Finite Group Databasebeta

← #20285550

Subgroups — #20285550

Order 6784.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811136C211yesmaximal, characteristic
2C2×#13133197C211yesmaximal, characteristic, complemented
3#16810966C212yesmaximal, complemented
4#16810941C211yesmaximal, characteristic, complemented
5#16810880C212yesmaximal, complemented
6#16810870C212yesmaximal, complemented
7#16810788C212yesmaximal, complemented
8#16810778C211yesmaximal, characteristic, complemented
9#16810764C211yesmaximal, characteristic, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#484C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15C2×#11719925C2212yescomplemented
16C2×#11719913C2212yescomplemented
17C2×#11719912C2212yes—
18#13133202—21no—
19#13133197—21no—
20#13133197—21no—
21#13133197C2212yes—
22#13133197C2212yes—
23#13133193C2211yescharacteristic
24#13133192C2211yescharacteristic
25#13133191C2211yescharacteristic
26#13133187—21no—
27#13133179—21no—
28#13133177—21no—
29#13133175C2211yescharacteristic
30C2×C4⋊Dic53—21no—
31C2×#11719921—21no—
32#13133148C2212yescomplemented
33#13133142C2212yescomplemented
34#13133140C2212yescomplemented
35#13133137C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38C22⋊Q8×C53C2211yescharacteristic
39C2×C4⋊C4×C53C2211yescharacteristic
40#107×C53C2212yescomplemented
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×Q8×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49C22×Dic53D411yescharacteristic
50#11719949—42no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212C2311yescharacteristic
54C2×C4×D53—21no—
55C2×C4×D53—42no—
56C2×C4×D53D411yescharacteristic
57C2×C4×D53D411yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×C4×D53C2311yescharacteristic
60C2×Dic106—21no—
61C2×Dic106—42no—
62C2×Dic106C2312yes—
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2312yes—
67C4⋊C4×C53C2311yescharacteristic
68C22⋊C4×C53—21no—
69#11719925—42no—
70#11719921—22no—
71#11719921—22no—
72#11719921—21no—
73#11719921—21no—
74#11719921—21no—
75#11719921—21no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53C2311yescharacteristic
79#11719919—21no—
80C4×Dic53—21no—
81C2×#88322C2312yes—
82#11719913—42no—
83#11719912—42no—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—21no—
87C22×D53—41no—
88C22×D53—41no—
89C22×D53C2×D411yescharacteristic
90C22×D53C2×D411yescharacteristic
91Q8×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—42no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2411yescharacteristic
99C22⋊D53—42no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—41no—
103C2×Dic53—42no—
104C2×Dic53C2×D411yescharacteristic
105C2×Dic53C2×D411yescharacteristic
106D212—22no—
107D212—42no—
108D212—42no—
109C4×D53—22no—
110C4×D53—22no—
111C4×D53—41no—
112C4×D53—41no—
113C4×D53—42no—
114Dic106—22no—
115Dic106—22no—
116Dic106—42no—
117C8×C53—22no—
118#88322—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#13712yes—
122C22×C53C22×D411yescharacteristic
123C2×D53—21no—
124C2×D53—21no—
125C2×D53—21no—
126C2×D53—21no—
127C2×D53—21no—
128C2×D53—21no—
129C2×D53—44no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—41no—
133C4×C53—21no—
134C4×C53—41no—
135C4×C53—41no—
136C4×C53—42no—
137C4×C53H(5,𝔽2)12yes—
138Dic53—21no—
139Dic53—21no—
140Dic53—42no—
141Dic53—41no—
142#3196—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53#53411yescharacteristic
145C2×C53C2×#13711yescharacteristic
146C2×C53#57511yescharacteristic
147D53—41no—
148D53—41no—
149D53—81no—
150#525—531no—
151C2×C22⋊Q8—531no—
152#485—532no—
153#484D5311yescharacteristic, complemented
154#481—531no—
155#470—532no—
156#468—532no—
157#452—532no—
158#451—531no—
159#448—531no—
160#407—531no—
161C53#319611yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C22×Q8—531no—
164C23×C4—531no—
165C2×Q16—532no—
166C2×SD16—532no—
167C2×M16—532no—
168#123—1061no—
169C22⋊Q8—531no—
170C22⋊Q8—532no—
171C22⋊Q8—532no—
172C22⋊Q8—1061no—
173C22⋊Q8—1061no—
174C22⋊Q8C2×D5311yescharacteristic
175#121—531no—
176#121—531no—
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#103C2×D5312yescomplemented
189#102—532no—
190C22⋊C8—531no—
191C22⋊C8—532no—
192C22⋊C8C2×D5311yescharacteristic
193C24—531no—
194#41—2122no—
195C2×Q8—532no—
196C2×Q8—1061no—
197C2×Q8—2122no—
198C2×Q8C22×D5311yescharacteristic
199C2×D4—531no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4—1061no—
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—
215M16—2122no—
216C2×C8—532no—
217C2×C8C22×D5312yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—531no—
221C4⋊C4—1062no—
222C4⋊C4—1062no—
223C4⋊C4—1061no—
224C4⋊C4C22×D5312yes—
225C4⋊C4C22×D5311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1061no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C42—1061no—
234C23—531no—
235C23—531no—
236C23—1062no—
237C23—1061no—
238C23—2121no—
239C23—2121no—
240C23D4×D5311yescharacteristic
241Q8—22no—
242Q8—1062no—
243Q8—1062no—
244Q8—2122no—
245D4—1062no—
246D4—2122no—
247D4—2122no—
248D4—2122no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—531no—
255C2×C4—531no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1061no—
259C2×C4—1062no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2122no—
265C2×C4D4×D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—21no—
270C22—21no—
271C22—1061no—
272C22—1061no—
273C22—1061no—
274C22—1061no—
275C22—1061no—
276C22—1061no—
277C22—2124no—
278C22—2121no—
279C22—2121no—
280C22—2121no—
281C22#1313324912yes—
282C22C2×D4×D5311yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—1061no—
288C4—1061no—
289C4—2121no—
290C4—2122no—
291C4#1313326312yes—
292C2—22no—
293C2—4241no—
294C2—2121no—
295C2—2121no—
296C2#1681118411yescharacteristic
297C2C2×#1313324911yescharacteristic
298C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3196
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#13133197C2semidirect
3. #16810966C2semidirect
4. #16810941C2semidirect
5. #16810880C2semidirect
6. #16810870C2semidirect
7. #16810788C2semidirect
8. #16810778C2semidirect
9. #16810764C2semidirect
10. #16810585C2semidirect
11. C53×#484C2semidirect
15. C2×#11719925C22semidirect
16. C2×#11719913C22semidirect
32. #13133148C22semidirect
33. #13133142C22semidirect
34. #13133140C22semidirect
35. #13133137C22semidirect
40. #107×C53C22semidirect
41. #103×C53C22semidirect
153. #484D53semidirect
161. C53#3196semidirect
184. #107C2×D53semidirect
188. #103C2×D53semidirect