Finite Group DatabasebetaFinite Group DatabasebetaOrder 13568.
494 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 88 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 | D53⋊#364 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20289723 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #493×D53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | D53⋊M32 | — | 2 | 2 | no | — |
| 5 | #16811546 | C22 | 1 | 4 | yes | complemented |
| 6 | #16811545 | — | 2 | 2 | no | — |
| 7 | C2×D8×D53 | — | 2 | 2 | no | — |
| 8 | #127×D53 | C4 | 1 | 2 | yes | — |
| 9 | #16811000 | C4 | 1 | 1 | yes | characteristic |
| 10 | #16810999 | C22 | 1 | 2 | yes | — |
| 11 | #16810998 | — | 2 | 2 | no | — |
| 12 | C4×C8×D53 | C22 | 1 | 1 | yes | characteristic |
| 13 | C53×#493 | C4 | 1 | 1 | yes | characteristic |
| 14 | #13133292 | — | 4 | 2 | no | — |
| 15 | #13133291 | — | 2 | 4 | no | — |
| 16 | D53⋊C16 | — | 4 | 2 | no | — |
| 17 | C2×D4×D53 | — | 2 | 2 | no | — |
| 18 | C2×D4×D53 | — | 4 | 2 | no | — |
| 19 | C2×D424 | — | 2 | 2 | no | — |
| 20 | C2×#11719922 | — | 2 | 4 | no | — |
| 21 | D8×D53 | — | 4 | 2 | no | — |
| 22 | D8×D53 | — | 4 | 2 | no | — |
| 23 | D8×D53 | — | 4 | 2 | no | — |
| 24 | D8×D53 | — | 4 | 2 | no | — |
| 25 | C2×C8×D53 | D4 | 1 | 2 | yes | — |
| 26 | #13133214 | C2×C4 | 1 | 2 | yes | — |
| 27 | #13133213 | — | 2 | 2 | no | — |
| 28 | #13133213 | — | 4 | 2 | no | — |
| 29 | C42×D53 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C4×#88322 | C23 | 1 | 1 | yes | characteristic |
| 31 | C8×Dic53 | D4 | 1 | 2 | yes | — |
| 32 | C2×D8×C53 | — | 2 | 2 | no | — |
| 33 | #127×C53 | C2×C4 | 1 | 2 | yes | — |
| 34 | C4×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C23×D53 | — | 4 | 2 | no | — |
| 36 | C2×D4×C53 | — | 2 | 2 | no | — |
| 37 | C2×D4×C53 | — | 4 | 2 | no | — |
| 38 | C2×C22⋊D53 | — | 4 | 4 | no | — |
| 39 | C2×C22⋊D53 | — | 4 | 2 | no | — |
| 40 | D4×D53 | — | 4 | 4 | no | — |
| 41 | D4×D53 | — | 4 | 4 | no | — |
| 42 | D4×D53 | — | 8 | 8 | no | — |
| 43 | C2×D212 | — | 2 | 2 | no | — |
| 44 | C2×D212 | — | 4 | 2 | no | — |
| 45 | C2×C4×D53 | — | 2 | 1 | no | — |
| 46 | C2×C4×D53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | D8×C53 | — | 4 | 2 | no | — |
| 48 | D8×C53 | — | 4 | 2 | no | — |
| 49 | C2×C8×C53 | C22⋊C4 | 1 | 2 | yes | — |
| 50 | C42×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 51 | #11719922 | — | 4 | 4 | no | — |
| 52 | #11719922 | — | 4 | 4 | no | — |
| 53 | C4×Dic53 | — | 2 | 1 | no | — |
| 54 | C4×Dic53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C2×#88322 | C2×D4 | 1 | 2 | yes | — |
| 56 | D424 | — | 4 | 2 | no | — |
| 57 | D424 | — | 4 | 2 | no | — |
| 58 | C8×D53 | D8 | 1 | 2 | yes | — |
| 59 | C8×D53 | D8 | 1 | 2 | yes | — |
| 60 | C8×D53 | SD16 | 1 | 2 | yes | — |
| 61 | C8×D53 | SD16 | 1 | 2 | yes | — |
| 62 | #11719910 | — | 4 | 4 | no | — |
| 63 | C23×C53 | — | 4 | 2 | no | — |
| 64 | C22×D53 | — | 4 | 2 | no | — |
| 65 | C22×D53 | — | 8 | 8 | no | — |
| 66 | C22×D53 | — | 8 | 2 | no | — |
| 67 | C22×D53 | — | 8 | 2 | no | — |
| 68 | C22×D53 | #100 | 1 | 1 | yes | characteristic |
| 69 | D4×C53 | — | 4 | 4 | no | — |
| 70 | D4×C53 | — | 8 | 4 | no | — |
| 71 | C2×C4×C53 | — | 2 | 1 | no | — |
| 72 | C2×C4×C53 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C22⋊D53 | — | 8 | 8 | no | — |
| 74 | C22⋊D53 | — | 8 | 4 | no | — |
| 75 | C2×Dic53 | — | 2 | 1 | no | — |
| 76 | C2×Dic53 | #100 | 1 | 1 | yes | characteristic |
| 77 | D212 | — | 4 | 4 | no | — |
| 78 | D212 | — | 8 | 4 | no | — |
| 79 | C4×D53 | — | 2 | 4 | no | — |
| 80 | C4×D53 | #102 | 1 | 2 | yes | — |
| 81 | C4×D53 | #102 | 1 | 2 | yes | — |
| 82 | C8×C53 | #102 | 1 | 2 | yes | — |
| 83 | C8×C53 | #102 | 1 | 2 | yes | — |
| 84 | #88322 | C2×D8 | 1 | 2 | yes | — |
| 85 | #88322 | C2×SD16 | 1 | 2 | yes | — |
| 86 | C2×#1357 | — | 53 | 1 | no | maximal, Sylow |
| 87 | C22×C53 | — | 8 | 4 | no | — |
| 88 | C22×C53 | — | 8 | 2 | no | — |
| 89 | C22×C53 | C2×#100 | 1 | 1 | yes | characteristic |
| 90 | C2×D53 | — | 2 | 2 | no | — |
| 91 | C2×D53 | — | 8 | 4 | no | — |
| 92 | C2×D53 | — | 8 | 2 | no | — |
| 93 | C2×D53 | — | 16 | 4 | no | — |
| 94 | C2×D53 | — | 16 | 4 | no | — |
| 95 | C2×D53 | #331 | 1 | 1 | yes | characteristic |
| 96 | C2×D53 | #331 | 1 | 1 | yes | characteristic |
| 97 | C4×C53 | — | 2 | 2 | no | — |
| 98 | C4×C53 | C2×#102 | 1 | 2 | yes | — |
| 99 | Dic53 | — | 2 | 2 | no | — |
| 100 | Dic53 | #331 | 1 | 2 | yes | — |
| 101 | C2×#493 | — | 53 | 1 | no | — |
| 102 | C2×#364 | — | 53 | 2 | no | — |
| 103 | #1357 | — | 53 | 4 | no | — |
| 104 | C2×C53 | — | 2 | 1 | no | — |
| 105 | C2×C53 | — | 16 | 2 | no | — |
| 106 | C2×C53 | C2×#331 | 1 | 1 | yes | characteristic |
| 107 | D53 | — | 2 | 1 | no | — |
| 108 | D53 | — | 16 | 2 | no | — |
| 109 | D53 | #1357 | 1 | 2 | yes | complemented |
| 110 | C22×D8 | — | 106 | 2 | no | — |
| 111 | C2×#127 | — | 53 | 2 | no | — |
| 112 | C2×M32 | — | 106 | 2 | no | — |
| 113 | #493 | — | 53 | 2 | no | — |
| 114 | #493 | — | 53 | 1 | no | — |
| 115 | #493 | — | 106 | 2 | no | — |
| 116 | #493 | #6157 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C8 | — | 53 | 1 | no | — |
| 118 | #364 | — | 53 | 4 | no | — |
| 119 | #364 | — | 106 | 2 | no | — |
| 120 | C53 | C2×#1357 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 121 | C22×D4 | — | 106 | 2 | no | — |
| 122 | C22×D4 | — | 212 | 2 | no | — |
| 123 | C2×D8 | — | 2 | 2 | no | — |
| 124 | C2×D8 | — | 106 | 4 | no | — |
| 125 | C2×D8 | — | 106 | 2 | no | — |
| 126 | C2×D8 | — | 212 | 2 | no | — |
| 127 | C2×D8 | — | 212 | 2 | no | — |
| 128 | C2×D8 | — | 212 | 2 | no | — |
| 129 | C2×D8 | — | 212 | 2 | no | — |
| 130 | C22×C8 | — | 53 | 2 | no | — |
| 131 | #127 | — | 53 | 2 | no | — |
| 132 | #127 | — | 106 | 2 | no | — |
| 133 | #127 | — | 212 | 2 | no | — |
| 134 | #127 | C2×#6157 | 1 | 2 | yes | — |
| 135 | C2×C42 | — | 53 | 1 | no | — |
| 136 | M32 | — | 106 | 4 | no | — |
| 137 | M32 | — | 212 | 2 | no | — |
| 138 | C2×C16 | — | 212 | 2 | no | — |
| 139 | C4×C8 | — | 53 | 2 | no | — |
| 140 | C4×C8 | — | 53 | 1 | no | — |
| 141 | C4×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 142 | C24 | — | 212 | 2 | no | — |
| 143 | C2×D4 | — | 2 | 2 | no | — |
| 144 | C2×D4 | — | 4 | 2 | no | — |
| 145 | C2×D4 | — | 106 | 2 | no | — |
| 146 | C2×D4 | — | 424 | 8 | no | — |
| 147 | C2×D4 | — | 212 | 2 | no | — |
| 148 | C2×D4 | — | 212 | 4 | no | — |
| 149 | C2×D4 | — | 212 | 4 | no | — |
| 150 | C2×D4 | — | 212 | 2 | no | — |
| 151 | C2×D4 | — | 212 | 4 | no | — |
| 152 | C22×C4 | — | 53 | 1 | no | — |
| 153 | C22×C4 | — | 106 | 1 | no | — |
| 154 | D8 | — | 4 | 2 | no | — |
| 155 | D8 | — | 4 | 2 | no | — |
| 156 | D8 | — | 212 | 4 | no | — |
| 157 | D8 | — | 212 | 4 | no | — |
| 158 | D8 | — | 212 | 2 | no | — |
| 159 | D8 | — | 212 | 2 | no | — |
| 160 | C2×C8 | — | 53 | 2 | no | — |
| 161 | C2×C8 | — | 53 | 2 | no | — |
| 162 | C2×C8 | — | 53 | 2 | no | — |
| 163 | C2×C8 | — | 53 | 2 | no | — |
| 164 | C2×C8 | — | 53 | 2 | no | — |
| 165 | C2×C8 | C22⋊#6157 | 1 | 2 | yes | — |
| 166 | C42 | — | 53 | 1 | no | — |
| 167 | C42 | — | 106 | 1 | no | — |
| 168 | C42 | C22×#6157 | 1 | 1 | yes | characteristic |
| 169 | C16 | — | 212 | 4 | no | — |
| 170 | C23 | — | 4 | 2 | no | — |
| 171 | C23 | — | 53 | 1 | no | — |
| 172 | C23 | — | 424 | 2 | no | — |
| 173 | C23 | — | 424 | 2 | no | — |
| 174 | C23 | — | 424 | 8 | no | — |
| 175 | C23 | — | 212 | 2 | no | — |
| 176 | D4 | — | 4 | 4 | no | — |
| 177 | D4 | — | 8 | 4 | no | — |
| 178 | D4 | — | 424 | 4 | no | — |
| 179 | D4 | — | 424 | 4 | no | — |
| 180 | D4 | — | 424 | 8 | no | — |
| 181 | D4 | — | 212 | 4 | no | — |
| 182 | C2×C4 | — | 2 | 1 | no | — |
| 183 | C2×C4 | — | 53 | 1 | no | — |
| 184 | C2×C4 | — | 53 | 2 | no | — |
| 185 | C2×C4 | — | 53 | 2 | no | — |
| 186 | C2×C4 | — | 106 | 4 | no | — |
| 187 | C2×C4 | — | 106 | 1 | no | — |
| 188 | C2×C4 | C2×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 189 | C8 | — | 53 | 2 | no | — |
| 190 | C8 | — | 53 | 2 | no | — |
| 191 | C8 | #13133279 | 1 | 2 | yes | — |
| 192 | C8 | #13133279 | 1 | 2 | yes | — |
| 193 | C22 | — | 8 | 4 | no | — |
| 194 | C22 | — | 8 | 2 | no | — |
| 195 | C22 | — | 53 | 1 | no | — |
| 196 | C22 | — | 53 | 1 | no | — |
| 197 | C22 | — | 848 | 4 | no | — |
| 198 | C22 | — | 848 | 4 | no | — |
| 199 | C22 | — | 106 | 2 | no | — |
| 200 | C22 | — | 424 | 4 | no | — |
| 201 | C22 | — | 424 | 2 | no | — |
| 202 | C22 | D53⋊#100 | 1 | 1 | yes | characteristic |
| 203 | C4 | — | 2 | 2 | no | — |
| 204 | C4 | — | 53 | 2 | no | — |
| 205 | C4 | — | 106 | 2 | no | — |
| 206 | C4 | C2×#13133279 | 1 | 2 | yes | — |
| 207 | C2 | — | 2 | 1 | no | — |
| 208 | C2 | — | 16 | 2 | no | — |
| 209 | C2 | — | 53 | 2 | no | — |
| 210 | C2 | — | 848 | 2 | no | — |
| 211 | C2 | — | 106 | 1 | no | — |
| 212 | C2 | D53⋊#331 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D53⋊#364 | C2 | semidirect |
| 2. #20289723 | C2 | semidirect |
| 5. #16811546 | C22 | semidirect |
| 109. D53 | #1357 | semidirect |
| 120. C53 | C2×#1357 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D53⋊#364 | 2 |
| #20289723 | 2 |
| #493×D53 | 2 |
| C2×#1357 | 53 |