Finite Group Databasebeta

← #20286957

Subgroups — #20286957

Order 6784.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#16811338C211yesmaximal, characteristic, complemented
2#16810969C211yesmaximal, characteristic, complemented
3#16810944C212yesmaximal, complemented
4#16810913C211yesmaximal, characteristic, complemented
5#16810874C212yesmaximal, complemented
6#16810815C211yesmaximal, characteristic, complemented
7#16810757C212yesmaximal, complemented
8C4×#11719922C211yesmaximal, characteristic, complemented
9#16810701C211yesmaximal, characteristic, complemented
10#16810628C212yesmaximal, complemented
11D42×C53C211yesmaximal, characteristic, complemented
12C2×#11719922C2211yescharacteristic
13C2×#11719922C2212yescomplemented
14C2×#11719917C2212yescomplemented
15#13133217C2211yescharacteristic, complemented
16#13133203—22no—
17#13133191C2212yescomplemented
18#13133188—21no—
19C4×D212C2211yescharacteristic, complemented
20#13133176C2212yescomplemented
21#13133175—21no—
22#13133150C2211yescharacteristic, complemented
23#13133148C2212yescomplemented
24C4⋊#88322C2211yescharacteristic, complemented
25#13133138C2212yes—
26#13133138C2211yescharacteristic
27#13133138C2212yescomplemented
28#13133138C2212yescomplemented
29#13133138C2212yescomplemented
30#13133137C2211yescharacteristic, complemented
31C22⋊#88322C2212yescomplemented
32C4×#88322C2211yescharacteristic, complemented
33C22×D4×C53—21no—
34C22×D4×C53C2212yes—
35#127×C53C2211yescharacteristic
36#121×C53—21no—
37#121×C53C2212yes—
38#120×C53—22no—
39C4×D4×C53C2211yescharacteristic
40C4×D4×C53C2211yescharacteristic
41C24×C53—22no—
42C2×D4×C53—24no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C2×D4×C53—21no—
46C2×D4×C53—21no—
47C2×D4×C53—42no—
48C2×D4×C53—44no—
49C2×D4×C53D411yescharacteristic
50C2×D4×C53C2312yes—
51C2×D4×C53C2311yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53C2312yes—
54C2×C22⋊D53—22no—
55C2×D212C2311yescharacteristic
56C2×C4×D53—21no—
57C2×Dic106—21no—
58C4⋊C4×C53D411yescharacteristic
59C4⋊C4×C53C2311yescharacteristic
60C22⋊C4×C53—21no—
61C22⋊C4×C53—22no—
62C22⋊C4×C53D412yes—
63C42×C53C2311yescharacteristic
64C22⋊Dic53—22no—
65#11719922—22no—
66#11719922—42no—
67#11719921—22no—
68#11719921—21no—
69C4⋊Dic53—21no—
70C4⋊Dic53C2312yes—
71C4⋊Dic53C2311yescharacteristic
72C4×Dic53—22no—
73#11719917—42no—
74C2×#88322C2311yescharacteristic
75C2×#88322C2311yescharacteristic
76C2×#88322C2312yes—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53—82no—
82C23×C53C2×D412yes—
83C22×D53—21no—
84D4×C53—22no—
85D4×C53—22no—
86D4×C53—21no—
87D4×C53—22no—
88D4×C53—44no—
89D4×C53—44no—
90D4×C53—42no—
91D4×C53—81no—
92D4×C53#4112yes—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—42no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D412yes—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—82no—
102C2×Dic53—22no—
103C2×Dic53—21no—
104D212—21no—
105D212—21no—
106C4×D53—42no—
107Dic106—81no—
108#88322—21no—
109#88322—21no—
110#88322—22no—
111C22×C53—22no—
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—88no—
121C22×C53C22×D411yescharacteristic
122C2×D53—41no—
123C2×D53—41no—
124C2×D53—41no—
125C4×C53—22no—
126C4×C53—21no—
127C4×C53—41no—
128C4×C53AGL(1,ℤ/8)12yes—
129C4×C53C2×#4111yescharacteristic
130C4×C53H(5,𝔽2)11yescharacteristic
131Dic53—41no—
132Dic53—42no—
133#3325—531nomaximal, Sylow
134C2×C53—42no—
135C2×C53—42no—
136C2×C53—82no—
137C2×C53#54611yescharacteristic
138C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
139C2×C53#57611yescharacteristic
140D53—81no—
141#555—531no—
142D42D5311yescharacteristic, complemented
143#486—531no—
144#482—532no—
145#476—531no—
146#469—532no—
147#459—531no—
148#447—532no—
149C4×D8—531no—
150#436—531no—
151#418—532no—
152C53#332511yescharacteristic, complemented, Sylow
153C22×D4—21no—
154C22×D4C2×D5312yes—
155C2×D8—531no—
156C2×D8—532no—
157C2×M16—532no—
158C4⋊Q8—531no—
159#127C2×D5311yescharacteristic
160#124—1062no—
161#121—21no—
162#121—532no—
163#121—1061no—
164#121C2×D5312yes—
165#120—22no—
166C4×D4—531no—
167C4×D4C2×D5311yescharacteristic
168C4×D4C2×D5311yescharacteristic
169#117—532no—
170#117—1061no—
171#107—531no—
172#106—532no—
173C4⋊C8—531no—
174#102—531no—
175#102—532no—
176#102—532no—
177#102—532no—
178#102—532no—
179#102—531no—
180C22⋊C8—532no—
181C4×C8—531no—
182C24—22no—
183C2×Q8—1061no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—531no—
192C2×D4—1062no—
193C2×D4C22⋊D5311yescharacteristic
194C2×D4C22×D5312yes—
195C2×D4C22×D5311yescharacteristic
196C22×C4—21no—
197C22×C4—1061no—
198C22×C4C22×D5312yes—
199D8—1062no—
200D8—2122no—
201M16—2122no—
202C2×C8—531no—
203C2×C8—531no—
204C2×C8—532no—
205C4⋊C4—532no—
206C4⋊C4—531no—
207C4⋊C4—1061no—
208C4⋊C4C22⋊D5311yescharacteristic
209C4⋊C4C22×D5311yescharacteristic
210C22⋊C4—21no—
211C22⋊C4—22no—
212C22⋊C4—1062no—
213C22⋊C4—1062no—
214C22⋊C4—1061no—
215C22⋊C4C22⋊D5312yes—
216C42—1062no—
217C42C22×D5311yescharacteristic
218C23—21no—
219C23—22no—
220C23—44no—
221C23—44no—
222C23—82no—
223C23—1061no—
224C23C2×C22⋊D5312yes—
225Q8—4241no—
226D4—22no—
227D4—22no—
228D4—21no—
229D4—22no—
230D4—44no—
231D4—44no—
232D4—42no—
233D4—81no—
234D4—1061no—
235D4—1061no—
236D4—4242no—
237D4#1171994712yes—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—1061no—
243C2×C4—1062no—
244C2×C4—2122no—
245C2×C4C2×C22⋊D5312yes—
246C2×C4C2×C22⋊D5311yescharacteristic
247C2×C4C2×C22⋊D5311yescharacteristic
248C2×C4C23×D5311yescharacteristic
249C8—1061no—
250C8—1061no—
251C8—1062no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—88no—
262C22—2121no—
263C22—2121no—
264C22—2121no—
265C22C22×C22⋊D5311yescharacteristic
266C4—22no—
267C4—21no—
268C4—41no—
269C4—2121no—
270C4—2122no—
271C4#1313324312yes—
272C4C2×#1171994711yescharacteristic
273C4#1313326211yescharacteristic
274C2—42no—
275C2—42no—
276C2—82no—
277C2—4241no—
278C2#1681126711yescharacteristic
279C2C2×#1313324311yescharacteristic
280C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingD42×C53C2
socleC22×C53C22×D4
hypercenterD42D53
hypocenterC53#3325
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #16811338C2semidirect
2. #16810969C2semidirect
3. #16810944C2semidirect
4. #16810913C2semidirect
5. #16810874C2semidirect
6. #16810815C2semidirect
7. #16810757C2semidirect
8. C4×#11719922C2semidirect
9. #16810701C2semidirect
10. #16810628C2semidirect
11. D42×C53C2semidirect
13. C2×#11719922C22semidirect
14. C2×#11719917C22semidirect
15. #13133217C22semidirect
17. #13133191C22semidirect
19. C4×D212C22semidirect
20. #13133176C22semidirect
22. #13133150C22semidirect
23. #13133148C22semidirect
24. C4⋊#88322C22semidirect
27. #13133138C22semidirect
28. #13133138C22semidirect
29. #13133138C22semidirect
30. #13133137C22semidirect
31. C22⋊#88322C22semidirect
32. C4×#88322C22semidirect
142. D42D53semidirect
152. C53#3325semidirect