Finite Group Databasebeta

← #20285525

Subgroups — #20285525

Order 6784.

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

normal subgroups only

This group has 466 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
1C2×#13133191C211yesmaximal, characteristic, complemented
2#16810951C211yesmaximal, characteristic, complemented
3#16810947C212yesmaximal, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810874C212yesmaximal, complemented
6#16810871C212yesmaximal, complemented
7#16810773C211yesmaximal, characteristic, complemented
8#16810768C212yesmaximal, complemented
9#16810756C211yesmaximal, characteristic, complemented
10#16810587C211yesmaximal, characteristic, complemented
11C53×#525C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53—21no—
14C2×#11719924C2212yescomplemented
15C2×#11719922C2212yescomplemented
16C2×#11719917C2212yescomplemented
17#13133191—21no—
18#13133191—21no—
19#13133191C2212yes—
20#13133191C2212yes—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719921—21no—
23#13133150C2212yescomplemented
24#13133148C2212yescomplemented
25#13133141C2212yescomplemented
26#13133138C2212yescomplemented
27#13133137C2212yescomplemented
28#13133137C2212yescomplemented
29C22⋊#88322C2211yescharacteristic
30C22⋊#88322C2211yescharacteristic
31C22⋊#88322C2212yescomplemented
32C2×#41×C53C2211yescharacteristic, complemented
33C23×C4×C53C2211yescharacteristic
34#123×C53—21no—
35C22⋊Q8×C53C2211yescharacteristic
36C22⋊Q8×C53C2211yescharacteristic, complemented
37#121×C53C2211yescharacteristic
38#121×C53C2211yescharacteristic, complemented
39#120×C53—21no—
40C4×D4×C53—21no—
41C4×D4×C53—21no—
42#117×C53C2211yescharacteristic, complemented
43C24×C53D411yescharacteristic
44C23×D53—21no—
45#41×C53—42no—
46C2×Q8×C53C2311yescharacteristic
47C2×D4×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53C2311yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—42no—
53C22×C4×C53D411yescharacteristic
54C22×C4×C53D411yescharacteristic
55C22×C4×C53D411yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C2×C22⋊D53—22no—
60C2×C22⋊D53—22no—
61C2×C22⋊D53—44no—
62C22×Dic53—21no—
63C2×D212—21no—
64C2×D212—42no—
65C2×D212C2312yes—
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C4⋊C4×C53C2311yescharacteristic
69C4⋊C4×C53C2311yescharacteristic
70C22⋊C4×C53—21no—
71C22⋊C4×C53—21no—
72C22⋊C4×C53—21no—
73C22⋊C4×C53—21no—
74C22⋊C4×C53—21no—
75C42×C53—21no—
76#11719924—42no—
77#11719922—42no—
78#11719921—22no—
79#11719921—22no—
80C4⋊Dic53—21no—
81C4⋊Dic53C2312yes—
82#11719917—42no—
83C2×#88322C2312yes—
84C2×#88322C2312yes—
85C23×C53—22no—
86C23×C53—21no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53C2×D411yescharacteristic
90C23×C53C2×D411yescharacteristic
91C23×C53C2×D411yescharacteristic
92C22×D53—22no—
93C22×D53—42no—
94C22×D53—42no—
95C22×D53—42no—
96Q8×C53—22no—
97D4×C53—22no—
98D4×C53—42no—
99D4×C53—42no—
100D4×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53C2×D411yescharacteristic
112C2×C4×C53C2×D411yescharacteristic
113C2×C4×C53C2×D411yescharacteristic
114C2×C4×C53C2411yescharacteristic
115C22⋊D53—44no—
116C22⋊D53—44no—
117C2×Dic53—22no—
118C2×Dic53—42no—
119D212—22no—
120D212—22no—
121D212—42no—
122#88322—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—44no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53AGL(1,ℤ/8)12yes—
137C22×C53C22×D411yescharacteristic
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—84no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53H(5,𝔽2)12yes—
149Dic53—42no—
150#3194—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53—41no—
153C2×C53—41no—
154C2×C53—81no—
155C2×C53#53411yescharacteristic
156C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C53#57511yescharacteristic
158D53—82no—
159#525D5311yescharacteristic, complemented
160C2×#121—531no—
161#483—531no—
162#482—532no—
163#480—531no—
164#469—532no—
165#468—532no—
166#450—531no—
167#449—532no—
168#447—531no—
169#407—531no—
170C53#319411yescharacteristic, complemented, Sylow
171C2×#41C2×D5311yescharacteristic, complemented
172C22×D4—531no—
173C22×D4—1061no—
174C23×C4C2×D5311yescharacteristic
175C2×SD16—532no—
176C2×D8—532no—
177C2×M16—532no—
178#123—21no—
179C22⋊Q8C2×D5311yescharacteristic
180C22⋊Q8C2×D5311yescharacteristic, complemented
181#121—532no—
182#121—532no—
183#121—1061no—
184#121—1061no—
185#121C2×D5311yescharacteristic
186#121C2×D5311yescharacteristic, complemented
187#120—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117C2×D5311yescharacteristic, complemented
191C2×C4⋊C4—531no—
192C2×C22⋊C4—1061no—
193#107—532no—
194#106—532no—
195#103—532no—
196#102—532no—
197#102—532no—
198#102—532no—
199C22⋊C8—531no—
200C22⋊C8—531no—
201C22⋊C8—532no—
202C24—1061no—
203C24C22⋊D5311yescharacteristic
204#41—42no—
205C2×Q8C22×D5311yescharacteristic
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—532no—
210C2×D4—1062no—
211C2×D4—1062no—
212C2×D4—1061no—
213C2×D4—2122no—
214C2×D4—2124no—
215C2×D4C22×D5311yescharacteristic
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—1061no—
219C22×C4C22⋊D5311yescharacteristic
220C22×C4C22⋊D5311yescharacteristic
221C22×C4C22⋊D5311yescharacteristic
222C22×C4C22×D5311yescharacteristic
223C22×C4C22×D5311yescharacteristic
224C22×C4C22×D5311yescharacteristic
225SD16—2122no—
226D8—2122no—
227M16—2122no—
228C2×C8—532no—
229C2×C8—532no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—532no—
233C4⋊C4—1061no—
234C4⋊C4C22×D5311yescharacteristic
235C4⋊C4C22×D5311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4—1062no—
242C22⋊C4—1062no—
243C42—21no—
244C23—22no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—1062no—
249C23—2122no—
250C23—2122no—
251C23—2122no—
252C23C2×C22⋊D5311yescharacteristic
253C23C2×C22⋊D5311yescharacteristic
254C23C2×C22⋊D5311yescharacteristic
255Q8—22no—
256D4—22no—
257D4—42no—
258D4—42no—
259D4—42no—
260D4—1062no—
261D4—1062no—
262D4—2122no—
263D4—2124no—
264D4—2124no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—1062no—
276C2×C4—2122no—
277C2×C4C2×C22⋊D5311yescharacteristic
278C2×C4C2×C22⋊D5311yescharacteristic
279C2×C4C2×C22⋊D5311yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—1062no—
282C8—1062no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—4244no—
296C22—2122no—
297C22—2122no—
298C22—2122no—
299C22#1313324212yes—
300C22C22×C22⋊D5311yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—2122no—
308C4#1313326212yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—81no—
313C2—4242no—
314C2#1681118211yescharacteristic
315C2C2×#1313324211yescharacteristic
316C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#525C2
socleC22×C53C22×D4
hypercenter#525D53
hypocenterC53#3194
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#13133191C2semidirect
2. #16810951C2semidirect
3. #16810947C2semidirect
4. #16810934C2semidirect
5. #16810874C2semidirect
6. #16810871C2semidirect
7. #16810773C2semidirect
8. #16810768C2semidirect
9. #16810756C2semidirect
10. #16810587C2semidirect
11. C53×#525C2semidirect
14. C2×#11719924C22semidirect
15. C2×#11719922C22semidirect
16. C2×#11719917C22semidirect
23. #13133150C22semidirect
24. #13133148C22semidirect
25. #13133141C22semidirect
26. #13133138C22semidirect
27. #13133137C22semidirect
28. #13133137C22semidirect
31. C22⋊#88322C22semidirect
32. C2×#41×C53C22semidirect
36. C22⋊Q8×C53C22semidirect
38. #121×C53C22semidirect
42. #117×C53C22semidirect
159. #525D53semidirect
170. C53#3194semidirect
171. C2×#41C2×D53semidirect
180. C22⋊Q8C2×D53semidirect
186. #121C2×D53semidirect
190. #117C2×D53semidirect