Finite Group DatabasebetaFinite Group DatabasebetaOrder 1600.
471 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 97 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 | Q8×D52 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | Q8×#1044 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C4×D52⋊C2 | C2 | 1 | 3 | yes | maximal |
| 4 | #11717380 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11717378 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×D52⋊C2 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q8×C5⋊D5 | C22 | 1 | 1 | yes | characteristic |
| 8 | C5×Q8×D5 | — | 2 | 2 | no | — |
| 9 | C4×D52 | C22 | 1 | 6 | yes | — |
| 10 | #87872 | C22 | 1 | 6 | yes | — |
| 11 | D5×Dic10 | — | 2 | 6 | no | — |
| 12 | C4⋊#1044 | C22 | 1 | 3 | yes | — |
| 13 | C4×#1044 | C22 | 1 | 3 | yes | — |
| 14 | #87836 | C22 | 1 | 6 | yes | — |
| 15 | #87836 | C22 | 1 | 3 | yes | — |
| 16 | #87835 | C22 | 1 | 6 | yes | — |
| 17 | C2×D52 | C23 | 1 | 2 | yes | — |
| 18 | C2×#1044 | C23 | 1 | 3 | yes | — |
| 19 | C2×#1044 | C23 | 1 | 1 | yes | characteristic |
| 20 | D52⋊C2 | Q8 | 1 | 4 | yes | direct product factor |
| 21 | C52×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 22 | C4×C5⋊D5 | C23 | 1 | 3 | yes | — |
| 23 | #6045 | D4 | 1 | 3 | yes | complemented |
| 24 | C4×C5×D5 | — | 2 | 6 | no | — |
| 25 | C5×Dic10 | — | 2 | 6 | no | — |
| 26 | #6039 | — | 2 | 6 | no | — |
| 27 | #6036 | C23 | 1 | 6 | yes | — |
| 28 | D5×Dic5 | — | 2 | 6 | no | — |
| 29 | C2×Q8×D5 | — | 10 | 2 | no | — |
| 30 | Q8×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 31 | C2×C5⋊D5 | C24 | 1 | 1 | yes | characteristic |
| 32 | C2×C5×D5 | — | 2 | 2 | no | — |
| 33 | D52 | — | 2 | 2 | no | — |
| 34 | D52 | C2×Q8 | 1 | 4 | yes | complemented |
| 35 | #1044 | — | 2 | 3 | no | — |
| 36 | #1044 | C2×Q8 | 1 | 2 | yes | complemented |
| 37 | C4×C52 | C2×D4 | 1 | 3 | yes | — |
| 38 | #1039 | C2×D4 | 1 | 3 | yes | — |
| 39 | C5×Dic5 | — | 2 | 6 | no | — |
| 40 | C2×C5×Q8 | — | 10 | 2 | no | — |
| 41 | Q8×D5 | — | 2 | 2 | no | — |
| 42 | Q8×D5 | — | 10 | 6 | no | — |
| 43 | Q8×D5 | — | 10 | 1 | no | — |
| 44 | Q8×D5 | — | 10 | 2 | no | — |
| 45 | Q8×D5 | — | 10 | 6 | no | — |
| 46 | C2×C4×D5 | — | 10 | 6 | no | — |
| 47 | C2×Dic10 | — | 10 | 6 | no | — |
| 48 | C4⋊AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 49 | C4×AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 50 | D4×Q8 | — | 25 | 1 | no | maximal, Sylow |
| 51 | C2×C52 | C22×D4 | 1 | 1 | yes | characteristic |
| 52 | C5⋊D5 | C22×Q8 | 1 | 1 | yes | characteristic |
| 53 | C5⋊D5 | 2−1+4 | 1 | 1 | yes | characteristic |
| 54 | C5×D5 | — | 2 | 4 | no | — |
| 55 | C22×D5 | — | 10 | 2 | no | — |
| 56 | C2×AGL(1,𝔽5) | — | 10 | 1 | no | — |
| 57 | C2×AGL(1,𝔽5) | — | 10 | 3 | no | — |
| 58 | C5×Q8 | — | 2 | 2 | no | — |
| 59 | C5×Q8 | — | 2 | 1 | no | — |
| 60 | C5×Q8 | — | 10 | 6 | no | — |
| 61 | C2×C4×C5 | — | 10 | 6 | no | — |
| 62 | C2×Dic5 | — | 10 | 6 | no | — |
| 63 | C4×D5 | — | 2 | 6 | no | — |
| 64 | C4×D5 | — | 10 | 6 | no | — |
| 65 | C4×D5 | — | 10 | 3 | no | — |
| 66 | C4×D5 | — | 10 | 6 | no | — |
| 67 | C4×D5 | — | 10 | 6 | no | — |
| 68 | Dic10 | — | 2 | 6 | no | — |
| 69 | Dic10 | — | 10 | 6 | no | — |
| 70 | Dic10 | — | 10 | 3 | no | — |
| 71 | Dic10 | — | 10 | 12 | no | — |
| 72 | C22×Q8 | — | 25 | 2 | no | — |
| 73 | C4⋊Q8 | — | 25 | 3 | no | — |
| 74 | C22⋊Q8 | — | 25 | 6 | no | — |
| 75 | C4×Q8 | — | 25 | 1 | no | — |
| 76 | C4×D4 | — | 25 | 3 | no | — |
| 77 | C52 | D4×Q8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 78 | C22×C5 | — | 10 | 2 | no | — |
| 79 | C2×D5 | — | 2 | 2 | no | — |
| 80 | C2×D5 | — | 10 | 4 | no | — |
| 81 | C2×D5 | — | 10 | 2 | no | — |
| 82 | C2×D5 | — | 10 | 1 | no | — |
| 83 | C2×D5 | — | 10 | 4 | no | — |
| 84 | AGL(1,𝔽5) | — | 10 | 2 | no | — |
| 85 | AGL(1,𝔽5) | — | 20 | 3 | no | — |
| 86 | C4×C5 | — | 2 | 6 | no | — |
| 87 | C4×C5 | — | 2 | 3 | no | — |
| 88 | C4×C5 | — | 10 | 6 | no | — |
| 89 | Dic5 | — | 2 | 6 | no | — |
| 90 | Dic5 | — | 10 | 6 | no | — |
| 91 | Dic5 | — | 10 | 3 | no | — |
| 92 | C2×Q8 | — | 10 | 2 | no | — |
| 93 | C2×Q8 | — | 25 | 6 | no | — |
| 94 | C2×Q8 | — | 25 | 1 | no | — |
| 95 | C2×Q8 | — | 50 | 6 | no | — |
| 96 | C2×D4 | — | 25 | 1 | no | — |
| 97 | C22×C4 | — | 25 | 6 | no | — |
| 98 | C4⋊C4 | — | 25 | 3 | no | — |
| 99 | C4⋊C4 | — | 25 | 6 | no | — |
| 100 | C4⋊C4 | — | 25 | 3 | no | — |
| 101 | C22⋊C4 | — | 25 | 6 | no | — |
| 102 | C42 | — | 25 | 3 | no | — |
| 103 | C2×C5 | — | 2 | 2 | no | — |
| 104 | C2×C5 | — | 2 | 1 | no | — |
| 105 | C2×C5 | — | 10 | 4 | no | — |
| 106 | D5 | — | 2 | 4 | no | — |
| 107 | D5 | — | 10 | 1 | no | — |
| 108 | D5 | — | 10 | 2 | no | — |
| 109 | D5 | — | 10 | 1 | no | — |
| 110 | D5 | — | 10 | 2 | no | — |
| 111 | C23 | — | 25 | 2 | no | — |
| 112 | Q8 | — | 10 | 6 | no | — |
| 113 | Q8 | — | 25 | 3 | no | — |
| 114 | Q8 | — | 50 | 6 | no | — |
| 115 | Q8 | D52⋊C2 | 1 | 1 | yes | characteristic, direct product factor |
| 116 | D4 | — | 25 | 4 | no | — |
| 117 | C2×C4 | — | 10 | 6 | no | — |
| 118 | C2×C4 | — | 25 | 6 | no | — |
| 119 | C2×C4 | — | 25 | 1 | no | — |
| 120 | C2×C4 | — | 25 | 3 | no | — |
| 121 | C2×C4 | — | 25 | 3 | no | — |
| 122 | C2×C4 | — | 50 | 6 | no | — |
| 123 | C5 | — | 2 | 2 | no | — |
| 124 | C5 | — | 2 | 1 | no | — |
| 125 | C22 | — | 10 | 2 | no | — |
| 126 | C22 | — | 25 | 4 | no | — |
| 127 | C22 | — | 25 | 1 | no | — |
| 128 | C22 | — | 50 | 2 | no | — |
| 129 | C4 | — | 10 | 6 | no | — |
| 130 | C4 | — | 25 | 2 | no | — |
| 131 | C4 | — | 25 | 3 | no | — |
| 132 | C4 | — | 50 | 3 | no | — |
| 133 | C4 | C2×D52⋊C2 | 1 | 3 | yes | — |
| 134 | C2 | — | 10 | 4 | no | — |
| 135 | C2 | — | 25 | 1 | no | — |
| 136 | C2 | — | 25 | 1 | no | — |
| 137 | C2 | C22×D52⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D52⋊C2 |
| commutator, cosocle | C2×C5⋊D5 | C24 |
| Fitting | C52×Q8 | D4 |
| socle | C2×C52 | C22×D4 |
| prime radical | C2×D52⋊C2 | C22 |
| hypercenter | Q8 | D52⋊C2 |
| hypocenter | C52 | D4×Q8 |
| LCS | G ▷ C2×C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D52 | C2 | semidirect |
| 2. Q8×#1044 | C2 | semidirect |
| 4. #11717380 | C2 | semidirect |
| 5. #11717378 | C2 | semidirect |
| 20. D52⋊C2 | Q8 | direct (N × H) |
| 21. C52×Q8 | D4 | semidirect |
| 23. #6045 | D4 | semidirect |
| 34. D52 | C2×Q8 | semidirect |
| 36. #1044 | C2×Q8 | semidirect |
| 77. C52 | D4×Q8 | semidirect |
| 115. Q8 | D52⋊C2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D52 | 2 |
| Q8×#1044 | 2 |
| C4×D52⋊C2 | 2 |
| #11717380 | 2 |
| #11717378 | 2 |
| D4×Q8 | 25 |