 1 | 3 | yes | complemented |
| 41 | #16810646 | C22 | 1 | 6 | yes | complemented |
| 42 | #16810645 | C22 | 1 | 6 | yes | — |
| 43 | #16810645 | C22 | 1 | 3 | yes | — |
| 44 | #16810645 | C22 | 1 | 1 | yes | characteristic |
| 45 | C4⋊C8×D53 | C22 | 1 | 6 | yes | complemented |
| 46 | C4⋊C8×D53 | C22 | 1 | 6 | yes | complemented |
| 47 | #16810585 | C22 | 1 | 6 | yes | — |
| 48 | C4×#11719912 | C22 | 1 | 6 | yes | — |
| 49 | C4×#41×C53 | C22 | 1 | 1 | yes | characteristic |
| 50 | C8×Q8×C53 | C22 | 1 | 2 | yes | complemented |
| 51 | C53×#436 | C22 | 1 | 6 | yes | complemented |
| 52 | C53×#431 | C22 | 1 | 3 | yes | complemented |
| 53 | C53×#423 | C22 | 1 | 3 | yes | complemented |
| 54 | C2×#11719949 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C2×#11719945 | C2×C4 | 1 | 3 | yes | — |
| 56 | C2×#11719947 | C2×C4 | 1 | 3 | yes | — |
| 57 | #41×D53 | — | 2 | 8 | no | — |
| 58 | C2×Q8×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C2×D4×D53 | C2×C4 | 1 | 3 | yes | — |
| 60 | C22×C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 61 | C22×C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 62 | C22×C4×D53 | C23 | 1 | 3 | yes | — |
| 63 | C2×#11719917 | C23 | 1 | 6 | yes | — |
| 64 | C2×#11719912 | C23 | 1 | 2 | yes | — |
| 65 | C2×#11719912 | C23 | 1 | 6 | yes | — |
| 66 | M16×D53 | — | 4 | 12 | no | — |
| 67 | C2×C8×D53 | C23 | 1 | 12 | yes | — |
| 68 | Q8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 69 | Q8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic106 | C2×C4 | 1 | 3 | yes | — |
| 71 | C4×Dic106 | C23 | 1 | 3 | yes | — |
| 72 | #13133182 | C2×C4 | 1 | 3 | yes | — |
| 73 | #13133182 | C2×C4 | 1 | 3 | yes | — |
| 74 | C4×D212 | C2×C4 | 1 | 3 | yes | — |
| 75 | C4×D212 | C23 | 1 | 3 | yes | — |
| 76 | D4×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 77 | D4×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 78 | #13133179 | C2×C4 | 1 | 3 | yes | — |
| 79 | #13133179 | C2×C4 | 1 | 3 | yes | — |
| 80 | C4×C22⋊D53 | C2×C4 | 1 | 6 | yes | — |
| 81 | C4×C22⋊D53 | C23 | 1 | 6 | yes | — |
| 82 | #13133177 | C2×C4 | 1 | 6 | yes | — |
| 83 | #13133177 | C2×C4 | 1 | 6 | yes | — |
| 84 | #13133176 | C2×C4 | 1 | 3 | yes | — |
| 85 | #13133175 | C2×C4 | 1 | 3 | yes | — |
| 86 | #13133175 | C2×C4 | 1 | 3 | yes | — |
| 87 | #13133174 | C2×C4 | 1 | 6 | yes | — |
| 88 | #13133173 | C2×C4 | 1 | 6 | yes | — |
| 89 | C4⋊C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 90 | C4⋊C4×D53 | C2×C4 | 1 | 3 | yes | — |
| 91 | C22⋊C4×D53 | C2×C4 | 1 | 6 | yes | — |
| 92 | C2×C4×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 93 | C2×C4×Dic53 | C2×C4 | 1 | 3 | yes | — |
| 94 | C2×C4×Dic53 | C23 | 1 | 3 | yes | — |
| 95 | C42×D53 | C2×C4 | 1 | 3 | yes | — |
| 96 | C42×D53 | C2×C4 | 1 | 3 | yes | — |
| 97 | C42×D53 | C2×C4 | 1 | 3 | yes | — |
| 98 | C42×D53 | C2×C4 | 1 | 3 | yes | — |
| 99 | C42×D53 | C23 | 1 | 1 | yes | characteristic |
| 100 | C42×D53 | C23 | 1 | 3 | yes | — |
| 101 | C42×D53 | C23 | 1 | 3 | yes | — |
| 102 | C42×D53 | C23 | 1 | 3 | yes | — |
| 103 | C4⋊#88322 | C23 | 1 | 6 | yes | — |
| 104 | C4⋊#88322 | C23 | 1 | 3 | yes | — |
| 105 | C4⋊#88322 | C23 | 1 | 1 | yes | characteristic |
| 106 | #13133146 | C23 | 1 | 12 | yes | — |
| 107 | #13133146 | C23 | 1 | 6 | yes | — |
| 108 | #13133146 | C23 | 1 | 2 | yes | — |
| 109 | C22⋊#88322 | C23 | 1 | 6 | yes | — |
| 110 | #13133129 | C23 | 1 | 6 | yes | — |
| 111 | #13133129 | C23 | 1 | 6 | yes | — |
| 112 | #13133127 | C23 | 1 | 6 | yes | — |
| 113 | #13133127 | C23 | 1 | 6 | yes | — |
| 114 | C4×#88322 | C23 | 1 | 6 | yes | — |
| 115 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 116 | C2×M16×C53 | C23 | 1 | 6 | yes | — |
| 117 | C4×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 118 | C4×Q8×C53 | C23 | 1 | 1 | yes | characteristic |
| 119 | C4×D4×C53 | C2×C4 | 1 | 3 | yes | — |
| 120 | C4×D4×C53 | C23 | 1 | 3 | yes | — |
| 121 | #117×C53 | C2×C4 | 1 | 3 | yes | — |
| 122 | C2×C42×C53 | C23 | 1 | 3 | yes | — |
| 123 | C4⋊C8×C53 | C23 | 1 | 3 | yes | — |
| 124 | C4⋊C8×C53 | C23 | 1 | 6 | yes | — |
| 125 | C4⋊C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 126 | C22⋊C8×C53 | C23 | 1 | 6 | yes | — |
| 127 | C4×C8×C53 | C23 | 1 | 6 | yes | — |
| 128 | C23×D53 | C22×C4 | 1 | 3 | yes | — |
| 129 | #41×C53 | — | 2 | 4 | no | — |
| 130 | C2×Q8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 131 | C2×D4×C53 | C22×C4 | 1 | 3 | yes | — |
| 132 | C22×C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 133 | C22×C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 134 | C22×C4×C53 | C24 | 1 | 3 | yes | — |
| 135 | C2×C22⋊D53 | C22×C4 | 1 | 6 | yes | — |
| 136 | C22×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 137 | C22×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 138 | C22×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 139 | #11719949 | M16 | 1 | 8 | yes | complemented |
| 140 | Q8×D53 | — | 2 | 4 | no | — |
| 141 | #11719947 | — | 2 | 12 | no | — |
| 142 | D4×D53 | — | 2 | 12 | no | — |
| 143 | #11719945 | — | 2 | 12 | no | — |
| 144 | C2×D212 | C22×C4 | 1 | 3 | yes | — |
| 145 | C2×C4×D53 | — | 2 | 12 | no | — |
| 146 | C2×C4×D53 | — | 2 | 6 | no | — |
| 147 | C2×C4×D53 | — | 2 | 6 | no | — |
| 148 | C2×C4×D53 | — | 4 | 6 | no | — |
| 149 | C2×C4×D53 | C22×C4 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×D53 | C22×C4 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 152 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 153 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 154 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 155 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 156 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 157 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 158 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 159 | C2×C4×D53 | C24 | 1 | 3 | yes | — |
| 160 | C2×C4×D53 | C24 | 1 | 1 | yes | characteristic |
| 161 | C2×Dic106 | C22×C4 | 1 | 3 | yes | — |
| 162 | M16×C53 | — | 4 | 6 | no | — |
| 163 | C2×C8×C53 | C24 | 1 | 2 | yes | — |
| 164 | C2×C8×C53 | C24 | 1 | 6 | yes | — |
| 165 | C4⋊C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 166 | C4⋊C4×C53 | C22×C4 | 1 | 3 | yes | — |
| 167 | C22⋊C4×C53 | C22×C4 | 1 | 6 | yes | — |
| 168 | C42×C53 | C22×C4 | 1 | 3 | yes | — |
| 169 | C42×C53 | C22×C4 | 1 | 3 | yes | — |
| 170 | C42×C53 | C24 | 1 | 3 | yes | — |
| 171 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 172 | C22⋊Dic53 | C22×C4 | 1 | 6 | yes | — |
| 173 | #11719921 | C22×C4 | 1 | 6 | yes | — |
| 174 | #11719921 | C22×C4 | 1 | 6 | yes | — |
| 175 | C4⋊Dic53 | C22×C4 | 1 | 3 | yes | — |
| 176 | C4⋊Dic53 | C22×C4 | 1 | 3 | yes | — |
| 177 | #11719919 | C22×C4 | 1 | 6 | yes | — |
| 178 | #11719919 | C22×C4 | 1 | 6 | yes | — |
| 179 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 180 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 181 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 182 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 183 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 184 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 185 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 186 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 187 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 188 | C4×Dic53 | C22×C4 | 1 | 3 | yes | — |
| 189 | C4×Dic53 | C24 | 1 | 3 | yes | — |
| 190 | C4×Dic53 | C24 | 1 | 1 | yes | characteristic |
| 191 | #11719917 | — | 4 | 6 | no | — |
| 192 | C2×#88322 | C24 | 1 | 2 | yes | — |
| 193 | C2×#88322 | C24 | 1 | 6 | yes | — |
| 194 | #11719912 | — | 2 | 12 | no | — |
| 195 | #11719912 | — | 2 | 4 | no | — |
| 196 | C8×D53 | — | 4 | 12 | no | — |
| 197 | C23×C53 | C23×C4 | 1 | 3 | yes | — |
| 198 | C22×D53 | — | 2 | 6 | no | — |
| 199 | C22×D53 | — | 2 | 6 | no | — |
| 200 | C22×D53 | — | 4 | 3 | no | — |
| 201 | C22×D53 | C23×C4 | 1 | 1 | yes | characteristic |
| 202 | C22×D53 | C23×C4 | 1 | 3 | yes | — |
| 203 | Q8×C53 | C2×M16 | 1 | 4 | yes | complemented |
| 204 | D4×C53 | — | 2 | 6 | no | — |
| 205 | C2×C4×C53 | — | 2 | 6 | no | — |
| 206 | C2×C4×C53 | — | 2 | 6 | no | — |
| 207 | C2×C4×C53 | — | 2 | 3 | no | — |
| 208 | C2×C4×C53 | — | 2 | 3 | no | — |
| 209 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 210 | C2×C4×C53 | C23×C4 | 1 | 3 | yes | — |
| 211 | C2×C4×C53 | C23×C4 | 1 | 3 | yes | — |
| 212 | C2×C4×C53 | C23×C4 | 1 | 3 | yes | — |
| 213 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 214 | C2×C4×C53 | C25 | 1 | 1 | yes | characteristic |
| 215 | C22⋊D53 | — | 2 | 12 | no | — |
| 216 | C2×Dic53 | — | 2 | 6 | no | — |
| 217 | C2×Dic53 | — | 2 | 6 | no | — |
| 218 | C2×Dic53 | — | 4 | 3 | no | — |
| 219 | C2×Dic53 | C23×C4 | 1 | 3 | yes | — |
| 220 | C2×Dic53 | C23×C4 | 1 | 1 | yes | characteristic |
| 221 | C2×Dic53 | C23×C4 | 1 | 3 | yes | — |
| 222 | C2×Dic53 | C23×C4 | 1 | 3 | yes | — |
| 223 | C2×Dic53 | C23×C4 | 1 | 1 | yes | characteristic |
| 224 | C2×Dic53 | C23×C4 | 1 | 1 | yes | characteristic |
| 225 | D212 | C2×M16 | 1 | 12 | yes | complemented |
| 226 | C4×D53 | — | 2 | 6 | no | — |
| 227 | C4×D53 | — | 2 | 6 | no | — |
| 228 | C4×D53 | — | 2 | 6 | no | — |
| 229 | C4×D53 | — | 2 | 2 | no | — |
| 230 | C4×D53 | — | 2 | 6 | no | — |
| 231 | C4×D53 | — | 2 | 3 | no | — |
| 232 | C4×D53 | — | 2 | 3 | no | — |
| 233 | C4×D53 | — | 2 | 6 | no | — |
| 234 | C4×D53 | — | 2 | 2 | no | — |
| 235 | C4×D53 | — | 4 | 3 | no | — |
| 236 | C4×D53 | — | 4 | 3 | no | — |
| 237 | C4×D53 | C2×M16 | 1 | 12 | yes | complemented |
| 238 | C4×D53 | H(5,𝔽2) | 1 | 2 | yes | — |
| 239 | C4×D53 | 2−1+4 | 1 | 2 | yes | — |
| 240 | Dic106 | — | 2 | 6 | no | — |
| 241 | C8×C53 | — | 2 | 6 | no | — |
| 242 | C8×C53 | — | 2 | 2 | no | — |
| 243 | #88322 | — | 2 | 6 | no | — |
| 244 | #88322 | — | 2 | 2 | no | — |
| 245 | #60263 | — | 53 | 1 | no | maximal, Sylow |
| 246 | C22×C53 | — | 2 | 3 | no | — |
| 247 | C22×C53 | — | 2 | 3 | no | — |
| 248 | C22×C53 | — | 2 | 3 | no | — |
| 249 | C22×C53 | C24×C4 | 1 | 1 | yes | characteristic |
| 250 | C2×D53 | — | 2 | 3 | no | — |
| 251 | C2×D53 | — | 2 | 1 | no | — |
| 252 | C2×D53 | — | 2 | 1 | no | — |
| 253 | C2×D53 | — | 2 | 3 | no | — |
| 254 | C2×D53 | — | 4 | 12 | no | — |
| 255 | C2×D53 | #520 | 1 | 2 | yes | — |
| 256 | C2×D53 | C22×M16 | 1 | 6 | yes | — |
| 257 | C4×C53 | — | 2 | 3 | no | — |
| 258 | C4×C53 | — | 2 | 1 | no | — |
| 259 | C4×C53 | — | 2 | 3 | no | — |
| 260 | C4×C53 | C22×M16 | 1 | 6 | yes | — |
| 261 | C4×C53 | #568 | 1 | 2 | yes | — |
| 262 | C4×C53 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 263 | C4×C53 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 264 | Dic53 | — | 2 | 3 | no | — |
| 265 | Dic53 | — | 2 | 3 | no | — |
| 266 | Dic53 | — | 2 | 1 | no | — |
| 267 | Dic53 | — | 2 | 3 | no | — |
| 268 | Dic53 | #520 | 1 | 2 | yes | — |
| 269 | Dic53 | C22×M16 | 1 | 2 | yes | — |
| 270 | C2×C4×#41 | — | 53 | 1 | no | — |
| 271 | #2991 | — | 53 | 1 | no | — |
| 272 | #2991 | D53 | 1 | 1 | yes | characteristic, complemented |
| 273 | #2990 | — | 53 | 3 | no | — |
| 274 | #2990 | — | 53 | 3 | no | — |
| 275 | #2989 | — | 53 | 6 | no | — |
| 276 | #2988 | — | 53 | 2 | no | — |
| 277 | Q8×M16 | — | 53 | 2 | no | — |
| 278 | D4×M16 | — | 53 | 6 | no | — |
| 279 | #2920 | — | 53 | 3 | no | — |
| 280 | C2×#423 | — | 53 | 3 | no | — |
| 281 | C2×C53 | — | 4 | 3 | no | — |
| 282 | C2×C53 | C2×#520 | 1 | 1 | yes | characteristic |
| 283 | C2×C53 | C23×M16 | 1 | 1 | yes | characteristic |
| 284 | C2×C53 | C2×#568 | 1 | 1 | yes | characteristic |
| 285 | D53 | — | 2 | 2 | no | — |
| 286 | D53 | — | 2 | 6 | no | — |
| 287 | C22×#41 | — | 53 | 1 | no | — |
| 288 | C22×M16 | — | 53 | 6 | no | — |
| 289 | C4×#41 | — | 53 | 1 | no | — |
| 290 | C4×#41 | — | 53 | 1 | no | — |
| 291 | C4×#41 | — | 53 | 1 | no | — |
| 292 | C4×#41 | — | 53 | 3 | no | — |
| 293 | C4×#41 | — | 53 | 3 | no | — |
| 294 | C4×#41 | — | 53 | 3 | no | — |
| 295 | C4×#41 | — | 53 | 3 | no | — |
| 296 | C4×#41 | C2×D53 | 1 | 1 | yes | characteristic |
| 297 | C2×C4×Q8 | — | 53 | 1 | no | — |
| 298 | C2×C4×Q8 | — | 53 | 1 | no | — |
| 299 | C2×C4×D4 | — | 53 | 3 | no | — |
| 300 | C2×C4×D4 | — | 53 | 3 | no | — |
| 301 | C2×#117 | — | 53 | 3 | no | — |
| 302 | C22×C42 | — | 53 | 3 | no | — |
| 303 | #446 | — | 53 | 6 | no | — |
| 304 | #446 | — | 53 | 6 | no | — |
| 305 | C8×Q8 | — | 53 | 2 | no | — |
| 306 | C8×Q8 | C2×D53 | 1 | 2 | yes | complemented |
| 307 | #436 | — | 53 | 6 | no | — |
| 308 | #436 | C2×D53 | 1 | 6 | yes | complemented |
| 309 | #435 | — | 53 | 12 | no | — |
| 310 | #435 | — | 53 | 12 | no | — |
| 311 | C8×D4 | — | 53 | 6 | no | — |
| 312 | C8×D4 | — | 53 | 6 | no | — |
| 313 | #432 | — | 53 | 3 | no | — |
| 314 | #432 | — | 53 | 3 | no | — |
| 315 | #432 | — | 53 | 6 | no | — |
| 316 | #431 | — | 53 | 3 | no | — |
| 317 | #431 | — | 53 | 6 | no | — |
| 318 | #431 | C2×D53 | 1 | 3 | yes | complemented |
| 319 | #423 | — | 53 | 6 | no | — |
| 320 | #423 | — | 53 | 3 | no | — |
| 321 | #423 | — | 53 | 3 | no | — |
| 322 | #423 | — | 53 | 6 | no | — |
| 323 | #423 | — | 53 | 1 | no | — |
| 324 | #423 | C2×D53 | 1 | 3 | yes | complemented |
| 325 | C2×C4⋊C8 | — | 53 | 6 | no | — |
| 326 | C2×C4⋊C8 | — | 53 | 6 | no | — |
| 327 | #407 | — | 53 | 6 | no | — |
| 328 | C4×M16 | — | 53 | 6 | no | — |
| 329 | C53 | #60263 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 330 | C2×#41 | — | 53 | 1 | no | — |
| 331 | C2×#41 | — | 53 | 3 | no | — |
| 332 | C2×#41 | — | 53 | 3 | no | — |
| 333 | C2×#41 | — | 106 | 8 | no | — |
| 334 | C2×#41 | C4×D53 | 1 | 1 | yes | characteristic |
| 335 | C22×Q8 | — | 53 | 1 | no | — |
| 336 | C22×D4 | — | 53 | 3 | no | — |
| 337 | C23×C4 | — | 53 | 3 | no | — |
| 338 | C23×C4 | — | 53 | 3 | no | — |
| 339 | C23×C4 | — | 53 | 3 | no | — |
| 340 | C2×M16 | — | 53 | 6 | no | — |
| 341 | C2×M16 | — | 53 | 6 | no | — |
| 342 | C2×M16 | — | 53 | 2 | no | — |
| 343 | C2×M16 | — | 212 | 12 | no | — |
| 344 | C2×M16 | C22×D53 | 1 | 6 | yes | — |
| 345 | C22×C8 | — | 53 | 12 | no | — |
| 346 | C4×Q8 | — | 53 | 1 | no | — |
| 347 | C4×Q8 | — | 53 | 1 | no | — |
| 348 | C4×Q8 | — | 53 | 3 | no | — |
| 349 | C4×Q8 | — | 53 | 3 | no | — |
| 350 | C4×Q8 | — | 53 | 3 | no | — |
| 351 | C4×Q8 | — | 53 | 3 | no | — |
| 352 | C4×Q8 | C4×D53 | 1 | 1 | yes | characteristic |
| 353 | C4×Q8 | C22×D53 | 1 | 1 | yes | characteristic |
| 354 | C4×D4 | — | 53 | 3 | no | — |
| 355 | C4×D4 | — | 53 | 3 | no | — |
| 356 | C4×D4 | — | 53 | 6 | no | — |
| 357 | C4×D4 | — | 53 | 6 | no | — |
| 358 | C4×D4 | — | 53 | 3 | no | — |
| 359 | C4×D4 | — | 53 | 3 | no | — |
| 360 | C4×D4 | — | 53 | 6 | no | — |
| 361 | C4×D4 | — | 53 | 6 | no | — |
| 362 | C4×D4 | — | 53 | 3 | no | — |
| 363 | C4×D4 | — | 53 | 3 | no | — |
| 364 | C4×D4 | C4×D53 | 1 | 3 | yes | — |
| 365 | C4×D4 | C22×D53 | 1 | 3 | yes | — |
| 366 | #117 | — | 53 | 3 | no | — |
| 367 | #117 | — | 53 | 3 | no | — |
| 368 | #117 | — | 53 | 3 | no | — |
| 369 | #117 | — | 53 | 6 | no | — |
| 370 | #117 | — | 53 | 6 | no | — |
| 371 | #117 | C4×D53 | 1 | 3 | yes | — |
| 372 | C2×C4⋊C4 | — | 53 | 3 | no | — |
| 373 | C2×C4⋊C4 | — | 53 | 3 | no | — |
| 374 | C2×C22⋊C4 | — | 53 | 6 | no | — |
| 375 | C2×C42 | — | 53 | 3 | no | — |
| 376 | C2×C42 | — | 53 | 3 | no | — |
| 377 | C2×C42 | — | 53 | 3 | no | — |
| 378 | C2×C42 | — | 53 | 3 | no | — |
| 379 | C2×C42 | — | 53 | 3 | no | — |
| 380 | C2×C42 | — | 53 | 3 | no | — |
| 381 | C2×C42 | — | 53 | 3 | no | — |
| 382 | C2×C42 | — | 53 | 3 | no | — |
| 383 | C2×C42 | — | 53 | 3 | no | — |
| 384 | C2×C42 | — | 53 | 3 | no | — |
| 385 | C2×C42 | — | 53 | 1 | no | — |
| 386 | C2×C42 | C22×D53 | 1 | 3 | yes | — |
| 387 | C4⋊C8 | — | 53 | 6 | no | — |
| 388 | C4⋊C8 | — | 53 | 12 | no | — |
| 389 | C4⋊C8 | — | 53 | 2 | no | — |
| 390 | C4⋊C8 | — | 53 | 3 | no | — |
| 391 | C4⋊C8 | — | 53 | 6 | no | — |
| 392 | C4⋊C8 | — | 53 | 1 | no | — |
| 393 | C4⋊C8 | C22×D53 | 1 | 3 | yes | — |
| 394 | C4⋊C8 | C22×D53 | 1 | 6 | yes | — |
| 395 | C4⋊C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 396 | C22⋊C8 | — | 53 | 6 | no | — |
| 397 | C22⋊C8 | — | 53 | 6 | no | — |
| 398 | C22⋊C8 | — | 53 | 6 | no | — |
| 399 | C22⋊C8 | C22×D53 | 1 | 6 | yes | — |
| 400 | #97 | — | 53 | 6 | no | — |
| 401 | #97 | — | 53 | 6 | no | — |
| 402 | C4×C8 | — | 53 | 6 | no | — |
| 403 | C4×C8 | C22×D53 | 1 | 6 | yes | — |
| 404 | C24 | — | 53 | 3 | no | — |
| 405 | #41 | — | 2 | 4 | no | — |
| 406 | #41 | — | 53 | 8 | no | — |
| 407 | #41 | — | 106 | 12 | no | — |
| 408 | #41 | — | 106 | 12 | no | — |
| 409 | C2×Q8 | — | 53 | 3 | no | — |
| 410 | C2×Q8 | — | 106 | 4 | no | — |
| 411 | C2×Q8 | C2×C4×D53 | 1 | 1 | yes | characteristic |
| 412 | C2×D4 | — | 53 | 6 | no | — |
| 413 | C2×D4 | — | 53 | 3 | no | — |
| 414 | C2×D4 | — | 106 | 12 | no | — |
| 415 | C2×D4 | C2×C4×D53 | 1 | 3 | yes | — |
| 416 | C22×C4 | — | 53 | 3 | no | — |
| 417 | C22×C4 | — | 53 | 1 | no | — |
| 418 | C22×C4 | — | 53 | 3 | no | — |
| 419 | C22×C4 | — | 53 | 3 | no | — |
| 420 | C22×C4 | — | 53 | 3 | no | — |
| 421 | C22×C4 | — | 53 | 1 | no | — |
| 422 | C22×C4 | — | 53 | 1 | no | — |
| 423 | C22×C4 | — | 53 | 3 | no | — |
| 424 | C22×C4 | — | 53 | 3 | no | — |
| 425 | C22×C4 | — | 53 | 3 | no | — |
| 426 | C22×C4 | — | 53 | 3 | no | — |
| 427 | C22×C4 | — | 53 | 3 | no | — |
| 428 | C22×C4 | — | 53 | 3 | no | — |
| 429 | C22×C4 | — | 53 | 3 | no | — |
| 430 | C22×C4 | — | 53 | 3 | no | — |
| 431 | C22×C4 | — | 106 | 6 | no | — |
| 432 | C22×C4 | — | 106 | 6 | no | — |
| 433 | C22×C4 | — | 106 | 12 | no | — |
| 434 | C22×C4 | — | 212 | 6 | no | — |
| 435 | C22×C4 | C2×C4×D53 | 1 | 3 | yes | — |
| 436 | C22×C4 | C2×C4×D53 | 1 | 3 | yes | — |
| 437 | C22×C4 | C23×D53 | 1 | 3 | yes | — |
| 438 | M16 | — | 4 | 6 | no | — |
| 439 | M16 | — | 106 | 12 | no | — |
| 440 | M16 | — | 106 | 4 | no | — |
| 441 | M16 | — | 212 | 6 | no | — |
| 442 | C2×C8 | — | 53 | 2 | no | — |
| 443 | C2×C8 | — | 53 | 6 | no | — |
| 444 | C2×C8 | — | 212 | 12 | no | — |
| 445 | C2×C8 | C23×D53 | 1 | 2 | yes | — |
| 446 | C2×C8 | C23×D53 | 1 | 6 | yes | — |
| 447 | C4⋊C4 | — | 53 | 6 | no | — |
| 448 | C4⋊C4 | — | 53 | 6 | no | — |
| 449 | C4⋊C4 | — | 53 | 3 | no | — |
| 450 | C4⋊C4 | — | 53 | 3 | no | — |
| 451 | C4⋊C4 | C2×C4×D53 | 1 | 3 | yes | — |
| 452 | C4⋊C4 | C2×C4×D53 | 1 | 3 | yes | — |
| 453 | C22⋊C4 | — | 53 | 6 | no | — |
| 454 | C22⋊C4 | — | 53 | 6 | no | — |
| 455 | C22⋊C4 | — | 53 | 6 | no | — |
| 456 | C22⋊C4 | C2×C4×D53 | 1 | 6 | yes | — |
| 457 | C42 | — | 53 | 1 | no | — |
| 458 | C42 | — | 53 | 1 | no | — |
| 459 | C42 | — | 53 | 3 | no | — |
| 460 | C42 | — | 53 | 3 | no | — |
| 461 | C42 | — | 53 | 3 | no | — |
| 462 | C42 | — | 53 | 3 | no | — |
| 463 | C42 | — | 53 | 3 | no | — |
| 464 | C42 | — | 53 | 3 | no | — |
| 465 | C42 | — | 53 | 3 | no | — |
| 466 | C42 | — | 53 | 3 | no | — |
| 467 | C42 | — | 53 | 3 | no | — |
| 468 | C42 | — | 53 | 1 | no | — |
| 469 | C42 | C2×C4×D53 | 1 | 3 | yes | — |
| 470 | C42 | C2×C4×D53 | 1 | 3 | yes | — |
| 471 | C42 | C23×D53 | 1 | 3 | yes | — |
| 472 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 473 | C23 | — | 53 | 3 | no | — |
| 474 | C23 | — | 53 | 1 | no | — |
| 475 | C23 | — | 106 | 6 | no | — |
| 476 | C23 | — | 106 | 6 | no | — |
| 477 | C23 | — | 212 | 3 | no | — |
| 478 | C23 | C22×C4×D53 | 1 | 3 | yes | — |
| 479 | Q8 | — | 106 | 6 | no | — |
| 480 | Q8 | M16×D53 | 1 | 4 | yes | complemented |
| 481 | D4 | — | 2 | 6 | no | — |
| 482 | D4 | — | 53 | 12 | no | — |
| 483 | D4 | — | 106 | 12 | no | — |
| 484 | C2×C4 | — | 2 | 6 | no | — |
| 485 | C2×C4 | — | 2 | 6 | no | — |
| 486 | C2×C4 | — | 2 | 3 | no | — |
| 487 | C2×C4 | — | 2 | 3 | no | — |
| 488 | C2×C4 | — | 53 | 1 | no | — |
| 489 | C2×C4 | — | 53 | 1 | no | — |
| 490 | C2×C4 | — | 53 | 12 | no | — |
| 491 | C2×C4 | — | 53 | 2 | no | — |
| 492 | C2×C4 | — | 53 | 2 | no | — |
| 493 | C2×C4 | — | 53 | 1 | no | — |
| 494 | C2×C4 | — | 53 | 3 | no | — |
| 495 | C2×C4 | — | 53 | 3 | no | — |
| 496 | C2×C4 | — | 53 | 3 | no | — |
| 497 | C2×C4 | — | 106 | 6 | no | — |
| 498 | C2×C4 | — | 106 | 2 | no | — |
| 499 | C2×C4 | — | 106 | 6 | no | — |
| 500 | C2×C4 | — | 106 | 6 | no | — |
| 501 | C2×C4 | — | 106 | 6 | no | — |
| 502 | C2×C4 | — | 106 | 6 | no | — |
| 503 | C2×C4 | — | 106 | 6 | no | — |
| 504 | C2×C4 | — | 106 | 6 | no | — |
| 505 | C2×C4 | — | 106 | 2 | no | — |
| 506 | C2×C4 | — | 106 | 3 | no | — |
| 507 | C2×C4 | — | 106 | 3 | no | — |
| 508 | C2×C4 | — | 212 | 3 | no | — |
| 509 | C2×C4 | — | 212 | 3 | no | — |
| 510 | C2×C4 | — | 212 | 3 | no | — |
| 511 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 512 | C2×C4 | C22×C4×D53 | 1 | 3 | yes | — |
| 513 | C2×C4 | C22×C4×D53 | 1 | 3 | yes | — |
| 514 | C2×C4 | C22×C4×D53 | 1 | 3 | yes | — |
| 515 | C2×C4 | C22×C4×D53 | 1 | 1 | yes | characteristic |
| 516 | C2×C4 | C24×D53 | 1 | 1 | yes | characteristic |
| 517 | C8 | — | 2 | 6 | no | — |
| 518 | C8 | — | 2 | 2 | no | — |
| 519 | C8 | — | 106 | 6 | no | — |
| 520 | C8 | — | 106 | 2 | no | — |
| 521 | C22 | — | 2 | 3 | no | — |
| 522 | C22 | — | 2 | 3 | no | — |
| 523 | C22 | — | 2 | 3 | no | — |
| 524 | C22 | — | 53 | 6 | no | — |
| 525 | C22 | — | 53 | 2 | no | — |
| 526 | C22 | — | 106 | 1 | no | — |
| 527 | C22 | — | 106 | 1 | no | — |
| 528 | C22 | — | 106 | 3 | no | — |
| 529 | C22 | — | 106 | 3 | no | — |
| 530 | C22 | — | 212 | 12 | no | — |
| 531 | C22 | C23×C4×D53 | 1 | 1 | yes | characteristic |
| 532 | C4 | — | 2 | 1 | no | — |
| 533 | C4 | — | 2 | 3 | no | — |
| 534 | C4 | — | 2 | 3 | no | — |
| 535 | C4 | — | 53 | 2 | no | — |
| 536 | C4 | — | 53 | 2 | no | — |
| 537 | C4 | — | 106 | 3 | no | — |
| 538 | C4 | — | 106 | 1 | no | — |
| 539 | C4 | — | 106 | 3 | no | — |
| 540 | C4 | — | 106 | 3 | no | — |
| 541 | C4 | C2×M16×D53 | 1 | 6 | yes | — |
| 542 | C4 | #16811388 | 1 | 2 | yes | — |
| 543 | C4 | H(5,𝔽2)×D53 | 1 | 1 | yes | characteristic |
| 544 | C4 | 2−1+4×D53 | 1 | 1 | yes | characteristic |
| 545 | C2 | — | 4 | 3 | no | — |
| 546 | C2 | — | 106 | 2 | no | — |
| 547 | C2 | — | 106 | 6 | no | — |
| 548 | C2 | #520×D53 | 1 | 1 | yes | characteristic |
| 549 | C2 | C22×M16×D53 | 1 | 1 | yes | characteristic |
| 550 | C2 | C2×#16811388 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta