Finite Group Databasebeta

← #20284071

Subgroups — #20284071

Order 6784.

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

normal subgroups only

This group has 762 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
1C2×#13133262C211yesmaximal, characteristic, complemented
2#16811421C212yesmaximal, complemented
3#16810800C214yesmaximal, complemented
4#16810790C214yesmaximal, complemented
5#16810641C212yesmaximal, complemented
6C2×#13133134C211yesmaximal, characteristic, complemented
7C53×#535C211yesmaximal, characteristic, complemented
8#13133262—22no—
9#13133262—24no—
10#13133262C2214yes—
11C2×#11719945C2212yes—
12C2×#11719947—22no—
13C22×C22⋊D53—22no—
14C2×D4×D53—22no—
15#13133248C2214yescomplemented
16#13133247—22no—
17#13133243C2214yescomplemented
18#13133242—22no—
19#13133237—24no—
20C2×#11719917C2212yescomplemented
21#13133145C2218yescomplemented
22#13133134C2214yescomplemented
23C2×#41×C53C2211yescharacteristic, complemented
24C22×D4×C53C2211yescharacteristic
25#127×C53C2212yescomplemented
26#124×C53C2212yescomplemented
27#121×C53—22no—
28#120×C53—22no—
29#117×C53C2211yescharacteristic, complemented
30C24×C53D412yes—
31C23×D53—22no—
32#41×C53—22no—
33#41×C53—22no—
34C2×Q8×C53C2311yescharacteristic, complemented
35C2×D4×C53—21no—
36C2×D4×C53—24no—
37C2×D4×C53—22no—
38C2×D4×C53—22no—
39C2×D4×C53D412yes—
40C2×D4×C53C2312yes—
41C2×D4×C53C2311yescharacteristic, complemented
42C22×C4×C53—21no—
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—24no—
45C2×C22⋊D53—28no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—28no—
48C22×Dic53—22no—
49#11719947—28no—
50#11719947—28no—
51D4×D53—28no—
52D4×D53—28no—
53#11719945—22no—
54#11719945—22no—
55#11719945D414yescomplemented
56#11719945C2314yes—
57C2×D212D412yescomplemented
58C2×C4×D53—22no—
59C2×Dic106D412yescomplemented
60C4⋊C4×C53—21no—
61C22⋊C4×C53—21no—
62C22⋊C4×C53—24no—
63C42×C53C2312yescomplemented
64#11719925—24no—
65#11719924—24no—
66#11719923—24no—
67#11719922—24no—
68#11719917—22no—
69#11719917C2314yescomplemented
70C2×#88322—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—21no—
74C23×C53—82no—
75C23×C53C2×D414yes—
76C23×C53C2×D411yescharacteristic
77C22×D53—22no—
78C22×D53—28no—
79C22×D53—84no—
80Q8×C53—22no—
81D4×C53—24no—
82D4×C53—24no—
83D4×C53—22no—
84D4×C53—41no—
85D4×C53—41no—
86D4×C53—84no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2411yescharacteristic
94C22⋊D53—216no—
95C22⋊D53—24no—
96C22⋊D53—24no—
97C2×Dic53—28no—
98C2×Dic53—22no—
99D212—22no—
100D212C2×D414yes—
101C4×D53—24no—
102C4×D53—24no—
103Dic106—22no—
104Dic106C2×D414yes—
105#88322—24no—
106C22×C53—24no—
107C22×C53—41no—
108C22×C53—44no—
109C22×C53—44no—
110C22×C53—42no—
111C22×C53—41no—
112C22×C53—82no—
113C22×C53#12011yescharacteristic
114C22×C53#12011yescharacteristic
115C22×C53C22×D411yescharacteristic
116C2×D53—24no—
117C2×D53—44no—
118C2×D53—88no—
119C4×C53—42no—
120C4×C53—41no—
121C4×C53#12012yes—
122C4×C53C22×D412yes—
123Dic53—24no—
124#3019—531nomaximal, Sylow
125C2×C53—21no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—44no—
129C2×C53—81no—
130C2×C53C2×#12011yescharacteristic
131D53—44no—
132C2×H(5,𝔽2)—531no—
133#575—532no—
134#535D5311yescharacteristic, complemented
135#455—534no—
136#453—534no—
137#421—532no—
138C2×#100—531no—
139C53#301911yescharacteristic, complemented, Sylow
140H(5,𝔽2)—534no—
141H(5,𝔽2)—1064no—
142H(5,𝔽2)—1062no—
143C2×#41—532no—
144C2×#41—1062no—
145C2×#41C2×D5311yescharacteristic, complemented
146C22×D4—1062no—
147C22×D4—1062no—
148C22×D4C2×D5311yescharacteristic
149#137—534no—
150#137—1062no—
151AGL(1,ℤ/8)—534no—
152AGL(1,ℤ/8)—1062no—
153#135—1064no—
154C2×M16—532no—
155#127C2×D5312yescomplemented
156#124C2×D5312yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D5311yescharacteristic, complemented
160#104—538no—
161#100—534no—
162C24—1062no—
163C24C22⋊D5312yes—
164#41—22no—
165#41—22no—
166#41—534no—
167#41—534no—
168#41—1068no—
169#41—1068no—
170#41—1062no—
171#41—1062no—
172C2×Q8—532no—
173C2×Q8C22×D5311yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—532no—
179C2×D4—1068no—
180C2×D4—1068no—
181C2×D4—1064no—
182C2×D4—1068no—
183C2×D4—1062no—
184C2×D4—1068no—
185C2×D4C22⋊D5312yes—
186C2×D4C22×D5312yes—
187C2×D4C22×D5311yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4C22×D5311yescharacteristic
192Q16—1064no—
193SD16—1064no—
194SD16—1064no—
195D8—1064no—
196M16—534no—
197M16—1062no—
198C2×C8—1062no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D5312yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—1068no—
208C23—1062no—
209C23—4244no—
210C23C2×C22⋊D5314yes—
211C23C2×C22⋊D5311yescharacteristic
212Q8—22no—
213Q8—534no—
214Q8—1062no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—534no—
222D4—10616no—
223D4—1064no—
224D4—1064no—
225D4—1062no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—1068no—
230C2×C4—1062no—
231C2×C4—1064no—
232C2×C4—1064no—
233C2×C4D4×D5312yes—
234C2×C4D4×D5312yes—
235C2×C4C2×C22⋊D5311yescharacteristic
236C2×C4C23×D5311yescharacteristic
237C8—1064no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—1064no—
246C22—4248no—
247C22—2124no—
248C22#1313318511yescharacteristic
249C22#1313318511yescharacteristic
250C22C22×C22⋊D5311yescharacteristic
251C4—42no—
252C4—41no—
253C4—1064no—
254C4#1313318512yes—
255C4C2×D4×D5312yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—2124no—
262C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC2×C53C2×#120
hypercenter#535D53
hypocenterC53#3019
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. C2×#13133262C2semidirect
2. #16811421C2semidirect
3. #16810800C2semidirect
4. #16810790C2semidirect
5. #16810641C2semidirect
6. C2×#13133134C2semidirect
7. C53×#535C2semidirect
15. #13133248C22semidirect
17. #13133243C22semidirect
20. C2×#11719917C22semidirect
21. #13133145C22semidirect
22. #13133134C22semidirect
23. C2×#41×C53C22semidirect
25. #127×C53C22semidirect
26. #124×C53C22semidirect
29. #117×C53C22semidirect
34. C2×Q8×C53C23semidirect
41. C2×D4×C53C23semidirect
55. #11719945D4semidirect
57. C2×D212D4semidirect
59. C2×Dic106D4semidirect
63. C42×C53C23semidirect
69. #11719917C23semidirect
134. #535D53semidirect
139. C53#3019semidirect
145. C2×#41C2×D53semidirect
155. #127C2×D53semidirect
156. #124C2×D53semidirect
159. #117C2×D53semidirect
173. C2×Q8C22×D53semidirect
187. C2×D4C22×D53semidirect
202. C42C22×D53semidirect