Finite Group Databasebeta

← #1659×C251 #44546246

Subgroups — #1659×C251 #44546246

Order 32128.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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
1H(3,𝔽4)×C251C211yesmaximal, characteristic
2#497×C251C211yesmaximal, characteristic, complemented
3#489×C251C212yesmaximal, complemented
4#329×C251C211yesmaximal, characteristic, complemented
5#329×C251C212yesmaximal, complemented
6C2×Q16×C251—21no—
7C2×SD16×C251—21no—
8C4⋊Q8×C251C2211yescharacteristic, complemented
9#124×C251—22no—
10#124×C251—21no—
11#124×C251C2212yes—
12#124×C251C2211yescharacteristic
13#120×C251—22no—
14#120×C251—21no—
15#103×C251—22no—
16#102×C251—22no—
17#97×C251C2212yescomplemented
18#97×C251C2211yescharacteristic, complemented
19C24×C251—21no—
20C2×Q8×C251—21no—
21C2×Q8×C251D412yes—
22C2×Q8×C251D411yescharacteristic
23C2×D4×C251—21no—
24C2×D4×C251—22no—
25C2×D4×C251D412yes—
26C2×D4×C251D411yescharacteristic
27Q16×C251—42no—
28SD16×C251—42no—
29C2×C8×C251—21no—
30C2×C8×C251—22no—
31C4⋊C4×C251—22no—
32C22⋊C4×C251—22no—
33C22⋊C4×C251—21no—
34C22⋊C4×C251—22no—
35C22⋊C4×C251—22no—
36C22⋊C4×C251—22no—
37C42×C251—21no—
38C42×C251C2311yescharacteristic
39C23×C251—21no—
40C23×C251—22no—
41C23×C251—81no—
42C23×C251—82no—
43Q8×C251—41no—
44Q8×C251—42no—
45Q8×C251—41no—
46Q8×C251—41no—
47D4×C251—41no—
48D4×C251—42no—
49D4×C251—81no—
50D4×C251—82no—
51C2×C4×C251—21no—
52C2×C4×C251—22no—
53C2×C4×C251—41no—
54C2×C4×C251C2×D412yes—
55C2×C4×C251C2×D411yescharacteristic
56C8×C251—22no—
57C8×C251—42no—
58C22×C251—41no—
59C22×C251—42no—
60C22×C251—42no—
61C22×C251—42no—
62C22×C251—42no—
63C22×C251—84no—
64C22×C251#12011yescharacteristic
65C4×C251—21no—
66C4×C251—22no—
67C4×C251—41no—
68C4×C251—42no—
69C4×C251—81no—
70C2×C251—81no—
71C2×C251—82no—
72C2×C251#45311yescharacteristic
73C2×C251#45512yes—
74C251#165911yescharacteristic, direct product factor, Sylow
75#1659C25111yesmaximal, characteristic, direct product factor, Sylow
76H(3,𝔽4)C2×C25111yescharacteristic
77#497C2×C25111yescharacteristic, complemented
78#489C2×C25112yescomplemented
79#329C2×C25111yescharacteristic, complemented
80#329C2×C25112yescomplemented
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C22×C25111yescharacteristic, complemented
84#124—22no—
85#124—21no—
86#124C22×C25112yes—
87#124C22×C25111yescharacteristic
88#120—22no—
89#120—21no—
90#103—22no—
91#102—22no—
92#97C22×C25112yescomplemented
93#97C22×C25111yescharacteristic, complemented
94C24—21no—
95C2×Q8—21no—
96C2×Q8D4×C25112yes—
97C2×Q8D4×C25111yescharacteristic
98C2×D4—21no—
99C2×D4—22no—
100C2×D4D4×C25112yes—
101C2×D4D4×C25111yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—22no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—21no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C42—21no—
113C42C23×C25111yescharacteristic
114C23—21no—
115C23—22no—
116C23—81no—
117C23—82no—
118Q8—41no—
119Q8—42no—
120Q8—41no—
121Q8—41no—
122D4—41no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4C2×D4×C25112yes—
130C2×C4C2×D4×C25111yescharacteristic
131C8—22no—
132C8—42no—
133C22—41no—
134C22—42no—
135C22—42no—
136C22—42no—
137C22—42no—
138C22—84no—
139C22#120×C25111yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—42no—
144C4—81no—
145C2—81no—
146C2—82no—
147C2#453×C25111yescharacteristic
148C2#455×C25112yes—
subgroupG/H
center, socleC22×C251#120
commutator, Frattini, cosocleC42C23×C251
prime radicalH(3,𝔽4)×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. #497×C251C2semidirect
3. #489×C251C2semidirect
4. #329×C251C2semidirect
5. #329×C251C2semidirect
8. C4⋊Q8×C251C22semidirect
17. #97×C251C22semidirect
18. #97×C251C22semidirect
74. C251#1659direct (N × H)
75. #1659C251direct (N × H)
77. #497C2×C251semidirect
78. #489C2×C251semidirect
79. #329C2×C251semidirect
80. #329C2×C251semidirect
83. C4⋊Q8C22×C251semidirect
92. #97C22×C251semidirect
93. #97C22×C251semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽4)×C2512
#497×C2512
#489×C2512
#329×C2512
#1659251