Finite Group Databasebeta

← #20285558

Subgroups — #20285558

Order 6784.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811136C212yesmaximal, complemented
2#16810962C212yesmaximal, complemented
3#16810949C212yesmaximal, complemented
4#16810876C211yesmaximal, characteristic, complemented
5#16810871C212yesmaximal, complemented
6#16810784C212yesmaximal, complemented
7#16810771C212yesmaximal, complemented
8#16810586C211yesmaximal, characteristic, complemented
9C53×#470C211yesmaximal, characteristic, complemented
10C2×#11719949C2212yes—
11C22×C4×D53C2211yescharacteristic
12C2×#11719924C2212yescomplemented
13C2×D424C2211yescharacteristic, complemented
14C2×#11719917C2211yescharacteristic
15#13133202—22no—
16#13133193C2212yes—
17#13133192C2212yes—
18#13133191C2212yes—
19#13133187—22no—
20#13133179—22no—
21#13133177—22no—
22#13133175C2212yes—
23#13133150C2212yescomplemented
24#13133149C2211yescharacteristic, complemented
25#13133141C2212yescomplemented
26#13133139C2212yescomplemented
27#13133137C2212yescomplemented
28#13133129C2212yes—
29#13133129C2212yes—
30C2×Q16×C53C2211yescharacteristic, complemented
31C2×M16×C53C2211yescharacteristic
32C22⋊Q8×C53C2212yes—
33#106×C53C2211yescharacteristic, complemented
34#103×C53C2212yescomplemented
35C23×D53D411yescharacteristic
36C2×Q8×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39C22×Dic53D411yescharacteristic
40#11719949—42no—
41#11719949—42no—
42C2×D212—22no—
43C2×D212—22no—
44C2×D212C2312yes—
45C2×C4×D53—22no—
46C2×C4×D53—41no—
47C2×C4×D53—41no—
48C2×C4×D53D412yes—
49C2×C4×D53C2312yes—
50Q16×C53—41no—
51M16×C53—41no—
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C4⋊C4×C53—22no—
55C4⋊C4×C53C2312yes—
56C22⋊C4×C53—22no—
57#11719924—42no—
58#11719921—22no—
59#11719921—22no—
60#11719921—22no—
61#11719921—22no—
62C4⋊Dic53C2312yes—
63#11719919—22no—
64C4×Dic53—22no—
65#11719917—41no—
66C2×#88322C2312yes—
67D424—41no—
68C23×C53C2×D411yescharacteristic
69C22×D53—22no—
70C22×D53—41no—
71C22×D53—42no—
72C22×D53C2×D412yes—
73Q8×C53—22no—
74Q8×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C22⋊D53—44no—
82C2×Dic53—22no—
83C2×Dic53—41no—
84C2×Dic53C2×D412yes—
85D212—22no—
86D212—22no—
87D212—44no—
88D212—44no—
89C4×D53—22no—
90C4×D53—22no—
91C4×D53—42no—
92C4×D53—44no—
93C8×C53—21no—
94C8×C53—21no—
95#88322—22no—
96C22×C53—21no—
97C22×C53—22no—
98C22×C53C22×D411yescharacteristic
99C2×D53—22no—
100C2×D53—22no—
101C2×D53—22no—
102C2×D53—44no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C4×C53—21no—
107C4×C53—42no—
108C4×C53—42no—
109C4×C53H(5,𝔽2)11yescharacteristic
110C4×C53H(5,𝔽2)11yescharacteristic
111Dic53—22no—
112Dic53—42no—
113#3197—531nomaximal, Sylow
114C2×C53—41no—
115C2×C53#53411yescharacteristic
116C2×C53#57512yes—
117D53—42no—
118D53—82no—
119#525—532no—
120#485—532no—
121#482—532no—
122#470D5311yescharacteristic, complemented
123#469—531no—
124#468—532no—
125#452—532no—
126#449—532no—
127#407—531no—
128C53#319711yescharacteristic, complemented, Sylow
129C2×#41—532no—
130C23×C4—531no—
131C2×Q16C2×D5311yescharacteristic, complemented
132C2×SD16—532no—
133C2×D8—531no—
134C2×M16—531no—
135C2×M16C2×D5311yescharacteristic
136#123—1062no—
137C22⋊Q8—532no—
138C22⋊Q8C2×D5312yes—
139#121—532no—
140#121—532no—
141#120—1062no—
142C4×D4—1062no—
143C4×D4—1062no—
144#117—532no—
145#107—532no—
146#106—531no—
147#106C2×D5311yescharacteristic, complemented
148#103—532no—
149#103C2×D5312yescomplemented
150#102—532no—
151#102—532no—
152C22⋊C8—532no—
153C22⋊C8—532no—
154C24—531no—
155#41—2122no—
156#41—2122no—
157C2×Q8C22×D5312yes—
158C2×D4—532no—
159C2×D4—1062no—
160C2×D4—1062no—
161C2×D4—1062no—
162C22×C4—531no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1062no—
166C22×C4—2121no—
167C22×C4—2121no—
168C22×C4C22×D5311yescharacteristic
169Q16—41no—
170SD16—2122no—
171D8—2121no—
172M16—41no—
173M16—2121no—
174C2×C8—532no—
175C2×C8C22×D5311yescharacteristic
176C2×C8C22×D5311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—532no—
179C4⋊C4—1062no—
180C4⋊C4C22×D5312yes—
181C22⋊C4—22no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—1062no—
187C23—532no—
188C23—1062no—
189C23—2121no—
190C23—2122no—
191C23D4×D5311yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—1062no—
195D4—1062no—
196D4—2124no—
197D4—2124no—
198D4—2124no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—532no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2122no—
208C2×C4—2124no—
209C2×C4—2121no—
210C2×C4D4×D5311yescharacteristic
211C2×C4C23×D5311yescharacteristic
212C8—21no—
213C8—21no—
214C8—1062no—
215C22—22no—
216C22—21no—
217C22—1062no—
218C22—1062no—
219C22—1062no—
220C22—2124no—
221C22—2122no—
222C22—2122no—
223C22—2122no—
224C22C2×D4×D5311yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—1062no—
229C4—2122no—
230C4#1313326211yescharacteristic
231C4#1313326311yescharacteristic
232C2—41no—
233C2—4242no—
234C2—2122no—
235C2#1681118111yescharacteristic
236C2#1681142212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#470C2
socleC22×C53C22×D4
hypercenter#470D53
hypocenterC53#3197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #16811136C2semidirect
2. #16810962C2semidirect
3. #16810949C2semidirect
4. #16810876C2semidirect
5. #16810871C2semidirect
6. #16810784C2semidirect
7. #16810771C2semidirect
8. #16810586C2semidirect
9. C53×#470C2semidirect
12. C2×#11719924C22semidirect
13. C2×D424C22semidirect
23. #13133150C22semidirect
24. #13133149C22semidirect
25. #13133141C22semidirect
26. #13133139C22semidirect
27. #13133137C22semidirect
30. C2×Q16×C53C22semidirect
33. #106×C53C22semidirect
34. #103×C53C22semidirect
122. #470D53semidirect
128. C53#3197semidirect
131. C2×Q16C2×D53semidirect
147. #106C2×D53semidirect
149. #103C2×D53semidirect