Finite Group Databasebeta

← #20284086

Subgroups — #20284086

Order 6784.

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

normal subgroups only

This group has 736 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, complemented
2C2×#13133243C212yesmaximal, complemented
3#16810795C214yesmaximal, complemented
4#16810790C214yesmaximal, complemented
5#16810641C212yesmaximal, complemented
6#16810607C211yesmaximal, characteristic, complemented
7C53×#535C211yesmaximal, characteristic, complemented
8#13133265C2212yescomplemented
9#13133264—22no—
10#13133263—22no—
11#13133262—22no—
12#13133262C2212yescomplemented
13C2×#11719945—22no—
14C2×#11719945C2212yes—
15#41×D53—24no—
16#13133243—24no—
17#13133243C2218yescomplemented
18C2×#11719923—22no—
19C2×#11719922—22no—
20C2×#11719917C2212yescomplemented
21#13133145C2218yescomplemented
22#13133136C2212yescomplemented
23#13133134C2212yescomplemented
24C2×#41×C53C2211yescharacteristic
25C22×D4×C53C2211yescharacteristic, complemented
26#127×C53C2212yescomplemented
27#124×C53C2212yescomplemented
28#121×C53—22no—
29#120×C53—22no—
30#117×C53C2211yescharacteristic, complemented
31C24×C53—21no—
32#41×C53—22no—
33#41×C53—22no—
34C2×Q8×C53C2311yescharacteristic
35C2×D4×C53—24no—
36C2×D4×C53—22no—
37C2×D4×C53—22no—
38C2×D4×C53—21no—
39C2×D4×C53D412yes—
40C2×D4×C53C2312yes—
41C2×D4×C53C2311yescharacteristic
42C22×C4×C53D412yes—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—24no—
45C2×C22⋊D53—22no—
46#11719949—24no—
47Q8×D53—24no—
48#11719947—28no—
49#11719947—24no—
50D4×D53—28no—
51D4×D53—24no—
52#11719945—24no—
53#11719945—24no—
54#11719945—22no—
55#11719945—24no—
56#11719945—22no—
57#11719945D414yescomplemented
58#11719945C2314yes—
59C2×D212—22no—
60C2×D212D412yescomplemented
61C2×C4×D53—24no—
62C2×C4×D53—22no—
63C2×Dic106—22no—
64C2×Dic106D412yescomplemented
65C4⋊C4×C53—21no—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—24no—
68C42×C53C2312yes—
69#11719923—28no—
70#11719922—28no—
71#11719917—22no—
72#11719917C2314yes—
73C2×#88322—22no—
74C23×C53—22no—
75C23×C53—42no—
76C23×C53—82no—
77C23×C53C2×D412yes—
78C23×C53C2×D411yescharacteristic
79C22×D53—24no—
80C22×D53—22no—
81Q8×C53—22no—
82D4×C53—24no—
83D4×C53—22no—
84D4×C53—24no—
85D4×C53—41no—
86D4×C53—41no—
87D4×C53—84no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D412yes—
93C2×C4×C53C2×D412yes—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2411yescharacteristic
96C22⋊D53—28no—
97C22⋊D53—24no—
98C22⋊D53—24no—
99C2×Dic53—24no—
100C2×Dic53—22no—
101D212—24no—
102D212—22no—
103D212C2×D414yes—
104C4×D53—28no—
105C4×D53—24no—
106C4×D53—24no—
107Dic106—24no—
108Dic106—22no—
109Dic106C2×D414yes—
110#88322—24no—
111C22×C53—22no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53—44no—
115C22×C53—42no—
116C22×C53—84no—
117C22×C53#12012yes—
118C22×C53C22×D411yescharacteristic
119C2×D53—24no—
120C2×D53—82no—
121C2×D53—84no—
122C4×C53—22no—
123C4×C53—42no—
124C4×C53#12011yescharacteristic
125C4×C53#12011yescharacteristic
126C4×C53C22×D412yes—
127Dic53—24no—
128#3021—531nomaximal, Sylow
129C2×C53—21no—
130C2×C53—22no—
131C2×C53—42no—
132C2×C53—82no—
133C2×C53C2×#12011yescharacteristic
134D53—44no—
135#585—531no—
136C2×AGL(1,ℤ/8)—532no—
137#535D5311yescharacteristic, complemented
138#454—534no—
139#453—534no—
140#421—532no—
141#413—531no—
142C53#302111yescharacteristic, complemented, Sylow
1432−1+4—532no—
1442−1+4—1062no—
145H(5,𝔽2)—532no—
146H(5,𝔽2)—1062no—
147H(5,𝔽2)—1062no—
148C2×#41—532no—
149C2×#41—1064no—
150C2×#41—1062no—
151C2×#41C2×D5311yescharacteristic
152C22×D4C2×D5311yescharacteristic, complemented
153AGL(1,ℤ/8)—538no—
154AGL(1,ℤ/8)—1064no—
155C2×SD16—1062no—
156C2×D8—1062no—
157C2×M16—532no—
158#127C2×D5312yescomplemented
159#124C2×D5312yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D5311yescharacteristic, complemented
163#104—538no—
164#101—532no—
165#100—532no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—534no—
170#41—534no—
171#41—1064no—
172#41—1064no—
173#41—1064no—
174#41—1068no—
175#41—1064no—
176#41—1062no—
177#41—1062no—
178#41—1064no—
179C2×Q8—532no—
180C2×Q8—1062no—
181C2×Q8—1064no—
182C2×Q8C22×D5311yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—532no—
188C2×D4—1062no—
189C2×D4—1064no—
190C2×D4—1068no—
191C2×D4—1062no—
192C2×D4—1064no—
193C2×D4C22⋊D5312yes—
194C2×D4C22×D5312yes—
195C2×D4C22×D5311yescharacteristic
196C22×C4—1064no—
197C22×C4—1062no—
198C22×C4C22⋊D5312yes—
199C22×C4C22×D5311yescharacteristic
200SD16—1068no—
201D8—1068no—
202M16—534no—
203M16—1062no—
204C2×C8—1062no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D5312yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—1064no—
213C23—1062no—
214C23C2×C22⋊D5312yes—
215C23C2×C22⋊D5311yescharacteristic
216Q8—22no—
217Q8—534no—
218Q8—1064no—
219Q8—1062no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—534no—
227D4—1064no—
228D4—1068no—
229D4—1064no—
230D4—1064no—
231D4—1062no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—1068no—
236C2×C4—1064no—
237C2×C4—1062no—
238C2×C4—1064no—
239C2×C4—1064no—
240C2×C4D4×D5312yes—
241C2×C4D4×D5312yes—
242C2×C4C2×C22⋊D5312yes—
243C2×C4C2×C22⋊D5311yescharacteristic
244C2×C4C23×D5311yescharacteristic
245C8—1064no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—1064no—
253C22—4242no—
254C22—4244no—
255C22#1313318512yes—
256C22C22×C22⋊D5311yescharacteristic
257C4—22no—
258C4—42no—
259C4—1064no—
260C4#1313318511yescharacteristic
261C4#1313318511yescharacteristic
262C4C2×D4×D5312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—2124no—
268C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC2×C53C2×#120
hypercenter#535D53
hypocenterC53#3021
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16811459C2semidirect
2. C2×#13133243C2semidirect
3. #16810795C2semidirect
4. #16810790C2semidirect
5. #16810641C2semidirect
6. #16810607C2semidirect
7. C53×#535C2semidirect
8. #13133265C22semidirect
12. #13133262C22semidirect
17. #13133243C22semidirect
20. C2×#11719917C22semidirect
21. #13133145C22semidirect
22. #13133136C22semidirect
23. #13133134C22semidirect
25. C22×D4×C53C22semidirect
26. #127×C53C22semidirect
27. #124×C53C22semidirect
30. #117×C53C22semidirect
57. #11719945D4semidirect
60. C2×D212D4semidirect
64. C2×Dic106D4semidirect
137. #535D53semidirect
142. C53#3021semidirect
152. C22×D4C2×D53semidirect
158. #127C2×D53semidirect
159. #124C2×D53semidirect
162. #117C2×D53semidirect