Finite Group Databasebeta

← #20284064

Subgroups — #20284064

Order 6784.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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#16811421C212yesmaximal, complemented
2#16811169C211yesmaximal, characteristic, complemented
3#16810806C212yesmaximal, complemented
4#16810801C212yesmaximal, complemented
5#16810796C212yesmaximal, complemented
6#16810791C212yesmaximal, complemented
7C2×#13133144C212yesmaximal, complemented
8#16810607C211yesmaximal, characteristic, complemented
9C53×#585C211yesmaximal, characteristic, complemented
10C2×#11719945C2211yescharacteristic, complemented
11#13133248—22no—
12#13133247C2214yescomplemented
13#13133243—22no—
14#13133242C2214yescomplemented
15#13133237—24no—
16C2×#11719917C2212yescomplemented
17#13133215C2211yescharacteristic, complemented
18#13133213C2211yescharacteristic, complemented
19#13133206C2211yescharacteristic, complemented
20#13133204C2211yescharacteristic, complemented
21#13133189—22no—
22C4×C22⋊D53—22no—
23C2×C4×Dic53C2211yescharacteristic, complemented
24#13133144C2214yescomplemented
25#13133144C2214yescomplemented
26#13133136C2212yescomplemented
27#13133134C2212yescomplemented
282−1+4×C53—22no—
292−1+4×C53C2212yes—
30H(5,𝔽2)×C53—22no—
31H(5,𝔽2)×C53—22no—
32H(5,𝔽2)×C53C2212yes—
33C2×#41×C53—24no—
34C2×#41×C53—22no—
35C2×#41×C53C2212yes—
36C2×#41×C53C2211yescharacteristic
37#41×C53—24no—
38#41×C53—28no—
39#41×C53—24no—
40#41×C53—24no—
41#41×C53—24no—
42#41×C53—22no—
43#41×C53—22no—
44#41×C53—24no—
45#41×C53—22no—
46#41×C53—22no—
47#41×C53D414yescomplemented
48#41×C53C2314yes—
49C2×Q8×C53—22no—
50C2×Q8×C53—24no—
51C2×Q8×C53D412yescomplemented
52C2×Q8×C53C2311yescharacteristic
53C2×D4×C53—28no—
54C2×D4×C53—22no—
55C2×D4×C53—24no—
56C2×D4×C53—22no—
57C2×D4×C53—24no—
58C2×D4×C53D412yes—
59C2×D4×C53D412yescomplemented
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—24no—
62C22×C4×C53—22no—
63C22×C4×C53D412yes—
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—22no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68#11719945—22no—
69#11719945—22no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×Dic106C2311yescharacteristic
73C22⋊Dic53—22no—
74#11719925—24no—
75#11719924—24no—
76#11719923—24no—
77#11719922—24no—
78#11719921—22no—
79#11719919—22no—
80C4×Dic53—21no—
81C4×Dic53C2311yescharacteristic
82C4×Dic53C2311yescharacteristic
83#11719917—22no—
84#11719917C2314yes—
85C2×#88322—22no—
86C23×C53—24no—
87C23×C53—22no—
88C23×C53C2×D412yes—
89C23×C53C2×D411yescharacteristic
90C22×D53—21no—
91Q8×C53—24no—
92Q8×C53—22no—
93Q8×C53—22no—
94Q8×C53C2×D414yes—
95D4×C53—28no—
96D4×C53—24no—
97D4×C53—24no—
98D4×C53—24no—
99D4×C53—24no—
100D4×C53—22no—
101D4×C53—22no—
102D4×C53C2×D414yes—
103C2×C4×C53—24no—
104C2×C4×C53—28no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—24no—
110C2×C4×C53C2×D412yes—
111C2×C4×C53C2×D411yescharacteristic
112C2×C4×C53C2×D412yes—
113C2×C4×C53C2×D412yes—
114C2×C4×C53C2411yescharacteristic
115C22⋊D53—44no—
116C22⋊D53—42no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—42no—
121D212—22no—
122C4×D53—41no—
123C4×D53—41no—
124Dic106—22no—
125#88322—24no—
126C22×C53—22no—
127C22×C53—24no—
128C22×C53—41no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53—84no—
132C22×C53#12012yes—
133C22×C53C22×D411yescharacteristic
134C2×D53—42no—
135C2×D53—41no—
136C4×C53—24no—
137C4×C53—22no—
138C4×C53#12011yescharacteristic
139C4×C53#12011yescharacteristic
140C4×C53C22×D412yes—
141Dic53—22no—
142Dic53—41no—
143Dic53—41no—
144#3018—531nomaximal, Sylow
145C2×C53—21no—
146C2×C53—22no—
147C2×C53—42no—
148C2×C53—44no—
149C2×C53C2×#12011yescharacteristic
150D53—81no—
151#585D5311yescharacteristic, complemented
152#575—532no—
153#532—531no—
154#456—532no—
155#455—532no—
156#454—532no—
157#453—532no—
158C2×#104—532no—
159#413—531no—
160C53#301811yescharacteristic, complemented, Sylow
1612−1+4—22no—
1622−1+4C2×D5312yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D5312yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—531no—
169C2×#41C2×D5312yes—
170C2×#41C2×D5311yescharacteristic
171#137—534no—
172#137—1062no—
173AGL(1,ℤ/8)—534no—
174AGL(1,ℤ/8)—1062no—
175#135—1064no—
176C2×M16—532no—
177C4⋊Q8—531no—
178#127—531no—
179#124—531no—
180#124—531no—
181#121—1062no—
182C4×D4—1062no—
183C2×C42—531no—
184#104—534no—
185#104—534no—
186#101—532no—
187#100—532no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—1062no—
199#41—1062no—
200#41C22⋊D5314yescomplemented
201#41C22×D5314yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—531no—
205C2×Q8C22⋊D5312yescomplemented
206C2×Q8C22×D5311yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—531no—
213C2×D4—1062no—
214C2×D4—1061no—
215C2×D4C22⋊D5312yes—
216C2×D4C22⋊D5312yescomplemented
217C2×D4C22×D5311yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4C22⋊D5312yes—
223C22×C4C22×D5311yescharacteristic
224Q16—1064no—
225SD16—1064no—
226SD16—1064no—
227D8—1064no—
228M16—534no—
229M16—1062no—
230C2×C8—1062no—
231C4⋊C4—1062no—
232C22⋊C4—1062no—
233C22⋊C4—1062no—
234C42—531no—
235C42—531no—
236C42—1061no—
237C23—24no—
238C23—22no—
239C23—1061no—
240C23C2×C22⋊D5312yes—
241C23C2×C22⋊D5311yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—1062no—
246Q8C2×C22⋊D5314yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—1062no—
255D4—2124no—
256D4—2122no—
257D4C2×C22⋊D5314yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4C2×C22⋊D5312yes—
272C2×C4C2×C22⋊D5311yescharacteristic
273C2×C4C2×C22⋊D5312yes—
274C2×C4C2×C22⋊D5312yes—
275C2×C4C23×D5311yescharacteristic
276C8—1064no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—2122no—
284C22—2121no—
285C22#1313318612yes—
286C22C22×C22⋊D5311yescharacteristic
287C4—24no—
288C4—22no—
289C4—1062no—
290C4—2121no—
291C4—2121no—
292C4#1313318611yescharacteristic
293C4#1313318611yescharacteristic
294C4C22×C22⋊D5312yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—4241no—
300C2C2×#1313318611yescharacteristic
subgroupG/H
centerC4#13133186
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#585C2
socleC2×C53C2×#120
hypercenter#585D53
hypocenterC53#3018
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #16811421C2semidirect
2. #16811169C2semidirect
3. #16810806C2semidirect
4. #16810801C2semidirect
5. #16810796C2semidirect
6. #16810791C2semidirect
7. C2×#13133144C2semidirect
8. #16810607C2semidirect
9. C53×#585C2semidirect
10. C2×#11719945C22semidirect
12. #13133247C22semidirect
14. #13133242C22semidirect
16. C2×#11719917C22semidirect
17. #13133215C22semidirect
18. #13133213C22semidirect
19. #13133206C22semidirect
20. #13133204C22semidirect
23. C2×C4×Dic53C22semidirect
24. #13133144C22semidirect
25. #13133144C22semidirect
26. #13133136C22semidirect
27. #13133134C22semidirect
47. #41×C53D4semidirect
51. C2×Q8×C53D4semidirect
59. C2×D4×C53D4semidirect
151. #585D53semidirect
160. C53#3018semidirect
200. #41C22⋊D53semidirect
205. C2×Q8C22⋊D53semidirect
216. C2×D4C22⋊D53semidirect