Finite Group Databasebeta

← #20285547

Subgroups — #20285547

Order 6784.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811134C211yesmaximal, characteristic
2C2×#13133196C211yesmaximal, characteristic, complemented
3#16810965C212yesmaximal, complemented
4#16810959C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810881C212yesmaximal, complemented
7#16810869C212yesmaximal, complemented
8#16810787C212yesmaximal, complemented
9#16810781C211yesmaximal, characteristic, complemented
10#16810764C211yesmaximal, characteristic, complemented
11C53×#407C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C2×Dic212C2212yescomplemented
15C2×#11719913C2212yescomplemented
16#13133200—21no—
17#13133196—21no—
18#13133196—21no—
19#13133196C2211yescharacteristic
20#13133196C2212yes—
21#13133196C2212yes—
22#13133196C2211yescharacteristic
23#13133191C2211yescharacteristic
24#13133191C2211yescharacteristic
25#13133186—21no—
26C4×C22⋊D53—21no—
27C4×C22⋊D53—21no—
28#13133176C2211yescharacteristic
29C2×C4⋊Dic53C2211yescharacteristic
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32#13133151C2212yescomplemented
33#13133149C2212yescomplemented
34#13133142C2212yescomplemented
35#13133142C2212yescomplemented
36#13133142C2212yescomplemented
37#13133139C2212yescomplemented
38C23×C4×C53C2211yescharacteristic
39C2×M16×C53C2212yes—
40C22⋊C8×C53C2211yescharacteristic
41C22⋊C8×C53C2211yescharacteristic
42C22⋊C8×C53C2212yes—
43C24×C53D411yescharacteristic
44C22×C4×C53—42no—
45C22×C4×C53D411yescharacteristic
46C22×C4×C53D411yescharacteristic
47C22×C4×C53D411yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—21no—
52C2×C22⋊D53—21no—
53C2×C22⋊D53—21no—
54C22×Dic53—21no—
55C22×Dic53—21no—
56#11719945—42no—
57C2×D212C2311yescharacteristic
58C2×C4×D53—21no—
59C2×Dic106—21no—
60C2×Dic106—42no—
61C2×Dic106C2312yes—
62C2×Dic106C2311yescharacteristic
63M16×C53—42no—
64C2×C8×C53C2312yes—
65C2×C8×C53C2312yes—
66C22⋊Dic53—22no—
67C22⋊Dic53—22no—
68C22⋊Dic53—21no—
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71#11719921—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53C2312yes—
75C4⋊Dic53C2311yescharacteristic
76C4⋊Dic53C2311yescharacteristic
77#11719919—22no—
78#11719919—22no—
79#11719919—21no—
80#11719919—21no—
81C4×Dic53—21no—
82Dic212—42no—
83#11719913—42no—
84C23×C53—22no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53C2×D411yescharacteristic
88C23×C53C2×D411yescharacteristic
89C23×C53C2×D411yescharacteristic
90C22×D53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—42no—
102C22⋊D53—42no—
103C22⋊D53—42no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111D212—22no—
112C4×D53—42no—
113Dic106—22no—
114Dic106—22no—
115Dic106—22no—
116Dic106—42no—
117C8×C53—22no—
118C8×C53—22no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—44no—
128C22×C53#13712yes—
129C22×C53C22×D411yescharacteristic
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—41no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53H(5,𝔽2)12yes—
137Dic53—42no—
138Dic53—41no—
139Dic53—41no—
140Dic53—42no—
141Dic53—41no—
142#3196—531nomaximal, Sylow
143C2×C53—22no—
144C2×C53—41no—
145C2×C53—41no—
146C2×C53#53411yescharacteristic
147C2×C53C2×#13711yescharacteristic
148C2×C53#57511yescharacteristic
149D53—81no—
150#525—531no—
151C2×C22⋊Q8—531no—
152#485—532no—
153#484—531no—
154#481—531no—
155#470—532no—
156#468—532no—
157#452—532no—
158#451—531no—
159#448—531no—
160#407D5311yescharacteristic, complemented
161C53#319611yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C22×Q8—531no—
164C23×C4C2×D5311yescharacteristic
165C2×Q16—532no—
166C2×SD16—532no—
167C2×M16C2×D5312yes—
168#123—1061no—
169C22⋊Q8—532no—
170C22⋊Q8—532no—
171C22⋊Q8—531no—
172C22⋊Q8—531no—
173C22⋊Q8—1061no—
174C22⋊Q8—1061no—
175#121—531no—
176#121—531no—
177#120—1061no—
178C4×D4—1061no—
179C4×D4—1061no—
180#117—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C22⋊C4—1061no—
184#107—532no—
185#106—532no—
186#103—532no—
187#103—532no—
188#103—532no—
189#102—532no—
190C22⋊C8C2×D5311yescharacteristic
191C22⋊C8C2×D5311yescharacteristic
192C22⋊C8C2×D5312yes—
193C24D21211yescharacteristic
194#41—2122no—
195C2×Q8—532no—
196C2×Q8—531no—
197C2×Q8—1061no—
198C2×Q8—2122no—
199C2×D4—531no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4—1061no—
203C22×C4—42no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4D21211yescharacteristic
208C22×C4D21211yescharacteristic
209C22×C4D21211yescharacteristic
210C22×C4C22×D5311yescharacteristic
211C22×C4C22×D5311yescharacteristic
212C22×C4C22×D5311yescharacteristic
213Q16—2122no—
214SD16—2122no—
215M16—42no—
216C2×C8C22×D5312yes—
217C2×C8C22×D5312yes—
218C4⋊C4—532no—
219C4⋊C4—531no—
220C4⋊C4—531no—
221C4⋊C4—1061no—
222C4⋊C4—1062no—
223C4⋊C4—1062no—
224C4⋊C4—1061no—
225C4⋊C4—1061no—
226C22⋊C4—1062no—
227C22⋊C4—1062no—
228C22⋊C4—1061no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C42—1061no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—1061no—
238C23C2×D21211yescharacteristic
239C23C2×D21211yescharacteristic
240C23C2×D21211yescharacteristic
241Q8—1062no—
242Q8—1062no—
243Q8—1062no—
244Q8—2122no—
245D4—1062no—
246D4—2122no—
247D4—2122no—
248D4—2122no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1062no—
258C2×C4—1061no—
259C2×C4—1062no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4C2×D21211yescharacteristic
264C2×C4C2×D21211yescharacteristic
265C2×C4C2×D21211yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—2121no—
279C22—2121no—
280C22—2121no—
281C22#1313325012yes—
282C22C22×D21211yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—2121no—
287C4—2122no—
288C4—2121no—
289C4—2121no—
290C4—2122no—
291C4#1313326212yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—4241no—
296C2#1681118511yescharacteristic
297C2C2×#1313325011yescharacteristic
298C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3196
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#13133196C2semidirect
3. #16810965C2semidirect
4. #16810959C2semidirect
5. #16810942C2semidirect
6. #16810881C2semidirect
7. #16810869C2semidirect
8. #16810787C2semidirect
9. #16810781C2semidirect
10. #16810764C2semidirect
11. C53×#407C2semidirect
14. C2×Dic212C22semidirect
15. C2×#11719913C22semidirect
32. #13133151C22semidirect
33. #13133149C22semidirect
34. #13133142C22semidirect
35. #13133142C22semidirect
36. #13133142C22semidirect
37. #13133139C22semidirect
160. #407D53semidirect
161. C53#3196semidirect