Finite Group Databasebeta

← #20286337

Subgroups — #20286337

Order 6784.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#13133245C212yesmaximal, complemented
2#16811294C211yesmaximal, characteristic, complemented
3D4×D212C211yesmaximal, characteristic, complemented
4#16811000C211yesmaximal, characteristic, complemented
5#16810876C212yesmaximal, complemented
6#16810824C211yesmaximal, characteristic, complemented
7#16810819C211yesmaximal, characteristic, complemented
8#16810770C212yesmaximal, complemented
9#16810759C212yesmaximal, complemented
10C4×#11719913C211yesmaximal, characteristic, complemented
11C53×#436C211yesmaximal, characteristic, complemented
12C2×#11719945C2212yescomplemented
13C22×D212C2212yes—
14C2×D4×D53—21no—
15#13133245—28no—
16C2×#11719913C2211yescharacteristic
17C2×#11719913C2212yescomplemented
18C2×D424C2212yes—
19C2×D424C2212yescomplemented
20C2×D424C2212yescomplemented
21#13133214C2211yescharacteristic
22#13133214C2211yescharacteristic, complemented
23#13133213—21no—
24#13133192—21no—
25#13133191C2212yes—
26#13133191C2212yescomplemented
27#13133190—22no—
28#13133187—22no—
29C4×Dic106C2211yescharacteristic, complemented
30C4×D212C2211yescharacteristic
31#13133179—21no—
32#13133149C2211yescharacteristic, complemented
33#13133142C2211yescharacteristic, complemented
34#13133139C2211yescharacteristic
35#13133139C2212yescomplemented
36#13133139C2212yescomplemented
37C2×M16×C53C2212yescomplemented
38C4×D4×C53C2211yescharacteristic
39C4⋊C8×C53C2211yescharacteristic, complemented
40C22⋊C8×C53C2212yescomplemented
41C4×C8×C53C2211yescharacteristic, complemented
42C23×D53—22no—
43C2×D4×C53D411yescharacteristic
44C22×C4×C53C2312yes—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48D4×D53—44no—
49#11719945—24no—
50#11719945—42no—
51C2×D212—24no—
52C2×D212—21no—
53C2×D212—22no—
54C2×D212—21no—
55C2×D212—21no—
56C2×D212—21no—
57C2×D212—42no—
58C2×D212C2312yes—
59C2×D212C2312yes—
60C2×D212C2311yescharacteristic
61C2×C4×D53—22no—
62C2×C4×D53—21no—
63C2×Dic106C2311yescharacteristic
64M16×C53—24no—
65C2×C8×C53C2311yescharacteristic
66C2×C8×C53C2312yes—
67C2×C8×C53C2311yescharacteristic
68C4⋊C4×C53D411yescharacteristic
69C22⋊C4×C53D412yes—
70C42×C53C2311yescharacteristic
71#11719921—22no—
72#11719921—21no—
73#11719921—22no—
74C4⋊Dic53C2311yescharacteristic
75C4⋊Dic53C2311yescharacteristic
76#11719919—21no—
77C4×Dic53—21no—
78D424—24no—
79D424—24no—
80D424—42no—
81#11719913—24no—
82#11719913D414yescomplemented
83C23×C53C2×D412yes—
84C22×D53—21no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—44no—
88C22×D53—44no—
89C22×D53—82no—
90D4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53C2×D412yes—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—44no—
98C22⋊D53—44no—
99C22⋊D53—82no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103D212—22no—
104D212—22no—
105D212—21no—
106D212—22no—
107D212—22no—
108D212—44no—
109D212—42no—
110D212—42no—
111D212—81no—
112D212—81no—
113D212C2×D412yes—
114C4×D53—44no—
115C4×D53—42no—
116Dic106—21no—
117Dic106C2×D412yescomplemented
118C8×C53—22no—
119C8×C53—21no—
120C8×C53C2×D412yescomplemented
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53C22×D411yescharacteristic
125C2×D53—22no—
126C2×D53—41no—
127C2×D53—41no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—88no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53AGL(1,ℤ/8)12yescomplemented
138C4×C53C22×D411yescharacteristic
139C4×C53H(5,𝔽2)11yescharacteristic
140Dic53—22no—
141Dic53—41no—
142Dic53—41no—
143#3274—531nomaximal, Sylow
144C2×C53—42no—
145C2×C53D4211yescharacteristic
146C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C53#57611yescharacteristic
148D53—42no—
149D53—82no—
150D53—82no—
151C2×AGL(1,ℤ/8)—532no—
152#550—531no—
153D42—531no—
154#493—531no—
155#469—532no—
156#460—531no—
157#459—531no—
158#449—532no—
159#447—532no—
160C4×SD16—531no—
161#436D5311yescharacteristic, complemented
162C53#327411yescharacteristic, complemented, Sylow
163C2×#41—532no—
164C22×D4—532no—
165C22×D4—1061no—
166AGL(1,ℤ/8)—1068no—
167C2×SD16—531no—
168C2×SD16—532no—
169C2×D8—532no—
170C2×D8—532no—
171C2×D8—532no—
172C2×M16C2×D5312yescomplemented
173#127—531no—
174#127—531no—
175#127—1061no—
176#121—532no—
177#121—532no—
178#121—1061no—
179#121—1062no—
180#120—1062no—
181C4×Q8—531no—
182C4×D4—531no—
183C4×D4—1061no—
184C4×D4C2×D5311yescharacteristic
185#106—531no—
186C4⋊C8C2×D5311yescharacteristic, complemented
187#103—531no—
188#102—531no—
189#102—532no—
190#102—532no—
191C22⋊C8C2×D5312yescomplemented
192C4×C8C2×D5311yescharacteristic, complemented
193C24—1062no—
194#41—1064no—
195#41—2122no—
196C2×Q8—531no—
197C2×D4—531no—
198C2×D4—532no—
199C2×D4—532no—
200C2×D4—1064no—
201C2×D4—1062no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—1061no—
207C2×D4—1062no—
208C2×D4—1061no—
209C2×D4—2122no—
210C2×D4—2124no—
211C2×D4D21211yescharacteristic
212C22×C4—1061no—
213C22×C4—1062no—
214C22×C4C22×D5312yes—
215SD16—534no—
216SD16—1064no—
217D8—1064no—
218D8—1064no—
219D8—2122no—
220M16—24no—
221C2×C8C22×D5311yescharacteristic
222C2×C8C22×D5312yes—
223C2×C8C22×D5311yescharacteristic
224C4⋊C4—531no—
225C4⋊C4—531no—
226C4⋊C4—1061no—
227C4⋊C4D21211yescharacteristic
228C22⋊C4—1061no—
229C22⋊C4—1062no—
230C22⋊C4—1062no—
231C22⋊C4D21212yes—
232C42—1061no—
233C42C22×D5311yescharacteristic
234C23—1061no—
235C23—1062no—
236C23—1062no—
237C23—4242no—
238C23—2124no—
239C23—2124no—
240C23C2×D21212yes—
241Q8—532no—
242Q8—1061no—
243D4—22no—
244D4—532no—
245D4—1062no—
246D4—1062no—
247D4—1061no—
248D4—1062no—
249D4—1062no—
250D4—4241no—
251D4—4241no—
252D4—4242no—
253D4—2124no—
254D4—2124no—
255D4—2124no—
256D4—2122no—
257D4—2122no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—2124no—
264C2×C4—2122no—
265C2×C4C2×D21211yescharacteristic
266C2×C4C2×D21212yes—
267C2×C4C2×D21211yescharacteristic
268C2×C4C23×D5311yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D5312yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—1062no—
276C22—4248no—
277C22—2122no—
278C22—2122no—
279C22—2122no—
280C22—2122no—
281C22—2122no—
282C22—2122no—
283C22—2121no—
284C22—2121no—
285C22C22×D21211yescharacteristic
286C4—21no—
287C4—22no—
288C4—1062no—
289C4—2121no—
290C4—2121no—
291C4#1313324512yescomplemented
292C4C2×D4×D5311yescharacteristic
293C4#1313326311yescharacteristic
294C2—42no—
295C2—4242no—
296C2—4242no—
297C2—2122no—
298C2D4×D21211yescharacteristic
299C2C2×#1313324511yescharacteristic
300C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#436C2
socleC22×C53C22×D4
hypercenter#436D53
hypocenterC53#3274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. #16811294C2semidirect
3. D4×D212C2semidirect
4. #16811000C2semidirect
5. #16810876C2semidirect
6. #16810824C2semidirect
7. #16810819C2semidirect
8. #16810770C2semidirect
9. #16810759C2semidirect
10. C4×#11719913C2semidirect
11. C53×#436C2semidirect
12. C2×#11719945C22semidirect
17. C2×#11719913C22semidirect
19. C2×D424C22semidirect
20. C2×D424C22semidirect
22. #13133214C22semidirect
26. #13133191C22semidirect
29. C4×Dic106C22semidirect
32. #13133149C22semidirect
33. #13133142C22semidirect
35. #13133139C22semidirect
36. #13133139C22semidirect
37. C2×M16×C53C22semidirect
39. C4⋊C8×C53C22semidirect
40. C22⋊C8×C53C22semidirect
41. C4×C8×C53C22semidirect
82. #11719913D4semidirect
117. Dic106C2×D4semidirect
120. C8×C53C2×D4semidirect
137. C4×C53AGL(1,ℤ/8)semidirect
161. #436D53semidirect
162. C53#3274semidirect
172. C2×M16C2×D53semidirect
186. C4⋊C8C2×D53semidirect
191. C22⋊C8C2×D53semidirect
192. C4×C8C2×D53semidirect
271. C8D4×D53semidirect
291. C4#13133245semidirect