Finite Group Databasebeta

← #1629×C251 #44546216

Subgroups — #1629×C251 #44546216

Order 32128.

314 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D42×C251C211yesmaximal, characteristic
2#492×C251C211yesmaximal, characteristic, complemented
3#476×C251C212yesmaximal, complemented
4#331×C251C211yesmaximal, characteristic, complemented
5D4⋊C8×C251C212yesmaximal, complemented
6C22×D4×C251—22no—
7C2×SD16×C251—22no—
8C4⋊Q8×C251C2211yescharacteristic, complemented
9#127×C251C2211yescharacteristic
10#121×C251—22no—
11#120×C251—22no—
12C4×D4×C251C2212yes—
13#106×C251—22no—
14C4⋊C8×C251C2212yescomplemented
15#102×C251—22no—
16C4×C8×C251C2211yescharacteristic, complemented
17C24×C251—22no—
18C2×Q8×C251—21no—
19C2×D4×C251—24no—
20C2×D4×C251—22no—
21C2×D4×C251—44no—
22C2×D4×C251—44no—
23C2×D4×C251D412yes—
24C2×D4×C251D412yes—
25C22×C4×C251—22no—
26SD16×C251—44no—
27C2×C8×C251—21no—
28C2×C8×C251—22no—
29C4⋊C4×C251—22no—
30C4⋊C4×C251D412yes—
31C22⋊C4×C251—22no—
32C22⋊C4×C251—22no—
33C42×C251C2311yescharacteristic
34C23×C251—22no—
35C23×C251—22no—
36C23×C251—48no—
37C23×C251—82no—
38Q8×C251—42no—
39D4×C251—22no—
40D4×C251—42no—
41D4×C251—48no—
42D4×C251—42no—
43D4×C251—42no—
44D4×C251SD1614yescomplemented
45C2×C4×C251—22no—
46C2×C4×C251—42no—
47C2×C4×C251—42no—
48C2×C4×C251—41no—
49C2×C4×C251C2×D411yescharacteristic
50C2×C4×C251C2×D412yes—
51C8×C251—22no—
52C8×C251—42no—
53C22×C251—24no—
54C22×C251—42no—
55C22×C251—42no—
56C22×C251—44no—
57C22×C251—42no—
58C22×C251—88no—
59C22×C251#12011yescharacteristic
60C4×C251—21no—
61C4×C251—42no—
62C4×C251—81no—
63C4×C251C2×SD1612yes—
64C4×C251AGL(1,ℤ/8)12yes—
65C2×C251—44no—
66C2×C251—82no—
67C2×C251#45012yes—
68C2×C251#45311yescharacteristic
69C251#162911yescharacteristic, direct product factor, Sylow
70#1629C25111yesmaximal, characteristic, direct product factor, Sylow
71D42C2×C25111yescharacteristic
72#492C2×C25111yescharacteristic, complemented
73#476C2×C25112yescomplemented
74#331C2×C25111yescharacteristic, complemented
75D4⋊C8C2×C25112yescomplemented
76C22×D4—22no—
77C2×SD16—22no—
78C4⋊Q8C22×C25111yescharacteristic, complemented
79#127C22×C25111yescharacteristic
80#121—22no—
81#120—22no—
82C4×D4C22×C25112yes—
83#106—22no—
84C4⋊C8C22×C25112yescomplemented
85#102—22no—
86C4×C8C22×C25111yescharacteristic, complemented
87C24—22no—
88C2×Q8—21no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4D4×C25112yes—
94C2×D4D4×C25112yes—
95C22×C4—22no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—22no—
99C4⋊C4—22no—
100C4⋊C4D4×C25112yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C23×C25111yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—42no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4SD16×C25114yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4C2×D4×C25111yescharacteristic
120C2×C4C2×D4×C25112yes—
121C8—22no—
122C8—42no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#120×C25111yescharacteristic
130C4—21no—
131C4—42no—
132C4—81no—
133C4C2×SD16×C25112yes—
134C4AGL(1,ℤ/8)×C25112yes—
135C2—44no—
136C2—82no—
137C2#450×C25112yes—
138C2#453×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radicalD42×C251C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ C42×C251 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #492×C251C2semidirect
3. #476×C251C2semidirect
4. #331×C251C2semidirect
5. D4⋊C8×C251C2semidirect
8. C4⋊Q8×C251C22semidirect
14. C4⋊C8×C251C22semidirect
16. C4×C8×C251C22semidirect
44. D4×C251SD16semidirect
69. C251#1629direct (N × H)
70. #1629C251direct (N × H)
72. #492C2×C251semidirect
73. #476C2×C251semidirect
74. #331C2×C251semidirect
75. D4⋊C8C2×C251semidirect
78. C4⋊Q8C22×C251semidirect
84. C4⋊C8C22×C251semidirect
86. C4×C8C22×C251semidirect
114. D4SD16×C251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D42×C2512
#492×C2512
#476×C2512
#331×C2512
D4⋊C8×C2512
#1629251