Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
290 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 42 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 | #20291007 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20290849 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20276377 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16811627 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×#13133281 | — | 2 | 1 | no | — |
| 6 | C2×#13133281 | — | 2 | 1 | no | — |
| 7 | #16811610 | — | 2 | 1 | no | — |
| 8 | C42×#6157 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #16811534 | C4 | 1 | 2 | yes | complemented |
| 10 | #16811531 | — | 2 | 1 | no | — |
| 11 | #16811171 | C22 | 1 | 1 | yes | characteristic |
| 12 | #16810595 | — | 2 | 1 | no | — |
| 13 | #16810417 | — | 2 | 1 | no | — |
| 14 | #16810416 | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | C53×#353 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#11719936 | — | 2 | 1 | no | — |
| 17 | C2×#11719940 | — | 4 | 1 | no | — |
| 18 | C2×C4×#6157 | — | 2 | 1 | no | — |
| 19 | C2×C4×#6157 | — | 4 | 1 | no | — |
| 20 | C4⋊#88324 | — | 4 | 1 | no | — |
| 21 | #13133281 | — | 4 | 2 | no | — |
| 22 | #13133281 | — | 4 | 2 | no | — |
| 23 | #13133278 | — | 2 | 2 | no | — |
| 24 | #13133277 | — | 4 | 1 | no | — |
| 25 | C2×#11719947 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×#11719947 | D4 | 1 | 1 | yes | characteristic |
| 27 | #13133217 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #13133204 | D4 | 1 | 1 | yes | characteristic |
| 29 | #13133204 | Q8 | 1 | 1 | yes | characteristic |
| 30 | #13133188 | — | 4 | 1 | no | — |
| 31 | D4×Dic53 | — | 4 | 1 | no | — |
| 32 | #13133173 | — | 4 | 1 | no | — |
| 33 | C42×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | #13133132 | — | 2 | 1 | no | — |
| 35 | #13133131 | — | 4 | 1 | no | — |
| 36 | #127×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #99×C53 | — | 2 | 1 | no | — |
| 38 | C22×#6157 | — | 4 | 1 | no | — |
| 39 | C2×D4×C53 | — | 4 | 1 | no | — |
| 40 | C2×D4×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 41 | C2×D4×C53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 43 | C2×C22⋊D53 | — | 2 | 1 | no | — |
| 44 | C22×Dic53 | — | 2 | 1 | no | — |
| 45 | C22×Dic53 | — | 2 | 1 | no | — |
| 46 | #11719947 | — | 4 | 1 | no | — |
| 47 | #11719947 | — | 4 | 1 | no | — |
| 48 | #11719947 | — | 4 | 1 | no | — |
| 49 | #11719947 | — | 4 | 1 | no | — |
| 50 | C2×C4×D53 | — | 2 | 1 | no | — |
| 51 | C2×C4×D53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C2×Dic106 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C2×Dic106 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 54 | #11719940 | — | 4 | 2 | no | — |
| 55 | #11719940 | — | 8 | 1 | no | — |
| 56 | C2×#88324 | — | 4 | 1 | no | — |
| 57 | C2×#88324 | — | 8 | 1 | no | — |
| 58 | C4×#6157 | — | 2 | 2 | no | — |
| 59 | C4×#6157 | — | 4 | 1 | no | — |
| 60 | C4×#6157 | — | 4 | 4 | no | — |
| 61 | #11719936 | — | 4 | 1 | no | — |
| 62 | #11719936 | — | 4 | 1 | no | — |
| 63 | C22⋊C4×C53 | — | 4 | 1 | no | — |
| 64 | C42×C53 | C42 | 1 | 1 | yes | characteristic |
| 65 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 66 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 67 | C22⋊Dic53 | — | 4 | 1 | no | — |
| 68 | C4⋊Dic53 | — | 4 | 1 | no | — |
| 69 | #11719919 | — | 8 | 1 | no | — |
| 70 | C4×Dic53 | — | 2 | 1 | no | — |
| 71 | C4×Dic53 | — | 8 | 1 | no | — |
| 72 | C4×Dic53 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C23×C53 | — | 2 | 1 | no | — |
| 74 | C23×C53 | — | 2 | 1 | no | — |
| 75 | C22×D53 | #99 | 1 | 1 | yes | characteristic |
| 76 | C2×#6157 | — | 4 | 2 | no | — |
| 77 | C2×#6157 | — | 4 | 4 | no | — |
| 78 | D4×C53 | — | 4 | 1 | no | — |
| 79 | D4×C53 | — | 4 | 1 | no | — |
| 80 | D4×C53 | — | 8 | 2 | no | — |
| 81 | C2×C4×C53 | — | 2 | 1 | no | — |
| 82 | C2×C4×C53 | — | 8 | 1 | no | — |
| 83 | C2×C4×C53 | #95 | 1 | 1 | yes | characteristic |
| 84 | C22⋊D53 | — | 8 | 1 | no | — |
| 85 | C22⋊D53 | — | 8 | 1 | no | — |
| 86 | C2×Dic53 | — | 2 | 1 | no | — |
| 87 | C2×Dic53 | — | 2 | 1 | no | — |
| 88 | C2×Dic53 | — | 2 | 1 | no | — |
| 89 | C2×Dic53 | — | 4 | 1 | no | — |
| 90 | C2×Dic53 | — | 4 | 1 | no | — |
| 91 | C2×Dic53 | — | 4 | 1 | no | — |
| 92 | C2×Dic53 | — | 4 | 1 | no | — |
| 93 | C2×Dic53 | — | 8 | 1 | no | — |
| 94 | C2×Dic53 | #99 | 1 | 1 | yes | characteristic |
| 95 | C4×D53 | — | 2 | 1 | no | — |
| 96 | C4×D53 | — | 2 | 1 | no | — |
| 97 | C4×D53 | — | 4 | 2 | no | — |
| 98 | Dic106 | — | 4 | 1 | no | — |
| 99 | Dic106 | — | 4 | 1 | no | — |
| 100 | #88324 | — | 8 | 2 | no | — |
| 101 | #88324 | — | 8 | 1 | no | — |
| 102 | #7497 | — | 53 | 1 | no | maximal, Sylow |
| 103 | C22×C53 | — | 4 | 1 | no | — |
| 104 | C22×C53 | — | 4 | 1 | no | — |
| 105 | C22×C53 | — | 8 | 1 | no | — |
| 106 | C22×C53 | — | 8 | 1 | no | — |
| 107 | C22×C53 | #342 | 1 | 1 | yes | characteristic |
| 108 | C2×D53 | — | 2 | 1 | no | — |
| 109 | C2×D53 | — | 4 | 1 | no | — |
| 110 | #6157 | — | 4 | 4 | no | — |
| 111 | C4×C53 | — | 2 | 2 | no | — |
| 112 | C4×C53 | — | 2 | 1 | no | — |
| 113 | C4×C53 | — | 16 | 1 | no | — |
| 114 | Dic53 | — | 4 | 1 | no | — |
| 115 | Dic53 | — | 4 | 1 | no | — |
| 116 | Dic53 | — | 4 | 1 | no | — |
| 117 | Dic53 | — | 16 | 1 | no | — |
| 118 | Dic53 | #353 | 1 | 1 | yes | characteristic |
| 119 | Dic53 | #354 | 1 | 1 | yes | characteristic |
| 120 | #2132 | — | 53 | 1 | no | — |
| 121 | #2126 | — | 53 | 1 | no | — |
| 122 | #1962 | — | 53 | 1 | no | — |
| 123 | C2×C53 | — | 2 | 1 | no | — |
| 124 | C2×C53 | — | 8 | 1 | no | — |
| 125 | C2×C53 | — | 8 | 1 | no | — |
| 126 | C2×C53 | #1393 | 1 | 1 | yes | characteristic |
| 127 | D53 | — | 4 | 1 | no | — |
| 128 | #532 | — | 53 | 1 | no | — |
| 129 | #423 | — | 53 | 1 | no | — |
| 130 | C2×#104 | — | 106 | 1 | no | — |
| 131 | C2×#104 | — | 106 | 1 | no | — |
| 132 | #413 | — | 106 | 1 | no | — |
| 133 | #410 | — | 106 | 1 | no | — |
| 134 | C43 | — | 53 | 1 | no | — |
| 135 | #356 | — | 53 | 2 | no | — |
| 136 | #355 | — | 106 | 1 | no | — |
| 137 | #354 | — | 106 | 1 | no | — |
| 138 | #353 | — | 53 | 1 | no | — |
| 139 | #353 | #6157 | 1 | 1 | yes | characteristic, complemented |
| 140 | C53 | #7497 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | C2×#41 | — | 53 | 1 | no | — |
| 142 | C2×#41 | — | 53 | 1 | no | — |
| 143 | C2×M16 | — | 106 | 1 | no | — |
| 144 | C2×M16 | — | 212 | 1 | no | — |
| 145 | C4⋊Q8 | — | 53 | 1 | no | — |
| 146 | #127 | C2×#6157 | 1 | 1 | yes | characteristic |
| 147 | #124 | — | 53 | 1 | no | — |
| 148 | #124 | — | 53 | 1 | no | — |
| 149 | #121 | — | 212 | 1 | no | — |
| 150 | C4×D4 | — | 212 | 1 | no | — |
| 151 | #117 | — | 212 | 1 | no | — |
| 152 | C2×C42 | — | 53 | 1 | no | — |
| 153 | C2×C42 | — | 106 | 1 | no | — |
| 154 | C2×C42 | — | 212 | 1 | no | — |
| 155 | C4⋊C8 | — | 212 | 1 | no | — |
| 156 | #104 | — | 212 | 2 | no | — |
| 157 | #104 | — | 212 | 2 | no | — |
| 158 | #101 | — | 106 | 2 | no | — |
| 159 | #100 | — | 212 | 1 | no | — |
| 160 | #99 | — | 2 | 1 | no | — |
| 161 | #99 | — | 106 | 1 | no | — |
| 162 | #99 | — | 212 | 1 | no | — |
| 163 | #41 | — | 212 | 1 | no | — |
| 164 | #41 | — | 212 | 1 | no | — |
| 165 | #41 | — | 212 | 1 | no | — |
| 166 | #41 | — | 212 | 1 | no | — |
| 167 | C2×Q8 | — | 53 | 1 | no | — |
| 168 | C2×Q8 | — | 53 | 1 | no | — |
| 169 | C2×D4 | — | 4 | 1 | no | — |
| 170 | C2×D4 | — | 106 | 1 | no | — |
| 171 | C2×D4 | — | 106 | 1 | no | — |
| 172 | C2×D4 | C4⋊#6157 | 1 | 1 | yes | characteristic |
| 173 | C2×D4 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 174 | C22×C4 | — | 53 | 1 | no | — |
| 175 | C22×C4 | — | 106 | 1 | no | — |
| 176 | C22×C4 | — | 106 | 1 | no | — |
| 177 | C22×C4 | — | 106 | 1 | no | — |
| 178 | C22×C4 | — | 212 | 1 | no | — |
| 179 | M16 | — | 424 | 1 | no | — |
| 180 | M16 | — | 212 | 1 | no | — |
| 181 | M16 | — | 212 | 1 | no | — |
| 182 | M16 | — | 212 | 2 | no | — |
| 183 | C2×C8 | — | 424 | 1 | no | — |
| 184 | C2×C8 | — | 212 | 1 | no | — |
| 185 | C4⋊C4 | — | 424 | 1 | no | — |
| 186 | C4⋊C4 | — | 212 | 1 | no | — |
| 187 | C22⋊C4 | — | 4 | 1 | no | — |
| 188 | C22⋊C4 | — | 212 | 1 | no | — |
| 189 | C22⋊C4 | — | 212 | 1 | no | — |
| 190 | C22⋊C4 | — | 212 | 1 | no | — |
| 191 | C42 | — | 53 | 1 | no | — |
| 192 | C42 | — | 106 | 1 | no | — |
| 193 | C42 | — | 106 | 2 | no | — |
| 194 | C42 | — | 424 | 1 | no | — |
| 195 | C42 | — | 212 | 1 | no | — |
| 196 | C42 | — | 212 | 4 | no | — |
| 197 | C42 | C4×#6157 | 1 | 1 | yes | characteristic |
| 198 | C23 | — | 2 | 1 | no | — |
| 199 | C23 | — | 2 | 1 | no | — |
| 200 | C23 | — | 53 | 1 | no | — |
| 201 | Q8 | — | 212 | 1 | no | — |
| 202 | Q8 | — | 212 | 1 | no | — |
| 203 | D4 | — | 4 | 1 | no | — |
| 204 | D4 | — | 4 | 1 | no | — |
| 205 | D4 | — | 8 | 2 | no | — |
| 206 | D4 | — | 424 | 1 | no | — |
| 207 | D4 | — | 424 | 1 | no | — |
| 208 | C2×C4 | — | 2 | 1 | no | — |
| 209 | C2×C4 | — | 8 | 1 | no | — |
| 210 | C2×C4 | — | 53 | 1 | no | — |
| 211 | C2×C4 | — | 106 | 1 | no | — |
| 212 | C2×C4 | — | 106 | 1 | no | — |
| 213 | C2×C4 | — | 106 | 1 | no | — |
| 214 | C2×C4 | — | 106 | 1 | no | — |
| 215 | C2×C4 | — | 106 | 1 | no | — |
| 216 | C2×C4 | — | 424 | 1 | no | — |
| 217 | C2×C4 | — | 212 | 2 | no | — |
| 218 | C2×C4 | — | 212 | 1 | no | — |
| 219 | C2×C4 | — | 212 | 1 | no | — |
| 220 | C2×C4 | — | 212 | 1 | no | — |
| 221 | C2×C4 | — | 212 | 1 | no | — |
| 222 | C2×C4 | — | 212 | 2 | no | — |
| 223 | C2×C4 | — | 212 | 4 | no | — |
| 224 | C2×C4 | (C2×#6157)⋊C4 | 1 | 1 | yes | characteristic |
| 225 | C8 | — | 424 | 1 | no | — |
| 226 | C8 | — | 424 | 2 | no | — |
| 227 | C22 | — | 4 | 1 | no | — |
| 228 | C22 | — | 4 | 1 | no | — |
| 229 | C22 | — | 8 | 1 | no | — |
| 230 | C22 | — | 8 | 1 | no | — |
| 231 | C22 | — | 106 | 1 | no | — |
| 232 | C22 | — | 212 | 1 | no | — |
| 233 | C22 | #16811500 | 1 | 1 | yes | characteristic |
| 234 | C4 | — | 2 | 2 | no | — |
| 235 | C4 | — | 2 | 1 | no | — |
| 236 | C4 | — | 16 | 1 | no | — |
| 237 | C4 | — | 53 | 1 | no | — |
| 238 | C4 | — | 53 | 1 | no | — |
| 239 | C4 | — | 848 | 1 | no | — |
| 240 | C4 | — | 212 | 1 | no | — |
| 241 | C4 | — | 212 | 4 | no | — |
| 242 | C4 | — | 212 | 1 | no | — |
| 243 | C4 | — | 212 | 1 | no | — |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 8 | 1 | no | — |
| 246 | C2 | — | 8 | 1 | no | — |
| 247 | C2 | — | 212 | 1 | no | — |
| 248 | C2 | #20289808 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #16811627 | C4 | semidirect |
| 8. C42×#6157 | C4 | semidirect |
| 9. #16811534 | C4 | semidirect |
| 14. #16810416 | C4 | semidirect |
| 15. C53×#353 | C4 | semidirect |
| 139. #353 | #6157 | semidirect |
| 140. C53 | #7497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20291007 | 2 |
| #20290849 | 2 |
| #20276377 | 2 |
| #7497 | 53 |