Finite Group Databasebeta

← #20284087

Subgroups — #20284087

Order 6784.

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

normal subgroups only

This group has 736 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
1C2×#13133242C212yesmaximal, complemented
2#16811187C211yesmaximal, characteristic, complemented
3#16810796C214yesmaximal, complemented
4#16810791C214yesmaximal, complemented
5#16810640C212yesmaximal, complemented
6#16810607C211yesmaximal, characteristic, complemented
7C53×#585C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic, complemented
9#13133242—24no—
10#13133242C2218yescomplemented
11C2×#11719924—22no—
12C2×#11719922—22no—
13C2×#11719917C2212yescomplemented
14#13133213C2212yescomplemented
15#13133206C2212yescomplemented
16#13133191—22no—
17#13133185—22no—
18#13133176C2211yescharacteristic, complemented
19#13133144C2218yescomplemented
20#13133136C2212yescomplemented
21#13133134C2212yescomplemented
222−1+4×C53—22no—
232−1+4×C53C2212yescomplemented
24H(5,𝔽2)×C53—22no—
25H(5,𝔽2)×C53—22no—
26H(5,𝔽2)×C53C2212yescomplemented
27C2×#41×C53—24no—
28C2×#41×C53—22no—
29C2×#41×C53C2212yes—
30C2×#41×C53C2211yescharacteristic
31C23×D53—21no—
32#41×C53—24no—
33#41×C53—28no—
34#41×C53—24no—
35#41×C53—24no—
36#41×C53—24no—
37#41×C53—22no—
38#41×C53—22no—
39#41×C53—24no—
40#41×C53—22no—
41#41×C53—22no—
42#41×C53D414yescomplemented
43#41×C53C2314yes—
44C2×Q8×C53—22no—
45C2×Q8×C53—24no—
46C2×Q8×C53D412yescomplemented
47C2×Q8×C53C2311yescharacteristic
48C2×D4×C53—28no—
49C2×D4×C53—22no—
50C2×D4×C53—24no—
51C2×D4×C53—22no—
52C2×D4×C53—24no—
53C2×D4×C53D412yes—
54C2×D4×C53D412yescomplemented
55C2×D4×C53C2311yescharacteristic
56C22×C4×C53—24no—
57C22×C4×C53—22no—
58C22×C4×C53D412yes—
59C22×C4×C53C2311yescharacteristic
60C2×C22⋊D53—24no—
61C2×D212—22no—
62C2×D212—22no—
63C2×D212—21no—
64C2×D212C2312yes—
65C22⋊Dic53—21no—
66#11719924—28no—
67#11719922—28no—
68#11719921—24no—
69C4⋊Dic53—21no—
70C4×Dic53C2312yes—
71#11719917—22no—
72#11719917C2314yes—
73C2×#88322—22no—
74C23×C53—24no—
75C23×C53—22no—
76C23×C53C2×D412yes—
77C23×C53C2×D411yescharacteristic
78C22×D53—22no—
79C22×D53—42no—
80C22×D53—82no—
81Q8×C53—24no—
82Q8×C53—22no—
83Q8×C53—22no—
84Q8×C53C2×D414yes—
85D4×C53—28no—
86D4×C53—24no—
87D4×C53—24no—
88D4×C53—24no—
89D4×C53—24no—
90D4×C53—22no—
91D4×C53—22no—
92D4×C53C2×D414yes—
93C2×C4×C53—24no—
94C2×C4×C53—28no—
95C2×C4×C53—22no—
96C2×C4×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—24no—
100C2×C4×C53C2×D412yes—
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2411yescharacteristic
105C22⋊D53—84no—
106C2×Dic53—22no—
107D212—24no—
108D212—41no—
109D212—41no—
110#88322—24no—
111C22×C53—22no—
112C22×C53—24no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—82no—
116C22×C53—84no—
117C22×C53#12012yes—
118C22×C53C22×D411yescharacteristic
119C2×D53—44no—
120C2×D53—42no—
121C2×D53—84no—
122C4×C53—24no—
123C4×C53—22no—
124C4×C53#12011yescharacteristic
125C4×C53#12011yescharacteristic
126C4×C53C22×D412yes—
127Dic53—42no—
128#3021—531nomaximal, Sylow
129C2×C53—21no—
130C2×C53—22no—
131C2×C53—42no—
132C2×C53—44no—
133C2×C53C2×#12011yescharacteristic
134D53—82no—
135#585D5311yescharacteristic, complemented
136C2×AGL(1,ℤ/8)—532no—
137#535—531no—
138#454—534no—
139#453—534no—
140#421—532no—
141#413—531no—
142C53#302111yescharacteristic, complemented, Sylow
1432−1+4—22no—
1442−1+4C2×D5312yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D5312yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D5312yes—
151C2×#41C2×D5311yescharacteristic
152C22×D4—531no—
153AGL(1,ℤ/8)—538no—
154AGL(1,ℤ/8)—1064no—
155C2×SD16—1062no—
156C2×D8—1062no—
157C2×M16—532no—
158#127—532no—
159#124—532no—
160#121—1062no—
161#120—1062no—
162#117—531no—
163#104—538no—
164#101—532no—
165#100—532no—
166C24—1061no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D5314yescomplemented
178#41C22×D5314yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D5312yescomplemented
182C2×Q8C22×D5311yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—532no—
189C2×D4—1064no—
190C2×D4—1062no—
191C2×D4—1062no—
192C2×D4—1061no—
193C2×D4C22⋊D5312yes—
194C2×D4C22⋊D5312yescomplemented
195C2×D4C22×D5311yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D5312yes—
199C22×C4C22×D5311yescharacteristic
200SD16—1068no—
201D8—1068no—
202M16—534no—
203M16—1062no—
204C2×C8—1062no—
205C4⋊C4—1061no—
206C22⋊C4—1061no—
207C22⋊C4—1064no—
208C42—532no—
209C23—24no—
210C23—22no—
211C23—1062no—
212C23—4242no—
213C23—2122no—
214C23C2×C22⋊D5312yes—
215C23C2×C22⋊D5311yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D5314yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—1064no—
228D4—4244no—
229D4—2121no—
230D4—2121no—
231D4C2×C22⋊D5314yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—1062no—
240C2×C4C2×C22⋊D5312yes—
241C2×C4C2×C22⋊D5311yescharacteristic
242C2×C4C2×C22⋊D5312yes—
243C2×C4C2×C22⋊D5312yes—
244C2×C4C23×D5311yescharacteristic
245C8—1064no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—4244no—
253C22—2124no—
254C22—2122no—
255C22#1313318612yes—
256C22C22×C22⋊D5311yescharacteristic
257C4—24no—
258C4—22no—
259C4—2122no—
260C4#1313318611yescharacteristic
261C4#1313318611yescharacteristic
262C4C22×C22⋊D5312yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—4242no—
268C2C2×#1313318611yescharacteristic
subgroupG/H
centerC2C2×#13133186
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#585C2
socleC2×C53C2×#120
hypercenter#585D53
hypocenterC53#3021
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#13133242C2semidirect
2. #16811187C2semidirect
3. #16810796C2semidirect
4. #16810791C2semidirect
5. #16810640C2semidirect
6. #16810607C2semidirect
7. C53×#585C2semidirect
8. C22×D212C22semidirect
10. #13133242C22semidirect
13. C2×#11719917C22semidirect
14. #13133213C22semidirect
15. #13133206C22semidirect
18. #13133176C22semidirect
19. #13133144C22semidirect
20. #13133136C22semidirect
21. #13133134C22semidirect
23. 2−1+4×C53C22semidirect
26. H(5,𝔽2)×C53C22semidirect
42. #41×C53D4semidirect
46. C2×Q8×C53D4semidirect
54. C2×D4×C53D4semidirect
135. #585D53semidirect
142. C53#3021semidirect
144. 2−1+4C2×D53semidirect
147. H(5,𝔽2)C2×D53semidirect
177. #41C22⋊D53semidirect
181. C2×Q8C22⋊D53semidirect
194. C2×D4C22⋊D53semidirect