Finite Group Databasebeta

← #20255010

Subgroups — #20255010

Order 6016.

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×#13094052C212yesmaximal, complemented
2#15689794C211yesmaximal, characteristic, complemented
3D4×D188C211yesmaximal, characteristic, complemented
4#15689500C211yesmaximal, characteristic, complemented
5#15689376C212yesmaximal, complemented
6#15689324C211yesmaximal, characteristic, complemented
7#15689319C211yesmaximal, characteristic, complemented
8#15689270C212yesmaximal, complemented
9#15689259C212yesmaximal, complemented
10C4×#10625325C211yesmaximal, characteristic, complemented
11C47×#436C211yesmaximal, characteristic, complemented
12C2×#10625350C2212yescomplemented
13C22×D188C2212yes—
14C2×D4×D47—21no—
15#13094052—28no—
16C2×#10625325C2211yescharacteristic
17C2×#10625325C2212yescomplemented
18C2×D376C2212yes—
19C2×D376C2212yescomplemented
20C2×D376C2212yescomplemented
21#13094021C2211yescharacteristic
22#13094021C2211yescharacteristic, complemented
23#13094020—21no—
24#13093999—21no—
25#13093998C2212yes—
26#13093998C2212yescomplemented
27#13093997—22no—
28#13093994—22no—
29C4×Dic94C2211yescharacteristic, complemented
30C4×D188C2211yescharacteristic
31#13093986—21no—
32#13093956C2211yescharacteristic, complemented
33#13093949C2211yescharacteristic, complemented
34#13093946C2211yescharacteristic
35#13093946C2212yescomplemented
36#13093946C2212yescomplemented
37C2×M16×C47C2212yescomplemented
38C4×D4×C47C2211yescharacteristic
39C4⋊C8×C47C2211yescharacteristic, complemented
40C22⋊C8×C47C2212yescomplemented
41C4×C8×C47C2211yescharacteristic, complemented
42C23×D47—22no—
43C2×D4×C47D411yescharacteristic
44C22×C4×C47C2312yes—
45C2×C22⋊D47—22no—
46C2×C22⋊D47—22no—
47C2×C22⋊D47—22no—
48D4×D47—44no—
49#10625350—24no—
50#10625350—42no—
51C2×D188—24no—
52C2×D188—21no—
53C2×D188—22no—
54C2×D188—21no—
55C2×D188—21no—
56C2×D188—21no—
57C2×D188—42no—
58C2×D188C2312yes—
59C2×D188C2312yes—
60C2×D188C2311yescharacteristic
61C2×C4×D47—22no—
62C2×C4×D47—21no—
63C2×Dic94C2311yescharacteristic
64M16×C47—24no—
65C2×C8×C47C2312yes—
66C2×C8×C47C2311yescharacteristic
67C2×C8×C47C2311yescharacteristic
68C4⋊C4×C47D411yescharacteristic
69C22⋊C4×C47D412yes—
70C42×C47C2311yescharacteristic
71(C2×D47)⋊C4—22no—
72(C2×D47)⋊C4—21no—
73(C2×D47)⋊C4—22no—
74C4⋊Dic47C2311yescharacteristic
75C4⋊Dic47C2311yescharacteristic
76Dic47⋊C4—21no—
77C4×Dic47—21no—
78D376—24no—
79D376—24no—
80D376—42no—
81#10625325—24no—
82#10625325D414yescomplemented
83C23×C47C2×D412yes—
84C22×D47—21no—
85C22×D47—22no—
86C22×D47—22no—
87C22×D47—44no—
88C22×D47—44no—
89C22×D47—82no—
90D4×C47—22no—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47C2×D411yescharacteristic
94C2×C4×C47C2×D412yes—
95C2×C4×C47C2×D411yescharacteristic
96C2×C4×C47C2411yescharacteristic
97C22⋊D47—44no—
98C22⋊D47—44no—
99C22⋊D47—82no—
100C2×Dic47—21no—
101C2×Dic47—21no—
102C2×Dic47—21no—
103D188—22no—
104D188—22no—
105D188—21no—
106D188—22no—
107D188—22no—
108D188—44no—
109D188—42no—
110D188—42no—
111D188—81no—
112D188—81no—
113D188C2×D412yes—
114C4×D47—44no—
115C4×D47—42no—
116Dic94—21no—
117Dic94C2×D412yescomplemented
118C8×C47—22no—
119C8×C47—21no—
120C8×C47C2×D412yescomplemented
121C22×C47—22no—
122C22×C47—22no—
123C22×C47—22no—
124C22×C47C22×D411yescharacteristic
125C2×D47—22no—
126C2×D47—41no—
127C2×D47—41no—
128C2×D47—42no—
129C2×D47—42no—
130C2×D47—42no—
131C2×D47—42no—
132C2×D47—42no—
133C2×D47—42no—
134C2×D47—88no—
135C4×C47—22no—
136C4×C47—21no—
137C4×C47AGL(1,ℤ/8)12yescomplemented
138C4×C47C22×D411yescharacteristic
139C4×C47H(5,𝔽2)11yescharacteristic
140Dic47—22no—
141Dic47—41no—
142Dic47—41no—
143#3274—471nomaximal, Sylow
144C2×C47—42no—
145C2×C47D4211yescharacteristic
146C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C47#57611yescharacteristic
148D47—42no—
149D47—82no—
150D47—82no—
151C2×AGL(1,ℤ/8)—472no—
152#550—471no—
153D42—471no—
154#493—471no—
155#469—472no—
156#460—471no—
157#459—471no—
158#449—472no—
159#447—472no—
160C4×SD16—471no—
161#436D4711yescharacteristic, complemented
162C47#327411yescharacteristic, complemented, Sylow
163C2×#41—472no—
164C22×D4—472no—
165C22×D4—941no—
166AGL(1,ℤ/8)—948no—
167C2×SD16—471no—
168C2×SD16—472no—
169C2×D8—472no—
170C2×D8—472no—
171C2×D8—472no—
172C2×M16C2×D4712yescomplemented
173#127—471no—
174#127—471no—
175#127—941no—
176#121—472no—
177#121—472no—
178#121—941no—
179#121—942no—
180#120—942no—
181C4×Q8—471no—
182C4×D4—471no—
183C4×D4—941no—
184C4×D4C2×D4711yescharacteristic
185#106—471no—
186C4⋊C8C2×D4711yescharacteristic, complemented
187#103—471no—
188#102—471no—
189#102—472no—
190#102—472no—
191C22⋊C8C2×D4712yescomplemented
192C4×C8C2×D4711yescharacteristic, complemented
193C24—942no—
194#41—944no—
195#41—1882no—
196C2×Q8—471no—
197C2×D4—471no—
198C2×D4—472no—
199C2×D4—472no—
200C2×D4—944no—
201C2×D4—942no—
202C2×D4—942no—
203C2×D4—942no—
204C2×D4—941no—
205C2×D4—941no—
206C2×D4—941no—
207C2×D4—942no—
208C2×D4—941no—
209C2×D4—1882no—
210C2×D4—1884no—
211C2×D4D18811yescharacteristic
212C22×C4—941no—
213C22×C4—942no—
214C22×C4C22×D4712yes—
215SD16—474no—
216SD16—944no—
217D8—944no—
218D8—944no—
219D8—1882no—
220M16—24no—
221C2×C8C22×D4711yescharacteristic
222C2×C8C22×D4712yes—
223C2×C8C22×D4711yescharacteristic
224C4⋊C4—471no—
225C4⋊C4—471no—
226C4⋊C4—941no—
227C4⋊C4D18811yescharacteristic
228C22⋊C4—941no—
229C22⋊C4—942no—
230C22⋊C4—942no—
231C22⋊C4D18812yes—
232C42—941no—
233C42C22×D4711yescharacteristic
234C23—941no—
235C23—942no—
236C23—942no—
237C23—3762no—
238C23—1884no—
239C23—1884no—
240C23C2×D18812yes—
241Q8—472no—
242Q8—941no—
243D4—22no—
244D4—472no—
245D4—942no—
246D4—942no—
247D4—941no—
248D4—942no—
249D4—942no—
250D4—3761no—
251D4—3761no—
252D4—3762no—
253D4—1884no—
254D4—1884no—
255D4—1884no—
256D4—1882no—
257D4—1882no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—941no—
261C2×C4—941no—
262C2×C4—941no—
263C2×C4—1884no—
264C2×C4—1882no—
265C2×C4C2×D18811yescharacteristic
266C2×C4C2×D18812yes—
267C2×C4C2×D18811yescharacteristic
268C2×C4C23×D4711yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D4712yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—942no—
276C22—3768no—
277C22—1882no—
278C22—1882no—
279C22—1882no—
280C22—1882no—
281C22—1882no—
282C22—1882no—
283C22—1881no—
284C22—1881no—
285C22C22×D18811yescharacteristic
286C4—22no—
287C4—21no—
288C4—942no—
289C4—1881no—
290C4—1881no—
291C4#1309405212yescomplemented
292C4C2×D4×D4711yescharacteristic
293C4#1309407011yescharacteristic
294C2—42no—
295C2—3762no—
296C2—3762no—
297C2—1882no—
298C2D4×D18811yescharacteristic
299C2C2×#1309405211yescharacteristic
300C2#1568992811yescharacteristic
subgroupG/H
centerC22C22×D188
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#436C2
socleC22×C47C22×D4
hypercenter#436D47
hypocenterC47#3274
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#13094052C2semidirect
2. #15689794C2semidirect
3. D4×D188C2semidirect
4. #15689500C2semidirect
5. #15689376C2semidirect
6. #15689324C2semidirect
7. #15689319C2semidirect
8. #15689270C2semidirect
9. #15689259C2semidirect
10. C4×#10625325C2semidirect
11. C47×#436C2semidirect
12. C2×#10625350C22semidirect
17. C2×#10625325C22semidirect
19. C2×D376C22semidirect
20. C2×D376C22semidirect
22. #13094021C22semidirect
26. #13093998C22semidirect
29. C4×Dic94C22semidirect
32. #13093956C22semidirect
33. #13093949C22semidirect
35. #13093946C22semidirect
36. #13093946C22semidirect
37. C2×M16×C47C22semidirect
39. C4⋊C8×C47C22semidirect
40. C22⋊C8×C47C22semidirect
41. C4×C8×C47C22semidirect
82. #10625325D4semidirect
117. Dic94C2×D4semidirect
120. C8×C47C2×D4semidirect
137. C4×C47AGL(1,ℤ/8)semidirect
161. #436D47semidirect
162. C47#3274semidirect
172. C2×M16C2×D47semidirect
186. C4⋊C8C2×D47semidirect
191. C22⋊C8C2×D47semidirect
192. C4×C8C2×D47semidirect
271. C8D4×D47semidirect
291. C4#13094052semidirect