Finite Group Databasebeta

← #20285576

Subgroups — #20285576

Order 6784.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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#16811187C211yesmaximal, characteristic, complemented
2#16810947C212yesmaximal, complemented
3#16810944C212yesmaximal, complemented
4#16810874C212yesmaximal, complemented
5#16810857C212yesmaximal, complemented
6#16810757C212yesmaximal, complemented
7#16810756C212yesmaximal, complemented
8#16810590C211yesmaximal, characteristic, complemented
9C53×#535C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic, complemented
11C2×#11719922C2212yes—
12C2×#11719922C2212yes—
13C2×#11719917C2211yescharacteristic, complemented
14C22×#88322C2211yescharacteristic, complemented
15#13133213—21no—
16#13133206—21no—
17#13133191C2212yes—
18#13133191C2212yescomplemented
19#13133185—21no—
20#13133185—21no—
21#13133176C2211yescharacteristic, complemented
22#13133150C2212yescomplemented
23#13133148C2212yescomplemented
24#13133138C2212yescomplemented
25#13133138C2212yescomplemented
26#13133137C2212yescomplemented
27#13133137C2212yescomplemented
28C22⋊#88322C2212yescomplemented
29C22⋊#88322C2212yescomplemented
30C2×#41×C53C2211yescharacteristic
31C22×D4×C53C2211yescharacteristic, complemented
32#127×C53—21no—
33#124×C53—21no—
34#121×C53C2212yes—
35#121×C53C2212yescomplemented
36#120×C53—21no—
37#120×C53—21no—
38#117×C53C2211yescharacteristic, complemented
39C24×C53—21no—
40C23×D53—21no—
41#41×C53—41no—
42#41×C53—41no—
43C2×Q8×C53D411yescharacteristic
44C2×D4×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—21no—
47C2×D4×C53—41no—
48C2×D4×C53—41no—
49C2×D4×C53D411yescharacteristic
50C2×D4×C53D411yescharacteristic
51C2×D4×C53D411yescharacteristic
52C2×D4×C53C2312yes—
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C2×C22⋊D53—22no—
57C2×C22⋊D53—22no—
58C2×D212—21no—
59C2×D212—41no—
60C2×D212—41no—
61C2×D212C2312yes—
62C4⋊C4×C53C2312yes—
63C22⋊C4×C53—21no—
64C22⋊C4×C53—22no—
65C22⋊C4×C53—22no—
66C42×C53—21no—
67C22⋊Dic53—21no—
68#11719922—42no—
69#11719922—42no—
70#11719921—22no—
71#11719921—22no—
72C4⋊Dic53C2312yes—
73C4×Dic53—21no—
74#11719917—41no—
75C2×#88322—41no—
76C2×#88322C2312yes—
77C2×#88322C2312yes—
78C23×C53—22no—
79C23×C53—42no—
80C23×C53—82no—
81C23×C53C2×D411yescharacteristic
82C23×C53C2×D411yescharacteristic
83C23×C53C2×D411yescharacteristic
84C22×D53—22no—
85C22×D53—42no—
86C22×D53—82no—
87Q8×C53—41no—
88D4×C53—22no—
89D4×C53—22no—
90D4×C53—41no—
91D4×C53—41no—
92D4×C53—41no—
93D4×C53—42no—
94D4×C53—82no—
95D4×C53—82no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—82no—
108C22⋊D53—82no—
109C2×Dic53—22no—
110D212—22no—
111D212—22no—
112D212—42no—
113#88322—22no—
114#88322—22no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—22no—
120C22×C53—22no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—84no—
125C22×C53C22×D411yescharacteristic
126C2×D53—42no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—84no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—21no—
133C4×C53—42no—
134C4×C53H(5,𝔽2)11yescharacteristic
135C4×C53H(5,𝔽2)11yescharacteristic
136Dic53—42no—
137#3199—531nomaximal, Sylow
138C2×C53—41no—
139C2×C53—41no—
140C2×C53—41no—
141C2×C53—82no—
142C2×C53#53411yescharacteristic
143C2×C53#57612yes—
144D53—82no—
145#535—531no—
146#535D5311yescharacteristic, complemented
147#482—532no—
148#482—532no—
149#469—532no—
150#466—532no—
151#447—532no—
152#447—532no—
153#408—531no—
154C53#319911yescharacteristic, complemented, Sylow
155C2×#41C2×D5311yescharacteristic
156C22×D4—531no—
157C22×D4C2×D5311yescharacteristic, complemented
158C2×D8—532no—
159C2×D8—532no—
160C2×M16—531no—
161C22×C8—531no—
162#127—21no—
163#127—1061no—
164#124—21no—
165#124—1061no—
166#121—532no—
167#121—532no—
168#121C2×D5312yes—
169#121C2×D5312yescomplemented
170#120—21no—
171#120—21no—
172#120—1061no—
173#120—1061no—
174#117—531no—
175#117C2×D5311yescharacteristic, complemented
176#107—532no—
177#106—532no—
178#102—532no—
179#102—532no—
180#102—532no—
181#102—532no—
182C22⋊C8—532no—
183C22⋊C8—532no—
184C24—21no—
185C24—1061no—
186#41—41no—
187#41—41no—
188C2×Q8C22⋊D5311yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—532no—
195C2×D4—1062no—
196C2×D4—1062no—
197C2×D4—1061no—
198C2×D4—2121no—
199C2×D4—2121no—
200C2×D4C22⋊D5311yescharacteristic
201C2×D4C22⋊D5311yescharacteristic
202C2×D4C22⋊D5311yescharacteristic
203C2×D4C22×D5312yes—
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5311yescharacteristic
206C22×C4C22×D5311yescharacteristic
207D8—2122no—
208D8—2122no—
209M16—2121no—
210C2×C8—532no—
211C2×C8—532no—
212C2×C8—2121no—
213C4⋊C4—532no—
214C4⋊C4C22×D5312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1061no—
219C22⋊C4—1062no—
220C22⋊C4—1062no—
221C42—21no—
222C42—1061no—
223C23—22no—
224C23—42no—
225C23—82no—
226C23—1062no—
227C23—4242no—
228C23—2122no—
229C23C2×C22⋊D5311yescharacteristic
230C23C2×C22⋊D5311yescharacteristic
231C23C2×C22⋊D5311yescharacteristic
232Q8—41no—
233D4—22no—
234D4—22no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—42no—
239D4—82no—
240D4—82no—
241D4—1062no—
242D4—1062no—
243D4—4242no—
244D4—4242no—
245D4—2122no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—1062no—
254C2×C4C2×C22⋊D5311yescharacteristic
255C2×C4C2×C22⋊D5311yescharacteristic
256C2×C4C2×C22⋊D5311yescharacteristic
257C2×C4C23×D5311yescharacteristic
258C8—1062no—
259C8—1062no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—84no—
270C22—4244no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22C22×C22⋊D5311yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—2122no—
280C4#1313326211yescharacteristic
281C4#1313326211yescharacteristic
282C2—41no—
283C2—41no—
284C2—41no—
285C2—82no—
286C2—4242no—
287C2#1681118211yescharacteristic
288C2#1681142612yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#535C2
socleC22×C53C22×D4
hypercenter#535D53
hypocenterC53#3199
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16811187C2semidirect
2. #16810947C2semidirect
3. #16810944C2semidirect
4. #16810874C2semidirect
5. #16810857C2semidirect
6. #16810757C2semidirect
7. #16810756C2semidirect
8. #16810590C2semidirect
9. C53×#535C2semidirect
10. C22×D212C22semidirect
13. C2×#11719917C22semidirect
14. C22×#88322C22semidirect
18. #13133191C22semidirect
21. #13133176C22semidirect
22. #13133150C22semidirect
23. #13133148C22semidirect
24. #13133138C22semidirect
25. #13133138C22semidirect
26. #13133137C22semidirect
27. #13133137C22semidirect
28. C22⋊#88322C22semidirect
29. C22⋊#88322C22semidirect
31. C22×D4×C53C22semidirect
35. #121×C53C22semidirect
38. #117×C53C22semidirect
146. #535D53semidirect
154. C53#3199semidirect
157. C22×D4C2×D53semidirect
169. #121C2×D53semidirect
175. #117C2×D53semidirect