Finite Group Databasebeta

← #20285526

Subgroups — #20285526

Order 6784.

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

normal subgroups only

This group has 466 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#16811133C211yesmaximal, characteristic, complemented
2C2×#13133188C211yesmaximal, characteristic, complemented
3#16810954C211yesmaximal, characteristic, complemented
4#16810946C212yesmaximal, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810873C212yesmaximal, complemented
7#16810868C212yesmaximal, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810765C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#447C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic, complemented
13C22×C22⋊D53—21no—
14C22×C4×D53C2211yescharacteristic
15C2×#11719923C2212yescomplemented
16C2×#11719912C2212yescomplemented
17#13133199—21no—
18#13133197C2211yescharacteristic, complemented
19#13133196C2211yescharacteristic
20#13133188—21no—
21#13133188—21no—
22#13133188C2212yes—
23#13133188C2212yes—
24#13133188C2211yescharacteristic, complemented
25#13133185—21no—
26D4×Dic53—21no—
27#13133177—21no—
28#13133175C2211yescharacteristic, complemented
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53—21no—
31#13133150C2212yescomplemented
32#13133149C2212yescomplemented
33#13133142C2212yescomplemented
34#13133138C2212yescomplemented
35#13133138C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yescomplemented
38C22×D4×C53C2211yescharacteristic
39C2×D8×C53C2212yescomplemented
40#121×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C24×C53—21no—
44C23×D53D411yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53C2312yes—
49C2×D4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—21no—
52C2×C22⋊D53—21no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—44no—
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53D411yescharacteristic
59#11719947—42no—
60C2×C4×D53—42no—
61C2×C4×D53D411yescharacteristic
62C2×C4×D53D411yescharacteristic
63C2×C4×D53C2311yescharacteristic
64C2×C4×D53C2311yescharacteristic
65C2×Dic106C2311yescharacteristic
66D8×C53—42no—
67C2×C8×C53C2312yes—
68C4⋊C4×C53C2311yescharacteristic
69C22⋊C4×C53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—22no—
72C22⋊Dic53—22no—
73C22⋊Dic53—21no—
74#11719923—42no—
75#11719921—21no—
76#11719921—21no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79C4⋊Dic53C2311yescharacteristic
80C4⋊Dic53C2312yes—
81#11719919—21no—
82C4×Dic53—21no—
83C2×#88322C2312yes—
84#11719912—42no—
85C23×C53—21no—
86C23×C53—22no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53C2×D411yescharacteristic
91C22×D53—22no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53C2×D411yescharacteristic
95C22×D53C2×D411yescharacteristic
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—22no—
99D4×C53—42no—
100D4×C53—42no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2411yescharacteristic
105C22⋊D53—42no—
106C22⋊D53—42no—
107C22⋊D53—44no—
108C22⋊D53—44no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53C2×D411yescharacteristic
116C2×Dic53C2×D411yescharacteristic
117C4×D53—22no—
118C4×D53—22no—
119C4×D53—41no—
120C4×D53—41no—
121Dic106—22no—
122C8×C53—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—41no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—42no—
132C22×C53—84no—
133C22×C53AGL(1,ℤ/8)12yes—
134C22×C53C22×D411yescharacteristic
135C2×D53—21no—
136C2×D53—21no—
137C2×D53—21no—
138C2×D53—21no—
139C2×D53—21no—
140C2×D53—21no—
141C2×D53—44no—
142C4×C53—21no—
143C4×C53—41no—
144C4×C53H(5,𝔽2)12yes—
145Dic53—21no—
146Dic53—21no—
147Dic53—41no—
148Dic53—42no—
149Dic53—41no—
150#3194—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53—81no—
153C2×C53—82no—
154C2×C53#53411yescharacteristic
155C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
156C2×C53#57511yescharacteristic
157D53—41no—
158D53—41no—
159#525—531no—
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#447D5311yescharacteristic, complemented
169#407—531no—
170C53#319411yescharacteristic, complemented, Sylow
171C2×#41—531no—
172C22×D4—1061no—
173C22×D4C2×D5311yescharacteristic
174C23×C4—531no—
175C2×SD16—532no—
176C2×D8C2×D5312yescomplemented
177C2×M16—532no—
178#123—1061no—
179C22⋊Q8—531no—
180C22⋊Q8—531no—
181#121—532no—
182#121—532no—
183#121—531no—
184#121—1061no—
185#121—1061no—
186#121C2×D5311yescharacteristic
187#120—1061no—
188C4×D4—1061no—
189C4×D4—1061no—
190#117—531no—
191C2×C4⋊C4—531no—
192C2×C22⋊C4—1061no—
193#107—532no—
194#106—532no—
195#103—532no—
196#102—532no—
197#102—532no—
198#102C2×D5312yescomplemented
199C22⋊C8—531no—
200C22⋊C8—532no—
201C22⋊C8C2×D5311yescharacteristic
202C24—21no—
203C24—531no—
204#41—2122no—
205C2×Q8—531no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—1061no—
210C2×D4—1061no—
211C2×D4—1062no—
212C2×D4—1062no—
213C2×D4—2124no—
214C2×D4C22×D5311yescharacteristic
215C2×D4C22×D5312yes—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—2122no—
224C22×C4C22×D5311yescharacteristic
225SD16—2122no—
226D8—42no—
227M16—2122no—
228C2×C8—532no—
229C2×C8C22×D5312yes—
230C4⋊C4—531no—
231C4⋊C4—532no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4—1061no—
235C4⋊C4C22×D5311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1061no—
238C22⋊C4—1061no—
239C22⋊C4—1062no—
240C22⋊C4—1062no—
241C22⋊C4—1061no—
242C22⋊C4—1061no—
243C42—1061no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—531no—
250C23—531no—
251C23—1062no—
252C23—2121no—
253C23—2121no—
254C23D4×D5311yescharacteristic
255Q8—1062no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2122no—
262D4—2124no—
263D4—2124no—
264D4—2122no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—531no—
268C2×C4—531no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1061no—
272C2×C4—1062no—
273C2×C4—1061no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4D4×D5311yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—22no—
282C8—1062no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—84no—
292C22—1061no—
293C22—1061no—
294C22—1061no—
295C22—1061no—
296C22—1061no—
297C22—1061no—
298C22—2124no—
299C22#1313324112yes—
300C22C2×D4×D5311yescharacteristic
301C4—21no—
302C4—41no—
303C4—1061no—
304C4—1061no—
305C4—2122no—
306C4—2121no—
307C4—2121no—
308C4#1313326212yes—
309C2—22no—
310C2—82no—
311C2—81no—
312C2—2121no—
313C2—2121no—
314C2#1681118311yescharacteristic
315C2C2×#1313324111yescharacteristic
316C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3194
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16811133C2semidirect
2. C2×#13133188C2semidirect
3. #16810954C2semidirect
4. #16810946C2semidirect
5. #16810934C2semidirect
6. #16810873C2semidirect
7. #16810868C2semidirect
8. #16810774C2semidirect
9. #16810765C2semidirect
10. #16810585C2semidirect
11. C53×#447C2semidirect
12. C2×#11719947C22semidirect
15. C2×#11719923C22semidirect
16. C2×#11719912C22semidirect
18. #13133197C22semidirect
24. #13133188C22semidirect
28. #13133175C22semidirect
31. #13133150C22semidirect
32. #13133149C22semidirect
33. #13133142C22semidirect
34. #13133138C22semidirect
35. #13133138C22semidirect
37. #13133129C22semidirect
39. C2×D8×C53C22semidirect
41. #102×C53C22semidirect
168. #447D53semidirect
170. C53#3194semidirect
176. C2×D8C2×D53semidirect
198. #102C2×D53semidirect