Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
196 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 25 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 | #26019366 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26018293 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C89×#1410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19152233 | C4 | 1 | 1 | yes | characteristic |
| 5 | #19151885 | — | 2 | 1 | no | — |
| 6 | #19151419 | — | 2 | 1 | no | — |
| 7 | #19151342 | C4 | 1 | 1 | yes | characteristic |
| 8 | #19151327 | — | 2 | 1 | no | — |
| 9 | #492×C89 | C22 | 1 | 1 | yes | characteristic |
| 10 | #356×C89 | — | 2 | 1 | no | — |
| 11 | C2×D4×D89 | — | 4 | 1 | no | — |
| 12 | #15667983 | — | 4 | 1 | no | — |
| 13 | #15667980 | — | 4 | 1 | no | — |
| 14 | C4×D356 | — | 2 | 1 | no | — |
| 15 | #15667952 | — | 4 | 1 | no | — |
| 16 | #15667943 | — | 4 | 1 | no | — |
| 17 | C4⋊#10623494 | — | 2 | 1 | no | — |
| 18 | #15667932 | — | 4 | 1 | no | — |
| 19 | C2×SD16×C89 | — | 4 | 1 | no | — |
| 20 | C4⋊Q8×C89 | D4 | 1 | 1 | yes | characteristic |
| 21 | #127×C89 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | #101×C89 | — | 4 | 1 | no | — |
| 23 | C4×C8×C89 | D4 | 1 | 1 | yes | characteristic |
| 24 | C23×D89 | — | 4 | 1 | no | — |
| 25 | C2×D356 | — | 2 | 1 | no | — |
| 26 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 27 | D4×D89 | — | 8 | 2 | no | — |
| 28 | D4×D89 | — | 8 | 2 | no | — |
| 29 | C2×C4×D89 | — | 4 | 1 | no | — |
| 30 | C2×#10623494 | — | 4 | 1 | no | — |
| 31 | C4⋊Dic89 | — | 2 | 1 | no | — |
| 32 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 33 | #13083551 | — | 4 | 1 | no | — |
| 34 | #13083549 | — | 8 | 1 | no | — |
| 35 | C2×Q8×C89 | — | 2 | 1 | no | — |
| 36 | C2×D4×C89 | — | 2 | 1 | no | — |
| 37 | C2×D4×C89 | — | 4 | 1 | no | — |
| 38 | SD16×C89 | — | 8 | 1 | no | — |
| 39 | SD16×C89 | — | 8 | 1 | no | — |
| 40 | M16×C89 | — | 8 | 1 | no | — |
| 41 | C2×C8×C89 | — | 2 | 1 | no | — |
| 42 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 43 | C42×C89 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | C23×C89 | — | 4 | 1 | no | — |
| 45 | C22×D89 | — | 4 | 1 | no | — |
| 46 | C22×D89 | — | 8 | 4 | no | — |
| 47 | C22×D89 | — | 16 | 1 | no | — |
| 48 | Q8×C89 | — | 8 | 1 | no | — |
| 49 | D4×C89 | — | 8 | 1 | no | — |
| 50 | D4×C89 | — | 8 | 1 | no | — |
| 51 | D4×C89 | — | 8 | 1 | no | — |
| 52 | C2×C4×C89 | — | 2 | 1 | no | — |
| 53 | C2×C4×C89 | — | 4 | 1 | no | — |
| 54 | C2×C4×C89 | #99 | 1 | 1 | yes | characteristic |
| 55 | C22⋊D89 | — | 8 | 4 | no | — |
| 56 | C2×Dic89 | — | 4 | 1 | no | — |
| 57 | D356 | — | 2 | 2 | no | — |
| 58 | D356 | — | 4 | 1 | no | — |
| 59 | C4×D89 | — | 8 | 1 | no | — |
| 60 | C4×D89 | — | 8 | 1 | no | — |
| 61 | C8×C89 | — | 2 | 1 | no | — |
| 62 | C8×C89 | — | 2 | 1 | no | — |
| 63 | C8×C89 | — | 16 | 1 | no | — |
| 64 | #10623494 | — | 8 | 1 | no | — |
| 65 | C22×C89 | — | 8 | 2 | no | — |
| 66 | C22×C89 | — | 8 | 1 | no | — |
| 67 | C22×C89 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 68 | C2×D89 | — | 4 | 2 | no | — |
| 69 | C2×D89 | — | 8 | 1 | no | — |
| 70 | C2×D89 | — | 8 | 1 | no | — |
| 71 | C2×D89 | — | 16 | 2 | no | — |
| 72 | C2×D89 | — | 16 | 2 | no | — |
| 73 | C4×C89 | — | 2 | 1 | no | — |
| 74 | C4×C89 | — | 2 | 1 | no | — |
| 75 | C4×C89 | — | 2 | 1 | no | — |
| 76 | C4×C89 | — | 8 | 1 | no | — |
| 77 | Dic89 | — | 8 | 1 | no | — |
| 78 | #7517 | — | 89 | 1 | no | maximal, Sylow |
| 79 | C2×C89 | — | 2 | 1 | no | — |
| 80 | C2×C89 | — | 16 | 1 | no | — |
| 81 | C2×C89 | #1402 | 1 | 1 | yes | characteristic |
| 82 | D89 | — | 8 | 2 | no | — |
| 83 | #1629 | — | 89 | 1 | no | — |
| 84 | #1410 | D89 | 1 | 1 | yes | characteristic, complemented |
| 85 | #1359 | — | 89 | 1 | no | — |
| 86 | C89 | #7517 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 87 | D42 | — | 89 | 1 | no | — |
| 88 | #492 | C2×D89 | 1 | 1 | yes | characteristic |
| 89 | #476 | — | 178 | 1 | no | — |
| 90 | #364 | — | 178 | 1 | no | — |
| 91 | #356 | — | 2 | 1 | no | — |
| 92 | #331 | — | 89 | 1 | no | — |
| 93 | D4⋊C8 | — | 178 | 1 | no | — |
| 94 | C22×D4 | — | 356 | 1 | no | — |
| 95 | C2×SD16 | — | 4 | 1 | no | — |
| 96 | C4⋊Q8 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 97 | #127 | C4×D89 | 1 | 1 | yes | characteristic |
| 98 | #121 | — | 356 | 1 | no | — |
| 99 | #120 | — | 356 | 1 | no | — |
| 100 | C4×D4 | — | 178 | 1 | no | — |
| 101 | M32 | — | 356 | 1 | no | — |
| 102 | #106 | — | 356 | 1 | no | — |
| 103 | C4⋊C8 | — | 178 | 1 | no | — |
| 104 | #102 | — | 356 | 1 | no | — |
| 105 | #101 | — | 4 | 1 | no | — |
| 106 | C4×C8 | D356 | 1 | 1 | yes | characteristic |
| 107 | C24 | — | 356 | 1 | no | — |
| 108 | C2×Q8 | — | 2 | 1 | no | — |
| 109 | C2×D4 | — | 2 | 1 | no | — |
| 110 | C2×D4 | — | 4 | 1 | no | — |
| 111 | C2×D4 | — | 356 | 2 | no | — |
| 112 | C2×D4 | — | 178 | 1 | no | — |
| 113 | C2×D4 | — | 712 | 2 | no | — |
| 114 | C2×D4 | — | 712 | 2 | no | — |
| 115 | C22×C4 | — | 356 | 1 | no | — |
| 116 | SD16 | — | 8 | 1 | no | — |
| 117 | SD16 | — | 8 | 1 | no | — |
| 118 | M16 | — | 8 | 1 | no | — |
| 119 | C2×C8 | — | 2 | 1 | no | — |
| 120 | C2×C8 | — | 356 | 1 | no | — |
| 121 | C4⋊C4 | — | 4 | 1 | no | — |
| 122 | C4⋊C4 | — | 178 | 1 | no | — |
| 123 | C22⋊C4 | — | 356 | 1 | no | — |
| 124 | C22⋊C4 | — | 356 | 1 | no | — |
| 125 | C42 | #13083551 | 1 | 1 | yes | characteristic |
| 126 | C16 | — | 712 | 1 | no | — |
| 127 | C23 | — | 4 | 1 | no | — |
| 128 | C23 | — | 356 | 1 | no | — |
| 129 | C23 | — | 1424 | 1 | no | — |
| 130 | C23 | — | 712 | 4 | no | — |
| 131 | Q8 | — | 8 | 1 | no | — |
| 132 | D4 | — | 8 | 1 | no | — |
| 133 | D4 | — | 8 | 1 | no | — |
| 134 | D4 | — | 8 | 1 | no | — |
| 135 | D4 | — | 356 | 1 | no | — |
| 136 | D4 | — | 178 | 2 | no | — |
| 137 | D4 | — | 712 | 4 | no | — |
| 138 | C2×C4 | — | 2 | 1 | no | — |
| 139 | C2×C4 | — | 4 | 1 | no | — |
| 140 | C2×C4 | — | 356 | 1 | no | — |
| 141 | C2×C4 | — | 712 | 1 | no | — |
| 142 | C2×C4 | — | 712 | 1 | no | — |
| 143 | C2×C4 | #15667926 | 1 | 1 | yes | characteristic |
| 144 | C8 | — | 2 | 1 | no | — |
| 145 | C8 | — | 2 | 1 | no | — |
| 146 | C8 | — | 16 | 1 | no | — |
| 147 | C8 | — | 712 | 1 | no | — |
| 148 | C22 | — | 8 | 2 | no | — |
| 149 | C22 | — | 8 | 1 | no | — |
| 150 | C22 | — | 356 | 2 | no | — |
| 151 | C22 | — | 1424 | 2 | no | — |
| 152 | C22 | — | 1424 | 2 | no | — |
| 153 | C22 | — | 712 | 1 | no | — |
| 154 | C22 | — | 712 | 1 | no | — |
| 155 | C22 | #19151384 | 1 | 1 | yes | characteristic |
| 156 | C4 | — | 2 | 1 | no | — |
| 157 | C4 | — | 2 | 1 | no | — |
| 158 | C4 | — | 2 | 1 | no | — |
| 159 | C4 | — | 8 | 1 | no | — |
| 160 | C4 | — | 712 | 1 | no | — |
| 161 | C2 | — | 2 | 1 | no | — |
| 162 | C2 | — | 16 | 1 | no | — |
| 163 | C2 | — | 712 | 2 | no | — |
| 164 | C2 | #26018396 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #26018293 | C2 | semidirect |
| 3. C89×#1410 | C2 | semidirect |
| 84. #1410 | D89 | semidirect |
| 86. C89 | #7517 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26019366 | 2 |
| #26018293 | 2 |
| C89×#1410 | 2 |
| #7517 | 89 |