Finite Group Databasebeta

← #20284118

Subgroups — #20284118

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#16811192C212yesmaximal, complemented
3#16811187C211yesmaximal, characteristic, complemented
4#16810811C218yesmaximal, complemented
5#16810596C212yesmaximal, complemented
6C53×#410C211yesmaximal, 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—
15C22×D212C2211yescharacteristic, complemented
16#13133214—22no—
17#13133213—21no—
18#13133206—21no—
19#13133205—22no—
20#13133192—24no—
21#13133191—22no—
22#13133187C2218yescomplemented
23#13133185C2214yescomplemented
24#13133176C2211yescharacteristic, complemented
25#13133131C2218yescomplemented
26C2×#41×C53C2211yescharacteristic
27#117×C53C2212yescomplemented
28#99×C53C2214yescomplemented
29C23×D53C2312yes—
30#41×C53—22no—
31#41×C53—22no—
32C2×Q8×C53D411yescharacteristic, complemented
33C2×D4×C53D411yescharacteristic, complemented
34C2×D4×C53C2312yes—
35C22×C4×C53D412yescomplemented
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—24no—
38C2×C22⋊D53—24no—
39C2×C22⋊D53C2314yes—
40#11719949—24no—
41Q8×D53—24no—
42#11719947—24no—
43#11719947—28no—
44D4×D53—24no—
45D4×D53—28no—
46#11719945—24no—
47#11719945—24no—
48#11719945—22no—
49#11719945—22no—
50#11719945—24no—
51#11719945—22no—
52#11719945—22no—
53C2×D212—22no—
54C2×D212—21no—
55C2×D212—22no—
56C2×D212—22no—
57C2×D212—21no—
58C2×D212—28no—
59C2×D212D412yescomplemented
60C2×C4×D53—24no—
61C2×C4×D53D414yescomplemented
62C2×Dic106—22no—
63C2×Dic106D412yescomplemented
64C4⋊C4×C53—22no—
65C22⋊C4×C53C2314yes—
66C42×C53—22no—
67C22⋊Dic53C2312yes—
68#11719921—24no—
69#11719921—28no—
70C4⋊Dic53—21no—
71C4×Dic53—21no—
72C23×C53C2×D412yes—
73C23×C53C2411yescharacteristic
74C22×D53—24no—
75C22×D53—24no—
76C22×D53—22no—
77C22×D53—82no—
78C22×D53C2×D414yes—
79Q8×C53—22no—
80D4×C53—24no—
81D4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D412yes—
90C22⋊D53—28no—
91C22⋊D53—24no—
92C22⋊D53—24no—
93C22⋊D53—84no—
94C2×Dic53—24no—
95C2×Dic53—22no—
96C2×Dic53C2×D414yes—
97D212—24no—
98D212—22no—
99D212—22no—
100D212—41no—
101D212—42no—
102D212—41no—
103D212—88no—
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×C53C22×D412yes—
114C22×C53C22×D411yescharacteristic
115C2×D53—24no—
116C2×D53—44no—
117C2×D53—48no—
118C2×D53—44no—
119C2×D53—42no—
120C2×D53—84no—
121C4×C53—22no—
122C4×C53—44no—
123C4×C53#12011yescharacteristic
124C4×C53#12012yes—
125C4×C53#12011yescharacteristic
126Dic53—24no—
127Dic53—42no—
128#3027—531nomaximal, Sylow
129C2×C53—21no—
130C2×C53—22no—
131C2×C53—42no—
132C2×C53C2×#12011yescharacteristic
133D53—44no—
134D53—82no—
135#585—531no—
136#535—531no—
137#535—532no—
138B(4,𝔽2)—538no—
139#410—532no—
140#410D5311yescharacteristic, complemented
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×D4—531no—
152#127—1062no—
153#127—1061no—
154#124—1062no—
155#124—1061no—
156#121—1064no—
157#121—1062no—
158#120—534no—
159#120—538no—
160#117—531no—
161#117C2×D5312yescomplemented
162#99—538no—
163#99C2×D5314yescomplemented
164C24—532no—
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—1064no—
175#41—1062no—
176#41—1062no—
177C2×Q8—532no—
178C2×Q8—1062no—
179C2×Q8—1064no—
180C2×Q8D21211yescharacteristic, complemented
181C2×D4—534no—
182C2×D4—532no—
183C2×D4—1062no—
184C2×D4—1068no—
185C2×D4—1068no—
186C2×D4—1064no—
187C2×D4—1064no—
188C2×D4—1064no—
189C2×D4—1061no—
190C2×D4—1061no—
191C2×D4—1062no—
192C2×D4—1062no—
193C2×D4D21211yescharacteristic, complemented
194C2×D4C22×D5312yes—
195C22×C4—534no—
196C22×C4—1064no—
197C22×C4D21212yescomplemented
198C22×C4C22×D5311yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—1061no—
201C22⋊C4—532no—
202C22⋊C4—1068no—
203C22⋊C4—1064no—
204C22⋊C4C22×D5314yes—
205C42—22no—
206C42—1061no—
207C23—534no—
208C23—1064no—
209C23—1064no—
210C23—1062no—
211C23—4242no—
212C23C2×D21212yes—
213C23C23×D5311yescharacteristic
214Q8—22no—
215Q8—1064no—
216Q8—1062no—
217Q8—1062no—
218D4—24no—
219D4—22no—
220D4—1064no—
221D4—1068no—
222D4—1064no—
223D4—1064no—
224D4—1062no—
225D4—1062no—
226D4—4248no—
227D4—4244no—
228D4—2121no—
229D4—2121no—
230D4—2122no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—534no—
235C2×C4—1068no—
236C2×C4—1064no—
237C2×C4—1064no—
238C2×C4—1064no—
239C2×C4—1062no—
240C2×C4C2×D21212yes—
241C2×C4C2×D21211yescharacteristic
242C2×C4C2×D21211yescharacteristic
243C2×C4D4×D5312yes—
244C2×C4D4×D5312yes—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—1064no—
249C22—4244no—
250C22—2128no—
251C22—2124no—
252C22—2124no—
253C22—2122no—
254C22C2×D4×D5312yes—
255C22C22×D21211yescharacteristic
256C4—22no—
257C4—44no—
258C4—1064no—
259C4—2122no—
260C4#1313318711yescharacteristic
261C4#1313318711yescharacteristic
262C4#1313318712yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—4242no—
267C2—2124no—
268C2C2×#1313318711yescharacteristic
subgroupG/H
centerC2C2×#13133187
commutatorC23×C53C24
FrattiniC23C23×D53
FittingC53×#410C2
socleC2×C53C2×#120
hypercenter#410D53
hypocenterC53#3027
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #16811459C2semidirect
2. #16811192C2semidirect
3. #16811187C2semidirect
4. #16810811C2semidirect
5. #16810596C2semidirect
6. C53×#410C2semidirect
15. C22×D212C22semidirect
22. #13133187C22semidirect
23. #13133185C22semidirect
24. #13133176C22semidirect
25. #13133131C22semidirect
27. #117×C53C22semidirect
28. #99×C53C22semidirect
32. C2×Q8×C53D4semidirect
33. C2×D4×C53D4semidirect
35. C22×C4×C53D4semidirect
59. C2×D212D4semidirect
61. C2×C4×D53D4semidirect
63. C2×Dic106D4semidirect
140. #410D53semidirect
141. C53#3027semidirect
161. #117C2×D53semidirect
163. #99C2×D53semidirect
180. C2×Q8D212semidirect
193. C2×D4D212semidirect
197. C22×C4D212semidirect