Finite Group Databasebeta

← #3093×C251 #44547680

Subgroups — #3093×C251 #44547680

Order 32128.

414 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 184 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
1#525×C251C211yesmaximal, characteristic
2C2×#117×C251C211yesmaximal, characteristic, complemented
3#485×C251C214yesmaximal, complemented
4#482×C251C214yesmaximal, complemented
5#428×C251C212yesmaximal, complemented
6#417×C251C212yesmaximal, complemented
7#407×C251C211yesmaximal, characteristic, complemented
8C2×#41×C251C2211yescharacteristic
9C23×C4×C251C2211yescharacteristic
10C2×M16×C251C2212yes—
11#123×C251—21no—
12C22⋊Q8×C251C2212yes—
13#121×C251C2212yes—
14#120×C251—21no—
15C4×D4×C251—22no—
16#117×C251—22no—
17#117×C251C2211yescharacteristic
18#117×C251C2214yes—
19C2×C4⋊C4×C251C2212yes—
20C2×C22⋊C4×C251—21no—
21C2×C42×C251—21no—
22#107×C251C2214yescomplemented
23#106×C251C2214yescomplemented
24#103×C251C2214yescomplemented
25#102×C251C2214yescomplemented
26C22⋊C8×C251C2214yes—
27C24×C251D411yescharacteristic
28#41×C251—24no—
29C2×Q8×C251C2311yescharacteristic
30C2×D4×C251—22no—
31C2×D4×C251—21no—
32C2×D4×C251C2311yescharacteristic
33C22×C4×C251—24no—
34C22×C4×C251—22no—
35C22×C4×C251—21no—
36C22×C4×C251D411yescharacteristic
37C22×C4×C251D412yes—
38C22×C4×C251C2312yes—
39C22×C4×C251C2311yescharacteristic
40M16×C251—24no—
41C2×C8×C251C2314yes—
42C4⋊C4×C251—22no—
43C4⋊C4×C251—22no—
44C4⋊C4×C251C2314yes—
45C4⋊C4×C251C2312yes—
46C22⋊C4×C251—24no—
47C22⋊C4×C251—21no—
48C22⋊C4×C251—22no—
49C22⋊C4×C251—22no—
50C42×C251—24no—
51C42×C251—21no—
52C23×C251—22no—
53C23×C251—21no—
54C23×C251—42no—
55C23×C251C2×D412yes—
56C23×C251C2×D411yescharacteristic
57Q8×C251—22no—
58D4×C251—22no—
59D4×C251—44no—
60D4×C251—42no—
61C2×C4×C251—22no—
62C2×C4×C251—24no—
63C2×C4×C251—24no—
64C2×C4×C251—24no—
65C2×C4×C251—22no—
66C2×C4×C251—21no—
67C2×C4×C251—44no—
68C2×C4×C251—42no—
69C2×C4×C251C2×D412yes—
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251#4114yes—
72C2×C4×C251C2411yescharacteristic
73C8×C251—24no—
74C22×C251—22no—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—44no—
79C22×C251—41no—
80C22×C251—42no—
81C22×C251#12312yes—
82C22×C251C22×D411yescharacteristic
83C4×C251—22no—
84C4×C251—41no—
85C4×C251—44no—
86C4×C251—42no—
87C4×C251#12312yes—
88C4×C251C2×#4112yes—
89C2×C251—22no—
90C2×C251—42no—
91C2×C251—81no—
92C2×C251C2×#12311yescharacteristic
93C2×C251#57512yes—
94C251#309311yescharacteristic, direct product factor, Sylow
95#3093C25111yesmaximal, characteristic, direct product factor, Sylow
96#525C2×C25111yescharacteristic
97C2×#117C2×C25111yescharacteristic, complemented
98#485C2×C25114yescomplemented
99#482C2×C25114yescomplemented
100#428C2×C25112yescomplemented
101#417C2×C25112yescomplemented
102#407C2×C25111yescharacteristic, complemented
103C2×#41C22×C25111yescharacteristic
104C23×C4C22×C25111yescharacteristic
105C2×M16C22×C25112yes—
106#123—21no—
107C22⋊Q8C22×C25112yes—
108#121C22×C25112yes—
109#120—21no—
110C4×D4—22no—
111#117—22no—
112#117C22×C25114yes—
113#117C22×C25111yescharacteristic
114C2×C4⋊C4C22×C25112yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107C22×C25114yescomplemented
118#106C22×C25114yescomplemented
119#103C22×C25114yescomplemented
120#102C22×C25114yescomplemented
121C22⋊C8C22×C25114yes—
122C24D4×C25111yescharacteristic
123#41—24no—
124C2×Q8C23×C25111yescharacteristic
125C2×D4—22no—
126C2×D4—21no—
127C2×D4C23×C25111yescharacteristic
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4D4×C25111yescharacteristic
132C22×C4D4×C25112yes—
133C22×C4C23×C25112yes—
134C22×C4C23×C25111yescharacteristic
135M16—24no—
136C2×C8C23×C25114yes—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4C23×C25114yes—
140C4⋊C4C23×C25112yes—
141C22⋊C4—24no—
142C22⋊C4—21no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C42—24no—
146C42—21no—
147C23—22no—
148C23—21no—
149C23—42no—
150C23C2×D4×C25112yes—
151C23C2×D4×C25111yescharacteristic
152Q8—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4C2×D4×C25112yes—
165C2×C4C2×D4×C25111yescharacteristic
166C2×C4#41×C25114yes—
167C2×C4C24×C25111yescharacteristic
168C8—24no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—41no—
175C22—42no—
176C22#123×C25112yes—
177C22C22×D4×C25111yescharacteristic
178C4—22no—
179C4—41no—
180C4—44no—
181C4—42no—
182C4#123×C25112yes—
183C4C2×#41×C25112yes—
184C2—22no—
185C2—42no—
186C2—81no—
187C2C2×#123×C25111yescharacteristic
188C2#575×C25112yes—
subgroupG/H
center, socleC22×C251C22×D4
commutator, Frattini, cosocleC2×C4C24×C251
prime radical#525×C251C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C251 ◁ C23×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#117×C251C2semidirect
3. #485×C251C2semidirect
4. #482×C251C2semidirect
5. #428×C251C2semidirect
6. #417×C251C2semidirect
7. #407×C251C2semidirect
22. #107×C251C22semidirect
23. #106×C251C22semidirect
24. #103×C251C22semidirect
25. #102×C251C22semidirect
94. C251#3093direct (N × H)
95. #3093C251direct (N × H)
97. C2×#117C2×C251semidirect
98. #485C2×C251semidirect
99. #482C2×C251semidirect
100. #428C2×C251semidirect
101. #417C2×C251semidirect
102. #407C2×C251semidirect
117. #107C22×C251semidirect
118. #106C22×C251semidirect
119. #103C22×C251semidirect
120. #102C22×C251semidirect

Maximal subgroups

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

subgroupindex
#525×C2512
C2×#117×C2512
#485×C2512
#482×C2512
#428×C2512
#417×C2512
#407×C2512
#3093251