← C53×#60648 #27268160
Subgroups — C53×#60648 #27268160
Order 13568.
3114 conjugacy classes of subgroups.
The subgroup lattice has 262 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C53×#578 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C53×#3499 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C53×#3156 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | C53×#3152 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C53×#3147 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×C53×#494 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C4×#131×C53 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×2−1+4×C53 | C22 | 1 | 4 | yes | — |
| 9 | C53×#578 | — | 2 | 32 | no | — |
| 10 | C2×#137×C53 | C22 | 1 | 24 | yes | — |
| 11 | C2×#135×C53 | C22 | 1 | 12 | yes | — |
| 12 | C22×Q16×C53 | C22 | 1 | 12 | yes | — |
| 13 | C2×#131×C53 | C22 | 1 | 2 | yes | — |
| 14 | D4×Q8×C53 | — | 2 | 4 | no | — |
| 15 | C53×#547 | — | 2 | 12 | no | — |
| 16 | C53×#536 | C22 | 1 | 12 | yes | — |
| 17 | C53×#532 | C22 | 1 | 6 | yes | — |
| 18 | C2×C4⋊Q8×C53 | C22 | 1 | 6 | yes | — |
| 19 | C4×#41×C53 | C22 | 1 | 1 | yes | characteristic |
| 20 | C53×#497 | C22 | 1 | 24 | yes | — |
| 21 | C53×#495 | C22 | 1 | 12 | yes | — |
| 22 | C53×#494 | C22 | 1 | 3 | yes | — |
| 23 | C53×#494 | C22 | 1 | 6 | yes | — |
| 24 | C53×#494 | C22 | 1 | 12 | yes | — |
| 25 | C53×#493 | C22 | 1 | 1 | yes | characteristic |
| 26 | C53×#492 | C22 | 1 | 6 | yes | — |
| 27 | C53×#405 | C22 | 1 | 6 | yes | — |
| 28 | C4×M16×C53 | C22 | 1 | 3 | yes | — |
| 29 | C2×C4×C8×C53 | C22 | 1 | 3 | yes | — |
| 30 | 2−1+4×C53 | — | 2 | 32 | no | — |
| 31 | C2×#41×C53 | — | 2 | 12 | no | — |
| 32 | C2×#41×C53 | C23 | 1 | 12 | yes | — |
| 33 | C2×#41×C53 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×Q8×C53 | — | 2 | 4 | no | — |
| 35 | C22×Q8×C53 | C23 | 1 | 12 | yes | — |
| 36 | #137×C53 | — | 2 | 96 | no | — |
| 37 | #135×C53 | — | 2 | 48 | no | — |
| 38 | C2×Q16×C53 | — | 2 | 48 | no | — |
| 39 | C2×Q16×C53 | C23 | 1 | 12 | yes | — |
| 40 | C2×Q16×C53 | C23 | 1 | 24 | yes | — |
| 41 | C2×SD16×C53 | C23 | 1 | 24 | yes | — |
| 42 | C2×D8×C53 | C23 | 1 | 4 | yes | — |
| 43 | #131×C53 | D4 | 1 | 16 | yes | complemented |
| 44 | C2×M16×C53 | C23 | 1 | 6 | yes | — |
| 45 | C22×C8×C53 | C23 | 1 | 6 | yes | — |
| 46 | C4⋊Q8×C53 | — | 2 | 12 | no | — |
| 47 | C4⋊Q8×C53 | C23 | 1 | 6 | yes | — |
| 48 | C4⋊Q8×C53 | C23 | 1 | 12 | yes | — |
| 49 | #127×C53 | C23 | 1 | 2 | yes | — |
| 50 | #124×C53 | C23 | 1 | 12 | yes | — |
| 51 | #123×C53 | — | 2 | 24 | no | — |
| 52 | C22⋊Q8×C53 | — | 2 | 24 | no | — |
| 53 | #121×C53 | — | 2 | 12 | no | — |
| 54 | C4×Q8×C53 | D4 | 1 | 2 | yes | — |
| 55 | C4×D4×C53 | — | 2 | 12 | no | — |
| 56 | C4×D4×C53 | D4 | 1 | 6 | yes | — |
| 57 | #117×C53 | C23 | 1 | 3 | yes | — |
| 58 | C2×C4⋊C4×C53 | — | 2 | 12 | no | — |
| 59 | C2×C42×C53 | C23 | 1 | 3 | yes | — |
| 60 | #97×C53 | C23 | 1 | 6 | yes | — |
| 61 | C4×C8×C53 | C23 | 1 | 3 | yes | — |
| 62 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 63 | C4×C8×C53 | C23 | 1 | 6 | yes | — |
| 64 | #41×C53 | — | 2 | 48 | no | — |
| 65 | #41×C53 | — | 4 | 48 | no | — |
| 66 | #41×C53 | C2×D4 | 1 | 8 | yes | — |
| 67 | C2×Q8×C53 | — | 2 | 12 | no | — |
| 68 | C2×Q8×C53 | — | 2 | 48 | no | — |
| 69 | C2×Q8×C53 | — | 4 | 16 | no | — |
| 70 | C2×Q8×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 71 | C2×Q8×C53 | C24 | 1 | 12 | yes | — |
| 72 | C2×D4×C53 | — | 2 | 12 | no | — |
| 73 | C2×D4×C53 | — | 2 | 4 | no | — |
| 74 | C2×D4×C53 | C2×D4 | 1 | 3 | yes | — |
| 75 | C2×D4×C53 | C24 | 1 | 4 | yes | — |
| 76 | C22×C4×C53 | — | 2 | 12 | no | — |
| 77 | C22×C4×C53 | — | 2 | 12 | no | — |
| 78 | C22×C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 79 | C22×C4×C53 | C24 | 1 | 3 | yes | — |
| 80 | Q16×C53 | — | 2 | 24 | no | — |
| 81 | Q16×C53 | — | 2 | 48 | no | — |
| 82 | SD16×C53 | — | 2 | 48 | no | — |
| 83 | D8×C53 | — | 2 | 8 | no | — |
| 84 | M16×C53 | C2×D4 | 1 | 24 | yes | — |
| 85 | C2×C8×C53 | C2×D4 | 1 | 24 | yes | — |
| 86 | C2×C8×C53 | C24 | 1 | 2 | yes | — |
| 87 | C2×C8×C53 | C24 | 1 | 6 | yes | — |
| 88 | C4⋊C4×C53 | — | 2 | 12 | no | — |
| 89 | C4⋊C4×C53 | — | 2 | 24 | no | — |
| 90 | C4⋊C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 91 | C22⋊C4×C53 | — | 2 | 24 | no | — |
| 92 | C22⋊C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 93 | C42×C53 | C2×D4 | 1 | 6 | yes | — |
| 94 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 95 | C42×C53 | C24 | 1 | 3 | yes | — |
| 96 | C23×C53 | — | 2 | 4 | no | — |
| 97 | C23×C53 | C22×D4 | 1 | 3 | yes | — |
| 98 | Q8×C53 | — | 2 | 24 | no | — |
| 99 | Q8×C53 | — | 4 | 24 | no | — |
| 100 | Q8×C53 | #127 | 1 | 4 | yes | — |
| 101 | D4×C53 | — | 2 | 8 | no | — |
| 102 | D4×C53 | — | 4 | 24 | no | — |
| 103 | D4×C53 | — | 4 | 8 | no | — |
| 104 | D4×C53 | #127 | 1 | 12 | yes | — |
| 105 | C2×C4×C53 | — | 2 | 12 | no | — |
| 106 | C2×C4×C53 | — | 2 | 12 | no | — |
| 107 | C2×C4×C53 | — | 4 | 24 | no | — |
| 108 | C2×C4×C53 | — | 4 | 24 | no | — |
| 109 | C2×C4×C53 | C22×D4 | 1 | 6 | yes | — |
| 110 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 111 | C2×C4×C53 | C22×D4 | 1 | 12 | yes | — |
| 112 | C2×C4×C53 | C22×D4 | 1 | 2 | yes | — |
| 113 | C2×C4×C53 | C25 | 1 | 1 | yes | characteristic |
| 114 | C8×C53 | C22×D4 | 1 | 4 | yes | — |
| 115 | C8×C53 | C22×D4 | 1 | 12 | yes | — |
| 116 | #60648 | C53 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 117 | C22×C53 | — | 2 | 6 | no | — |
| 118 | C22×C53 | — | 4 | 4 | no | — |
| 119 | C22×C53 | — | 4 | 8 | no | — |
| 120 | C22×C53 | C2×#127 | 1 | 6 | yes | — |
| 121 | C22×C53 | C23×D4 | 1 | 1 | yes | characteristic |
| 122 | C4×C53 | — | 2 | 6 | no | — |
| 123 | C4×C53 | — | 4 | 12 | no | — |
| 124 | C4×C53 | C2×#127 | 1 | 6 | yes | — |
| 125 | C4×C53 | #578 | 1 | 4 | yes | complemented |
| 126 | C4×C53 | C23×D4 | 1 | 2 | yes | — |
| 127 | C2×#578 | C2×C53 | 1 | 4 | yes | complemented |
| 128 | #3499 | C2×C53 | 1 | 2 | yes | complemented |
| 129 | #3156 | C2×C53 | 1 | 12 | yes | complemented |
| 130 | #3152 | C2×C53 | 1 | 6 | yes | complemented |
| 131 | #3147 | C2×C53 | 1 | 3 | yes | complemented |
| 132 | C2×#494 | C2×C53 | 1 | 3 | yes | complemented |
| 133 | C4×#131 | C2×C53 | 1 | 1 | yes | characteristic, complemented |
| 134 | C2×C53 | — | 2 | 6 | no | — |
| 135 | C2×C53 | — | 8 | 4 | no | — |
| 136 | C2×C53 | C22×#127 | 1 | 1 | yes | characteristic |
| 137 | C2×C53 | C2×#578 | 1 | 2 | yes | — |
| 138 | C2×2−1+4 | C22×C53 | 1 | 4 | yes | — |
| 139 | #578 | — | 2 | 32 | no | — |
| 140 | C2×#137 | C22×C53 | 1 | 24 | yes | — |
| 141 | C2×#135 | C22×C53 | 1 | 12 | yes | — |
| 142 | C22×Q16 | C22×C53 | 1 | 12 | yes | — |
| 143 | C2×#131 | C22×C53 | 1 | 2 | yes | — |
| 144 | D4×Q8 | — | 2 | 4 | no | — |
| 145 | #547 | — | 2 | 12 | no | — |
| 146 | #536 | C22×C53 | 1 | 12 | yes | — |
| 147 | #532 | C22×C53 | 1 | 6 | yes | — |
| 148 | C2×C4⋊Q8 | C22×C53 | 1 | 6 | yes | — |
| 149 | C4×#41 | C22×C53 | 1 | 1 | yes | characteristic |
| 150 | #497 | C22×C53 | 1 | 24 | yes | — |
| 151 | #495 | C22×C53 | 1 | 12 | yes | — |
| 152 | #494 | C22×C53 | 1 | 3 | yes | — |
| 153 | #494 | C22×C53 | 1 | 6 | yes | — |
| 154 | #494 | C22×C53 | 1 | 12 | yes | — |
| 155 | #493 | C22×C53 | 1 | 1 | yes | characteristic |
| 156 | #492 | C22×C53 | 1 | 6 | yes | — |
| 157 | #405 | C22×C53 | 1 | 6 | yes | — |
| 158 | C4×M16 | C22×C53 | 1 | 3 | yes | — |
| 159 | C2×C4×C8 | C22×C53 | 1 | 3 | yes | — |
| 160 | C53 | #60648 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 161 | 2−1+4 | — | 2 | 32 | no | — |
| 162 | C2×#41 | — | 2 | 12 | no | — |
| 163 | C2×#41 | C23×C53 | 1 | 12 | yes | — |
| 164 | C2×#41 | C23×C53 | 1 | 1 | yes | characteristic |
| 165 | C22×Q8 | — | 2 | 4 | no | — |
| 166 | C22×Q8 | C23×C53 | 1 | 12 | yes | — |
| 167 | #137 | — | 2 | 96 | no | — |
| 168 | #135 | — | 2 | 48 | no | — |
| 169 | C2×Q16 | — | 2 | 48 | no | — |
| 170 | C2×Q16 | C23×C53 | 1 | 12 | yes | — |
| 171 | C2×Q16 | C23×C53 | 1 | 24 | yes | — |
| 172 | C2×SD16 | C23×C53 | 1 | 24 | yes | — |
| 173 | C2×D8 | C23×C53 | 1 | 4 | yes | — |
| 174 | #131 | D4×C53 | 1 | 16 | yes | complemented |
| 175 | C2×M16 | C23×C53 | 1 | 6 | yes | — |
| 176 | C22×C8 | C23×C53 | 1 | 6 | yes | — |
| 177 | C4⋊Q8 | — | 2 | 12 | no | — |
| 178 | C4⋊Q8 | C23×C53 | 1 | 6 | yes | — |
| 179 | C4⋊Q8 | C23×C53 | 1 | 12 | yes | — |
| 180 | #127 | C23×C53 | 1 | 2 | yes | — |
| 181 | #124 | C23×C53 | 1 | 12 | yes | — |
| 182 | #123 | — | 2 | 24 | no | — |
| 183 | C22⋊Q8 | — | 2 | 24 | no | — |
| 184 | #121 | — | 2 | 12 | no | — |
| 185 | C4×Q8 | D4×C53 | 1 | 2 | yes | — |
| 186 | C4×D4 | — | 2 | 12 | no | — |
| 187 | C4×D4 | D4×C53 | 1 | 6 | yes | — |
| 188 | #117 | C23×C53 | 1 | 3 | yes | — |
| 189 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 190 | C2×C42 | C23×C53 | 1 | 3 | yes | — |
| 191 | #97 | C23×C53 | 1 | 6 | yes | — |
| 192 | C4×C8 | C23×C53 | 1 | 1 | yes | characteristic |
| 193 | C4×C8 | C23×C53 | 1 | 3 | yes | — |
| 194 | C4×C8 | C23×C53 | 1 | 6 | yes | — |
| 195 | #41 | — | 2 | 48 | no | — |
| 196 | #41 | — | 4 | 48 | no | — |
| 197 | #41 | C2×D4×C53 | 1 | 8 | yes | — |
| 198 | C2×Q8 | — | 2 | 12 | no | — |
| 199 | C2×Q8 | — | 2 | 48 | no | — |
| 200 | C2×Q8 | — | 4 | 16 | no | — |
| 201 | C2×Q8 | C2×D4×C53 | 1 | 1 | yes | characteristic |
| 202 | C2×Q8 | C24×C53 | 1 | 12 | yes | — |
| 203 | C2×D4 | — | 2 | 12 | no | — |
| 204 | C2×D4 | — | 2 | 4 | no | — |
| 205 | C2×D4 | C2×D4×C53 | 1 | 3 | yes | — |
| 206 | C2×D4 | C24×C53 | 1 | 4 | yes | — |
| 207 | C22×C4 | — | 2 | 12 | no | — |
| 208 | C22×C4 | — | 2 | 12 | no | — |
| 209 | C22×C4 | C2×D4×C53 | 1 | 6 | yes | — |
| 210 | C22×C4 | C24×C53 | 1 | 3 | yes | — |
| 211 | Q16 | — | 2 | 24 | no | — |
| 212 | Q16 | — | 2 | 48 | no | — |
| 213 | SD16 | — | 2 | 48 | no | — |
| 214 | D8 | — | 2 | 8 | no | — |
| 215 | M16 | C2×D4×C53 | 1 | 24 | yes | — |
| 216 | C2×C8 | C2×D4×C53 | 1 | 24 | yes | — |
| 217 | C2×C8 | C24×C53 | 1 | 2 | yes | — |
| 218 | C2×C8 | C24×C53 | 1 | 6 | yes | — |
| 219 | C4⋊C4 | — | 2 | 12 | no | — |
| 220 | C4⋊C4 | — | 2 | 24 | no | — |
| 221 | C4⋊C4 | C2×D4×C53 | 1 | 6 | yes | — |
| 222 | C22⋊C4 | — | 2 | 24 | no | — |
| 223 | C22⋊C4 | C2×D4×C53 | 1 | 6 | yes | — |
| 224 | C42 | C2×D4×C53 | 1 | 6 | yes | — |
| 225 | C42 | C24×C53 | 1 | 1 | yes | characteristic |
| 226 | C42 | C24×C53 | 1 | 3 | yes | — |
| 227 | C23 | — | 2 | 4 | no | — |
| 228 | C23 | C22×D4×C53 | 1 | 3 | yes | — |
| 229 | Q8 | — | 2 | 24 | no | — |
| 230 | Q8 | — | 4 | 24 | no | — |
| 231 | Q8 | #127×C53 | 1 | 4 | yes | — |
| 232 | D4 | — | 2 | 8 | no | — |
| 233 | D4 | — | 4 | 24 | no | — |
| 234 | D4 | — | 4 | 8 | no | — |
| 235 | D4 | #127×C53 | 1 | 12 | yes | — |
| 236 | C2×C4 | — | 2 | 12 | no | — |
| 237 | C2×C4 | — | 2 | 12 | no | — |
| 238 | C2×C4 | — | 4 | 24 | no | — |
| 239 | C2×C4 | — | 4 | 24 | no | — |
| 240 | C2×C4 | C22×D4×C53 | 1 | 6 | yes | — |
| 241 | C2×C4 | C22×D4×C53 | 1 | 3 | yes | — |
| 242 | C2×C4 | C22×D4×C53 | 1 | 12 | yes | — |
| 243 | C2×C4 | C22×D4×C53 | 1 | 2 | yes | — |
| 244 | C2×C4 | C25×C53 | 1 | 1 | yes | characteristic |
| 245 | C8 | C22×D4×C53 | 1 | 4 | yes | — |
| 246 | C8 | C22×D4×C53 | 1 | 12 | yes | — |
| 247 | C22 | — | 2 | 6 | no | — |
| 248 | C22 | — | 4 | 8 | no | — |
| 249 | C22 | — | 4 | 4 | no | — |
| 250 | C22 | C2×#127×C53 | 1 | 6 | yes | — |
| 251 | C22 | C23×D4×C53 | 1 | 1 | yes | characteristic |
| 252 | C4 | — | 2 | 6 | no | — |
| 253 | C4 | — | 4 | 12 | no | — |
| 254 | C4 | C2×#127×C53 | 1 | 6 | yes | — |
| 255 | C4 | C53×#578 | 1 | 4 | yes | complemented |
| 256 | C4 | C23×D4×C53 | 1 | 2 | yes | — |
| 257 | C2 | — | 2 | 6 | no | — |
| 258 | C2 | — | 8 | 4 | no | — |
| 259 | C2 | C22×#127×C53 | 1 | 1 | yes | characteristic |
| 260 | C2 | C2×C53×#578 | 1 | 2 | yes | — |
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