Finite Group Databasebeta

← #20284089

Subgroups — #20284089

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×#13133245C212yesmaximal, complemented
3#16811187C211yesmaximal, characteristic, complemented
4#16810798C214yesmaximal, complemented
5#16810793C214yesmaximal, complemented
6#16810639C212yesmaximal, complemented
7C53×#413C211yesmaximal, characteristic, complemented
8#13133265C2212yescomplemented
9#13133264—22no—
10#13133263—22no—
11#13133262—22no—
12#13133262C2212yescomplemented
13C2×#11719945—22no—
14C2×#11719945C2212yes—
15#41×D53—24no—
16C22×D212C2211yescharacteristic, complemented
17#13133245—24no—
18#13133245C2218yescomplemented
19C2×#11719913—22no—
20C2×D424—22no—
21#13133213C2212yescomplemented
22#13133206C2212yescomplemented
23#13133191—22no—
24#13133185—22no—
25#13133176C2211yescharacteristic, complemented
26#13133143C2218yescomplemented
27C2×#41×C53C2211yescharacteristic
28C2×M16×C53C2212yescomplemented
29#101×C53C2212yescomplemented
30#100×C53C2212yescomplemented
31C23×D53—21no—
32#41×C53—22no—
33#41×C53—22no—
34C2×Q8×C53C2311yescharacteristic
35C2×D4×C53D412yes—
36C2×D4×C53C2311yescharacteristic
37C22×C4×C53D412yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—24no—
40C2×C22⋊D53—22no—
41C2×C22⋊D53—24no—
42#11719949—24no—
43Q8×D53—24no—
44#11719947—28no—
45#11719947—24no—
46D4×D53—28no—
47D4×D53—24no—
48#11719945—24no—
49#11719945—24no—
50#11719945—22no—
51#11719945—24no—
52#11719945—22no—
53#11719945D414yescomplemented
54#11719945C2314yes—
55C2×D212—22no—
56C2×D212—22no—
57C2×D212—22no—
58C2×D212—21no—
59C2×D212D412yescomplemented
60C2×D212C2312yes—
61C2×C4×D53—24no—
62C2×C4×D53—22no—
63C2×Dic106—22no—
64C2×Dic106D412yescomplemented
65M16×C53—22no—
66M16×C53C2314yes—
67C2×C8×C53—22no—
68C22⋊Dic53—21no—
69#11719921—24no—
70C4⋊Dic53—21no—
71C4×Dic53C2312yes—
72D424—28no—
73#11719913—28no—
74C23×C53C2×D412yes—
75C23×C53C2×D411yescharacteristic
76C22×D53—24no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53—42no—
80C22×D53—82no—
81Q8×C53—22no—
82D4×C53—24no—
83D4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53C2×D412yes—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2411yescharacteristic
91C22⋊D53—28no—
92C22⋊D53—24no—
93C22⋊D53—24no—
94C22⋊D53—84no—
95C2×Dic53—24no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98D212—24no—
99D212—22no—
100D212—24no—
101D212—41no—
102D212—41no—
103D212C2×D414yes—
104C4×D53—28no—
105C4×D53—24no—
106C4×D53—24no—
107Dic106—24no—
108Dic106—22no—
109Dic106C2×D414yes—
110C8×C53—24no—
111C22×C53—22no—
112C22×C53—41no—
113C22×C53—42no—
114C22×C53#12012yes—
115C22×C53C22×D411yescharacteristic
116C2×D53—24no—
117C2×D53—44no—
118C2×D53—42no—
119C2×D53—82no—
120C2×D53—84no—
121C2×D53—84no—
122C4×C53—22no—
123C4×C53#12011yescharacteristic
124C4×C53#12011yescharacteristic
125C4×C53C22×D412yes—
126Dic53—24no—
127Dic53—42no—
128#3021—531nomaximal, Sylow
129C2×C53—21no—
130C2×C53—22no—
131C2×C53—42no—
132C2×C53C2×#12011yescharacteristic
133D53—44no—
134D53—82no—
135#585—531no—
136C2×AGL(1,ℤ/8)—532no—
137#535—531no—
138#454—534no—
139#453—534no—
140#421—532no—
141#413D5311yescharacteristic, complemented
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×D4—531no—
153AGL(1,ℤ/8)—538no—
154AGL(1,ℤ/8)—1064no—
155C2×SD16—1062no—
156C2×D8—1062no—
157C2×M16C2×D5312yescomplemented
158#127—532no—
159#124—532no—
160#121—1062no—
161#120—1062no—
162#117—531no—
163#104—538no—
164#101C2×D5312yescomplemented
165#100C2×D5312yescomplemented
166C24—1061no—
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—1064no—
178#41—1062no—
179C2×Q8—532no—
180C2×Q8—1062no—
181C2×Q8—1064no—
182C2×Q8C22×D5311yescharacteristic
183C2×D4—532no—
184C2×D4—532no—
185C2×D4—1062no—
186C2×D4—1064no—
187C2×D4—1068no—
188C2×D4—1062no—
189C2×D4—1064no—
190C2×D4—1064no—
191C2×D4—1061no—
192C2×D4—1062no—
193C2×D4—1062no—
194C2×D4D21212yes—
195C2×D4C22×D5311yescharacteristic
196C22×C4—1064no—
197C22×C4—1062no—
198C22×C4D21212yes—
199C22×C4C22×D5311yescharacteristic
200SD16—1068no—
201D8—1068no—
202M16—22no—
203M16C22×D5314yes—
204C2×C8—22no—
205C4⋊C4—1061no—
206C22⋊C4—1061no—
207C22⋊C4—1064no—
208C42—532no—
209C23—1064no—
210C23—1062no—
211C23—1062no—
212C23—4242no—
213C23—2122no—
214C23C2×D21212yes—
215C23C2×D21211yescharacteristic
216Q8—22no—
217Q8—534no—
218Q8—1064no—
219Q8—1062no—
220D4—24no—
221D4—22no—
222D4—534no—
223D4—1064no—
224D4—1068no—
225D4—1064no—
226D4—1064no—
227D4—1062no—
228D4—1064no—
229D4—4244no—
230D4—2121no—
231D4—2121no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—1068no—
235C2×C4—1064no—
236C2×C4—1062no—
237C2×C4—1064no—
238C2×C4—1064no—
239C2×C4—1062no—
240C2×C4C2×D21212yes—
241C2×C4C2×D21211yescharacteristic
242C2×C4D4×D5312yes—
243C2×C4D4×D5312yes—
244C2×C4C23×D5311yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—1064no—
250C22—4244no—
251C22—4242no—
252C22—4244no—
253C22—2124no—
254C22—2122no—
255C22#1313318712yes—
256C22C22×D21211yescharacteristic
257C4—22no—
258C4—1064no—
259C4—2122no—
260C4#1313318711yescharacteristic
261C4#1313318711yescharacteristic
262C4C2×D4×D5312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—4242no—
267C2—2124no—
268C2C2×#1313318711yescharacteristic
subgroupG/H
centerC2C2×#13133187
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#413C2
socleC2×C53C2×#120
hypercenter#413D53
hypocenterC53#3021
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #16811459C2semidirect
2. C2×#13133245C2semidirect
3. #16811187C2semidirect
4. #16810798C2semidirect
5. #16810793C2semidirect
6. #16810639C2semidirect
7. C53×#413C2semidirect
8. #13133265C22semidirect
12. #13133262C22semidirect
16. C22×D212C22semidirect
18. #13133245C22semidirect
21. #13133213C22semidirect
22. #13133206C22semidirect
25. #13133176C22semidirect
26. #13133143C22semidirect
28. C2×M16×C53C22semidirect
29. #101×C53C22semidirect
30. #100×C53C22semidirect
53. #11719945D4semidirect
59. C2×D212D4semidirect
64. C2×Dic106D4semidirect
141. #413D53semidirect
142. C53#3021semidirect
157. C2×M16C2×D53semidirect
164. #101C2×D53semidirect
165. #100C2×D53semidirect