Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
872 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 95 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 | #20275785 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #20272536 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C53×#2679 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×#11719922 | — | 2 | 3 | no | — |
| 5 | C2×#13133214 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#13133137 | — | 2 | 3 | no | — |
| 7 | #16810378 | C22 | 1 | 3 | yes | complemented |
| 8 | C2×#120×C53 | — | 2 | 3 | no | — |
| 9 | C53×#394 | — | 2 | 3 | no | — |
| 10 | C53×#390 | C22 | 1 | 3 | yes | — |
| 11 | C22×D212 | — | 2 | 3 | no | — |
| 12 | C2×#11719922 | — | 4 | 6 | no | — |
| 13 | C2×#11719922 | — | 4 | 6 | no | — |
| 14 | C2×#11719922 | — | 4 | 6 | no | — |
| 15 | C22×#88322 | — | 2 | 3 | no | — |
| 16 | #13133214 | — | 4 | 2 | no | — |
| 17 | #13133137 | — | 4 | 6 | no | — |
| 18 | C25×C53 | — | 2 | 1 | no | — |
| 19 | C22×D4×C53 | — | 2 | 3 | no | — |
| 20 | C22×D4×C53 | D4 | 1 | 3 | yes | — |
| 21 | #120×C53 | — | 4 | 6 | no | — |
| 22 | #120×C53 | — | 4 | 6 | no | — |
| 23 | C2×C4⋊C4×C53 | D4 | 1 | 3 | yes | — |
| 24 | C2×C22⋊C4×C53 | — | 2 | 3 | no | — |
| 25 | C2×C22⋊C4×C53 | — | 2 | 6 | no | — |
| 26 | C2×C42×C53 | C23 | 1 | 1 | yes | characteristic |
| 27 | #95×C53 | — | 2 | 1 | no | — |
| 28 | C24×C53 | — | 2 | 3 | no | — |
| 29 | C24×C53 | — | 4 | 6 | no | — |
| 30 | C24×C53 | — | 8 | 3 | no | — |
| 31 | C24×C53 | — | 8 | 1 | no | — |
| 32 | C23×D53 | — | 4 | 1 | no | — |
| 33 | C2×D4×C53 | — | 2 | 3 | no | — |
| 34 | C2×D4×C53 | — | 4 | 3 | no | — |
| 35 | C2×D4×C53 | — | 4 | 3 | no | — |
| 36 | C2×D4×C53 | — | 4 | 12 | no | — |
| 37 | C2×D4×C53 | — | 8 | 3 | no | — |
| 38 | C2×D4×C53 | D8 | 1 | 6 | yes | complemented |
| 39 | C22×C4×C53 | — | 2 | 3 | no | — |
| 40 | C22×C4×C53 | C2×D4 | 1 | 3 | yes | — |
| 41 | C2×D212 | — | 2 | 6 | no | — |
| 42 | C2×D212 | — | 4 | 3 | no | — |
| 43 | C2×D212 | — | 4 | 3 | no | — |
| 44 | C2×D212 | — | 4 | 3 | no | — |
| 45 | C2×D212 | — | 4 | 3 | no | — |
| 46 | C2×D212 | — | 4 | 3 | no | — |
| 47 | C4⋊C4×C53 | — | 4 | 3 | no | — |
| 48 | C22⋊C4×C53 | — | 4 | 6 | no | — |
| 49 | C22⋊C4×C53 | — | 4 | 6 | no | — |
| 50 | C22⋊C4×C53 | — | 4 | 6 | no | — |
| 51 | C42×C53 | — | 4 | 1 | no | — |
| 52 | #11719922 | — | 4 | 24 | no | — |
| 53 | C2×#88322 | — | 2 | 6 | no | — |
| 54 | C2×#88322 | — | 4 | 3 | no | — |
| 55 | C2×#88322 | — | 4 | 3 | no | — |
| 56 | C23×C53 | — | 2 | 6 | no | — |
| 57 | C23×C53 | — | 4 | 6 | no | — |
| 58 | C23×C53 | — | 4 | 3 | no | — |
| 59 | C23×C53 | — | 4 | 6 | no | — |
| 60 | C23×C53 | — | 4 | 3 | no | — |
| 61 | C23×C53 | — | 8 | 12 | no | — |
| 62 | C23×C53 | — | 8 | 12 | no | — |
| 63 | C23×C53 | — | 8 | 4 | no | — |
| 64 | C23×C53 | #120 | 1 | 1 | yes | characteristic |
| 65 | C22×D53 | — | 4 | 2 | no | — |
| 66 | C22×D53 | — | 8 | 3 | no | — |
| 67 | C22×D53 | — | 8 | 3 | no | — |
| 68 | D4×C53 | — | 4 | 6 | no | — |
| 69 | D4×C53 | — | 8 | 3 | no | — |
| 70 | D4×C53 | — | 8 | 12 | no | — |
| 71 | C2×C4×C53 | — | 2 | 3 | no | — |
| 72 | C2×C4×C53 | — | 2 | 3 | no | — |
| 73 | C2×C4×C53 | — | 4 | 6 | no | — |
| 74 | C2×C4×C53 | — | 4 | 3 | no | — |
| 75 | C2×C4×C53 | C2×D8 | 1 | 3 | yes | — |
| 76 | C2×C4×C53 | AGL(1,ℤ/8) | 1 | 3 | yes | — |
| 77 | D212 | — | 4 | 6 | no | — |
| 78 | D212 | — | 4 | 6 | no | — |
| 79 | D212 | — | 8 | 3 | no | — |
| 80 | D212 | — | 8 | 3 | no | — |
| 81 | #88322 | — | 4 | 6 | no | — |
| 82 | #7734 | — | 53 | 1 | no | maximal, Sylow |
| 83 | C22×C53 | — | 4 | 6 | no | — |
| 84 | C22×C53 | — | 4 | 12 | no | — |
| 85 | C22×C53 | — | 8 | 16 | no | — |
| 86 | C22×C53 | — | 8 | 6 | no | — |
| 87 | C22×C53 | — | 8 | 3 | no | — |
| 88 | C22×C53 | — | 8 | 3 | no | — |
| 89 | C22×C53 | #447 | 1 | 3 | yes | — |
| 90 | C22×C53 | #453 | 1 | 3 | yes | — |
| 91 | C22×C53 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 92 | C2×D53 | — | 8 | 6 | no | — |
| 93 | C2×D53 | — | 16 | 3 | no | — |
| 94 | C2×D53 | — | 16 | 1 | no | — |
| 95 | C4×C53 | — | 2 | 3 | no | — |
| 96 | C4×C53 | — | 2 | 3 | no | — |
| 97 | C4×C53 | — | 8 | 3 | no | — |
| 98 | #2679 | D53 | 1 | 1 | yes | characteristic, complemented |
| 99 | #2011 | — | 53 | 3 | no | — |
| 100 | #1270 | — | 53 | 3 | no | — |
| 101 | C2×C53 | — | 8 | 6 | no | — |
| 102 | C2×C53 | #1595 | 1 | 3 | yes | — |
| 103 | C2×C53 | #1619 | 1 | 3 | yes | — |
| 104 | C2×C53 | #1655 | 1 | 1 | yes | characteristic |
| 105 | D53 | — | 16 | 2 | no | — |
| 106 | C22×D8 | — | 106 | 3 | no | — |
| 107 | C2×#127 | — | 53 | 1 | no | — |
| 108 | C2×#120 | — | 2 | 3 | no | — |
| 109 | C2×#102 | — | 106 | 3 | no | — |
| 110 | #394 | — | 2 | 3 | no | — |
| 111 | #390 | C2×D53 | 1 | 3 | yes | — |
| 112 | #336 | — | 53 | 3 | no | — |
| 113 | C53 | #7734 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 114 | C25 | — | 2 | 1 | no | — |
| 115 | C22×D4 | — | 2 | 3 | no | — |
| 116 | C22×D4 | — | 106 | 3 | no | — |
| 117 | C22×D4 | C22⋊D53 | 1 | 3 | yes | — |
| 118 | C2×D8 | — | 212 | 6 | no | — |
| 119 | C2×D8 | — | 212 | 6 | no | — |
| 120 | C2×D8 | — | 212 | 6 | no | — |
| 121 | C22×C8 | — | 106 | 3 | no | — |
| 122 | #127 | — | 212 | 2 | no | — |
| 123 | #120 | — | 4 | 6 | no | — |
| 124 | #120 | — | 4 | 6 | no | — |
| 125 | C2×C4⋊C4 | C22⋊D53 | 1 | 3 | yes | — |
| 126 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 127 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 128 | C2×C42 | C22×D53 | 1 | 1 | yes | characteristic |
| 129 | #102 | — | 212 | 6 | no | — |
| 130 | #95 | — | 2 | 1 | no | — |
| 131 | C24 | — | 2 | 3 | no | — |
| 132 | C24 | — | 4 | 6 | no | — |
| 133 | C24 | — | 8 | 3 | no | — |
| 134 | C24 | — | 8 | 1 | no | — |
| 135 | C24 | — | 212 | 1 | no | — |
| 136 | C2×D4 | — | 2 | 3 | no | — |
| 137 | C2×D4 | — | 4 | 3 | no | — |
| 138 | C2×D4 | — | 4 | 3 | no | — |
| 139 | C2×D4 | — | 4 | 12 | no | — |
| 140 | C2×D4 | — | 8 | 3 | no | — |
| 141 | C2×D4 | — | 106 | 6 | no | — |
| 142 | C2×D4 | — | 212 | 3 | no | — |
| 143 | C2×D4 | — | 212 | 3 | no | — |
| 144 | C2×D4 | — | 212 | 3 | no | — |
| 145 | C2×D4 | — | 212 | 3 | no | — |
| 146 | C2×D4 | — | 212 | 3 | no | — |
| 147 | C2×D4 | #11719922 | 1 | 6 | yes | complemented |
| 148 | C22×C4 | — | 2 | 3 | no | — |
| 149 | C22×C4 | C2×C22⋊D53 | 1 | 3 | yes | — |
| 150 | D8 | — | 212 | 24 | no | — |
| 151 | C2×C8 | — | 106 | 6 | no | — |
| 152 | C2×C8 | — | 212 | 3 | no | — |
| 153 | C2×C8 | — | 212 | 3 | no | — |
| 154 | C4⋊C4 | — | 4 | 3 | no | — |
| 155 | C22⋊C4 | — | 4 | 6 | no | — |
| 156 | C22⋊C4 | — | 4 | 6 | no | — |
| 157 | C22⋊C4 | — | 4 | 6 | no | — |
| 158 | C42 | — | 4 | 1 | no | — |
| 159 | C23 | — | 2 | 6 | no | — |
| 160 | C23 | — | 4 | 6 | no | — |
| 161 | C23 | — | 4 | 3 | no | — |
| 162 | C23 | — | 4 | 6 | no | — |
| 163 | C23 | — | 4 | 3 | no | — |
| 164 | C23 | — | 8 | 12 | no | — |
| 165 | C23 | — | 8 | 12 | no | — |
| 166 | C23 | — | 8 | 4 | no | — |
| 167 | C23 | — | 424 | 3 | no | — |
| 168 | C23 | — | 424 | 3 | no | — |
| 169 | C23 | — | 212 | 2 | no | — |
| 170 | C23 | #13133186 | 1 | 1 | yes | characteristic |
| 171 | D4 | — | 4 | 6 | no | — |
| 172 | D4 | — | 8 | 3 | no | — |
| 173 | D4 | — | 8 | 12 | no | — |
| 174 | D4 | — | 424 | 3 | no | — |
| 175 | D4 | — | 424 | 3 | no | — |
| 176 | D4 | — | 212 | 6 | no | — |
| 177 | D4 | — | 212 | 6 | no | — |
| 178 | C2×C4 | — | 2 | 3 | no | — |
| 179 | C2×C4 | — | 2 | 3 | no | — |
| 180 | C2×C4 | — | 4 | 6 | no | — |
| 181 | C2×C4 | — | 4 | 3 | no | — |
| 182 | C2×C4 | C2×#11719922 | 1 | 3 | yes | — |
| 183 | C2×C4 | #13133243 | 1 | 3 | yes | — |
| 184 | C8 | — | 212 | 6 | no | — |
| 185 | C22 | — | 4 | 12 | no | — |
| 186 | C22 | — | 4 | 6 | no | — |
| 187 | C22 | — | 8 | 16 | no | — |
| 188 | C22 | — | 8 | 3 | no | — |
| 189 | C22 | — | 8 | 3 | no | — |
| 190 | C22 | — | 8 | 6 | no | — |
| 191 | C22 | — | 848 | 1 | no | — |
| 192 | C22 | — | 848 | 3 | no | — |
| 193 | C22 | — | 424 | 6 | no | — |
| 194 | C22 | #16810757 | 1 | 3 | yes | — |
| 195 | C22 | #16810791 | 1 | 3 | yes | — |
| 196 | C22 | #16810810 | 1 | 1 | yes | characteristic |
| 197 | C4 | — | 2 | 3 | no | — |
| 198 | C4 | — | 2 | 3 | no | — |
| 199 | C4 | — | 8 | 3 | no | — |
| 200 | C2 | — | 8 | 6 | no | — |
| 201 | C2 | — | 848 | 2 | no | — |
| 202 | C2 | #20273633 | 1 | 3 | yes | — |
| 203 | C2 | #20273769 | 1 | 3 | yes | — |
| 204 | C2 | #20274004 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20275785 | C2 | semidirect |
| 2. #20272536 | C2 | semidirect |
| 3. C53×#2679 | C2 | semidirect |
| 5. C2×#13133214 | C22 | semidirect |
| 7. #16810378 | C22 | semidirect |
| 38. C2×D4×C53 | D8 | semidirect |
| 98. #2679 | D53 | semidirect |
| 113. C53 | #7734 | semidirect |
| 147. C2×D4 | #11719922 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20275785 | 2 |
| #20272536 | 2 |
| C53×#2679 | 2 |
| #7734 | 53 |