Finite Group Databasebeta

← #20285495

Subgroups — #20285495

Order 6784.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811133C211yesmaximal, characteristic
2C2×#13133197C211yesmaximal, characteristic, complemented
3#16810966C212yesmaximal, complemented
4#16810953C211yesmaximal, characteristic, complemented
5#16810865C212yesmaximal, complemented
6#16810851C212yesmaximal, complemented
7#16810786C212yesmaximal, complemented
8#16810781C211yesmaximal, characteristic, complemented
9#16810761C211yesmaximal, characteristic, complemented
10C22⋊C8×D53C211yesmaximal, characteristic, complemented
11C53×#480C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C2×Dic212C2212yescomplemented
16C2×#11719923C2212yescomplemented
17C2×C8×D53C2212yes—
18#13133199—21no—
19#13133197—21no—
20#13133197—21no—
21#13133197C2212yes—
22#13133197C2212yes—
23#13133197C2211yescharacteristic
24#13133196C2211yescharacteristic
25#13133188C2211yescharacteristic
26#13133185—21no—
27D4×Dic53—21no—
28#13133177—21no—
29#13133175C2211yescharacteristic
30C2×C4⋊Dic53—21no—
31C2×#11719921—21no—
32#13133148C2212yescomplemented
33#13133142C2212yescomplemented
34#13133140C2212yescomplemented
35#13133140C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38#121×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×D4×C53—21no—
45C2×D4×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—21no—
49C2×C22⋊D53—21no—
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53D411yescharacteristic
53#11719947—42no—
54C2×C4×D53—42no—
55C2×C4×D53D411yescharacteristic
56C2×C4×D53D411yescharacteristic
57C2×C4×D53C2311yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×Dic106—21no—
60C2×Dic106—42no—
61C2×Dic106C2311yescharacteristic
62C2×Dic106C2312yes—
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53C2312yes—
66C4⋊C4×C53C2311yescharacteristic
67C22⋊C4×C53—21no—
68C22⋊Dic53—21no—
69C22⋊Dic53—21no—
70#11719923—42no—
71#11719921—22no—
72#11719921—22no—
73#11719921—21no—
74#11719921—21no—
75C4⋊Dic53—22no—
76C4⋊Dic53—22no—
77C4⋊Dic53—21no—
78C4⋊Dic53C2311yescharacteristic
79#11719919—21no—
80C4×Dic53—21no—
81C2×#88322C2312yes—
82Dic212—42no—
83C8×D53—42no—
84C23×C53—21no—
85C23×C53C2×D411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—41no—
89C22×D53C2×D411yescharacteristic
90C22×D53C2×D411yescharacteristic
91D4×C53—22no—
92D4×C53—42no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—42no—
100C22⋊D53—42no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53C2×D411yescharacteristic
108C2×Dic53C2×D411yescharacteristic
109C4×D53—22no—
110C4×D53—22no—
111C4×D53—41no—
112C4×D53—41no—
113Dic106—22no—
114Dic106—22no—
115Dic106—22no—
116Dic106—42no—
117C8×C53—22no—
118#88322—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53#13512yes—
125C22×C53C22×D411yescharacteristic
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—21no—
130C2×D53—21no—
131C2×D53—44no—
132C2×D53#13712yes—
133C4×C53—21no—
134C4×C53—42no—
135C4×C53—41no—
136C4×C53H(5,𝔽2)12yes—
137Dic53—21no—
138Dic53—21no—
139Dic53—42no—
140Dic53—41no—
141Dic53—41no—
142#3190—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53—81no—
145C2×C53#53411yescharacteristic
146C2×C53C2×#13511yescharacteristic
147C2×C53C2×#13711yescharacteristic
148D53—22no—
149D53—41no—
150#525—531no—
151C2×C22⋊Q8—531no—
152#485—532no—
153#483—531no—
154#480D5311yescharacteristic, complemented
155#467—532no—
156#465—532no—
157#452—532no—
158#451—531no—
159#448—531no—
160C2×C22⋊C8—531no—
161C53#319011yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C22×Q8—531no—
164C23×C4—531no—
165C2×Q16—532no—
166C2×SD16—532no—
167C22×C8—532no—
168#123—1061no—
169C22⋊Q8—531no—
170C22⋊Q8—532no—
171C22⋊Q8—532no—
172C22⋊Q8—531no—
173C22⋊Q8—1061no—
174C22⋊Q8—1061no—
175#121—531no—
176#121C2×D5311yescharacteristic
177#120—1061no—
178C4×D4—1061no—
179C4×D4—1061no—
180#117—531no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4C2×D5311yescharacteristic
183C2×C22⋊C4—1061no—
184#107C2×D5312yescomplemented
185#106—532no—
186#103—532no—
187#103—532no—
188#103—532no—
189#102C2×D5312yescomplemented
190C22⋊C8—531no—
191C22⋊C8—532no—
192C22⋊C8C2×D5311yescharacteristic
193C24—531no—
194#41—2122no—
195C2×Q8—532no—
196C2×Q8—531no—
197C2×Q8—1061no—
198C2×Q8—2122no—
199C2×D4—21no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4C22×D5311yescharacteristic
203C22×C4—21no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—2122no—
212C22×C4C22×D5311yescharacteristic
213Q16—2122no—
214SD16—2122no—
215C2×C8—532no—
216C2×C8—2122no—
217C2×C8C22×D5312yes—
218C4⋊C4—21no—
219C4⋊C4—531no—
220C4⋊C4—1062no—
221C4⋊C4—1062no—
222C4⋊C4—1061no—
223C4⋊C4—1061no—
224C4⋊C4C22×D5312yes—
225C4⋊C4C22×D5311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C42—1061no—
234C23—21no—
235C23—531no—
236C23—531no—
237C23—1062no—
238C23—1062no—
239C23—2121no—
240C23D4×D5311yescharacteristic
241Q8—1062no—
242Q8—1062no—
243Q8—1062no—
244Q8—2122no—
245D4—22no—
246D4—42no—
247D4—2122no—
248D4—2122no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—531no—
254C2×C4—531no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1062no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4D4×D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—532no—
275C22—1061no—
276C22—1061no—
277C22—1061no—
278C22—1061no—
279C22—1061no—
280C22—2124no—
281C22#1313323612yes—
282C22C2×D4×D5311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—1061no—
287C4—1061no—
288C4—2121no—
289C4—2121no—
290C4—2122no—
291C4#1313326312yes—
292C2—22no—
293C2—81no—
294C2—1062no—
295C2—2121no—
296C2#1681118411yescharacteristic
297C2C2×#1313323611yescharacteristic
298C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3190
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#13133197C2semidirect
3. #16810966C2semidirect
4. #16810953C2semidirect
5. #16810865C2semidirect
6. #16810851C2semidirect
7. #16810786C2semidirect
8. #16810781C2semidirect
9. #16810761C2semidirect
10. C22⋊C8×D53C2semidirect
11. C53×#480C2semidirect
15. C2×Dic212C22semidirect
16. C2×#11719923C22semidirect
32. #13133148C22semidirect
33. #13133142C22semidirect
34. #13133140C22semidirect
35. #13133140C22semidirect
40. #107×C53C22semidirect
41. #102×C53C22semidirect
154. #480D53semidirect
161. C53#3190semidirect
184. #107C2×D53semidirect
189. #102C2×D53semidirect