Finite Group Databasebeta

← #20284093

Subgroups — #20284093

Order 6784.

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#16811459C211yesmaximal, characteristic
2C2×#13133248C212yesmaximal, complemented
3#16810805C214yesmaximal, complemented
4#16810800C214yesmaximal, complemented
5#16810641C212yesmaximal, complemented
6#16810607C211yesmaximal, characteristic, complemented
7C53×#536C211yesmaximal, characteristic, complemented
8#13133265C2212yes—
9#13133264—22no—
10#13133263—22no—
11#13133262—22no—
12#13133262C2212yes—
13C2×#11719945—22no—
14C2×#11719945C2212yes—
15#41×D53—24no—
16#13133248—24no—
17#13133248C2218yescomplemented
18C2×#11719925—22no—
19C2×#11719924—22no—
20C2×#11719917C2212yescomplemented
21#13133145C2218yescomplemented
22#13133136C2212yescomplemented
23#13133134C2212yescomplemented
24C2×#41×C53C2211yescharacteristic
25C22×Q8×C53C2211yescharacteristic, complemented
26C4⋊Q8×C53C2212yescomplemented
27#124×C53C2212yescomplemented
28#123×C53—22no—
29C22⋊Q8×C53—22no—
30#117×C53C2211yescharacteristic, complemented
31#41×C53—22no—
32#41×C53—22no—
33C2×Q8×C53—21no—
34C2×Q8×C53—22no—
35C2×Q8×C53—22no—
36C2×Q8×C53C2312yes—
37C2×Q8×C53C2311yescharacteristic
38C2×D4×C53D412yes—
39C2×D4×C53C2311yescharacteristic
40C22×C4×C53—21no—
41C22×C4×C53D412yes—
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—24no—
44C2×C22⋊D53—22no—
45#11719949—24no—
46Q8×D53—24no—
47#11719947—28no—
48#11719947—24no—
49D4×D53—28no—
50D4×D53—24no—
51#11719945—24no—
52#11719945—24no—
53#11719945—22no—
54#11719945—24no—
55#11719945—22no—
56#11719945D414yes—
57#11719945C2314yes—
58C2×D212—22no—
59C2×D212D412yes—
60C2×C4×D53—24no—
61C2×C4×D53—22no—
62C2×Dic106—22no—
63C2×Dic106D412yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53—24no—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—24no—
68C42×C53C2312yes—
69#11719925—28no—
70#11719924—28no—
71#11719917—22no—
72#11719917C2314yes—
73C2×#88322—22no—
74C23×C53C2×D412yes—
75C23×C53C2×D411yescharacteristic
76C22×D53—24no—
77C22×D53—22no—
78Q8×C53—22no—
79Q8×C53—24no—
80Q8×C53—41no—
81Q8×C53—41no—
82D4×C53—24no—
83D4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2411yescharacteristic
94C22⋊D53—28no—
95C22⋊D53—24no—
96C22⋊D53—24no—
97C2×Dic53—24no—
98C2×Dic53—22no—
99D212—24no—
100D212—22no—
101D212C2×D414yes—
102C4×D53—28no—
103C4×D53—24no—
104C4×D53—24no—
105Dic106—24no—
106Dic106—22no—
107Dic106C2×D414yes—
108#88322—24no—
109C22×C53—22no—
110C22×C53—41no—
111C22×C53—42no—
112C22×C53#12012yes—
113C22×C53C22×D411yescharacteristic
114C2×D53—24no—
115C2×D53—82no—
116C2×D53—84no—
117C4×C53—22no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53#12011yescharacteristic
121C4×C53#12011yescharacteristic
122C4×C53C22×D412yes—
123Dic53—24no—
124#3022—531nomaximal, Sylow
125C2×C53—21no—
126C2×C53—22no—
127C2×C53—42no—
128C2×C53C2×#12011yescharacteristic
129D53—44no—
130#585—531no—
131C2×#137—532no—
132#536D5311yescharacteristic, complemented
133#456—534no—
134#455—534no—
135#421—532no—
136#413—531no—
137C53#302211yescharacteristic, complemented, Sylow
1382−1+4—532no—
1392−1+4—1062no—
140H(5,𝔽2)—532no—
141H(5,𝔽2)—1062no—
142H(5,𝔽2)—1062no—
143C2×#41—532no—
144C2×#41—1064no—
145C2×#41—1062no—
146C2×#41C2×D5311yescharacteristic
147C22×Q8C2×D5311yescharacteristic, complemented
148#137—538no—
149#137—1064no—
150C2×Q16—1062no—
151C2×SD16—1062no—
152C2×M16—532no—
153C4⋊Q8C2×D5312yescomplemented
154#124C2×D5312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D5311yescharacteristic, complemented
158#104—538no—
159#101—532no—
160#100—532no—
161#41—22no—
162#41—22no—
163#41—534no—
164#41—534no—
165#41—1064no—
166#41—1064no—
167#41—1064no—
168#41—1068no—
169#41—1064no—
170#41—1062no—
171#41—1062no—
172#41—1064no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—532no—
177C2×Q8—1062no—
178C2×Q8—1064no—
179C2×Q8C22×D5312yes—
180C2×Q8C22×D5311yescharacteristic
181C2×D4—532no—
182C2×D4—1062no—
183C2×D4—1064no—
184C2×D4—1068no—
185C2×D4—1062no—
186C2×D4—1064no—
187C2×D4C22⋊D5312yes—
188C2×D4C22×D5311yescharacteristic
189C22×C4—21no—
190C22×C4—1064no—
191C22×C4—1062no—
192C22×C4C22⋊D5312yes—
193C22×C4C22×D5311yescharacteristic
194Q16—1068no—
195SD16—1068no—
196M16—534no—
197M16—1062no—
198C2×C8—1062no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D5312yes—
204C23—1064no—
205C23—1062no—
206C23C2×C22⋊D5312yes—
207C23C2×C22⋊D5311yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—534no—
213Q8—1064no—
214Q8—1062no—
215D4—24no—
216D4—22no—
217D4—534no—
218D4—1064no—
219D4—1068no—
220D4—1064no—
221D4—1064no—
222D4—1062no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1068no—
229C2×C4—1064no—
230C2×C4—1062no—
231C2×C4—1064no—
232C2×C4—1064no—
233C2×C4D4×D5312yes—
234C2×C4D4×D5312yes—
235C2×C4C2×C22⋊D5312yes—
236C2×C4C2×C22⋊D5311yescharacteristic
237C2×C4C23×D5311yescharacteristic
238C8—1064no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—1064no—
243C22—4242no—
244C22—4244no—
245C22#1313318512yes—
246C22C22×C22⋊D5311yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—1064no—
251C4#1313318511yescharacteristic
252C4#1313318511yescharacteristic
253C4C2×D4×D5312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—2124no—
258C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#536C2
socleC2×C53C2×#120
hypercenter#536D53
hypocenterC53#3022
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. #16810805C2semidirect
4. #16810800C2semidirect
5. #16810641C2semidirect
6. #16810607C2semidirect
7. C53×#536C2semidirect
17. #13133248C22semidirect
20. C2×#11719917C22semidirect
21. #13133145C22semidirect
22. #13133136C22semidirect
23. #13133134C22semidirect
25. C22×Q8×C53C22semidirect
26. C4⋊Q8×C53C22semidirect
27. #124×C53C22semidirect
30. #117×C53C22semidirect
132. #536D53semidirect
137. C53#3022semidirect
147. C22×Q8C2×D53semidirect
153. C4⋊Q8C2×D53semidirect
154. #124C2×D53semidirect
157. #117C2×D53semidirect