Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
318 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 39 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 | #20274678 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20272641 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C53×#1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #16811459 | C4 | 1 | 1 | yes | characteristic |
| 5 | #16810639 | — | 2 | 1 | no | — |
| 6 | #16810620 | — | 2 | 1 | no | — |
| 7 | #16810597 | C4 | 1 | 1 | yes | characteristic |
| 8 | C22⋊#11719908 | — | 2 | 1 | no | — |
| 9 | C53×#408 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#349 | — | 2 | 1 | no | — |
| 11 | #13133265 | C8 | 1 | 2 | yes | — |
| 12 | #13133264 | — | 4 | 1 | no | — |
| 13 | #13133263 | — | 4 | 1 | no | — |
| 14 | #13133262 | — | 2 | 1 | no | — |
| 15 | #13133262 | — | 4 | 1 | no | — |
| 16 | C2×#11719945 | — | 2 | 1 | no | — |
| 17 | C2×#11719945 | — | 4 | 1 | no | — |
| 18 | #41×D53 | — | 4 | 2 | no | — |
| 19 | #13133176 | — | 2 | 1 | no | — |
| 20 | C2×#11719908 | — | 4 | 1 | no | — |
| 21 | #13133143 | — | 4 | 2 | no | — |
| 22 | #13133142 | — | 4 | 1 | no | — |
| 23 | #13133139 | — | 4 | 1 | no | — |
| 24 | #13133132 | — | 4 | 1 | no | — |
| 25 | C2×#41×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×M16×C53 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C8×C53 | D4 | 1 | 1 | yes | characteristic |
| 28 | M32×C53 | — | 4 | 1 | no | — |
| 29 | C22⋊C8×C53 | — | 4 | 1 | no | — |
| 30 | #41×C53 | — | 4 | 1 | no | — |
| 31 | #41×C53 | — | 4 | 1 | no | — |
| 32 | C2×Q8×C53 | C2×C8 | 1 | 1 | yes | characteristic |
| 33 | C2×D4×C53 | — | 2 | 1 | no | — |
| 34 | C2×D4×C53 | M16 | 1 | 1 | yes | characteristic |
| 35 | C22×C4×C53 | — | 2 | 1 | no | — |
| 36 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 37 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 38 | C2×C22⋊D53 | — | 4 | 1 | no | — |
| 39 | #11719949 | — | 4 | 2 | no | — |
| 40 | Q8×D53 | — | 4 | 2 | no | — |
| 41 | #11719947 | — | 4 | 2 | no | — |
| 42 | #11719947 | — | 4 | 4 | no | — |
| 43 | D4×D53 | — | 4 | 2 | no | — |
| 44 | D4×D53 | — | 4 | 4 | no | — |
| 45 | #11719945 | — | 2 | 2 | no | — |
| 46 | #11719945 | — | 2 | 2 | no | — |
| 47 | #11719945 | — | 4 | 1 | no | — |
| 48 | #11719945 | — | 4 | 2 | no | — |
| 49 | #11719945 | — | 4 | 1 | no | — |
| 50 | #11719945 | — | 4 | 2 | no | — |
| 51 | #11719945 | — | 4 | 2 | no | — |
| 52 | C2×D212 | — | 2 | 1 | no | — |
| 53 | C2×D212 | — | 4 | 1 | no | — |
| 54 | C2×C4×D53 | — | 4 | 1 | no | — |
| 55 | C2×C4×D53 | — | 4 | 2 | no | — |
| 56 | C2×Dic106 | — | 2 | 1 | no | — |
| 57 | C2×Dic106 | — | 4 | 1 | no | — |
| 58 | M16×C53 | — | 4 | 1 | no | — |
| 59 | C2×C8×C53 | — | 2 | 1 | no | — |
| 60 | C2×C8×C53 | — | 2 | 1 | no | — |
| 61 | C2×C8×C53 | — | 4 | 1 | no | — |
| 62 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 63 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 64 | C4×Dic53 | — | 4 | 1 | no | — |
| 65 | C16×C53 | — | 8 | 1 | no | — |
| 66 | #11719908 | — | 8 | 1 | no | — |
| 67 | C23×C53 | — | 2 | 1 | no | — |
| 68 | C23×C53 | #100 | 1 | 1 | yes | characteristic |
| 69 | C22×D53 | — | 4 | 1 | no | — |
| 70 | C22×D53 | — | 4 | 2 | no | — |
| 71 | Q8×C53 | — | 4 | 1 | no | — |
| 72 | D4×C53 | — | 4 | 1 | no | — |
| 73 | D4×C53 | — | 4 | 2 | no | — |
| 74 | C2×C4×C53 | — | 2 | 1 | no | — |
| 75 | C2×C4×C53 | — | 2 | 1 | no | — |
| 76 | C2×C4×C53 | — | 2 | 1 | no | — |
| 77 | C2×C4×C53 | — | 4 | 1 | no | — |
| 78 | C2×C4×C53 | — | 4 | 1 | no | — |
| 79 | C2×C4×C53 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 80 | C2×C4×C53 | #99 | 1 | 1 | yes | characteristic |
| 81 | C22⋊D53 | — | 4 | 2 | no | — |
| 82 | C22⋊D53 | — | 4 | 4 | no | — |
| 83 | C22⋊D53 | — | 4 | 2 | no | — |
| 84 | C2×Dic53 | — | 4 | 1 | no | — |
| 85 | C2×Dic53 | — | 4 | 2 | no | — |
| 86 | C2×Dic53 | — | 8 | 1 | no | — |
| 87 | D212 | — | 2 | 2 | no | — |
| 88 | D212 | — | 4 | 2 | no | — |
| 89 | D212 | — | 4 | 1 | no | — |
| 90 | C4×D53 | — | 4 | 2 | no | — |
| 91 | C4×D53 | — | 4 | 4 | no | — |
| 92 | C4×D53 | — | 4 | 2 | no | — |
| 93 | Dic106 | — | 2 | 2 | no | — |
| 94 | Dic106 | — | 4 | 1 | no | — |
| 95 | Dic106 | — | 4 | 2 | no | — |
| 96 | C8×C53 | — | 4 | 1 | no | — |
| 97 | C8×C53 | — | 4 | 1 | no | — |
| 98 | #7347 | — | 53 | 1 | no | maximal, Sylow |
| 99 | C22×C53 | — | 2 | 1 | no | — |
| 100 | C22×C53 | — | 4 | 1 | no | — |
| 101 | C22×C53 | — | 4 | 1 | no | — |
| 102 | C22×C53 | — | 8 | 1 | no | — |
| 103 | C22×C53 | #323 | 1 | 1 | yes | characteristic |
| 104 | C2×D53 | — | 4 | 2 | no | — |
| 105 | C2×D53 | — | 16 | 1 | no | — |
| 106 | C2×D53 | — | 16 | 2 | no | — |
| 107 | C4×C53 | — | 2 | 1 | no | — |
| 108 | C4×C53 | — | 4 | 1 | no | — |
| 109 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 110 | C4×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 111 | Dic53 | — | 4 | 2 | no | — |
| 112 | Dic53 | — | 16 | 1 | no | — |
| 113 | #1794 | — | 53 | 1 | no | — |
| 114 | #1321 | D53 | 1 | 1 | yes | characteristic, complemented |
| 115 | #1315 | — | 53 | 1 | no | — |
| 116 | C2×C53 | — | 2 | 1 | no | — |
| 117 | C2×C53 | — | 4 | 1 | no | — |
| 118 | C2×C53 | — | 8 | 1 | no | — |
| 119 | C2×C53 | #1316 | 1 | 1 | yes | characteristic |
| 120 | D53 | — | 8 | 2 | no | — |
| 121 | #585 | — | 53 | 1 | no | — |
| 122 | #421 | — | 106 | 1 | no | — |
| 123 | #416 | — | 106 | 1 | no | — |
| 124 | #410 | — | 53 | 1 | no | — |
| 125 | #408 | C2×D53 | 1 | 1 | yes | characteristic |
| 126 | #349 | — | 2 | 1 | no | — |
| 127 | C22⋊C16 | — | 106 | 1 | no | — |
| 128 | C53 | #7347 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | 2−1+4 | — | 53 | 2 | no | — |
| 130 | 2−1+4 | — | 212 | 1 | no | — |
| 131 | H(5,𝔽2) | — | 106 | 1 | no | — |
| 132 | H(5,𝔽2) | — | 212 | 1 | no | — |
| 133 | H(5,𝔽2) | — | 212 | 1 | no | — |
| 134 | C2×#41 | — | 106 | 1 | no | — |
| 135 | C2×#41 | — | 212 | 1 | no | — |
| 136 | C2×#41 | — | 212 | 2 | no | — |
| 137 | C2×#41 | C4×D53 | 1 | 1 | yes | characteristic |
| 138 | C2×M16 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 139 | C22×C8 | D212 | 1 | 1 | yes | characteristic |
| 140 | #117 | — | 106 | 1 | no | — |
| 141 | M32 | — | 4 | 1 | no | — |
| 142 | C2×C16 | — | 212 | 1 | no | — |
| 143 | #104 | — | 212 | 2 | no | — |
| 144 | #103 | — | 212 | 1 | no | — |
| 145 | #102 | — | 212 | 1 | no | — |
| 146 | #99 | — | 212 | 1 | no | — |
| 147 | C22⋊C8 | — | 4 | 1 | no | — |
| 148 | #41 | — | 4 | 1 | no | — |
| 149 | #41 | — | 4 | 1 | no | — |
| 150 | #41 | — | 106 | 2 | no | — |
| 151 | #41 | — | 106 | 2 | no | — |
| 152 | #41 | — | 212 | 1 | no | — |
| 153 | #41 | — | 212 | 2 | no | — |
| 154 | #41 | — | 212 | 2 | no | — |
| 155 | #41 | — | 212 | 2 | no | — |
| 156 | #41 | — | 212 | 2 | no | — |
| 157 | #41 | — | 212 | 4 | no | — |
| 158 | #41 | — | 212 | 2 | no | — |
| 159 | #41 | — | 212 | 1 | no | — |
| 160 | C2×Q8 | — | 106 | 1 | no | — |
| 161 | C2×Q8 | — | 212 | 1 | no | — |
| 162 | C2×Q8 | — | 212 | 2 | no | — |
| 163 | C2×Q8 | C8×D53 | 1 | 1 | yes | characteristic |
| 164 | C2×D4 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 106 | 1 | no | — |
| 166 | C2×D4 | — | 212 | 1 | no | — |
| 167 | C2×D4 | — | 212 | 2 | no | — |
| 168 | C2×D4 | — | 212 | 4 | no | — |
| 169 | C2×D4 | — | 212 | 2 | no | — |
| 170 | C2×D4 | — | 212 | 1 | no | — |
| 171 | C2×D4 | #11719912 | 1 | 1 | yes | characteristic |
| 172 | C22×C4 | — | 2 | 1 | no | — |
| 173 | C22×C4 | — | 212 | 1 | no | — |
| 174 | C22×C4 | — | 212 | 2 | no | — |
| 175 | C22×C4 | #11719921 | 1 | 1 | yes | characteristic |
| 176 | M16 | — | 4 | 1 | no | — |
| 177 | C2×C8 | — | 2 | 1 | no | — |
| 178 | C2×C8 | — | 2 | 1 | no | — |
| 179 | C2×C8 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 212 | 1 | no | — |
| 181 | C22⋊C4 | — | 212 | 1 | no | — |
| 182 | C42 | — | 212 | 1 | no | — |
| 183 | C16 | — | 8 | 1 | no | — |
| 184 | C16 | — | 424 | 1 | no | — |
| 185 | C23 | — | 2 | 1 | no | — |
| 186 | C23 | — | 212 | 1 | no | — |
| 187 | C23 | — | 212 | 2 | no | — |
| 188 | C23 | #13133133 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 4 | 1 | no | — |
| 190 | Q8 | — | 106 | 2 | no | — |
| 191 | Q8 | — | 212 | 2 | no | — |
| 192 | Q8 | — | 212 | 1 | no | — |
| 193 | D4 | — | 4 | 1 | no | — |
| 194 | D4 | — | 4 | 2 | no | — |
| 195 | D4 | — | 106 | 2 | no | — |
| 196 | D4 | — | 212 | 2 | no | — |
| 197 | D4 | — | 212 | 2 | no | — |
| 198 | D4 | — | 212 | 4 | no | — |
| 199 | D4 | — | 212 | 2 | no | — |
| 200 | D4 | — | 212 | 1 | no | — |
| 201 | C2×C4 | — | 2 | 1 | no | — |
| 202 | C2×C4 | — | 2 | 1 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 4 | 1 | no | — |
| 205 | C2×C4 | — | 4 | 1 | no | — |
| 206 | C2×C4 | — | 424 | 1 | no | — |
| 207 | C2×C4 | — | 212 | 2 | no | — |
| 208 | C2×C4 | — | 212 | 4 | no | — |
| 209 | C2×C4 | — | 212 | 1 | no | — |
| 210 | C2×C4 | — | 212 | 2 | no | — |
| 211 | C2×C4 | — | 212 | 2 | no | — |
| 212 | C2×C4 | #13133129 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #13133131 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 4 | 1 | no | — |
| 215 | C8 | — | 4 | 1 | no | — |
| 216 | C22 | — | 2 | 1 | no | — |
| 217 | C22 | — | 4 | 1 | no | — |
| 218 | C22 | — | 4 | 1 | no | — |
| 219 | C22 | — | 8 | 1 | no | — |
| 220 | C22 | — | 848 | 1 | no | — |
| 221 | C22 | — | 848 | 2 | no | — |
| 222 | C22 | — | 212 | 2 | no | — |
| 223 | C22 | #16810346 | 1 | 1 | yes | characteristic |
| 224 | C4 | — | 2 | 1 | no | — |
| 225 | C4 | — | 4 | 1 | no | — |
| 226 | C4 | — | 848 | 1 | no | — |
| 227 | C4 | — | 212 | 2 | no | — |
| 228 | C4 | #16810409 | 1 | 1 | yes | characteristic |
| 229 | C4 | #16810411 | 1 | 1 | yes | characteristic |
| 230 | C2 | — | 2 | 1 | no | — |
| 231 | C2 | — | 4 | 1 | no | — |
| 232 | C2 | — | 8 | 1 | no | — |
| 233 | C2 | — | 424 | 2 | no | — |
| 234 | C2 | #20272642 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20272641 | C2 | semidirect |
| 3. C53×#1321 | C2 | semidirect |
| 114. #1321 | D53 | semidirect |
| 128. C53 | #7347 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20274678 | 2 |
| #20272641 | 2 |
| C53×#1321 | 2 |
| #7347 | 53 |