Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
304 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 36 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 | #20289812 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20289646 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20275805 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#13133278 | — | 2 | 1 | no | — |
| 5 | #16811468 | — | 2 | 1 | no | — |
| 6 | C22×#11719913 | — | 2 | 1 | no | — |
| 7 | C2×#13133214 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×#13133139 | — | 2 | 1 | no | — |
| 9 | #16810529 | C22 | 1 | 1 | yes | characteristic |
| 10 | C53×#336 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#11719940 | — | 4 | 1 | no | — |
| 12 | #13133278 | — | 2 | 2 | no | — |
| 13 | #13133278 | — | 4 | 1 | no | — |
| 14 | C22⋊#88324 | — | 4 | 1 | no | — |
| 15 | C22×Dic106 | D4 | 1 | 1 | yes | characteristic |
| 16 | C22×D212 | — | 2 | 1 | no | — |
| 17 | C22×D212 | — | 4 | 1 | no | — |
| 18 | C2×#11719913 | — | 4 | 2 | no | — |
| 19 | C2×#11719913 | — | 4 | 2 | no | — |
| 20 | C2×#11719913 | — | 4 | 1 | no | — |
| 21 | C2×#11719913 | — | 4 | 1 | no | — |
| 22 | #13133214 | — | 4 | 2 | no | — |
| 23 | C2×C4⋊Dic53 | D4 | 1 | 1 | yes | characteristic |
| 24 | #13133139 | — | 4 | 2 | no | — |
| 25 | #13133124 | — | 4 | 1 | no | — |
| 26 | C22×C8×C53 | — | 2 | 1 | no | — |
| 27 | C2×C42×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C23×D53 | — | 4 | 1 | no | — |
| 29 | C22×C4×C53 | — | 2 | 1 | no | — |
| 30 | C22×C4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 31 | C22×Dic53 | — | 2 | 1 | no | — |
| 32 | C22×Dic53 | — | 2 | 1 | no | — |
| 33 | C2×D212 | — | 2 | 2 | no | — |
| 34 | C2×D212 | — | 4 | 1 | no | — |
| 35 | C2×D212 | — | 4 | 1 | no | — |
| 36 | C2×D212 | — | 4 | 1 | no | — |
| 37 | C2×D212 | — | 4 | 1 | no | — |
| 38 | C2×D212 | — | 4 | 1 | no | — |
| 39 | C2×D212 | — | 4 | 2 | no | — |
| 40 | C2×D212 | — | 4 | 2 | no | — |
| 41 | C2×D212 | — | 8 | 2 | no | — |
| 42 | C2×D212 | — | 8 | 2 | no | — |
| 43 | C2×Dic106 | — | 2 | 1 | no | — |
| 44 | C2×Dic106 | — | 4 | 1 | no | — |
| 45 | C2×Dic106 | — | 4 | 1 | no | — |
| 46 | C2×Dic106 | D8 | 1 | 1 | yes | characteristic |
| 47 | C2×Dic106 | SD16 | 1 | 1 | yes | characteristic |
| 48 | #11719940 | — | 4 | 2 | no | — |
| 49 | #11719940 | — | 8 | 1 | no | — |
| 50 | C2×#88324 | — | 8 | 1 | no | — |
| 51 | C2×#88324 | — | 8 | 1 | no | — |
| 52 | C2×C8×C53 | — | 2 | 2 | no | — |
| 53 | C2×C8×C53 | — | 4 | 1 | no | — |
| 54 | C2×C8×C53 | — | 4 | 1 | no | — |
| 55 | C42×C53 | — | 4 | 1 | no | — |
| 56 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 57 | #11719913 | — | 4 | 4 | no | — |
| 58 | #11719913 | — | 4 | 4 | no | — |
| 59 | C23×C53 | #99 | 1 | 1 | yes | characteristic |
| 60 | C22×D53 | — | 4 | 2 | no | — |
| 61 | C22×D53 | — | 8 | 2 | no | — |
| 62 | C22×D53 | — | 8 | 2 | no | — |
| 63 | C22×D53 | — | 8 | 1 | no | — |
| 64 | C22×D53 | — | 8 | 1 | no | — |
| 65 | C2×C4×C53 | — | 2 | 1 | no | — |
| 66 | C2×C4×C53 | — | 2 | 1 | no | — |
| 67 | C2×C4×C53 | — | 2 | 2 | no | — |
| 68 | C2×C4×C53 | — | 4 | 1 | no | — |
| 69 | C2×C4×C53 | — | 4 | 1 | no | — |
| 70 | C2×C4×C53 | #102 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×C53 | #104 | 1 | 1 | yes | characteristic |
| 72 | C2×Dic53 | — | 2 | 1 | no | — |
| 73 | C2×Dic53 | — | 2 | 1 | no | — |
| 74 | C2×Dic53 | — | 4 | 1 | no | — |
| 75 | C2×Dic53 | — | 4 | 1 | no | — |
| 76 | C2×Dic53 | — | 4 | 1 | no | — |
| 77 | C2×Dic53 | — | 8 | 1 | no | — |
| 78 | D212 | — | 4 | 2 | no | — |
| 79 | D212 | — | 4 | 2 | no | — |
| 80 | D212 | — | 8 | 1 | no | — |
| 81 | D212 | — | 8 | 1 | no | — |
| 82 | D212 | — | 8 | 4 | no | — |
| 83 | D212 | — | 8 | 4 | no | — |
| 84 | Dic106 | — | 4 | 1 | no | — |
| 85 | Dic106 | — | 4 | 1 | no | — |
| 86 | Dic106 | — | 8 | 1 | no | — |
| 87 | #88324 | — | 8 | 2 | no | — |
| 88 | #88324 | — | 16 | 1 | no | — |
| 89 | C8×C53 | — | 4 | 2 | no | — |
| 90 | #7404 | — | 53 | 1 | no | maximal, Sylow |
| 91 | C22×C53 | — | 2 | 1 | no | — |
| 92 | C22×C53 | — | 2 | 1 | no | — |
| 93 | C22×C53 | #327 | 1 | 1 | yes | characteristic |
| 94 | C22×C53 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 95 | C22×C53 | #353 | 1 | 1 | yes | characteristic |
| 96 | C2×D53 | — | 8 | 4 | no | — |
| 97 | C2×D53 | — | 8 | 2 | no | — |
| 98 | C2×D53 | — | 16 | 2 | no | — |
| 99 | C2×D53 | — | 16 | 1 | no | — |
| 100 | C2×D53 | — | 16 | 1 | no | — |
| 101 | C4×C53 | — | 2 | 1 | no | — |
| 102 | C4×C53 | — | 2 | 1 | no | — |
| 103 | C4×C53 | — | 4 | 2 | no | — |
| 104 | Dic53 | — | 4 | 1 | no | — |
| 105 | Dic53 | — | 4 | 1 | no | — |
| 106 | Dic53 | — | 8 | 1 | no | — |
| 107 | #2014 | — | 53 | 1 | no | — |
| 108 | #1394 | — | 53 | 1 | no | — |
| 109 | #1319 | — | 53 | 1 | no | — |
| 110 | C2×C53 | — | 2 | 1 | no | — |
| 111 | C2×C53 | — | 2 | 1 | no | — |
| 112 | C2×C53 | #1343 | 1 | 1 | yes | characteristic |
| 113 | C2×C53 | #1406 | 1 | 1 | yes | characteristic |
| 114 | C2×C53 | #1410 | 1 | 1 | yes | characteristic |
| 115 | D53 | — | 16 | 2 | no | — |
| 116 | C22×SD16 | — | 106 | 1 | no | — |
| 117 | C2×#127 | — | 53 | 1 | no | — |
| 118 | C2×#102 | — | 106 | 1 | no | — |
| 119 | C2×#101 | — | 106 | 1 | no | — |
| 120 | #391 | — | 53 | 1 | no | — |
| 121 | #336 | #6157 | 1 | 1 | yes | characteristic |
| 122 | #324 | — | 106 | 1 | no | — |
| 123 | C53 | #7404 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 124 | C22×Q8 | — | 53 | 1 | no | — |
| 125 | C22×D4 | — | 106 | 1 | no | — |
| 126 | C22×D4 | — | 212 | 1 | no | — |
| 127 | C2×SD16 | — | 212 | 2 | no | — |
| 128 | C2×SD16 | — | 212 | 2 | no | — |
| 129 | C2×SD16 | — | 212 | 1 | no | — |
| 130 | C2×SD16 | — | 212 | 1 | no | — |
| 131 | C2×M16 | — | 212 | 1 | no | — |
| 132 | C22×C8 | — | 2 | 1 | no | — |
| 133 | #127 | — | 212 | 2 | no | — |
| 134 | C2×C4⋊C4 | — | 53 | 1 | no | — |
| 135 | C2×C42 | C2×#6157 | 1 | 1 | yes | characteristic |
| 136 | #102 | — | 212 | 2 | no | — |
| 137 | #101 | — | 106 | 2 | no | — |
| 138 | #101 | — | 212 | 1 | no | — |
| 139 | C22⋊C8 | — | 212 | 1 | no | — |
| 140 | #95 | — | 212 | 1 | no | — |
| 141 | C24 | — | 212 | 1 | no | — |
| 142 | C2×Q8 | — | 53 | 1 | no | — |
| 143 | C2×Q8 | — | 53 | 1 | no | — |
| 144 | C2×Q8 | — | 106 | 1 | no | — |
| 145 | C2×Q8 | — | 212 | 1 | no | — |
| 146 | C2×Q8 | — | 212 | 1 | no | — |
| 147 | C2×D4 | — | 106 | 2 | no | — |
| 148 | C2×D4 | — | 424 | 2 | no | — |
| 149 | C2×D4 | — | 424 | 2 | no | — |
| 150 | C2×D4 | — | 212 | 1 | no | — |
| 151 | C2×D4 | — | 212 | 1 | no | — |
| 152 | C2×D4 | — | 212 | 1 | no | — |
| 153 | C2×D4 | — | 212 | 2 | no | — |
| 154 | C2×D4 | — | 212 | 2 | no | — |
| 155 | C2×D4 | — | 212 | 1 | no | — |
| 156 | C2×D4 | — | 212 | 1 | no | — |
| 157 | C22×C4 | — | 2 | 1 | no | — |
| 158 | C22×C4 | — | 106 | 1 | no | — |
| 159 | C22×C4 | — | 106 | 1 | no | — |
| 160 | C22×C4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 161 | SD16 | — | 212 | 4 | no | — |
| 162 | SD16 | — | 212 | 4 | no | — |
| 163 | M16 | — | 424 | 1 | no | — |
| 164 | M16 | — | 212 | 2 | no | — |
| 165 | C2×C8 | — | 2 | 2 | no | — |
| 166 | C2×C8 | — | 4 | 1 | no | — |
| 167 | C2×C8 | — | 4 | 1 | no | — |
| 168 | C2×C8 | — | 424 | 1 | no | — |
| 169 | C2×C8 | — | 424 | 1 | no | — |
| 170 | C4⋊C4 | — | 212 | 1 | no | — |
| 171 | C42 | — | 4 | 1 | no | — |
| 172 | C23 | — | 424 | 2 | no | — |
| 173 | C23 | — | 424 | 2 | no | — |
| 174 | C23 | — | 424 | 1 | no | — |
| 175 | C23 | — | 424 | 1 | no | — |
| 176 | C23 | — | 212 | 2 | no | — |
| 177 | C23 | #13133276 | 1 | 1 | yes | characteristic |
| 178 | Q8 | — | 424 | 1 | no | — |
| 179 | Q8 | — | 212 | 1 | no | — |
| 180 | Q8 | — | 212 | 1 | no | — |
| 181 | D4 | — | 424 | 1 | no | — |
| 182 | D4 | — | 424 | 1 | no | — |
| 183 | D4 | — | 424 | 4 | no | — |
| 184 | D4 | — | 424 | 4 | no | — |
| 185 | D4 | — | 212 | 2 | no | — |
| 186 | D4 | — | 212 | 2 | no | — |
| 187 | C2×C4 | — | 2 | 1 | no | — |
| 188 | C2×C4 | — | 2 | 1 | no | — |
| 189 | C2×C4 | — | 2 | 2 | no | — |
| 190 | C2×C4 | — | 4 | 1 | no | — |
| 191 | C2×C4 | — | 4 | 1 | no | — |
| 192 | C2×C4 | — | 106 | 1 | no | — |
| 193 | C2×C4 | — | 106 | 1 | no | — |
| 194 | C2×C4 | — | 424 | 1 | no | — |
| 195 | C2×C4 | — | 212 | 1 | no | — |
| 196 | C2×C4 | — | 212 | 1 | no | — |
| 197 | C2×C4 | — | 212 | 1 | no | — |
| 198 | C2×C4 | #13133279 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #13133282 | 1 | 1 | yes | characteristic |
| 200 | C8 | — | 4 | 2 | no | — |
| 201 | C8 | — | 848 | 1 | no | — |
| 202 | C8 | — | 424 | 2 | no | — |
| 203 | C22 | — | 2 | 1 | no | — |
| 204 | C22 | — | 2 | 1 | no | — |
| 205 | C22 | — | 848 | 2 | no | — |
| 206 | C22 | — | 848 | 1 | no | — |
| 207 | C22 | — | 848 | 1 | no | — |
| 208 | C22 | — | 424 | 4 | no | — |
| 209 | C22 | — | 424 | 2 | no | — |
| 210 | C22 | #16811475 | 1 | 1 | yes | characteristic |
| 211 | C22 | #16811524 | 1 | 1 | yes | characteristic |
| 212 | C22 | #16811528 | 1 | 1 | yes | characteristic |
| 213 | C4 | — | 2 | 1 | no | — |
| 214 | C4 | — | 2 | 1 | no | — |
| 215 | C4 | — | 4 | 2 | no | — |
| 216 | C4 | — | 424 | 1 | no | — |
| 217 | C4 | — | 212 | 1 | no | — |
| 218 | C4 | — | 212 | 1 | no | — |
| 219 | C2 | — | 2 | 1 | no | — |
| 220 | C2 | — | 2 | 1 | no | — |
| 221 | C2 | — | 848 | 2 | no | — |
| 222 | C2 | #20289694 | 1 | 1 | yes | characteristic |
| 223 | C2 | #20289843 | 1 | 1 | yes | characteristic |
| 224 | C2 | #20289851 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20289812 | C2 | semidirect |
| 2. #20289646 | C2 | semidirect |
| 123. C53 | #7404 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289812 | 2 |
| #20289646 | 2 |
| #20275805 | 2 |
| #7404 | 53 |