Finite Group Databasebeta

← #20283945

Subgroups — #20283945

Order 6784.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D212C211yesmaximal, characteristic, complemented
2C2×#13133245C212yesmaximal, complemented
3#16811134C211yesmaximal, characteristic, complemented
4#16810776C214yesmaximal, complemented
5#16810759C214yesmaximal, complemented
6#16810629C212yesmaximal, complemented
7C53×#407C211yesmaximal, characteristic, complemented
8C24×D53—21no—
9C2×#11719945C2211yescharacteristic, complemented
10C22×D212—22no—
11C22×D212—21no—
12C22×D212—28no—
13C22×D212C2214yes—
14C22×D212C2212yes—
15#13133245—28no—
16C2×#11719913C2214yescomplemented
17C2×D424C2214yescomplemented
18#13133200—21no—
19#13133196C2212yescomplemented
20#13133191C2212yescomplemented
21#13133186—21no—
22C4×C22⋊D53—22no—
23#13133176C2211yescharacteristic, complemented
24#13133139C2218yescomplemented
25C23×C4×C53C2211yescharacteristic
26C2×M16×C53C2212yescomplemented
27C22⋊C8×C53C2214yescomplemented
28C24×C53D411yescharacteristic
29C23×D53—28no—
30C23×D53—24no—
31C23×D53—22no—
32C23×D53—48no—
33C22×C4×C53—24no—
34C22×C4×C53D412yes—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39C2×C22⋊D53—21no—
40#11719945—24no—
41C2×D212—28no—
42C2×D212—216no—
43C2×D212—24no—
44C2×D212—22no—
45C2×D212—24no—
46C2×D212—216no—
47C2×D212D418yescomplemented
48C2×D212C2314yes—
49C2×D212C2311yescharacteristic, complemented
50C2×C4×D53—21no—
51C2×Dic106C2311yescharacteristic, complemented
52M16×C53—24no—
53C2×C8×C53C2314yescomplemented
54C22⋊Dic53—21no—
55C22⋊Dic53—22no—
56#11719921—22no—
57C4⋊Dic53C2312yescomplemented
58#11719919—22no—
59C4×Dic53—21no—
60D424—28no—
61#11719913—28no—
62C23×C53—22no—
63C23×C53—24no—
64C23×C53C2×D411yescharacteristic
65C23×C53C2×D412yes—
66C22×D53—216no—
67C22×D53—28no—
68C22×D53—24no—
69C22×D53—21no—
70C22×D53—432no—
71C22×D53—48no—
72C22×D53—48no—
73C22×D53—48no—
74C2×C4×C53—22no—
75C2×C4×C53—24no—
76C2×C4×C53—24no—
77C2×C4×C53C2×D412yes—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D414yes—
80C2×C4×C53C2411yescharacteristic
81C22⋊D53—42no—
82C22⋊D53—44no—
83C2×Dic53—21no—
84C2×Dic53—22no—
85D212—216no—
86D212—28no—
87D212—24no—
88D212—22no—
89D212C2×D418yes—
90C4×D53—42no—
91Dic106—22no—
92C8×C53—24no—
93C22×C53—28no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—22no—
97C22×C53#12012yes—
98C22×C53AGL(1,ℤ/8)14yescomplemented
99C22×C53C22×D411yescharacteristic
100C2×D53—28no—
101C2×D53—416no—
102C2×D53—432no—
103C2×D53—48no—
104C2×D53—41no—
105C2×D53—42no—
106C4×C53—22no—
107C4×C53#12012yes—
108C4×C53C22×D412yes—
109Dic53—41no—
110Dic53—42no—
111#3003—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53—24no—
114C2×C53C2×#12011yescharacteristic
115C2×C53C2×AGL(1,ℤ/8)12yes—
116D53—48no—
117D53—81no—
118C23×D4—531no—
119C2×AGL(1,ℤ/8)—532no—
120#525—531no—
121#450—534no—
122#447—534no—
123#418—532no—
124#407D5311yescharacteristic, complemented
125C53#300311yescharacteristic, complemented, Sylow
126C25—1061no—
127C2×#41—531no—
128C22×D4—532no—
129C22×D4—534no—
130C22×D4—1068no—
131C22×D4—1062no—
132C22×D4—1061no—
133C23×C4C2×D5311yescharacteristic
134AGL(1,ℤ/8)—1068no—
135C2×SD16—534no—
136C2×D8—534no—
137C2×M16C2×D5312yescomplemented
138#123—1061no—
139C22⋊Q8—532no—
140#121—532no—
141#120—1061no—
142C4×D4—1062no—
143#117—531no—
144#102—538no—
145C22⋊C8C2×D5314yescomplemented
146C24—1068no—
147C24—1064no—
148C24—1062no—
149C24—2128no—
150C24D21211yescharacteristic
151#41—1064no—
152C2×Q8—531no—
153C2×D4—538no—
154C2×D4—534no—
155C2×D4—531no—
156C2×D4—10616no—
157C2×D4—1068no—
158C2×D4—10616no—
159C2×D4—1062no—
160C2×D4—1061no—
161C2×D4—1064no—
162C2×D4—1062no—
163C2×D4—1064no—
164C22×C4—24no—
165C22×C4—1061no—
166C22×C4D21212yes—
167C22×C4D21211yescharacteristic
168C22×C4C22×D5312yes—
169C22×C4C22×D5311yescharacteristic
170SD16—1068no—
171D8—1068no—
172M16—24no—
173C2×C8C22×D5314yescomplemented
174C4⋊C4—532no—
175C4⋊C4—1062no—
176C22⋊C4—1061no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C42—1061no—
180C23—22no—
181C23—24no—
182C23—10616no—
183C23—1068no—
184C23—1064no—
185C23—1061no—
186C23—21232no—
187C23—2128no—
188C23—2128no—
189C23—2128no—
190C23C2×D21211yescharacteristic
191C23C2×D21212yes—
192Q8—1062no—
193D4—538no—
194D4—10616no—
195D4—1068no—
196D4—1064no—
197D4—1062no—
198D4—2122no—
199D4—2124no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—1062no—
204C2×C4—1061no—
205C2×C4—2122no—
206C2×C4C2×D21212yes—
207C2×C4C2×D21211yescharacteristic
208C2×C4D4×D5314yes—
209C2×C4C23×D5311yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—1068no—
216C22—21216no—
217C22—21232no—
218C22—2121no—
219C22—2122no—
220C22—2128no—
221C22#1313318712yes—
222C22#1313324514yescomplemented
223C22C22×D21211yescharacteristic
224C4—22no—
225C4—2121no—
226C4—2122no—
227C4#1313318712yes—
228C4C2×D4×D5312yes—
229C2—22no—
230C2—24no—
231C2—4241no—
232C2—2128no—
233C2C2×#1313318711yescharacteristic
234C2C2×#1313324512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3003
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. C23×D212C2semidirect
2. C2×#13133245C2semidirect
3. #16811134C2semidirect
4. #16810776C2semidirect
5. #16810759C2semidirect
6. #16810629C2semidirect
7. C53×#407C2semidirect
9. C2×#11719945C22semidirect
16. C2×#11719913C22semidirect
17. C2×D424C22semidirect
19. #13133196C22semidirect
20. #13133191C22semidirect
23. #13133176C22semidirect
24. #13133139C22semidirect
26. C2×M16×C53C22semidirect
27. C22⋊C8×C53C22semidirect
47. C2×D212D4semidirect
49. C2×D212C23semidirect
51. C2×Dic106C23semidirect
53. C2×C8×C53C23semidirect
57. C4⋊Dic53C23semidirect
98. C22×C53AGL(1,ℤ/8)semidirect
124. #407D53semidirect
125. C53#3003semidirect
137. C2×M16C2×D53semidirect
145. C22⋊C8C2×D53semidirect
173. C2×C8C22×D53semidirect
222. C22#13133245semidirect