Finite Group Databasebeta

← #20285545

Subgroups — #20285545

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
2#16810961C212yesmaximal, complemented
3#16810958C211yesmaximal, characteristic, complemented
4#16810941C211yesmaximal, characteristic, complemented
5#16810879C212yesmaximal, complemented
6#16810871C212yesmaximal, complemented
7#16810783C212yesmaximal, complemented
8#16810779C211yesmaximal, characteristic, complemented
9#16810762C211yesmaximal, characteristic, complemented
10#16810587C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic
13C2×#11719925C2212yescomplemented
14C2×#11719924C2212yescomplemented
15C2×#11719917C2212yes—
16#13133200—21no—
17#13133196C2211yescharacteristic
18#13133196C2211yescharacteristic
19#13133191C2211yescharacteristic
20#13133191C2211yescharacteristic
21#13133186—21no—
22C4×C22⋊D53—21no—
23C4×C22⋊D53—21no—
24#13133176C2211yescharacteristic
25#13133150C2212yescomplemented
26#13133148C2212yescomplemented
27#13133141C2212yescomplemented
28#13133141C2212yescomplemented
29#13133140C2212yescomplemented
30#13133137C2212yescomplemented
31C22⋊#88322C2211yescharacteristic
32C22⋊#88322C2211yescharacteristic
33C22⋊#88322C2212yes—
34C22×Q8×C53C2211yescharacteristic
35C23×C4×C53C2211yescharacteristic
36C22⋊Q8×C53—21no—
37C22⋊Q8×C53—21no—
38C22⋊Q8×C53C2212yes—
39C22⋊Q8×C53C2212yes—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2211yescharacteristic
42C2×C22⋊C4×C53—21no—
43C24×C53D411yescharacteristic
44C2×Q8×C53—21no—
45C2×Q8×C53—42no—
46C2×Q8×C53C2312yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—42no—
50C22×C4×C53D411yescharacteristic
51C22×C4×C53D411yescharacteristic
52C22×C4×C53D411yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C2×C22⋊D53—21no—
57C2×C22⋊D53—21no—
58C2×C22⋊D53—21no—
59#11719945—42no—
60C2×D212C2311yescharacteristic
61C2×C4×D53—21no—
62C2×Dic106C2311yescharacteristic
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—21no—
66C4⋊C4×C53C2312yes—
67C22⋊C4×C53—22no—
68C22⋊C4×C53—22no—
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719925—42no—
73#11719924—42no—
74#11719921—21no—
75#11719921—21no—
76C4⋊Dic53C2311yescharacteristic
77C4⋊Dic53C2311yescharacteristic
78#11719919—21no—
79#11719919—21no—
80C4×Dic53—21no—
81#11719917—42no—
82C2×#88322C2312yes—
83C2×#88322C2312yes—
84C23×C53—22no—
85C23×C53—41no—
86C23×C53—41no—
87C23×C53C2×D411yescharacteristic
88C23×C53C2×D411yescharacteristic
89C23×C53C2×D411yescharacteristic
90C22×D53—21no—
91Q8×C53—22no—
92Q8×C53—22no—
93Q8×C53—42no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2411yescharacteristic
108C22⋊D53—42no—
109C22⋊D53—42no—
110C22⋊D53—42no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114D212—22no—
115C4×D53—42no—
116Dic106—22no—
117#88322—22no—
118#88322—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×C53—42no—
137C4×C53—42no—
138C4×C53H(5,𝔽2)12yes—
139Dic53—41no—
140Dic53—41no—
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⋊Q8D5311yescharacteristic, complemented
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#407—531no—
161C53#319611yescharacteristic, complemented, Sylow
162C2×#41—531no—
163C22×Q8C2×D5311yescharacteristic
164C23×C4C2×D5311yescharacteristic
165C2×Q16—532no—
166C2×SD16—532no—
167C2×M16—532no—
168#123—1061no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—531no—
172C22⋊Q8—531no—
173C22⋊Q8C2×D5312yes—
174C22⋊Q8C2×D5312yes—
175#121—531no—
176#121—531no—
177#120—1061no—
178C4×D4—1061no—
179C4×D4—1061no—
180#117—531no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D5311yescharacteristic
183C2×C22⋊C4—21no—
184#107—532no—
185#106—532no—
186#103—532no—
187#103—532no—
188#103—532no—
189#102—532no—
190C22⋊C8—531no—
191C22⋊C8—531no—
192C22⋊C8—532no—
193C24C22⋊D5311yescharacteristic
194#41—2122no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—531no—
198C2×Q8C22×D5312yes—
199C2×D4—531no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4—1061no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—1061no—
207C22×C4C22⋊D5311yescharacteristic
208C22×C4C22⋊D5311yescharacteristic
209C22×C4C22⋊D5311yescharacteristic
210C22×C4C22×D5311yescharacteristic
211C22×C4C22×D5311yescharacteristic
212C22×C4C22×D5311yescharacteristic
213Q16—2122no—
214SD16—2122no—
215M16—2122no—
216C2×C8—532no—
217C2×C8—532no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—531no—
222C4⋊C4—531no—
223C4⋊C4—1061no—
224C4⋊C4—1061no—
225C4⋊C4C22×D5312yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
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×C22⋊D5311yescharacteristic
239C23C2×C22⋊D5311yescharacteristic
240C23C2×C22⋊D5311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1062no—
245D4—1062no—
246D4—2122no—
247D4—2122no—
248D4—2122no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—2122no—
263C2×C4C2×C22⋊D5311yescharacteristic
264C2×C4C2×C22⋊D5311yescharacteristic
265C2×C4C2×C22⋊D5311yescharacteristic
266C2×C4C23×D5311yescharacteristic
267C8—1062no—
268C8—1062no—
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#1313324812yes—
282C22C22×C22⋊D5311yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—2121no—
289C4—2121no—
290C4—2121no—
291C4#1313326212yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—4241no—
296C2#1681118211yescharacteristic
297C2C2×#1313324811yescharacteristic
298C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C22⋊Q8×C53C2
socleC22×C53C22×D4
hypercenterC2×C22⋊Q8D53
hypocenterC53#3196
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16810961C2semidirect
3. #16810958C2semidirect
4. #16810941C2semidirect
5. #16810879C2semidirect
6. #16810871C2semidirect
7. #16810783C2semidirect
8. #16810779C2semidirect
9. #16810762C2semidirect
10. #16810587C2semidirect
11. C2×C22⋊Q8×C53C2semidirect
13. C2×#11719925C22semidirect
14. C2×#11719924C22semidirect
25. #13133150C22semidirect
26. #13133148C22semidirect
27. #13133141C22semidirect
28. #13133141C22semidirect
29. #13133140C22semidirect
30. #13133137C22semidirect
151. C2×C22⋊Q8D53semidirect
161. C53#3196semidirect