Finite Group Databasebeta

← #20285551

Subgroups — #20285551

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#16811133C212yesmaximal, complemented
2#16810967C212yesmaximal, complemented
3#16810946C212yesmaximal, complemented
4#16810881C211yesmaximal, characteristic, complemented
5#16810867C212yesmaximal, complemented
6#16810786C212yesmaximal, complemented
7#16810765C212yesmaximal, complemented
8#16810586C211yesmaximal, characteristic, complemented
9C53×#469C211yesmaximal, characteristic, complemented
10C2×#11719947C2212yes—
11C22×C4×D53C2211yescharacteristic
12C2×Dic212C2211yescharacteristic, complemented
13C2×#11719923C2212yescomplemented
14C2×#11719917C2211yescharacteristic
15#13133199—22no—
16#13133197C2212yes—
17#13133196C2212yes—
18#13133188C2212yes—
19#13133185—22no—
20D4×Dic53—22no—
21#13133177—22no—
22#13133175C2212yes—
23#13133150C2212yescomplemented
24#13133149C2211yescharacteristic, complemented
25#13133142C2212yescomplemented
26#13133140C2212yescomplemented
27#13133138C2212yescomplemented
28#13133129C2212yes—
29#13133129C2212yes—
30C2×D8×C53C2211yescharacteristic, complemented
31C2×M16×C53C2211yescharacteristic
32#121×C53C2212yes—
33#106×C53C2211yescharacteristic, complemented
34#102×C53C2212yescomplemented
35C23×D53D411yescharacteristic
36C2×D4×C53—22no—
37C2×D4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—22no—
40C2×C22⋊D53—22no—
41C22×Dic53—22no—
42C22×Dic53D411yescharacteristic
43#11719947—42no—
44#11719947—42no—
45C2×C4×D53—41no—
46C2×C4×D53—41no—
47C2×C4×D53D412yes—
48C2×C4×D53C2312yes—
49C2×Dic106C2312yes—
50D8×C53—41no—
51M16×C53—41no—
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—22no—
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58#11719923—42no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53C2312yes—
63#11719919—22no—
64C4×Dic53—22no—
65#11719917—41no—
66C2×#88322C2312yes—
67Dic212—41no—
68C23×C53—22no—
69C23×C53C2×D411yescharacteristic
70C22×D53—41no—
71C22×D53—42no—
72C22×D53C2×D412yes—
73D4×C53—22no—
74D4×C53—22no—
75D4×C53—44no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C22⋊D53—44no—
82C22⋊D53—44no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—41no—
86C2×Dic53—44no—
87C2×Dic53C2×D412yes—
88C4×D53—22no—
89C4×D53—22no—
90C4×D53—42no—
91Dic106—22no—
92Dic106—22no—
93C8×C53—21no—
94C8×C53—21no—
95#88322—22no—
96C22×C53—21no—
97C22×C53—22no—
98C22×C53—42no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53C22×D411yescharacteristic
102C2×D53—22no—
103C2×D53—22no—
104C2×D53—22no—
105C2×D53—44no—
106C4×C53—21no—
107C4×C53—42no—
108C4×C53H(5,𝔽2)11yescharacteristic
109C4×C53H(5,𝔽2)11yescharacteristic
110Dic53—22no—
111Dic53—42no—
112Dic53—42no—
113#3197—531nomaximal, Sylow
114C2×C53—41no—
115C2×C53—82no—
116C2×C53#53411yescharacteristic
117C2×C53#57512yes—
118D53—42no—
119#525—532no—
120#485—532no—
121#482—532no—
122#470—531no—
123#469D5311yescharacteristic, complemented
124#468—532no—
125#452—532no—
126#449—532no—
127#407—531no—
128C53#319711yescharacteristic, complemented, Sylow
129C2×#41—532no—
130C23×C4—531no—
131C2×Q16—531no—
132C2×SD16—532no—
133C2×D8C2×D5311yescharacteristic, complemented
134C2×M16—531no—
135C2×M16C2×D5311yescharacteristic
136#123—1062no—
137C22⋊Q8—532no—
138C22⋊Q8—532no—
139#121—532no—
140#121C2×D5312yes—
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#103—532no—
150#102—532no—
151#102C2×D5312yescomplemented
152C22⋊C8—532no—
153C22⋊C8—532no—
154C24—531no—
155#41—2122no—
156#41—2122no—
157C2×Q8—532no—
158C2×D4—22no—
159C2×D4—1062no—
160C2×D4—1062no—
161C2×D4C22×D5312yes—
162C22×C4—531no—
163C22×C4—532no—
164C22×C4—532no—
165C22×C4—1062no—
166C22×C4—2121no—
167C22×C4—2121no—
168C22×C4C22×D5311yescharacteristic
169Q16—2121no—
170SD16—2122no—
171D8—41no—
172M16—41no—
173M16—2121no—
174C2×C8—532no—
175C2×C8C22×D5311yescharacteristic
176C2×C8C22×D5311yescharacteristic
177C4⋊C4—532no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4C22×D5312yes—
181C22⋊C4—22no—
182C22⋊C4—1062no—
183C22⋊C4—1062no—
184C22⋊C4—1062no—
185C22⋊C4—1062no—
186C42—1062no—
187C23—22no—
188C23—532no—
189C23—2121no—
190C23—2122no—
191C23D4×D5311yescharacteristic
192Q8—1062no—
193Q8—1062no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—2124no—
198D4—2124no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—532no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2122no—
208C2×C4—2121no—
209C2×C4—2124no—
210C2×C4D4×D5311yescharacteristic
211C2×C4C23×D5311yescharacteristic
212C8—21no—
213C8—21no—
214C8—1062no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—1062no—
221C22—1062no—
222C22—1062no—
223C22—2124no—
224C22C2×D4×D5311yescharacteristic
225C4—21no—
226C4—42no—
227C4—1062no—
228C4—2122no—
229C4—2122no—
230C4#1313326211yescharacteristic
231C4#1313326311yescharacteristic
232C2—41no—
233C2—82no—
234C2—2122no—
235C2#1681118111yescharacteristic
236C2#1681141912yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#469C2
socleC22×C53C22×D4
hypercenter#469D53
hypocenterC53#3197
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16811133C2semidirect
2. #16810967C2semidirect
3. #16810946C2semidirect
4. #16810881C2semidirect
5. #16810867C2semidirect
6. #16810786C2semidirect
7. #16810765C2semidirect
8. #16810586C2semidirect
9. C53×#469C2semidirect
12. C2×Dic212C22semidirect
13. C2×#11719923C22semidirect
23. #13133150C22semidirect
24. #13133149C22semidirect
25. #13133142C22semidirect
26. #13133140C22semidirect
27. #13133138C22semidirect
30. C2×D8×C53C22semidirect
33. #106×C53C22semidirect
34. #102×C53C22semidirect
123. #469D53semidirect
128. C53#3197semidirect
133. C2×D8C2×D53semidirect
147. #106C2×D53semidirect
151. #102C2×D53semidirect