Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
364 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #20286514 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20276883 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20272752 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #20272692 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C53×#1715 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #16811424 | — | 2 | 2 | no | — |
| 7 | #16810890 | — | 2 | 2 | no | — |
| 8 | #16810790 | — | 2 | 1 | no | — |
| 9 | #16810790 | — | 2 | 1 | no | — |
| 10 | #16810741 | C22 | 1 | 2 | yes | — |
| 11 | #16810445 | — | 2 | 2 | no | — |
| 12 | #16810444 | C22 | 1 | 2 | yes | complemented |
| 13 | #16810432 | — | 2 | 2 | no | — |
| 14 | C53×#498 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C53×#493 | C22 | 1 | 1 | yes | characteristic |
| 16 | C53×#492 | — | 2 | 1 | no | — |
| 17 | C53×#486 | — | 2 | 1 | no | — |
| 18 | C53×#322 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #13133262 | — | 2 | 1 | no | — |
| 20 | #13133262 | — | 2 | 1 | no | — |
| 21 | #13133243 | — | 2 | 2 | no | — |
| 22 | #13133243 | — | 2 | 2 | no | — |
| 23 | #13133241 | — | 4 | 2 | no | — |
| 24 | #13133241 | — | 4 | 2 | no | — |
| 25 | #13133239 | D4 | 1 | 2 | yes | — |
| 26 | #13133236 | — | 4 | 2 | no | — |
| 27 | D8×D53 | — | 4 | 2 | no | — |
| 28 | #13133223 | — | 2 | 2 | no | — |
| 29 | #13133157 | — | 2 | 2 | no | — |
| 30 | #13133153 | D4 | 1 | 2 | yes | — |
| 31 | #13133145 | — | 2 | 2 | no | — |
| 32 | #13133134 | — | 2 | 1 | no | — |
| 33 | #13133134 | — | 2 | 1 | no | — |
| 34 | C2×SD16×C53 | — | 4 | 2 | no | — |
| 35 | C2×D8×C53 | — | 2 | 2 | no | — |
| 36 | C4⋊Q8×C53 | — | 2 | 1 | no | — |
| 37 | #127×C53 | D4 | 1 | 1 | yes | characteristic |
| 38 | #127×C53 | D4 | 1 | 1 | yes | characteristic |
| 39 | #106×C53 | — | 2 | 2 | no | — |
| 40 | #102×C53 | — | 4 | 2 | no | — |
| 41 | C4×C8×C53 | — | 2 | 1 | no | — |
| 42 | C4×C8×C53 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×Q8×C53 | — | 4 | 1 | no | — |
| 44 | C2×D4×C53 | — | 2 | 1 | no | — |
| 45 | C2×D4×C53 | — | 2 | 1 | no | — |
| 46 | C2×D4×C53 | — | 4 | 1 | no | — |
| 47 | C2×D4×C53 | — | 4 | 1 | no | — |
| 48 | C2×C22⋊D53 | — | 8 | 1 | no | — |
| 49 | C2×C22⋊D53 | — | 8 | 1 | no | — |
| 50 | #11719947 | — | 4 | 2 | no | — |
| 51 | #11719947 | — | 4 | 2 | no | — |
| 52 | D4×D53 | — | 4 | 2 | no | — |
| 53 | D4×D53 | — | 4 | 2 | no | — |
| 54 | #11719945 | — | 2 | 2 | no | — |
| 55 | SD16×C53 | — | 8 | 4 | no | — |
| 56 | D8×C53 | — | 4 | 2 | no | — |
| 57 | D8×C53 | — | 4 | 2 | no | — |
| 58 | C2×C8×C53 | — | 2 | 2 | no | — |
| 59 | C2×C8×C53 | C2×D4 | 1 | 2 | yes | — |
| 60 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 61 | C4⋊C4×C53 | — | 4 | 1 | no | — |
| 62 | C42×C53 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | #11719923 | — | 4 | 2 | no | — |
| 64 | #11719923 | — | 4 | 2 | no | — |
| 65 | #11719922 | — | 4 | 2 | no | — |
| 66 | #11719922 | — | 4 | 2 | no | — |
| 67 | #11719917 | — | 2 | 2 | no | — |
| 68 | Dic212 | SD16 | 1 | 2 | yes | complemented |
| 69 | D424 | SD16 | 1 | 2 | yes | complemented |
| 70 | #11719913 | — | 2 | 2 | no | — |
| 71 | #11719912 | — | 4 | 2 | no | — |
| 72 | C8×D53 | — | 4 | 2 | no | — |
| 73 | #11719908 | — | 4 | 2 | no | — |
| 74 | C23×C53 | — | 4 | 1 | no | — |
| 75 | C23×C53 | — | 4 | 1 | no | — |
| 76 | C22×D53 | — | 8 | 1 | no | — |
| 77 | C22×D53 | — | 8 | 1 | no | — |
| 78 | Q8×C53 | — | 8 | 2 | no | — |
| 79 | D4×C53 | — | 4 | 2 | no | — |
| 80 | D4×C53 | — | 4 | 2 | no | — |
| 81 | D4×C53 | — | 8 | 2 | no | — |
| 82 | D4×C53 | — | 8 | 2 | no | — |
| 83 | C2×C4×C53 | — | 2 | 1 | no | — |
| 84 | C2×C4×C53 | — | 8 | 1 | no | — |
| 85 | C2×C4×C53 | #120 | 1 | 1 | yes | characteristic |
| 86 | C22⋊D53 | — | 4 | 2 | no | — |
| 87 | C22⋊D53 | — | 8 | 2 | no | — |
| 88 | C22⋊D53 | — | 8 | 2 | no | — |
| 89 | C2×Dic53 | — | 8 | 1 | no | — |
| 90 | C2×Dic53 | — | 8 | 1 | no | — |
| 91 | D212 | — | 2 | 2 | no | — |
| 92 | C4×D53 | — | 4 | 2 | no | — |
| 93 | Dic106 | — | 2 | 2 | no | — |
| 94 | C8×C53 | — | 4 | 2 | no | — |
| 95 | C8×C53 | C2×SD16 | 1 | 2 | yes | — |
| 96 | C8×C53 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 97 | #88322 | — | 4 | 2 | no | — |
| 98 | #12366 | — | 53 | 1 | no | maximal, Sylow |
| 99 | C22×C53 | — | 8 | 1 | no | — |
| 100 | C22×C53 | — | 8 | 1 | no | — |
| 101 | C22×C53 | — | 16 | 1 | no | — |
| 102 | C22×C53 | — | 16 | 1 | no | — |
| 103 | C22×C53 | #453 | 1 | 1 | yes | characteristic |
| 104 | C2×D53 | — | 4 | 2 | no | — |
| 105 | C2×D53 | — | 32 | 1 | no | — |
| 106 | C2×D53 | — | 32 | 1 | no | — |
| 107 | C4×C53 | — | 2 | 2 | no | — |
| 108 | C4×C53 | — | 16 | 1 | no | — |
| 109 | C4×C53 | #450 | 1 | 2 | yes | — |
| 110 | Dic53 | — | 4 | 2 | no | — |
| 111 | #3292 | — | 53 | 1 | no | — |
| 112 | #2230 | — | 53 | 2 | no | — |
| 113 | #1715 | D53 | 1 | 1 | yes | characteristic, complemented |
| 114 | #1357 | — | 53 | 1 | no | — |
| 115 | #1336 | — | 53 | 2 | no | — |
| 116 | C2×C53 | — | 2 | 1 | no | — |
| 117 | C2×C53 | — | 16 | 1 | no | — |
| 118 | C2×C53 | — | 16 | 1 | no | — |
| 119 | C2×C53 | #1629 | 1 | 1 | yes | characteristic |
| 120 | D53 | — | 8 | 2 | no | — |
| 121 | #576 | — | 106 | 2 | no | — |
| 122 | #498 | C2×D53 | 1 | 1 | yes | characteristic, complemented |
| 123 | #493 | C2×D53 | 1 | 1 | yes | characteristic |
| 124 | #492 | — | 2 | 1 | no | — |
| 125 | #486 | — | 2 | 1 | no | — |
| 126 | #472 | — | 106 | 2 | no | — |
| 127 | #453 | — | 106 | 1 | no | — |
| 128 | #453 | — | 106 | 1 | no | — |
| 129 | #443 | — | 53 | 2 | no | — |
| 130 | #365 | — | 106 | 2 | no | — |
| 131 | #364 | — | 53 | 2 | no | — |
| 132 | #360 | — | 106 | 2 | no | — |
| 133 | #322 | C2×D53 | 1 | 1 | yes | characteristic, complemented |
| 134 | C53 | #12366 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 135 | H(5,𝔽2) | — | 106 | 1 | no | — |
| 136 | H(5,𝔽2) | — | 106 | 1 | no | — |
| 137 | AGL(1,ℤ/8) | — | 106 | 2 | no | — |
| 138 | AGL(1,ℤ/8) | — | 106 | 2 | no | — |
| 139 | AGL(1,ℤ/8) | — | 212 | 2 | no | — |
| 140 | AGL(1,ℤ/8) | — | 212 | 2 | no | — |
| 141 | #135 | — | 53 | 2 | no | — |
| 142 | #135 | — | 212 | 2 | no | — |
| 143 | C2×SD16 | — | 4 | 2 | no | — |
| 144 | C2×D8 | — | 2 | 2 | no | — |
| 145 | C2×D8 | — | 212 | 2 | no | — |
| 146 | #131 | — | 106 | 2 | no | — |
| 147 | C4⋊Q8 | — | 2 | 1 | no | — |
| 148 | #127 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 149 | #127 | C22⋊D53 | 1 | 1 | yes | characteristic |
| 150 | M32 | — | 106 | 2 | no | — |
| 151 | #108 | — | 53 | 2 | no | — |
| 152 | #106 | — | 2 | 2 | no | — |
| 153 | #104 | — | 106 | 2 | no | — |
| 154 | #102 | — | 4 | 2 | no | — |
| 155 | #100 | — | 106 | 1 | no | — |
| 156 | #100 | — | 106 | 1 | no | — |
| 157 | C4×C8 | — | 2 | 1 | no | — |
| 158 | C4×C8 | C22×D53 | 1 | 1 | yes | characteristic |
| 159 | #41 | — | 106 | 2 | no | — |
| 160 | #41 | — | 212 | 2 | no | — |
| 161 | #41 | — | 212 | 2 | no | — |
| 162 | C2×Q8 | — | 4 | 1 | no | — |
| 163 | C2×D4 | — | 2 | 1 | no | — |
| 164 | C2×D4 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 4 | 1 | no | — |
| 166 | C2×D4 | — | 4 | 1 | no | — |
| 167 | C2×D4 | — | 424 | 1 | no | — |
| 168 | C2×D4 | — | 424 | 1 | no | — |
| 169 | C2×D4 | — | 212 | 2 | no | — |
| 170 | C2×D4 | — | 212 | 2 | no | — |
| 171 | Q16 | — | 53 | 2 | no | — |
| 172 | SD16 | — | 8 | 4 | no | — |
| 173 | SD16 | — | 106 | 2 | no | — |
| 174 | SD16 | — | 212 | 2 | no | — |
| 175 | SD16 | — | 212 | 2 | no | — |
| 176 | D8 | — | 4 | 2 | no | — |
| 177 | D8 | — | 4 | 2 | no | — |
| 178 | D8 | — | 53 | 2 | no | — |
| 179 | D8 | — | 212 | 2 | no | — |
| 180 | D8 | — | 212 | 2 | no | — |
| 181 | M16 | — | 106 | 2 | no | — |
| 182 | M16 | — | 212 | 2 | no | — |
| 183 | C2×C8 | — | 2 | 2 | no | — |
| 184 | C2×C8 | — | 212 | 2 | no | — |
| 185 | C2×C8 | D4×D53 | 1 | 2 | yes | — |
| 186 | C4⋊C4 | — | 4 | 1 | no | — |
| 187 | C4⋊C4 | — | 4 | 1 | no | — |
| 188 | C42 | C2×C22⋊D53 | 1 | 1 | yes | characteristic |
| 189 | C16 | — | 212 | 2 | no | — |
| 190 | C23 | — | 4 | 1 | no | — |
| 191 | C23 | — | 4 | 1 | no | — |
| 192 | C23 | — | 424 | 1 | no | — |
| 193 | C23 | — | 424 | 1 | no | — |
| 194 | Q8 | — | 8 | 2 | no | — |
| 195 | Q8 | — | 106 | 2 | no | — |
| 196 | D4 | — | 4 | 2 | no | — |
| 197 | D4 | — | 4 | 2 | no | — |
| 198 | D4 | — | 8 | 2 | no | — |
| 199 | D4 | — | 8 | 2 | no | — |
| 200 | D4 | — | 106 | 2 | no | — |
| 201 | D4 | — | 424 | 2 | no | — |
| 202 | D4 | — | 424 | 2 | no | — |
| 203 | D4 | — | 212 | 2 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 8 | 1 | no | — |
| 206 | C2×C4 | — | 424 | 1 | no | — |
| 207 | C2×C4 | — | 424 | 1 | no | — |
| 208 | C2×C4 | — | 212 | 2 | no | — |
| 209 | C2×C4 | #13133185 | 1 | 1 | yes | characteristic |
| 210 | C8 | — | 4 | 2 | no | — |
| 211 | C8 | — | 212 | 2 | no | — |
| 212 | C8 | SD16×D53 | 1 | 2 | yes | — |
| 213 | C8 | #13133244 | 1 | 2 | yes | — |
| 214 | C22 | — | 8 | 1 | no | — |
| 215 | C22 | — | 8 | 1 | no | — |
| 216 | C22 | — | 16 | 1 | no | — |
| 217 | C22 | — | 16 | 1 | no | — |
| 218 | C22 | — | 1696 | 1 | no | — |
| 219 | C22 | — | 1696 | 1 | no | — |
| 220 | C22 | — | 212 | 2 | no | — |
| 221 | C22 | #16810790 | 1 | 1 | yes | characteristic |
| 222 | C4 | — | 2 | 2 | no | — |
| 223 | C4 | — | 16 | 1 | no | — |
| 224 | C4 | — | 212 | 2 | no | — |
| 225 | C4 | #16810772 | 1 | 2 | yes | — |
| 226 | C2 | — | 2 | 1 | no | — |
| 227 | C2 | — | 16 | 1 | no | — |
| 228 | C2 | — | 16 | 1 | no | — |
| 229 | C2 | — | 424 | 2 | no | — |
| 230 | C2 | #20273829 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20276883 | C2 | semidirect |
| 3. #20272752 | C2 | semidirect |
| 4. #20272692 | C2 | semidirect |
| 5. C53×#1715 | C2 | semidirect |
| 12. #16810444 | C22 | semidirect |
| 14. C53×#498 | C22 | semidirect |
| 18. C53×#322 | C22 | semidirect |
| 68. Dic212 | SD16 | semidirect |
| 69. D424 | SD16 | semidirect |
| 113. #1715 | D53 | semidirect |
| 122. #498 | C2×D53 | semidirect |
| 133. #322 | C2×D53 | semidirect |
| 134. C53 | #12366 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20286514 | 2 |
| #20276883 | 2 |
| #20272752 | 2 |
| #20272692 | 2 |
| C53×#1715 | 2 |
| #12366 | 53 |