Finite Group Databasebeta

← #28399952

Subgroups — #28399952

Order 14464.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#20299434C211yesmaximal, characteristic
2C2×#16813653C212yesmaximal, complemented
3#20298780C214yesmaximal, complemented
4#20298775C214yesmaximal, complemented
5#20298616C212yesmaximal, complemented
6#20298582C211yesmaximal, characteristic, complemented
7#536×C113C211yesmaximal, characteristic, complemented
8#16813670C2212yes—
9#16813669—22no—
10#16813668—22no—
11#16813667—22no—
12#16813667C2212yes—
13C2×#14268647—22no—
14C2×#14268647C2212yes—
15#41×D113—24no—
16#16813653—24no—
17#16813653C2218yescomplemented
18C2×#14268637—22no—
19C2×#14268634—22no—
20C2×#14268631C2212yescomplemented
21#16813550C2218yescomplemented
22#16813541C2212yescomplemented
23#16813539C2212yescomplemented
24C2×#41×C113C2211yescharacteristic
25C22×Q8×C113C2211yescharacteristic, complemented
26C4⋊Q8×C113C2212yescomplemented
27#124×C113C2212yescomplemented
28#123×C113—22no—
29C22⋊Q8×C113—22no—
30#117×C113C2211yescharacteristic, complemented
31#14268648—24no—
32#14268647—24no—
33#14268647—24no—
34#14268647—22no—
35#14268647—24no—
36#14268647—22no—
37#14268647D414yes—
38#14268647C2314yes—
39#14268646—28no—
40#14268646—24no—
41C2×Dic226—22no—
42C2×Dic226D412yes—
43Q8×D113—24no—
44C2×D452—22no—
45C2×D452D412yes—
46C2×C22⋊D113—24no—
47C2×C22⋊D113—22no—
48D4×D113—28no—
49D4×D113—24no—
50C2×C4×D113—24no—
51C2×C4×D113—22no—
52#14268637—28no—
53#14268634—28no—
54#14268631—22no—
55#14268631C2314yes—
56C2×#11744751—22no—
57#41×C113—22no—
58#41×C113—22no—
59C2×Q8×C113—21no—
60C2×Q8×C113—22no—
61C2×Q8×C113—22no—
62C2×Q8×C113C2312yes—
63C2×Q8×C113C2311yescharacteristic
64C2×D4×C113D412yes—
65C2×D4×C113C2311yescharacteristic
66C22×C4×C113—21no—
67C22×C4×C113D412yes—
68C22×C4×C113C2311yescharacteristic
69C4⋊C4×C113—21no—
70C4⋊C4×C113—24no—
71C22⋊C4×C113—21no—
72C22⋊C4×C113—24no—
73C42×C113C2312yes—
74C23×C113C2×D412yes—
75C23×C113C2×D411yescharacteristic
76C22×D113—24no—
77C22×D113—22no—
78Q8×C113—22no—
79Q8×C113—24no—
80Q8×C113—41no—
81Q8×C113—41no—
82D4×C113—24no—
83D4×C113—22no—
84C2×C4×C113—22no—
85C2×C4×C113—22no—
86C2×C4×C113—22no—
87C2×C4×C113—22no—
88C2×C4×C113—42no—
89C2×C4×C113C2×D412yes—
90C2×C4×C113C2×D412yes—
91C2×C4×C113C2×D412yes—
92C2×C4×C113C2×D411yescharacteristic
93C2×C4×C113C2411yescharacteristic
94C22⋊D113—28no—
95C22⋊D113—24no—
96C22⋊D113—24no—
97C2×Dic113—24no—
98C2×Dic113—22no—
99D452—24no—
100D452—22no—
101D452C2×D414yes—
102C4×D113—28no—
103C4×D113—24no—
104C4×D113—24no—
105Dic226—24no—
106Dic226—22no—
107Dic226C2×D414yes—
108#11744751—24no—
109C22×C113—22no—
110C22×C113—41no—
111C22×C113—42no—
112C22×C113#12012yes—
113C22×C113C22×D411yescharacteristic
114C2×D113—24no—
115C2×D113—82no—
116C2×D113—84no—
117C4×C113—22no—
118C4×C113—42no—
119C4×C113—42no—
120C4×C113#12011yescharacteristic
121C4×C113#12011yescharacteristic
122C4×C113C22×D412yes—
123Dic113—24no—
124C2×C113—21no—
125C2×C113—22no—
126C2×C113—42no—
127C2×C113C2×#12011yescharacteristic
128D113—44no—
129#3022—1131nomaximal, Sylow
130C113#302211yescharacteristic, complemented, Sylow
131#585—1131no—
132C2×#137—1132no—
133#536D11311yescharacteristic, complemented
134#456—1134no—
135#455—1134no—
136#421—1132no—
137#413—1131no—
1382−1+4—1132no—
1392−1+4—2262no—
140H(5,𝔽2)—1132no—
141H(5,𝔽2)—2262no—
142H(5,𝔽2)—2262no—
143C2×#41—1132no—
144C2×#41—2264no—
145C2×#41—2262no—
146C2×#41C2×D11311yescharacteristic
147C22×Q8C2×D11311yescharacteristic, complemented
148#137—1138no—
149#137—2264no—
150C2×Q16—2262no—
151C2×SD16—2262no—
152C2×M16—1132no—
153C4⋊Q8C2×D11312yescomplemented
154#124C2×D11312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D11311yescharacteristic, complemented
158#104—1138no—
159#101—1132no—
160#100—1132no—
161#41—22no—
162#41—22no—
163#41—1134no—
164#41—1134no—
165#41—2264no—
166#41—2264no—
167#41—2264no—
168#41—2268no—
169#41—2264no—
170#41—2262no—
171#41—2262no—
172#41—2264no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—1132no—
177C2×Q8—2262no—
178C2×Q8—2264no—
179C2×Q8C22×D11312yes—
180C2×Q8C22×D11311yescharacteristic
181C2×D4—1132no—
182C2×D4—2262no—
183C2×D4—2264no—
184C2×D4—2268no—
185C2×D4—2262no—
186C2×D4—2264no—
187C2×D4C22⋊D11312yes—
188C2×D4C22×D11311yescharacteristic
189C22×C4—21no—
190C22×C4—2264no—
191C22×C4—2262no—
192C22×C4C22⋊D11312yes—
193C22×C4C22×D11311yescharacteristic
194Q16—2268no—
195SD16—2268no—
196M16—1134no—
197M16—2262no—
198C2×C8—2262no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D11312yes—
204C23—2264no—
205C23—2262no—
206C23C2×C22⋊D11312yes—
207C23C2×C22⋊D11311yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—1134no—
213Q8—2264no—
214Q8—2262no—
215D4—24no—
216D4—22no—
217D4—1134no—
218D4—2264no—
219D4—2268no—
220D4—2264no—
221D4—2264no—
222D4—2262no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—2268no—
229C2×C4—2264no—
230C2×C4—2262no—
231C2×C4—2264no—
232C2×C4—2264no—
233C2×C4D4×D11312yes—
234C2×C4D4×D11312yes—
235C2×C4C2×C22⋊D11312yes—
236C2×C4C2×C22⋊D11311yescharacteristic
237C2×C4C23×D11311yescharacteristic
238C8—2264no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—9042no—
243C22—9044no—
244C22—2264no—
245C22#1681359012yes—
246C22C22×C22⋊D11311yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—2264no—
251C4#1681359011yescharacteristic
252C4#1681359011yescharacteristic
253C4C2×D4×D11312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—4524no—
258C2C2×#1681359011yescharacteristic
subgroupG/H
centerC2C2×#16813590
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#536×C113C2
socleC2×C113C2×#120
prime radical#20299434C2
hypercenter#536D113
hypocenterC113#3022
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#16813653C2semidirect
3. #20298780C2semidirect
4. #20298775C2semidirect
5. #20298616C2semidirect
6. #20298582C2semidirect
7. #536×C113C2semidirect
17. #16813653C22semidirect
20. C2×#14268631C22semidirect
21. #16813550C22semidirect
22. #16813541C22semidirect
23. #16813539C22semidirect
25. C22×Q8×C113C22semidirect
26. C4⋊Q8×C113C22semidirect
27. #124×C113C22semidirect
30. #117×C113C22semidirect
130. C113#3022semidirect
133. #536D113semidirect
147. C22×Q8C2×D113semidirect
153. C4⋊Q8C2×D113semidirect
154. #124C2×D113semidirect
157. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202994342
C2×#168136532
#202987802
#202987752
#202986162
#202985822
#536×C1132
#3022113