Finite Group Databasebeta

← #20284116

Subgroups — #20284116

Order 6784.

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

normal subgroups only

This group has 772 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
2#16811190C212yesmaximal, complemented
3#16810809C218yesmaximal, complemented
4#16810597C211yesmaximal, characteristic, complemented
5#16810596C212yesmaximal, complemented
6C53×#535C211yesmaximal, characteristic, complemented
7#13133265—21no—
8#13133264—22no—
9#13133263—22no—
10#13133262—21no—
11#13133262C2214yes—
12C2×#11719945—22no—
13C2×#11719945C2212yes—
14#41×D53—24no—
15#13133213—22no—
16#13133204—22no—
17#13133188—24no—
18#13133186C2218yescomplemented
19#13133176C2212yescomplemented
20#13133132C2214yescomplemented
21#13133131C2218yescomplemented
22C2×#41×C53C2211yescharacteristic
23C22×D4×C53C2211yescharacteristic, complemented
24#127×C53—21no—
25#124×C53—21no—
26#121×C53—22no—
27#120×C53C2214yescomplemented
28#117×C53C2211yescharacteristic, complemented
29C24×C53C2312yes—
30#41×C53—22no—
31#41×C53—22no—
32C2×Q8×C53D411yescharacteristic, complemented
33C2×D4×C53—24no—
34C2×D4×C53—21no—
35C2×D4×C53—21no—
36C2×D4×C53—22no—
37C2×D4×C53—22no—
38C2×D4×C53D411yescharacteristic, complemented
39C2×D4×C53C2312yes—
40C22×C4×C53D412yescomplemented
41C22×C4×C53C2311yescharacteristic
42C2×C22⋊D53—24no—
43C2×C22⋊D53—28no—
44C2×C22⋊D53C2314yes—
45#11719949—24no—
46Q8×D53—24no—
47#11719947—24no—
48#11719947—28no—
49D4×D53—24no—
50D4×D53—28no—
51#11719945—24no—
52#11719945—24no—
53#11719945—22no—
54#11719945—22no—
55#11719945—24no—
56#11719945—22no—
57#11719945—22no—
58C2×D212—22no—
59C2×D212D412yescomplemented
60C2×C4×D53—24no—
61C2×C4×D53D414yescomplemented
62C2×Dic106—22no—
63C2×Dic106D412yescomplemented
64C4⋊C4×C53—21no—
65C22⋊C4×C53—24no—
66C22⋊C4×C53C2312yes—
67C42×C53—21no—
68C22⋊Dic53—28no—
69C22⋊Dic53C2314yes—
70C4⋊Dic53—22no—
71C4×Dic53—22no—
72C23×C53—24no—
73C23×C53—22no—
74C23×C53—82no—
75C23×C53C2×D412yes—
76C23×C53C2411yescharacteristic
77C22×D53—24no—
78C22×D53C2×D414yes—
79Q8×C53—22no—
80D4×C53—24no—
81D4×C53—22no—
82D4×C53—41no—
83D4×C53—42no—
84D4×C53—41no—
85D4×C53—84no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C22⋊D53—28no—
95C22⋊D53—24no—
96C22⋊D53—24no—
97C22⋊D53—88no—
98C2×Dic53—24no—
99C2×Dic53—24no—
100C2×Dic53C2×D414yes—
101D212—24no—
102D212—22no—
103D212—22no—
104C4×D53—28no—
105C4×D53—24no—
106C4×D53—24no—
107Dic106—24no—
108Dic106—22no—
109Dic106—22no—
110C22×C53—22no—
111C22×C53—41no—
112C22×C53—42no—
113C22×C53—48no—
114C22×C53—44no—
115C22×C53—42no—
116C22×C53C22×D412yes—
117C22×C53C22×D411yescharacteristic
118C2×D53—24no—
119C2×D53—44no—
120C2×D53—84no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53#12011yescharacteristic
124C4×C53#12012yes—
125C4×C53#12011yescharacteristic
126Dic53—24no—
127Dic53—44no—
128#3027—531nomaximal, Sylow
129C2×C53—22no—
130C2×C53—21no—
131C2×C53—42no—
132C2×C53—82no—
133C2×C53C2×#12011yescharacteristic
134D53—44no—
135#585—531no—
136#535—532no—
137#535D5311yescharacteristic, complemented
138B(4,𝔽2)—538no—
139#410—531no—
140#410—532no—
141C53#302711yescharacteristic, complemented, Sylow
1422−1+4—1062no—
1432−1+4—1061no—
144H(5,𝔽2)—534no—
145H(5,𝔽2)—1062no—
146H(5,𝔽2)—1061no—
147C2×#41—532no—
148C2×#41—1064no—
149C2×#41—1062no—
150C2×#41C2×D5311yescharacteristic
151C22×D4C2×D5311yescharacteristic, complemented
152#127—21no—
153#127—1062no—
154#124—21no—
155#124—1062no—
156#121—22no—
157#121—1064no—
158#120—538no—
159#120C2×D5314yescomplemented
160#117—532no—
161#117C2×D5311yescharacteristic, complemented
162#99—534no—
163#99—538no—
164C24C22×D5312yes—
165#41—22no—
166#41—22no—
167#41—1064no—
168#41—1064no—
169#41—1068no—
170#41—1064no—
171#41—1064no—
172#41—1062no—
173#41—1062no—
174#41—1062no—
175#41—1062no—
176#41—1064no—
177C2×Q8—532no—
178C2×Q8—1062no—
179C2×Q8—1064no—
180C2×Q8C22⋊D5311yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—534no—
187C2×D4—532no—
188C2×D4—1062no—
189C2×D4—1068no—
190C2×D4—1064no—
191C2×D4—1064no—
192C2×D4—1068no—
193C2×D4C22⋊D5311yescharacteristic, complemented
194C2×D4C22×D5312yes—
195C22×C4—534no—
196C22×C4—1064no—
197C22×C4C22⋊D5312yescomplemented
198C22×C4C22×D5311yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—1062no—
201C22⋊C4—24no—
202C22⋊C4—534no—
203C22⋊C4—1068no—
204C22⋊C4C22×D5312yes—
205C42—21no—
206C42—1062no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—534no—
211C23—1064no—
212C23C2×C22⋊D5312yes—
213C23C23×D5311yescharacteristic
214Q8—22no—
215Q8—1064no—
216Q8—1062no—
217Q8—1062no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—1064no—
225D4—1068no—
226D4—1064no—
227D4—1064no—
228D4—1062no—
229D4—1062no—
230D4—4248no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—534no—
235C2×C4—1068no—
236C2×C4—1064no—
237C2×C4—1064no—
238C2×C4—1064no—
239C2×C4—1064no—
240C2×C4D4×D5312yes—
241C2×C4D4×D5312yes—
242C2×C4C2×C22⋊D5312yes—
243C2×C4C2×C22⋊D5311yescharacteristic
244C2×C4C2×C22⋊D5311yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—1064no—
252C22—4244no—
253C22—2124no—
254C22C2×D4×D5312yes—
255C22C22×C22⋊D5311yescharacteristic
256C4—22no—
257C4—42no—
258C4—1064no—
259C4—2124no—
260C4#1313318511yescharacteristic
261C4#1313318511yescharacteristic
262C4#1313318512yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—2124no—
268C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC23×C53C24
FrattiniC23C23×D53
FittingC53×#535C2
socleC2×C53C2×#120
hypercenter#535D53
hypocenterC53#3027
LCSG ▷ C23×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. #16811190C2semidirect
3. #16810809C2semidirect
4. #16810597C2semidirect
5. #16810596C2semidirect
6. C53×#535C2semidirect
18. #13133186C22semidirect
19. #13133176C22semidirect
20. #13133132C22semidirect
21. #13133131C22semidirect
23. C22×D4×C53C22semidirect
27. #120×C53C22semidirect
28. #117×C53C22semidirect
32. C2×Q8×C53D4semidirect
38. C2×D4×C53D4semidirect
40. C22×C4×C53D4semidirect
59. C2×D212D4semidirect
61. C2×C4×D53D4semidirect
63. C2×Dic106D4semidirect
137. #535D53semidirect
141. C53#3027semidirect
151. C22×D4C2×D53semidirect
159. #120C2×D53semidirect
161. #117C2×D53semidirect
180. C2×Q8C22⋊D53semidirect
193. C2×D4C22⋊D53semidirect
197. C22×C4C22⋊D53semidirect