Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
876 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 145 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 | #20289209 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20285088 | C2 | 1 | 3 | yes | maximal |
| 3 | #20274018 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20274017 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #20274016 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #20273308 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C53×#1658 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C53⋊#564 | C22 | 1 | 3 | yes | complemented |
| 9 | #16811350 | — | 2 | 6 | no | — |
| 10 | #16811232 | — | 2 | 6 | no | — |
| 11 | #16811179 | C22 | 1 | 3 | yes | — |
| 12 | #127×D53 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16810989 | C22 | 1 | 3 | yes | — |
| 14 | #16810988 | C22 | 1 | 3 | yes | — |
| 15 | #16810987 | C22 | 1 | 3 | yes | — |
| 16 | #16810968 | — | 2 | 6 | no | — |
| 17 | #16810626 | — | 2 | 6 | no | — |
| 18 | #16810575 | C22 | 1 | 3 | yes | — |
| 19 | #16810365 | C22 | 1 | 3 | yes | complemented |
| 20 | #16810364 | C22 | 1 | 6 | yes | complemented |
| 21 | C53×#564 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | C53×#490 | C22 | 1 | 3 | yes | — |
| 23 | C53×#330 | C22 | 1 | 3 | yes | complemented |
| 24 | C2×D4×D53 | — | 2 | 3 | no | — |
| 25 | C2×#11719912 | — | 2 | 3 | no | — |
| 26 | #13133216 | — | 2 | 6 | no | — |
| 27 | #13133216 | D4 | 1 | 6 | yes | — |
| 28 | #13133214 | C23 | 1 | 1 | yes | characteristic |
| 29 | #13133213 | — | 2 | 3 | no | — |
| 30 | #13133212 | — | 2 | 6 | no | — |
| 31 | #13133211 | — | 2 | 2 | no | — |
| 32 | #13133209 | C23 | 1 | 3 | yes | — |
| 33 | #13133208 | — | 2 | 6 | no | — |
| 34 | #13133207 | — | 2 | 6 | no | — |
| 35 | #13133207 | — | 2 | 6 | no | — |
| 36 | #13133207 | — | 2 | 6 | no | — |
| 37 | #13133195 | — | 2 | 6 | no | — |
| 38 | C4×Dic106 | — | 2 | 6 | no | — |
| 39 | #13133182 | — | 2 | 6 | no | — |
| 40 | #13133182 | — | 2 | 6 | no | — |
| 41 | #13133175 | D4 | 1 | 6 | yes | — |
| 42 | C42×D53 | C23 | 1 | 1 | yes | characteristic |
| 43 | #13133139 | — | 2 | 6 | no | — |
| 44 | #13133138 | — | 2 | 6 | no | — |
| 45 | #13133137 | — | 2 | 6 | no | — |
| 46 | #13133128 | C23 | 1 | 3 | yes | — |
| 47 | C8×Dic53 | — | 2 | 3 | no | — |
| 48 | #127×C53 | C23 | 1 | 1 | yes | characteristic |
| 49 | #125×C53 | — | 2 | 6 | no | — |
| 50 | #125×C53 | C23 | 1 | 3 | yes | — |
| 51 | #102×C53 | — | 2 | 6 | no | — |
| 52 | #97×C53 | C23 | 1 | 3 | yes | — |
| 53 | C23×D53 | — | 4 | 1 | no | — |
| 54 | C2×D4×C53 | — | 2 | 3 | no | — |
| 55 | C2×C22⋊D53 | — | 4 | 3 | no | — |
| 56 | D4×D53 | — | 4 | 12 | no | — |
| 57 | C2×D212 | — | 2 | 3 | no | — |
| 58 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 59 | C2×Dic106 | — | 2 | 6 | no | — |
| 60 | C2×C8×C53 | — | 2 | 3 | no | — |
| 61 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 62 | C4⋊C4×C53 | — | 2 | 6 | no | — |
| 63 | C4⋊C4×C53 | C2×D4 | 1 | 6 | yes | — |
| 64 | C42×C53 | — | 2 | 2 | no | — |
| 65 | C42×C53 | C24 | 1 | 1 | yes | characteristic |
| 66 | #11719921 | — | 2 | 6 | no | — |
| 67 | C4⋊Dic53 | — | 2 | 6 | no | — |
| 68 | C4⋊Dic53 | C2×D4 | 1 | 6 | yes | — |
| 69 | #11719919 | — | 2 | 6 | no | — |
| 70 | #11719919 | — | 2 | 6 | no | — |
| 71 | #11719919 | — | 2 | 6 | no | — |
| 72 | #11719919 | — | 2 | 6 | no | — |
| 73 | C4×Dic53 | — | 2 | 6 | no | — |
| 74 | C4×Dic53 | — | 2 | 6 | no | — |
| 75 | C4×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 76 | C2×#88322 | — | 2 | 3 | no | — |
| 77 | #11719912 | — | 4 | 6 | no | — |
| 78 | C23×C53 | — | 4 | 1 | no | — |
| 79 | C22×D53 | — | 4 | 1 | no | — |
| 80 | C22×D53 | — | 8 | 6 | no | — |
| 81 | C22×D53 | #120 | 1 | 1 | yes | characteristic |
| 82 | D4×C53 | — | 4 | 6 | no | — |
| 83 | C2×C4×C53 | — | 2 | 6 | no | — |
| 84 | C2×C4×C53 | C22×D4 | 1 | 3 | yes | — |
| 85 | C22⋊D53 | — | 8 | 6 | no | — |
| 86 | C2×Dic53 | — | 2 | 6 | no | — |
| 87 | C2×Dic53 | #120 | 1 | 3 | yes | — |
| 88 | D212 | — | 4 | 6 | no | — |
| 89 | C4×D53 | — | 2 | 6 | no | — |
| 90 | Dic106 | — | 4 | 12 | no | — |
| 91 | C8×C53 | — | 4 | 3 | no | — |
| 92 | #88322 | — | 4 | 3 | no | — |
| 93 | #19083 | — | 53 | 1 | no | maximal, Sylow |
| 94 | C22×C53 | — | 8 | 3 | no | — |
| 95 | C22×C53 | C2×#120 | 1 | 1 | yes | characteristic |
| 96 | C2×D53 | — | 2 | 3 | no | — |
| 97 | C2×D53 | — | 8 | 3 | no | — |
| 98 | C2×D53 | — | 16 | 4 | no | — |
| 99 | C4×C53 | — | 2 | 3 | no | — |
| 100 | C4×C53 | — | 4 | 6 | no | — |
| 101 | Dic53 | — | 4 | 6 | no | — |
| 102 | Dic53 | #454 | 1 | 6 | yes | — |
| 103 | #3556 | — | 53 | 1 | no | — |
| 104 | #3138 | — | 53 | 3 | no | — |
| 105 | #1658 | — | 53 | 3 | no | — |
| 106 | #1658 | — | 53 | 3 | no | — |
| 107 | #1658 | — | 53 | 1 | no | — |
| 108 | #1658 | D53 | 1 | 1 | yes | characteristic, complemented |
| 109 | #1526 | — | 53 | 3 | no | — |
| 110 | C2×C53 | — | 16 | 1 | no | — |
| 111 | C2×C53 | C2×#454 | 1 | 3 | yes | — |
| 112 | D53 | — | 4 | 1 | no | — |
| 113 | D53 | — | 16 | 1 | no | — |
| 114 | #564 | — | 53 | 3 | no | — |
| 115 | #564 | C2×D53 | 1 | 1 | yes | characteristic, complemented |
| 116 | #557 | — | 106 | 6 | no | — |
| 117 | #541 | — | 106 | 6 | no | — |
| 118 | #533 | — | 53 | 3 | no | — |
| 119 | C2×#127 | — | 53 | 1 | no | — |
| 120 | #490 | — | 53 | 3 | no | — |
| 121 | #490 | — | 53 | 3 | no | — |
| 122 | #490 | — | 53 | 3 | no | — |
| 123 | #490 | C2×D53 | 1 | 3 | yes | — |
| 124 | #486 | — | 106 | 6 | no | — |
| 125 | #418 | — | 106 | 6 | no | — |
| 126 | C4×M16 | — | 53 | 3 | no | — |
| 127 | #330 | — | 53 | 3 | no | — |
| 128 | #330 | — | 53 | 6 | no | — |
| 129 | #330 | C2×D53 | 1 | 3 | yes | complemented |
| 130 | C53 | #19083 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 131 | C22×D4 | — | 106 | 3 | no | — |
| 132 | C2×M16 | — | 106 | 3 | no | — |
| 133 | C4⋊Q8 | — | 53 | 6 | no | — |
| 134 | C4⋊Q8 | — | 106 | 6 | no | — |
| 135 | #127 | — | 53 | 1 | no | — |
| 136 | #127 | — | 106 | 3 | no | — |
| 137 | #127 | C22×D53 | 1 | 1 | yes | characteristic |
| 138 | #126 | — | 106 | 2 | no | — |
| 139 | #126 | — | 106 | 6 | no | — |
| 140 | #125 | — | 2 | 6 | no | — |
| 141 | #125 | — | 53 | 3 | no | — |
| 142 | #125 | — | 106 | 6 | no | — |
| 143 | #125 | — | 106 | 6 | no | — |
| 144 | #125 | — | 106 | 6 | no | — |
| 145 | #125 | — | 106 | 6 | no | — |
| 146 | #125 | C22×D53 | 1 | 3 | yes | — |
| 147 | C22⋊Q8 | — | 106 | 6 | no | — |
| 148 | C4×Q8 | — | 106 | 6 | no | — |
| 149 | C4×Q8 | — | 106 | 6 | no | — |
| 150 | C4×Q8 | — | 106 | 6 | no | — |
| 151 | #117 | — | 53 | 6 | no | — |
| 152 | C2×C42 | — | 53 | 1 | no | — |
| 153 | #102 | — | 2 | 6 | no | — |
| 154 | #102 | — | 106 | 6 | no | — |
| 155 | #102 | — | 106 | 6 | no | — |
| 156 | #102 | — | 106 | 6 | no | — |
| 157 | #97 | — | 53 | 3 | no | — |
| 158 | #97 | C22×D53 | 1 | 3 | yes | — |
| 159 | C4×C8 | — | 106 | 3 | no | — |
| 160 | C24 | — | 212 | 1 | no | — |
| 161 | C2×Q8 | — | 106 | 6 | no | — |
| 162 | C2×D4 | — | 2 | 3 | no | — |
| 163 | C2×D4 | — | 106 | 3 | no | — |
| 164 | C2×D4 | — | 212 | 3 | no | — |
| 165 | C2×D4 | — | 212 | 12 | no | — |
| 166 | C22×C4 | — | 53 | 3 | no | — |
| 167 | M16 | — | 212 | 6 | no | — |
| 168 | C2×C8 | — | 2 | 3 | no | — |
| 169 | C2×C8 | — | 106 | 3 | no | — |
| 170 | C4⋊C4 | — | 2 | 6 | no | — |
| 171 | C4⋊C4 | — | 2 | 6 | no | — |
| 172 | C4⋊C4 | — | 53 | 6 | no | — |
| 173 | C4⋊C4 | — | 106 | 6 | no | — |
| 174 | C4⋊C4 | — | 106 | 6 | no | — |
| 175 | C4⋊C4 | — | 106 | 6 | no | — |
| 176 | C4⋊C4 | — | 106 | 6 | no | — |
| 177 | C4⋊C4 | — | 106 | 6 | no | — |
| 178 | C4⋊C4 | D4×D53 | 1 | 6 | yes | — |
| 179 | C22⋊C4 | — | 106 | 6 | no | — |
| 180 | C42 | — | 2 | 2 | no | — |
| 181 | C42 | — | 53 | 3 | no | — |
| 182 | C42 | — | 106 | 6 | no | — |
| 183 | C42 | — | 106 | 6 | no | — |
| 184 | C42 | C23×D53 | 1 | 1 | yes | characteristic |
| 185 | C23 | — | 4 | 1 | no | — |
| 186 | C23 | — | 53 | 1 | no | — |
| 187 | C23 | — | 424 | 6 | no | — |
| 188 | C23 | — | 212 | 1 | no | — |
| 189 | Q8 | — | 212 | 12 | no | — |
| 190 | D4 | — | 4 | 6 | no | — |
| 191 | D4 | — | 424 | 6 | no | — |
| 192 | D4 | — | 212 | 6 | no | — |
| 193 | C2×C4 | — | 2 | 6 | no | — |
| 194 | C2×C4 | — | 53 | 3 | no | — |
| 195 | C2×C4 | — | 106 | 6 | no | — |
| 196 | C2×C4 | — | 106 | 6 | no | — |
| 197 | C2×C4 | C2×D4×D53 | 1 | 3 | yes | — |
| 198 | C8 | — | 4 | 3 | no | — |
| 199 | C8 | — | 212 | 3 | no | — |
| 200 | C22 | — | 8 | 3 | no | — |
| 201 | C22 | — | 848 | 4 | no | — |
| 202 | C22 | — | 106 | 3 | no | — |
| 203 | C22 | — | 424 | 3 | no | — |
| 204 | C22 | #120×D53 | 1 | 1 | yes | characteristic |
| 205 | C4 | — | 2 | 3 | no | — |
| 206 | C4 | — | 4 | 6 | no | — |
| 207 | C4 | — | 53 | 6 | no | — |
| 208 | C4 | — | 212 | 6 | no | — |
| 209 | C2 | — | 16 | 1 | no | — |
| 210 | C2 | — | 848 | 1 | no | — |
| 211 | C2 | — | 212 | 1 | no | — |
| 212 | C2 | #454×D53 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20289209 | C2 | semidirect |
| 3. #20274018 | C2 | semidirect |
| 4. #20274017 | C2 | semidirect |
| 5. #20274016 | C2 | semidirect |
| 6. #20273308 | C2 | semidirect |
| 7. C53×#1658 | C2 | semidirect |
| 8. C53⋊#564 | C22 | semidirect |
| 19. #16810365 | C22 | semidirect |
| 20. #16810364 | C22 | semidirect |
| 21. C53×#564 | C22 | semidirect |
| 23. C53×#330 | C22 | semidirect |
| 108. #1658 | D53 | semidirect |
| 115. #564 | C2×D53 | semidirect |
| 129. #330 | C2×D53 | semidirect |
| 130. C53 | #19083 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20289209 | 2 |
| #20285088 | 2 |
| #20274018 | 2 |
| #20274017 | 2 |
| #20274016 | 2 |
| #20273308 | 2 |
| C53×#1658 | 2 |
| #19083 | 53 |