Finite Group Databasebeta

← #20285557

Subgroups — #20285557

Order 6784.

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

normal subgroups only

This group has 418 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#16811134C212yesmaximal, complemented
2#16810965C212yesmaximal, complemented
3#16810948C212yesmaximal, complemented
4#16810881C211yesmaximal, characteristic, complemented
5#16810876C211yesmaximal, characteristic, complemented
6#16810869C212yesmaximal, complemented
7#16810787C212yesmaximal, complemented
8#16810770C212yesmaximal, complemented
9C53×#407C211yesmaximal, characteristic, complemented
10C2×#11719945C2212yes—
11C2×Dic212C2211yescharacteristic, complemented
12C2×#11719913C2212yescomplemented
13C2×D424C2211yescharacteristic, complemented
14#13133200—22no—
15#13133196C2212yes—
16#13133196C2212yes—
17#13133191C2212yes—
18#13133191C2212yes—
19#13133186—22no—
20C4×C22⋊D53—22no—
21C4×C22⋊D53—22no—
22#13133176C2212yes—
23#13133151C2212yescomplemented
24#13133149C2211yescharacteristic, complemented
25#13133149C2211yescharacteristic, complemented
26#13133142C2212yescomplemented
27#13133142C2212yescomplemented
28#13133139C2212yescomplemented
29#13133139C2212yescomplemented
30C23×C4×C53C2211yescharacteristic
31C2×M16×C53C2211yescharacteristic
32C2×M16×C53C2211yescharacteristic
33C22⋊C8×C53C2212yes—
34C22⋊C8×C53C2212yes—
35C24×C53D411yescharacteristic
36C22×C4×C53—41no—
37C22×C4×C53—41no—
38C22×C4×C53D411yescharacteristic
39C22×C4×C53D412yes—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C2×C22⋊D53—22no—
43C2×C22⋊D53—22no—
44C2×C22⋊D53—22no—
45#11719945—42no—
46#11719945—42no—
47C2×D212C2312yes—
48C2×C4×D53—22no—
49C2×Dic106C2312yes—
50M16×C53—41no—
51M16×C53—41no—
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2312yes—
54C2×C8×C53C2311yescharacteristic
55C22⋊Dic53—22no—
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58#11719921—22no—
59#11719921—22no—
60C4⋊Dic53C2312yes—
61C4⋊Dic53C2312yes—
62#11719919—22no—
63#11719919—22no—
64C4×Dic53—22no—
65Dic212—41no—
66D424—41no—
67#11719913—42no—
68C23×C53—41no—
69C23×C53—42no—
70C23×C53C2×D412yes—
71C23×C53C2×D411yescharacteristic
72C22×D53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53—42no—
79C2×C4×C53C2×D412yes—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—44no—
83C22⋊D53—44no—
84C22⋊D53—44no—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88D212—22no—
89D212—22no—
90C4×D53—44no—
91Dic106—22no—
92Dic106—22no—
93C8×C53—22no—
94C8×C53—21no—
95C8×C53—21no—
96C22×C53—22no—
97C22×C53—21no—
98C22×C53—22no—
99C22×C53—22no—
100C22×C53—22no—
101C22×C53—44no—
102C22×C53C22×D411yescharacteristic
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—42no—
106C4×C53—22no—
107C4×C53—21no—
108C4×C53H(5,𝔽2)11yescharacteristic
109C4×C53H(5,𝔽2)11yescharacteristic
110Dic53—42no—
111Dic53—42no—
112Dic53—42no—
113#3197—531nomaximal, Sylow
114C2×C53—42no—
115C2×C53—41no—
116C2×C53#53411yescharacteristic
117C2×C53#57512yes—
118D53—82no—
119#525—532no—
120#485—532no—
121#482—532no—
122#470—531no—
123#469—531no—
124#468—532no—
125#452—532no—
126#449—532no—
127#407D5311yescharacteristic, complemented
128C53#319711yescharacteristic, complemented, Sylow
129C2×#41—532no—
130C23×C4C2×D5311yescharacteristic
131C2×Q16—531no—
132C2×SD16—532no—
133C2×D8—531no—
134C2×M16C2×D5311yescharacteristic
135C2×M16C2×D5311yescharacteristic
136#123—1062no—
137C22⋊Q8—532no—
138C22⋊Q8—532no—
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#106—531no—
148#103—532no—
149#103—532no—
150#102—532no—
151#102—532no—
152C22⋊C8C2×D5312yes—
153C22⋊C8C2×D5312yes—
154C24D21211yescharacteristic
155#41—2122no—
156#41—2122no—
157C2×Q8—532no—
158C2×D4—532no—
159C2×D4—1062no—
160C2×D4—1062no—
161C2×D4—1062no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—1062no—
165C22×C4D21211yescharacteristic
166C22×C4D21212yes—
167C22×C4C22×D5312yes—
168C22×C4C22×D5311yescharacteristic
169Q16—2121no—
170SD16—2122no—
171D8—2121no—
172M16—41no—
173M16—41no—
174C2×C8C22×D5311yescharacteristic
175C2×C8C22×D5312yes—
176C2×C8C22×D5311yescharacteristic
177C4⋊C4—532no—
178C4⋊C4—532no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—1062no—
187C23—41no—
188C23—42no—
189C23—1062no—
190C23C2×D21211yescharacteristic
191C23C2×D21212yes—
192Q8—1062no—
193Q8—1062no—
194D4—1062no—
195D4—1062no—
196D4—2124no—
197D4—2124no—
198D4—2124no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—2124no—
209C2×C4C2×D21212yes—
210C2×C4C2×D21211yescharacteristic
211C2×C4C23×D5311yescharacteristic
212C8—21no—
213C8—22no—
214C8—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—44no—
221C22—2122no—
222C22—2122no—
223C22—2122no—
224C22C22×D21211yescharacteristic
225C4—22no—
226C4—21no—
227C4—2122no—
228C4—2122no—
229C4—2122no—
230C4#1313326211yescharacteristic
231C4#1313326211yescharacteristic
232C2—41no—
233C2—42no—
234C2—4242no—
235C2#1681118511yescharacteristic
236C2#1681142312yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16811134C2semidirect
2. #16810965C2semidirect
3. #16810948C2semidirect
4. #16810881C2semidirect
5. #16810876C2semidirect
6. #16810869C2semidirect
7. #16810787C2semidirect
8. #16810770C2semidirect
9. C53×#407C2semidirect
11. C2×Dic212C22semidirect
12. C2×#11719913C22semidirect
13. C2×D424C22semidirect
23. #13133151C22semidirect
24. #13133149C22semidirect
25. #13133149C22semidirect
26. #13133142C22semidirect
27. #13133142C22semidirect
28. #13133139C22semidirect
29. #13133139C22semidirect
127. #407D53semidirect
128. C53#3197semidirect