← #2991×C251 #44547578
Subgroups — #2991×C251 #44547578
Order 32128.
394 conjugacy classes of subgroups.
The subgroup lattice has 132 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×#41×C251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C8×Q8×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #436×C251 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #431×C251 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #423×C251 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#41×C251 | C4 | 1 | 1 | yes | characteristic |
| 7 | C2×M16×C251 | C22 | 1 | 6 | yes | — |
| 8 | C4×Q8×C251 | C4 | 1 | 1 | yes | characteristic |
| 9 | C4×Q8×C251 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×D4×C251 | C4 | 1 | 3 | yes | — |
| 11 | C4×D4×C251 | C22 | 1 | 3 | yes | — |
| 12 | #117×C251 | C4 | 1 | 3 | yes | — |
| 13 | C2×C42×C251 | C22 | 1 | 3 | yes | — |
| 14 | C4⋊C8×C251 | C22 | 1 | 3 | yes | — |
| 15 | C4⋊C8×C251 | C22 | 1 | 6 | yes | — |
| 16 | C4⋊C8×C251 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22⋊C8×C251 | C22 | 1 | 6 | yes | — |
| 18 | C4×C8×C251 | C22 | 1 | 6 | yes | — |
| 19 | #41×C251 | — | 2 | 4 | no | — |
| 20 | C2×Q8×C251 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×D4×C251 | C2×C4 | 1 | 3 | yes | — |
| 22 | C22×C4×C251 | C2×C4 | 1 | 3 | yes | — |
| 23 | C22×C4×C251 | C2×C4 | 1 | 3 | yes | — |
| 24 | C22×C4×C251 | C23 | 1 | 3 | yes | — |
| 25 | M16×C251 | — | 4 | 6 | no | — |
| 26 | C2×C8×C251 | C23 | 1 | 2 | yes | — |
| 27 | C2×C8×C251 | C23 | 1 | 6 | yes | — |
| 28 | C4⋊C4×C251 | C2×C4 | 1 | 3 | yes | — |
| 29 | C4⋊C4×C251 | C2×C4 | 1 | 3 | yes | — |
| 30 | C22⋊C4×C251 | C2×C4 | 1 | 6 | yes | — |
| 31 | C42×C251 | C2×C4 | 1 | 3 | yes | — |
| 32 | C42×C251 | C2×C4 | 1 | 3 | yes | — |
| 33 | C42×C251 | C23 | 1 | 3 | yes | — |
| 34 | C42×C251 | C23 | 1 | 1 | yes | characteristic |
| 35 | C23×C251 | C22×C4 | 1 | 3 | yes | — |
| 36 | Q8×C251 | M16 | 1 | 4 | yes | complemented |
| 37 | D4×C251 | — | 2 | 6 | no | — |
| 38 | C2×C4×C251 | — | 2 | 6 | no | — |
| 39 | C2×C4×C251 | — | 2 | 6 | no | — |
| 40 | C2×C4×C251 | — | 2 | 3 | no | — |
| 41 | C2×C4×C251 | — | 2 | 3 | no | — |
| 42 | C2×C4×C251 | C22×C4 | 1 | 1 | yes | characteristic |
| 43 | C2×C4×C251 | C22×C4 | 1 | 3 | yes | — |
| 44 | C2×C4×C251 | C22×C4 | 1 | 3 | yes | — |
| 45 | C2×C4×C251 | C22×C4 | 1 | 3 | yes | — |
| 46 | C2×C4×C251 | C22×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×C251 | C24 | 1 | 1 | yes | characteristic |
| 48 | C8×C251 | — | 2 | 6 | no | — |
| 49 | C8×C251 | — | 2 | 2 | no | — |
| 50 | C22×C251 | — | 2 | 3 | no | — |
| 51 | C22×C251 | — | 2 | 3 | no | — |
| 52 | C22×C251 | — | 2 | 3 | no | — |
| 53 | C22×C251 | C23×C4 | 1 | 1 | yes | characteristic |
| 54 | C4×C251 | — | 2 | 3 | no | — |
| 55 | C4×C251 | — | 2 | 1 | no | — |
| 56 | C4×C251 | — | 2 | 3 | no | — |
| 57 | C4×C251 | C2×M16 | 1 | 6 | yes | — |
| 58 | C4×C251 | #131 | 1 | 2 | yes | — |
| 59 | C4×C251 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 60 | C4×C251 | 2−1+4 | 1 | 1 | yes | characteristic |
| 61 | C2×C251 | — | 4 | 3 | no | — |
| 62 | C2×C251 | #520 | 1 | 1 | yes | characteristic |
| 63 | C2×C251 | C22×M16 | 1 | 1 | yes | characteristic |
| 64 | C2×C251 | C2×#131 | 1 | 1 | yes | characteristic |
| 65 | C251 | #2991 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 66 | #2991 | C251 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 67 | C4×#41 | C2×C251 | 1 | 1 | yes | characteristic |
| 68 | C8×Q8 | C2×C251 | 1 | 2 | yes | complemented |
| 69 | #436 | C2×C251 | 1 | 6 | yes | complemented |
| 70 | #431 | C2×C251 | 1 | 3 | yes | complemented |
| 71 | #423 | C2×C251 | 1 | 3 | yes | complemented |
| 72 | C2×#41 | C4×C251 | 1 | 1 | yes | characteristic |
| 73 | C2×M16 | C22×C251 | 1 | 6 | yes | — |
| 74 | C4×Q8 | C4×C251 | 1 | 1 | yes | characteristic |
| 75 | C4×Q8 | C22×C251 | 1 | 1 | yes | characteristic |
| 76 | C4×D4 | C4×C251 | 1 | 3 | yes | — |
| 77 | C4×D4 | C22×C251 | 1 | 3 | yes | — |
| 78 | #117 | C4×C251 | 1 | 3 | yes | — |
| 79 | C2×C42 | C22×C251 | 1 | 3 | yes | — |
| 80 | C4⋊C8 | C22×C251 | 1 | 3 | yes | — |
| 81 | C4⋊C8 | C22×C251 | 1 | 6 | yes | — |
| 82 | C4⋊C8 | C22×C251 | 1 | 1 | yes | characteristic |
| 83 | C22⋊C8 | C22×C251 | 1 | 6 | yes | — |
| 84 | C4×C8 | C22×C251 | 1 | 6 | yes | — |
| 85 | #41 | — | 2 | 4 | no | — |
| 86 | C2×Q8 | C2×C4×C251 | 1 | 1 | yes | characteristic |
| 87 | C2×D4 | C2×C4×C251 | 1 | 3 | yes | — |
| 88 | C22×C4 | C2×C4×C251 | 1 | 3 | yes | — |
| 89 | C22×C4 | C2×C4×C251 | 1 | 3 | yes | — |
| 90 | C22×C4 | C23×C251 | 1 | 3 | yes | — |
| 91 | M16 | — | 4 | 6 | no | — |
| 92 | C2×C8 | C23×C251 | 1 | 2 | yes | — |
| 93 | C2×C8 | C23×C251 | 1 | 6 | yes | — |
| 94 | C4⋊C4 | C2×C4×C251 | 1 | 3 | yes | — |
| 95 | C4⋊C4 | C2×C4×C251 | 1 | 3 | yes | — |
| 96 | C22⋊C4 | C2×C4×C251 | 1 | 6 | yes | — |
| 97 | C42 | C2×C4×C251 | 1 | 3 | yes | — |
| 98 | C42 | C2×C4×C251 | 1 | 3 | yes | — |
| 99 | C42 | C23×C251 | 1 | 3 | yes | — |
| 100 | C42 | C23×C251 | 1 | 1 | yes | characteristic |
| 101 | C23 | C22×C4×C251 | 1 | 3 | yes | — |
| 102 | Q8 | M16×C251 | 1 | 4 | yes | complemented |
| 103 | D4 | — | 2 | 6 | no | — |
| 104 | C2×C4 | — | 2 | 6 | no | — |
| 105 | C2×C4 | — | 2 | 6 | no | — |
| 106 | C2×C4 | — | 2 | 3 | no | — |
| 107 | C2×C4 | — | 2 | 3 | no | — |
| 108 | C2×C4 | C22×C4×C251 | 1 | 1 | yes | characteristic |
| 109 | C2×C4 | C22×C4×C251 | 1 | 3 | yes | — |
| 110 | C2×C4 | C22×C4×C251 | 1 | 3 | yes | — |
| 111 | C2×C4 | C22×C4×C251 | 1 | 3 | yes | — |
| 112 | C2×C4 | C22×C4×C251 | 1 | 1 | yes | characteristic |
| 113 | C2×C4 | C24×C251 | 1 | 1 | yes | characteristic |
| 114 | C8 | — | 2 | 6 | no | — |
| 115 | C8 | — | 2 | 2 | no | — |
| 116 | C22 | — | 2 | 3 | no | — |
| 117 | C22 | — | 2 | 3 | no | — |
| 118 | C22 | — | 2 | 3 | no | — |
| 119 | C22 | C23×C4×C251 | 1 | 1 | yes | characteristic |
| 120 | C4 | — | 2 | 3 | no | — |
| 121 | C4 | — | 2 | 1 | no | — |
| 122 | C4 | — | 2 | 3 | no | — |
| 123 | C4 | C2×M16×C251 | 1 | 6 | yes | — |
| 124 | C4 | #131×C251 | 1 | 2 | yes | — |
| 125 | C4 | H(5,𝔽2)×C251 | 1 | 1 | yes | characteristic |
| 126 | C4 | 2−1+4×C251 | 1 | 1 | yes | characteristic |
| 127 | C2 | — | 4 | 3 | no | — |
| 128 | C2 | #520×C251 | 1 | 1 | yes | characteristic |
| 129 | C2 | C22×M16×C251 | 1 | 1 | yes | characteristic |
| 130 | C2 | C2×#131×C251 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta