Finite Group Databasebeta

← #20285719

Subgroups — #20285719

Order 6784.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#16811178C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic
3#16811018C211yesmaximal, characteristic, complemented
4#16811017C211yesmaximal, characteristic, complemented
5#16810985C211yesmaximal, characteristic, complemented
6#16810848C212yesmaximal, complemented
7#16810844C212yesmaximal, complemented
8#16810780C212yesmaximal, complemented
9#16810775C212yesmaximal
10#16810678C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×Dic212C2212yescomplemented
15C2×#11719925C2212yescomplemented
16C2×#11719913C2212yes—
17C2×#11719923C2212yes—
18#13133217C2211yescharacteristic, complemented
19#13133208—21no—
20#13133206C2211yescharacteristic
21#13133205C2211yescharacteristic
22#13133204C2211yescharacteristic
23#13133203—21no—
24#13133203—21no—
25#13133197C2212yes—
26#13133193—21no—
27Q8×Dic53—21no—
28#13133182C2212yescomplemented
29#13133175—21no—
30C22⋊C4×D53—21no—
31C22⋊C4×D53—21no—
32C42×D53C2211yescharacteristic
33#13133146C2212yescomplemented
34#13133140C2212yescomplemented
35#13133137C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic, complemented
38C4⋊Q8×C53C2211yescharacteristic, complemented
39#124×C53C2211yescharacteristic
40#103×C53C2212yescomplemented
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic, complemented
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48Q8×D53—42no—
49D4×D53—42no—
50C2×D212C2311yescharacteristic
51C2×C4×D53—21no—
52C2×C4×D53D411yescharacteristic
53C2×C4×D53D411yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×Dic106—21no—
56C2×Dic106C2311yescharacteristic
57C2×Dic106C2312yes—
58C2×C8×C53C2312yes—
59C4⋊C4×C53—21no—
60C4⋊C4×C53C2312yes—
61C22⋊C4×C53—21no—
62C22⋊C4×C53—21no—
63C42×C53C2311yescharacteristic
64C22⋊Dic53—21no—
65C22⋊Dic53—21no—
66#11719925—42no—
67#11719923—42no—
68#11719921—22no—
69#11719921—21no—
70#11719921—21no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53—22no—
76#11719919—22no—
77C4×Dic53—22no—
78C4×Dic53D411yescharacteristic
79C4×Dic53D411yescharacteristic
80C4×Dic53C2311yescharacteristic
81C2×#88322C2312yes—
82Dic212—42no—
83#11719913—42no—
84C23×C53—21no—
85C22×D53—21no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—42no—
89C22×D53C2×D411yescharacteristic
90Q8×C53—22no—
91Q8×C53—42no—
92D4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—42no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104D212—22no—
105C4×D53—22no—
106C4×D53—22no—
107C4×D53—41no—
108C4×D53—42no—
109Dic106—22no—
110Dic106—22no—
111Dic106—22no—
112Dic106—42no—
113C8×C53—22no—
114#88322—22no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53C22×D411yescharacteristic
119C2×D53—21no—
120C2×D53—21no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—84no—
125C2×D53#13712yes—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—42no—
130C4×C53H(5,𝔽2)12yes—
131Dic53—21no—
132Dic53—21no—
133Dic53—21no—
134Dic53—41no—
135Dic53—42no—
136#3215—531nomaximal, Sylow
137C2×C53—81no—
138C2×C53#53511yescharacteristic
139C2×C53C2×#13711yescharacteristic
140C2×C53#57511yescharacteristic
141D53—22no—
142D53—81no—
143#533—531no—
144C2×#124—531no—
145#497—531no—
146#497—531no—
147#489—531no—
148#489D5311yescharacteristic, complemented
149#464—532no—
150#464—532no—
151#451—532no—
152#450—532no—
153#432—531no—
154C53#321511yescharacteristic, complemented, Sylow
155C22×Q8—531no—
156C22×D4—531no—
157C2×Q16—532no—
158C2×Q16—532no—
159C2×SD16—532no—
160C2×SD16—532no—
161C4⋊Q8—531no—
162C4⋊Q8C2×D5311yescharacteristic, complemented
163#125—1061no—
164#124—531no—
165#124—531no—
166#124—531no—
167#124—1061no—
168#124—1061no—
169#124C2×D5311yescharacteristic
170C22⋊Q8—532no—
171C22⋊Q8—1061no—
172C4×Q8—532no—
173C4×Q8—1061no—
174#117—1061no—
175C2×C22⋊C4—1061no—
176C2×C22⋊C4—1061no—
177C2×C42—531no—
178C4⋊C8—532no—
179#103—532no—
180#103C2×D5312yescomplemented
181#102—532no—
182#102C2×D5312yes—
183C22⋊C8—532no—
184#97—531no—
185#97C2×D5311yescharacteristic, complemented
186C24—1061no—
187C2×Q8—21no—
188C2×Q8—531no—
189C2×Q8—532no—
190C2×Q8—1061no—
191C2×Q8—2122no—
192C2×Q8C22×D5311yescharacteristic
193C2×D4—531no—
194C2×D4—1061no—
195C2×D4—2122no—
196C2×D4C22×D5311yescharacteristic
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—1061no—
201Q16—2122no—
202Q16—2122no—
203SD16—2122no—
204SD16—2122no—
205C2×C8—532no—
206C2×C8C22×D5312yes—
207C4⋊C4—21no—
208C4⋊C4—1062no—
209C4⋊C4—1062no—
210C4⋊C4—1061no—
211C4⋊C4—1061no—
212C4⋊C4C22×D5312yes—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—1061no—
216C22⋊C4—1061no—
217C22⋊C4—1062no—
218C22⋊C4—1061no—
219C22⋊C4—1061no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C42—531no—
223C42—531no—
224C42—531no—
225C42—1062no—
226C42C22×D5311yescharacteristic
227C23—21no—
228C23—531no—
229C23—1061no—
230C23—2122no—
231C23—2122no—
232C23—2122no—
233Q8—22no—
234Q8—42no—
235Q8—1062no—
236Q8—1062no—
237Q8—1062no—
238Q8—2122no—
239D4—22no—
240D4—1062no—
241D4—2122no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—531no—
245C2×C4—531no—
246C2×C4—531no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2121no—
252C2×C4—2122no—
253C2×C4D4×D5311yescharacteristic
254C2×C4D4×D5311yescharacteristic
255C2×C4C23×D5311yescharacteristic
256C8—22no—
257C8—1062no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—532no—
262C22—1061no—
263C22—1061no—
264C22—4244no—
265C22—2121no—
266C22—2121no—
267C22—2121no—
268C22C2×D4×D5311yescharacteristic
269C4—21no—
270C4—21no—
271C4—42no—
272C4—41no—
273C4—1061no—
274C4—1061no—
275C4—1061no—
276C4—2121no—
277C4—2122no—
278C4#1313326312yes—
279C2—81no—
280C2—1062no—
281C2—4241no—
282C2#1681119111yescharacteristic
283C2#137×D5311yescharacteristic
284C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3215
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811018C2semidirect
4. #16811017C2semidirect
5. #16810985C2semidirect
6. #16810848C2semidirect
7. #16810844C2semidirect
8. #16810780C2semidirect
10. #16810678C2semidirect
11. C53×#489C2semidirect
14. C2×Dic212C22semidirect
15. C2×#11719925C22semidirect
18. #13133217C22semidirect
28. #13133182C22semidirect
33. #13133146C22semidirect
34. #13133140C22semidirect
37. #13133128C22semidirect
38. C4⋊Q8×C53C22semidirect
40. #103×C53C22semidirect
42. #97×C53C22semidirect
148. #489D53semidirect
154. C53#3215semidirect
162. C4⋊Q8C2×D53semidirect
180. #103C2×D53semidirect
185. #97C2×D53semidirect