Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
318 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 39 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 | #26020215 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26018178 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C89×#1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19152434 | C4 | 1 | 1 | yes | characteristic |
| 5 | #19151614 | — | 2 | 1 | no | — |
| 6 | #19151595 | — | 2 | 1 | no | — |
| 7 | #19151572 | C4 | 1 | 1 | yes | characteristic |
| 8 | C22⋊#13083549 | — | 2 | 1 | no | — |
| 9 | #408×C89 | C22 | 1 | 1 | yes | characteristic |
| 10 | #349×C89 | — | 2 | 1 | no | — |
| 11 | #15668060 | C8 | 1 | 2 | yes | — |
| 12 | #15668059 | — | 4 | 1 | no | — |
| 13 | #15668058 | — | 4 | 1 | no | — |
| 14 | #15668057 | — | 2 | 1 | no | — |
| 15 | #15668057 | — | 4 | 1 | no | — |
| 16 | C2×#13083574 | — | 2 | 1 | no | — |
| 17 | C2×#13083574 | — | 4 | 1 | no | — |
| 18 | #41×D89 | — | 4 | 2 | no | — |
| 19 | #15667971 | — | 2 | 1 | no | — |
| 20 | C2×#13083549 | — | 4 | 1 | no | — |
| 21 | #15667938 | — | 4 | 2 | no | — |
| 22 | #15667937 | — | 4 | 1 | no | — |
| 23 | #15667934 | — | 4 | 1 | no | — |
| 24 | #15667927 | — | 4 | 1 | no | — |
| 25 | C2×#41×C89 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×M16×C89 | D4 | 1 | 1 | yes | characteristic |
| 27 | C22×C8×C89 | D4 | 1 | 1 | yes | characteristic |
| 28 | M32×C89 | — | 4 | 1 | no | — |
| 29 | C22⋊C8×C89 | — | 4 | 1 | no | — |
| 30 | #13083575 | — | 4 | 2 | no | — |
| 31 | #13083574 | — | 2 | 2 | no | — |
| 32 | #13083574 | — | 2 | 2 | no | — |
| 33 | #13083574 | — | 4 | 1 | no | — |
| 34 | #13083574 | — | 4 | 2 | no | — |
| 35 | #13083574 | — | 4 | 1 | no | — |
| 36 | #13083574 | — | 4 | 2 | no | — |
| 37 | #13083574 | — | 4 | 2 | no | — |
| 38 | #13083573 | — | 4 | 2 | no | — |
| 39 | #13083573 | — | 4 | 4 | no | — |
| 40 | C2×Dic178 | — | 2 | 1 | no | — |
| 41 | C2×Dic178 | — | 4 | 1 | no | — |
| 42 | Q8×D89 | — | 4 | 2 | no | — |
| 43 | C2×D356 | — | 2 | 1 | no | — |
| 44 | C2×D356 | — | 4 | 1 | no | — |
| 45 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 46 | C2×C22⋊D89 | — | 4 | 1 | no | — |
| 47 | D4×D89 | — | 4 | 2 | no | — |
| 48 | D4×D89 | — | 4 | 4 | no | — |
| 49 | C2×C4×D89 | — | 4 | 1 | no | — |
| 50 | C2×C4×D89 | — | 4 | 2 | no | — |
| 51 | C4⋊Dic89 | — | 4 | 1 | no | — |
| 52 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 53 | C4×Dic89 | — | 4 | 1 | no | — |
| 54 | #13083549 | — | 8 | 1 | no | — |
| 55 | #41×C89 | — | 4 | 1 | no | — |
| 56 | #41×C89 | — | 4 | 1 | no | — |
| 57 | C2×Q8×C89 | C2×C8 | 1 | 1 | yes | characteristic |
| 58 | C2×D4×C89 | — | 2 | 1 | no | — |
| 59 | C2×D4×C89 | M16 | 1 | 1 | yes | characteristic |
| 60 | C22×C4×C89 | — | 2 | 1 | no | — |
| 61 | C22×C4×C89 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | M16×C89 | — | 4 | 1 | no | — |
| 63 | C2×C8×C89 | — | 2 | 1 | no | — |
| 64 | C2×C8×C89 | — | 2 | 1 | no | — |
| 65 | C2×C8×C89 | — | 4 | 1 | no | — |
| 66 | C16×C89 | — | 8 | 1 | no | — |
| 67 | C23×C89 | — | 2 | 1 | no | — |
| 68 | C23×C89 | #100 | 1 | 1 | yes | characteristic |
| 69 | C22×D89 | — | 4 | 1 | no | — |
| 70 | C22×D89 | — | 4 | 2 | no | — |
| 71 | Q8×C89 | — | 4 | 1 | no | — |
| 72 | D4×C89 | — | 4 | 1 | no | — |
| 73 | D4×C89 | — | 4 | 2 | no | — |
| 74 | C2×C4×C89 | — | 2 | 1 | no | — |
| 75 | C2×C4×C89 | — | 2 | 1 | no | — |
| 76 | C2×C4×C89 | — | 2 | 1 | no | — |
| 77 | C2×C4×C89 | — | 4 | 1 | no | — |
| 78 | C2×C4×C89 | — | 4 | 1 | no | — |
| 79 | C2×C4×C89 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 80 | C2×C4×C89 | #99 | 1 | 1 | yes | characteristic |
| 81 | C22⋊D89 | — | 4 | 2 | no | — |
| 82 | C22⋊D89 | — | 4 | 4 | no | — |
| 83 | C22⋊D89 | — | 4 | 2 | no | — |
| 84 | C2×Dic89 | — | 4 | 1 | no | — |
| 85 | C2×Dic89 | — | 4 | 2 | no | — |
| 86 | C2×Dic89 | — | 8 | 1 | no | — |
| 87 | D356 | — | 2 | 2 | no | — |
| 88 | D356 | — | 4 | 2 | no | — |
| 89 | D356 | — | 4 | 1 | no | — |
| 90 | C4×D89 | — | 4 | 2 | no | — |
| 91 | C4×D89 | — | 4 | 4 | no | — |
| 92 | C4×D89 | — | 4 | 2 | no | — |
| 93 | Dic178 | — | 2 | 2 | no | — |
| 94 | Dic178 | — | 4 | 1 | no | — |
| 95 | Dic178 | — | 4 | 2 | no | — |
| 96 | C8×C89 | — | 4 | 1 | no | — |
| 97 | C8×C89 | — | 4 | 1 | no | — |
| 98 | C22×C89 | — | 2 | 1 | no | — |
| 99 | C22×C89 | — | 4 | 1 | no | — |
| 100 | C22×C89 | — | 4 | 1 | no | — |
| 101 | C22×C89 | — | 8 | 1 | no | — |
| 102 | C22×C89 | #323 | 1 | 1 | yes | characteristic |
| 103 | C2×D89 | — | 4 | 2 | no | — |
| 104 | C2×D89 | — | 16 | 1 | no | — |
| 105 | C2×D89 | — | 16 | 2 | no | — |
| 106 | C4×C89 | — | 2 | 1 | no | — |
| 107 | C4×C89 | — | 4 | 1 | no | — |
| 108 | C4×C89 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 109 | C4×C89 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 110 | Dic89 | — | 4 | 2 | no | — |
| 111 | Dic89 | — | 16 | 1 | no | — |
| 112 | #7347 | — | 89 | 1 | no | maximal, Sylow |
| 113 | C2×C89 | — | 2 | 1 | no | — |
| 114 | C2×C89 | — | 4 | 1 | no | — |
| 115 | C2×C89 | — | 8 | 1 | no | — |
| 116 | C2×C89 | #1316 | 1 | 1 | yes | characteristic |
| 117 | D89 | — | 8 | 2 | no | — |
| 118 | #1794 | — | 89 | 1 | no | — |
| 119 | #1321 | D89 | 1 | 1 | yes | characteristic, complemented |
| 120 | #1315 | — | 89 | 1 | no | — |
| 121 | C89 | #7347 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 122 | #585 | — | 89 | 1 | no | — |
| 123 | #421 | — | 178 | 1 | no | — |
| 124 | #416 | — | 178 | 1 | no | — |
| 125 | #410 | — | 89 | 1 | no | — |
| 126 | #408 | C2×D89 | 1 | 1 | yes | characteristic |
| 127 | #349 | — | 2 | 1 | no | — |
| 128 | C22⋊C16 | — | 178 | 1 | no | — |
| 129 | 2−1+4 | — | 89 | 2 | no | — |
| 130 | 2−1+4 | — | 356 | 1 | no | — |
| 131 | H(5,𝔽2) | — | 356 | 1 | no | — |
| 132 | H(5,𝔽2) | — | 356 | 1 | no | — |
| 133 | H(5,𝔽2) | — | 178 | 1 | no | — |
| 134 | C2×#41 | — | 356 | 1 | no | — |
| 135 | C2×#41 | — | 356 | 2 | no | — |
| 136 | C2×#41 | — | 178 | 1 | no | — |
| 137 | C2×#41 | C4×D89 | 1 | 1 | yes | characteristic |
| 138 | C2×M16 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 139 | C22×C8 | D356 | 1 | 1 | yes | characteristic |
| 140 | #117 | — | 178 | 1 | no | — |
| 141 | M32 | — | 4 | 1 | no | — |
| 142 | C2×C16 | — | 356 | 1 | no | — |
| 143 | #104 | — | 356 | 2 | no | — |
| 144 | #103 | — | 356 | 1 | no | — |
| 145 | #102 | — | 356 | 1 | no | — |
| 146 | #99 | — | 356 | 1 | no | — |
| 147 | C22⋊C8 | — | 4 | 1 | no | — |
| 148 | #41 | — | 4 | 1 | no | — |
| 149 | #41 | — | 4 | 1 | no | — |
| 150 | #41 | — | 356 | 1 | no | — |
| 151 | #41 | — | 356 | 2 | no | — |
| 152 | #41 | — | 356 | 2 | no | — |
| 153 | #41 | — | 356 | 2 | no | — |
| 154 | #41 | — | 356 | 4 | no | — |
| 155 | #41 | — | 356 | 2 | no | — |
| 156 | #41 | — | 356 | 2 | no | — |
| 157 | #41 | — | 356 | 1 | no | — |
| 158 | #41 | — | 178 | 2 | no | — |
| 159 | #41 | — | 178 | 2 | no | — |
| 160 | C2×Q8 | — | 356 | 1 | no | — |
| 161 | C2×Q8 | — | 356 | 2 | no | — |
| 162 | C2×Q8 | — | 178 | 1 | no | — |
| 163 | C2×Q8 | C8×D89 | 1 | 1 | yes | characteristic |
| 164 | C2×D4 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 356 | 1 | no | — |
| 166 | C2×D4 | — | 356 | 4 | no | — |
| 167 | C2×D4 | — | 356 | 2 | no | — |
| 168 | C2×D4 | — | 356 | 2 | no | — |
| 169 | C2×D4 | — | 356 | 1 | no | — |
| 170 | C2×D4 | — | 178 | 1 | no | — |
| 171 | C2×D4 | #13083557 | 1 | 1 | yes | characteristic |
| 172 | C22×C4 | — | 2 | 1 | no | — |
| 173 | C22×C4 | — | 356 | 1 | no | — |
| 174 | C22×C4 | — | 356 | 2 | no | — |
| 175 | C22×C4 | #13083551 | 1 | 1 | yes | characteristic |
| 176 | M16 | — | 4 | 1 | no | — |
| 177 | C2×C8 | — | 2 | 1 | no | — |
| 178 | C2×C8 | — | 2 | 1 | no | — |
| 179 | C2×C8 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 356 | 1 | no | — |
| 181 | C22⋊C4 | — | 356 | 1 | no | — |
| 182 | C42 | — | 356 | 1 | no | — |
| 183 | C16 | — | 8 | 1 | no | — |
| 184 | C16 | — | 712 | 1 | no | — |
| 185 | C23 | — | 2 | 1 | no | — |
| 186 | C23 | — | 356 | 1 | no | — |
| 187 | C23 | — | 356 | 2 | no | — |
| 188 | C23 | #15667928 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 4 | 1 | no | — |
| 190 | Q8 | — | 356 | 2 | no | — |
| 191 | Q8 | — | 356 | 1 | no | — |
| 192 | Q8 | — | 178 | 2 | no | — |
| 193 | D4 | — | 4 | 1 | no | — |
| 194 | D4 | — | 4 | 2 | no | — |
| 195 | D4 | — | 356 | 2 | no | — |
| 196 | D4 | — | 356 | 2 | no | — |
| 197 | D4 | — | 356 | 4 | no | — |
| 198 | D4 | — | 356 | 2 | no | — |
| 199 | D4 | — | 356 | 1 | no | — |
| 200 | D4 | — | 178 | 2 | no | — |
| 201 | C2×C4 | — | 2 | 1 | no | — |
| 202 | C2×C4 | — | 2 | 1 | no | — |
| 203 | C2×C4 | — | 2 | 1 | no | — |
| 204 | C2×C4 | — | 4 | 1 | no | — |
| 205 | C2×C4 | — | 4 | 1 | no | — |
| 206 | C2×C4 | — | 356 | 2 | no | — |
| 207 | C2×C4 | — | 356 | 4 | no | — |
| 208 | C2×C4 | — | 356 | 1 | no | — |
| 209 | C2×C4 | — | 356 | 2 | no | — |
| 210 | C2×C4 | — | 356 | 2 | no | — |
| 211 | C2×C4 | — | 712 | 1 | no | — |
| 212 | C2×C4 | #15667924 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #15667926 | 1 | 1 | yes | characteristic |
| 214 | C8 | — | 4 | 1 | no | — |
| 215 | C8 | — | 4 | 1 | no | — |
| 216 | C22 | — | 2 | 1 | no | — |
| 217 | C22 | — | 4 | 1 | no | — |
| 218 | C22 | — | 4 | 1 | no | — |
| 219 | C22 | — | 8 | 1 | no | — |
| 220 | C22 | — | 356 | 2 | no | — |
| 221 | C22 | — | 1424 | 1 | no | — |
| 222 | C22 | — | 1424 | 2 | no | — |
| 223 | C22 | #19151321 | 1 | 1 | yes | characteristic |
| 224 | C4 | — | 2 | 1 | no | — |
| 225 | C4 | — | 4 | 1 | no | — |
| 226 | C4 | — | 356 | 2 | no | — |
| 227 | C4 | — | 1424 | 1 | no | — |
| 228 | C4 | #19151384 | 1 | 1 | yes | characteristic |
| 229 | C4 | #19151386 | 1 | 1 | yes | characteristic |
| 230 | C2 | — | 2 | 1 | no | — |
| 231 | C2 | — | 4 | 1 | no | — |
| 232 | C2 | — | 8 | 1 | no | — |
| 233 | C2 | — | 712 | 2 | no | — |
| 234 | C2 | #26018179 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #26018178 | C2 | semidirect |
| 3. C89×#1321 | C2 | semidirect |
| 119. #1321 | D89 | semidirect |
| 121. C89 | #7347 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26020215 | 2 |
| #26018178 | 2 |
| C89×#1321 | 2 |
| #7347 | 89 |