Finite Group Databasebeta

← #20285704

Subgroups — #20285704

Order 6784.

444 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811173C211yesmaximal, characteristic, complemented
2C2×#13133205C211yesmaximal, characteristic, complemented
3#16811012C212yesmaximal, complemented
4#16810983C212yesmaximal, complemented
5#16810839C214yesmaximal, complemented
6#16810764C212yesmaximal, complemented
7#16810759C212yesmaximal, complemented
8C53×#432C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12C2×#11719913C2214yescomplemented
13C2×D424C2214yes—
14#13133217C2211yescharacteristic, complemented
15#13133214C2211yescharacteristic, complemented
16#13133205—21no—
17#13133205—21no—
18#13133205—21no—
19#13133205C2212yes—
20#13133205C2212yes—
21#13133191—21no—
22#13133191C2212yes—
23C4×D212—21no—
24C4×D212C2212yescomplemented
25C2×#11719921—21no—
26C2×#11719921—21no—
27#13133142C2214yescomplemented
28#13133139C2214yes—
29C2×C42×C53C2211yescharacteristic
30C4⋊C8×C53C2212yescomplemented
31C22⋊C8×C53C2212yes—
32#97×C53C2212yescomplemented
33C23×D53—21no—
34C22×C4×C53D411yescharacteristic
35C22×C4×C53D411yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—22no—
38C22×Dic53—21no—
39#11719945—44no—
40C2×D212—21no—
41C2×D212—21no—
42C2×D212—21no—
43C2×D212—42no—
44C2×D212C2312yes—
45C2×D212C2312yes—
46C2×C4×D53—22no—
47C2×Dic106—21no—
48C2×Dic106—21no—
49C2×Dic106—42no—
50C2×Dic106C2312yes—
51C2×C8×C53C2314yes—
52C42×C53D411yescharacteristic
53C42×C53D411yescharacteristic
54C42×C53C2311yescharacteristic
55C42×C53C2311yescharacteristic
56#11719921—22no—
57#11719921—22no—
58#11719921—22no—
59#11719921—22no—
60#11719921—22no—
61#11719921—22no—
62C4⋊Dic53—21no—
63C4⋊Dic53C2312yes—
64D424—44no—
65#11719913—44no—
66C23×C53C2×D411yescharacteristic
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—42no—
70C22×D53—42no—
71C22×D53—42no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—41no—
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2411yescharacteristic
81C22⋊D53—44no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—42no—
85D212—24no—
86D212—24no—
87D212—42no—
88D212—42no—
89D212—42no—
90C4×D53—44no—
91Dic106—24no—
92Dic106—42no—
93Dic106—42no—
94C8×C53—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53AGL(1,ℤ/8)12yes—
98C22×C53C22×D411yescharacteristic
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53—42no—
104C2×D53—42no—
105C2×D53—84no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53H(5,𝔽2)12yes—
112Dic53—42no—
113Dic53—42no—
114#3213—531nomaximal, Sylow
115C2×C53—22no—
116C2×C53#53511yescharacteristic
117C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
118C2×C53#57511yescharacteristic
119D53—82no—
120D53—82no—
121#532—531no—
122C2×#124—531no—
123#496—532no—
124#489—532no—
125#463—534no—
126#448—532no—
127#447—532no—
128#432D5311yescharacteristic, complemented
129C53#321311yescharacteristic, complemented, Sylow
130C2×#41—1061no—
131C22×Q8—531no—
132C22×D4—531no—
133C2×SD16—534no—
134C2×D8—534no—
135C4⋊Q8—531no—
136#127—531no—
137#124—532no—
138#124—532no—
139#124—1061no—
140#124—1061no—
141#124—1061no—
142#121—532no—
143#121—1061no—
144C4×D4—532no—
145C4×D4—1061no—
146C2×C22⋊C4—1061no—
147C2×C22⋊C4—1061no—
148C2×C42C2×D5311yescharacteristic
149C4⋊C8C2×D5312yescomplemented
150#103—534no—
151#102—534no—
152C22⋊C8C2×D5312yes—
153#97C2×D5312yescomplemented
154C24—1061no—
155#41—2124no—
156C2×Q8—532no—
157C2×Q8—1061no—
158C2×Q8—1061no—
159C2×Q8—2122no—
160C2×D4—532no—
161C2×D4—532no—
162C2×D4—1062no—
163C2×D4—1061no—
164C2×D4—1061no—
165C2×D4—1061no—
166C2×D4—2122no—
167C22×C4—1061no—
168C22×C4—1062no—
169C22×C4D21211yescharacteristic
170C22×C4D21211yescharacteristic
171C22×C4C22×D5311yescharacteristic
172SD16—2124no—
173D8—2124no—
174C2×C8C22×D5314yes—
175C4⋊C4—532no—
176C4⋊C4—1061no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C22⋊C4—1062no—
180C22⋊C4—1062no—
181C22⋊C4—1062no—
182C22⋊C4—1062no—
183C42D21211yescharacteristic
184C42D21211yescharacteristic
185C42C22×D5311yescharacteristic
186C42C22×D5311yescharacteristic
187C23—1062no—
188C23—1062no—
189C23—2122no—
190C23—2122no—
191C23—2122no—
192C23C2×D21211yescharacteristic
193Q8—1064no—
194Q8—2122no—
195Q8—2122no—
196D4—1064no—
197D4—1064no—
198D4—2122no—
199D4—2122no—
200D4—2122no—
201D4—2124no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—1062no—
206C2×C4—1062no—
207C2×C4—2124no—
208C2×C4—2122no—
209C2×C4C2×D21211yescharacteristic
210C2×C4C2×D21211yescharacteristic
211C2×C4C2×D21211yescharacteristic
212C2×C4C2×D21211yescharacteristic
213C2×C4C2×D21211yescharacteristic
214C2×C4C23×D5311yescharacteristic
215C8—24no—
216C22—21no—
217C22—21no—
218C22—4244no—
219C22—2122no—
220C22—2122no—
221C22—2122no—
222C22—2122no—
223C22—2122no—
224C22—2122no—
225C22#1313324512yes—
226C22C22×D21211yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—2122no—
233C4—2122no—
234C4#1313326312yes—
235C2—22no—
236C2—4242no—
237C2—4242no—
238C2#1681119211yescharacteristic
239C2C2×#1313324511yescharacteristic
240C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#432C2
socleC22×C53C22×D4
hypercenter#432D53
hypocenterC53#3213
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #16811173C2semidirect
2. C2×#13133205C2semidirect
3. #16811012C2semidirect
4. #16810983C2semidirect
5. #16810839C2semidirect
6. #16810764C2semidirect
7. #16810759C2semidirect
8. C53×#432C2semidirect
12. C2×#11719913C22semidirect
14. #13133217C22semidirect
15. #13133214C22semidirect
24. C4×D212C22semidirect
27. #13133142C22semidirect
30. C4⋊C8×C53C22semidirect
32. #97×C53C22semidirect
128. #432D53semidirect
129. C53#3213semidirect
149. C4⋊C8C2×D53semidirect
153. #97C2×D53semidirect