Finite Group Databasebeta

← #20286314

Subgroups — #20286314

Order 6784.

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

normal subgroups only

This group has 546 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
1C2×#13133244C212yesmaximal, complemented
2#16811276C211yesmaximal, characteristic, complemented
3#16811258C211yesmaximal, characteristic, complemented
4#16810866C212yesmaximal, complemented
5#16810825C211yesmaximal, characteristic, complemented
6#16810815C211yesmaximal, characteristic, complemented
7#16810772C212yesmaximal, complemented
8#16810767C212yesmaximal, complemented
9C4×D424C211yesmaximal, characteristic, complemented
10#16810702C211yesmaximal, characteristic, complemented
11C53×#492C211yesmaximal, characteristic, complemented
12C2×#11719949C2212yescomplemented
13C2×D4×D53—21no—
14C2×D4×D53C2212yes—
15#13133244—28no—
16C2×#11719924C2212yes—
17C2×D424C2211yescharacteristic
18C2×#11719922C2212yescomplemented
19C2×#11719912C2212yescomplemented
20#13133202—22no—
21#13133193C2212yescomplemented
22#13133192—21no—
23#13133188—21no—
24#13133188C2212yes—
25#13133185—22no—
26C4×D212C2211yescharacteristic
27C4×D212C2211yescharacteristic
28C4×D212C2211yescharacteristic, complemented
29C4⋊C4×D53—21no—
30#13133151C2211yescharacteristic, complemented
31C4⋊#88322C2211yescharacteristic, complemented
32#13133141C2212yescomplemented
33#13133139C2211yescharacteristic
34#13133139C2211yescharacteristic, complemented
35#13133138C2212yescomplemented
36#13133129C2212yescomplemented
37C2×SD16×C53C2212yescomplemented
38C2×SD16×C53C2212yescomplemented
39C4⋊Q8×C53C2211yescharacteristic, complemented
40#127×C53C2211yescharacteristic
41C4×C8×C53C2211yescharacteristic, complemented
42C23×D53—22no—
43C2×Q8×C53C2312yes—
44C2×D4×C53—21no—
45C2×D4×C53—21no—
46C2×D4×C53C2312yes—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49#11719949—24no—
50#11719949—42no—
51D4×D53—24no—
52D4×D53—42no—
53D4×D53—44no—
54C2×D212—22no—
55C2×D212D411yescharacteristic
56C2×D212C2311yescharacteristic
57C2×D212C2311yescharacteristic
58C2×C4×D53—21no—
59C2×C4×D53—21no—
60C2×C4×D53—22no—
61C2×C4×D53C2312yes—
62SD16×C53—24no—
63SD16×C53—42no—
64C2×C8×C53C2311yescharacteristic
65C2×C8×C53C2311yescharacteristic
66C4⋊C4×C53—21no—
67C4⋊C4×C53—21no—
68C42×C53C2311yescharacteristic
69C22⋊Dic53—22no—
70#11719924—24no—
71#11719922—24no—
72#11719921—22no—
73#11719921—21no—
74#11719921—21no—
75#11719921D412yes—
76C4⋊Dic53D411yescharacteristic
77C4⋊Dic53C2311yescharacteristic
78C4⋊Dic53C2311yescharacteristic
79#11719919—22no—
80C2×#88322C2312yes—
81D424D414yescomplemented
82#11719912—24no—
83C23×C53—22no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—44no—
87C22×D53—44no—
88C22×D53—82no—
89C22×D53C2×D412yes—
90Q8×C53—22no—
91Q8×C53—22no—
92D4×C53—22no—
93D4×C53—22no—
94D4×C53—42no—
95D4×C53—81no—
96C2×C4×C53—22no—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2411yescharacteristic
100C22⋊D53—44no—
101C22⋊D53—44no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53C2×D412yes—
105D212—22no—
106D212—21no—
107D212—21no—
108D212—44no—
109D212C2×D412yes—
110D212C2×D412yescomplemented
111C4×D53—22no—
112C4×D53—22no—
113C4×D53—44no—
114C4×D53—42no—
115C4×D53—42no—
116C8×C53—21no—
117C8×C53C2×D412yescomplemented
118#88322—22no—
119C22×C53—42no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53C22×D411yescharacteristic
123C2×D53—22no—
124C2×D53—22no—
125C2×D53—22no—
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—41no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—88no—
133C4×C53—21no—
134C4×C53—42no—
135C4×C53AGL(1,ℤ/8)12yescomplemented
136C4×C53C22×D411yescharacteristic
137C4×C53H(5,𝔽2)11yescharacteristic
138Dic53—22no—
139Dic53—41no—
140Dic53—41no—
141#3272—531nomaximal, Sylow
142C2×C53—82no—
143C2×C53D4211yescharacteristic
144C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C53#57711yescharacteristic
146D53—42no—
147D53—42no—
148D53—42no—
149C2×AGL(1,ℤ/8)—532no—
150#547—531no—
151D42—531no—
152#492D5311yescharacteristic, complemented
153#468—532no—
154#461—531no—
155#459—531no—
156#450—532no—
157#449—532no—
158C4×D8—531no—
159#436—531no—
160C53#327211yescharacteristic, complemented, Sylow
161C2×#41—532no—
162C22×D4—532no—
163C22×D4—1061no—
164AGL(1,ℤ/8)—1068no—
165C2×SD16—532no—
166C2×SD16C2×D5312yescomplemented
167C2×SD16C2×D5312yescomplemented
168C2×D8—532no—
169C2×D8—531no—
170C2×M16—532no—
171C4⋊Q8C2×D5311yescharacteristic, complemented
172#127C2×D5311yescharacteristic
173#123—1062no—
174C22⋊Q8—532no—
175#121—532no—
176#121—1061no—
177#121—1061no—
178#120—1062no—
179C4×D4—531no—
180C4×D4—531no—
181C4×D4—531no—
182C2×C4⋊C4—1061no—
183#107—531no—
184C4⋊C8—531no—
185#103—532no—
186#102—532no—
187#102—531no—
188#102—531no—
189C22⋊C8—532no—
190C4×C8C2×D5311yescharacteristic, complemented
191C24—1062no—
192#41—1064no—
193#41—2122no—
194C2×Q8C22×D5312yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—531no—
198C2×D4—531no—
199C2×D4—531no—
200C2×D4—1062no—
201C2×D4—1062no—
202C2×D4—1064no—
203C2×D4—1062no—
204C2×D4—2122no—
205C2×D4—2124no—
206C2×D4C22×D5312yes—
207C22×C4—532no—
208C22×C4—1062no—
209C22×C4—1061no—
210C22×C4—1061no—
211SD16—24no—
212SD16—42no—
213SD16—1064no—
214D8—534no—
215D8—1064no—
216M16—1064no—
217C2×C8—532no—
218C2×C8C22×D5311yescharacteristic
219C2×C8C22×D5311yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—531no—
223C4⋊C4—531no—
224C4⋊C4—531no—
225C4⋊C4—1062no—
226C22⋊C4—532no—
227C22⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1061no—
230C22⋊C4—1061no—
231C42C22×D5311yescharacteristic
232C23—22no—
233C23—532no—
234C23—1061no—
235C23—1061no—
236C23—4242no—
237C23—2124no—
238C23—2124no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—532no—
246D4—532no—
247D4—1062no—
248D4—1061no—
249D4—1061no—
250D4—2124no—
251D4—2124no—
252D4—2124no—
253C2×C4—22no—
254C2×C4—532no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—2124no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4D4×D5311yescharacteristic
263C2×C4D4×D5311yescharacteristic
264C2×C4C23×D5311yescharacteristic
265C8—21no—
266C8—1062no—
267C8D4×D5312yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—1062no—
272C22—1062no—
273C22—1062no—
274C22—1062no—
275C22—1062no—
276C22—4248no—
277C22—2121no—
278C22—2121no—
279C22—2121no—
280C22—2121no—
281C22C2×D4×D5311yescharacteristic
282C4—21no—
283C4—42no—
284C4—1062no—
285C4—2121no—
286C4—2121no—
287C4#1313324412yescomplemented
288C4C2×D4×D5311yescharacteristic
289C4#1313326211yescharacteristic
290C2—82no—
291C2—2122no—
292C2—2122no—
293C2—2122no—
294C2#1681125811yescharacteristic
295C2C2×#1313324411yescharacteristic
296C2#1681142911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#492C2
socleC22×C53C22×D4
hypercenter#492D53
hypocenterC53#3272
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #16811276C2semidirect
3. #16811258C2semidirect
4. #16810866C2semidirect
5. #16810825C2semidirect
6. #16810815C2semidirect
7. #16810772C2semidirect
8. #16810767C2semidirect
9. C4×D424C2semidirect
10. #16810702C2semidirect
11. C53×#492C2semidirect
12. C2×#11719949C22semidirect
18. C2×#11719922C22semidirect
19. C2×#11719912C22semidirect
21. #13133193C22semidirect
28. C4×D212C22semidirect
30. #13133151C22semidirect
31. C4⋊#88322C22semidirect
32. #13133141C22semidirect
34. #13133139C22semidirect
35. #13133138C22semidirect
36. #13133129C22semidirect
37. C2×SD16×C53C22semidirect
38. C2×SD16×C53C22semidirect
39. C4⋊Q8×C53C22semidirect
41. C4×C8×C53C22semidirect
81. D424D4semidirect
110. D212C2×D4semidirect
117. C8×C53C2×D4semidirect
135. C4×C53AGL(1,ℤ/8)semidirect
152. #492D53semidirect
160. C53#3272semidirect
166. C2×SD16C2×D53semidirect
167. C2×SD16C2×D53semidirect
171. C4⋊Q8C2×D53semidirect
190. C4×C8C2×D53semidirect
267. C8D4×D53semidirect
287. C4#13133244semidirect