Finite Group Databasebeta

← #20254142

Subgroups — #20254142

Order 6016.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D47C211yesmaximal, characteristic, complemented
2C2×#13093995C211yesmaximal, characteristic
3#15689454C211yesmaximal, characteristic, complemented
4#15689451C211yesmaximal, characteristic, complemented
5#15689440C212yesmaximal, complemented
6#15689351C212yesmaximal, complemented
7#15689349C212yesmaximal, complemented
8#15689274C211yesmaximal, characteristic, complemented
9#15689260C212yesmaximal, complemented
10C22⋊C8×D47C211yesmaximal, characteristic, complemented
11C47×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D47C2211yescharacteristic, complemented
13C22×C22⋊D47—21no—
14C22×C4×D47C2211yescharacteristic
15C2×#10625335C2212yescomplemented
16C2×C8×D47C2212yescomplemented
17#13094004C2211yescharacteristic, complemented
18#13094003C2211yescharacteristic
19#13094002—21no—
20#13094001—21no—
21#13094000C2211yescharacteristic, complemented
22#13093995—21no—
23#13093995—21no—
24#13093995C2212yes—
25#13093995C2212yes—
26C2×C4⋊Dic47C2211yescharacteristic
27C4⋊C4×D47—21no—
28C4⋊C4×D47C2211yescharacteristic, complemented
29C2×C22⋊Dic47—21no—
30C22⋊C4×D47—21no—
31#13093956C2212yescomplemented
32#13093955C2212yescomplemented
33#13093949C2212yescomplemented
34#13093948C2212yescomplemented
35#13093945C2212yescomplemented
36C22⋊(C47⋊C8)C2211yescharacteristic
37(C2×D47)⋊C8C2212yescomplemented
38C22×D4×C47C2211yescharacteristic
39C2×SD16×C47C2212yescomplemented
40C22⋊Q8×C47C2211yescharacteristic
41#102×C47C2212yescomplemented
42C22⋊C8×C47C2211yescharacteristic
43C24×C47—21no—
44C23×D47D411yescharacteristic
45C2×Q8×C47C2311yescharacteristic
46C2×D4×C47—21no—
47C2×D4×C47—42no—
48C2×D4×C47C2312yes—
49C22×C4×C47C2311yescharacteristic
50C2×C22⋊D47—22no—
51C2×C22⋊D47—22no—
52C2×C22⋊D47—44no—
53C22×Dic47—21no—
54C22×Dic47D411yescharacteristic
55Q8×D47—42no—
56C2×C4×D47—21no—
57C2×C4×D47—21no—
58C2×C4×D47—42no—
59C2×C4×D47D411yescharacteristic
60C2×C4×D47D411yescharacteristic
61C2×C4×D47C2311yescharacteristic
62C2×C4×D47C2311yescharacteristic
63C2×Dic94—21no—
64C2×Dic94C2311yescharacteristic
65SD16×C47—42no—
66C2×C8×C47C2312yes—
67C4⋊C4×C47—21no—
68C4⋊C4×C47C2311yescharacteristic
69C22⋊C4×C47—21no—
70C22⋊Dic47—21no—
71C22⋊Dic47—22no—
72C22⋊Dic47—22no—
73#10625335—42no—
74(C2×D47)⋊C4—21no—
75(C2×D47)⋊C4—21no—
76C4⋊Dic47—21no—
77C4⋊Dic47—21no—
78C4⋊Dic47C2311yescharacteristic
79C4⋊Dic47C2312yes—
80Dic47⋊C4—21no—
81Dic47⋊C4—21no—
82Dic47⋊C4—21no—
83C2×C47⋊C8C2312yes—
84C8×D47—42no—
85C23×C47—22no—
86C23×C47—42no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47C2×D411yescharacteristic
90C22×D47—41no—
91C22×D47—41no—
92C22×D47SD1614yescomplemented
93C22×D47C2×D411yescharacteristic
94C22×D47C2×D411yescharacteristic
95Q8×C47—22no—
96D4×C47—22no—
97D4×C47—22no—
98D4×C47—42no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47C2×D411yescharacteristic
103C2×C4×C47C2411yescharacteristic
104C22⋊D47—44no—
105C22⋊D47—44no—
106C2×Dic47—21no—
107C2×Dic47—21no—
108C2×Dic47—22no—
109C2×Dic47—41no—
110C2×Dic47—42no—
111C2×Dic47C2×D411yescharacteristic
112C2×Dic47C2×D411yescharacteristic
113C4×D47—22no—
114C4×D47—22no—
115C4×D47—41no—
116C4×D47—41no—
117C4×D47—42no—
118C4×D47—42no—
119Dic94—22no—
120Dic94—42no—
121C8×C47—22no—
122C47⋊C8—22no—
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—42no—
126C22×C47—42no—
127C22×C47—42no—
128C22×C47—84no—
129C22×C47C2×SD1612yes—
130C22×C47C22×D411yescharacteristic
131C2×D47—21no—
132C2×D47—21no—
133C2×D47—21no—
134C2×D47—21no—
135C2×D47—44no—
136C2×D47C2×SD1612yes—
137C2×D47C2×SD1612yes—
138C4×C47—21no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47H(5,𝔽2)12yes—
142Dic47—21no—
143Dic47—21no—
144Dic47—41no—
145Dic47—42no—
146Dic47—41no—
147#3187—471nomaximal, Sylow
148C2×C47—22no—
149C2×C47—82no—
150C2×C47#53411yescharacteristic
151C2×C47C22×SD1611yescharacteristic
152C2×C47#57511yescharacteristic
153D47—22no—
154D47—22no—
155C2×C22⋊Q8—471no—
156C2×#121—471no—
157#483—471no—
158#483—471no—
159#481—472no—
160#465—472no—
161#465—472no—
162#450—471no—
163#450D4711yescharacteristic, complemented
164#448—472no—
165C2×C22⋊C8—471no—
166C47#318711yescharacteristic, complemented, Sylow
167C22×Q8—471no—
168C22×D4—941no—
169C22×D4C2×D4711yescharacteristic
170C23×C4—471no—
171C2×SD16—472no—
172C2×SD16C2×D4712yescomplemented
173C22×C8—472no—
174C22⋊Q8—471no—
175C22⋊Q8—471no—
176C22⋊Q8—471no—
177C22⋊Q8—941no—
178C22⋊Q8—941no—
179C22⋊Q8C2×D4711yescharacteristic
180#121—472no—
181#121—472no—
182#121—941no—
183#121—941no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—941no—
187C2×C22⋊C4—941no—
188C2×C22⋊C4—941no—
189#106—472no—
190#106—472no—
191#103—472no—
192#103—472no—
193#102—472no—
194#102C2×D4712yescomplemented
195C22⋊C8—471no—
196C22⋊C8—472no—
197C22⋊C8C2×D4711yescharacteristic
198C24—21no—
199C24—471no—
200C2×Q8—471no—
201C2×Q8—941no—
202C2×Q8—1882no—
203C2×Q8C22×D4711yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—942no—
207C2×D4—942no—
208C2×D4—1884no—
209C2×D4C22×D4712yes—
210C22×C4—471no—
211C22×C4—471no—
212C22×C4—471no—
213C22×C4—471no—
214C22×C4—471no—
215C22×C4—941no—
216C22×C4—941no—
217C22×C4—941no—
218C22×C4—1882no—
219C22×C4C22×D4711yescharacteristic
220SD16—42no—
221SD16—1882no—
222C2×C8—472no—
223C2×C8—1882no—
224C2×C8C22×D4712yes—
225C4⋊C4—21no—
226C4⋊C4—471no—
227C4⋊C4—472no—
228C4⋊C4—941no—
229C4⋊C4—941no—
230C4⋊C4—941no—
231C4⋊C4—941no—
232C4⋊C4—941no—
233C4⋊C4C22×D4711yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—941no—
236C22⋊C4—941no—
237C22⋊C4—942no—
238C22⋊C4—942no—
239C22⋊C4—941no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—471no—
245C23—471no—
246C23—474no—
247C23—1881no—
248C23—1881no—
249C23D4×D4711yescharacteristic
250Q8—22no—
251Q8—942no—
252Q8—1882no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—1884no—
257D4—1884no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—471no—
262C2×C4—471no—
263C2×C4—942no—
264C2×C4—942no—
265C2×C4—941no—
266C2×C4—942no—
267C2×C4—941no—
268C2×C4—1881no—
269C2×C4—1881no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4—1881no—
273C2×C4—1882no—
274C2×C4D4×D4711yescharacteristic
275C2×C4C23×D4711yescharacteristic
276C8—22no—
277C8—942no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—472no—
285C22—472no—
286C22—941no—
287C22—941no—
288C22—941no—
289C22—941no—
290C22—1884no—
291C22SD16×D4712yes—
292C22C2×D4×D4711yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—941no—
297C4—941no—
298C4—1882no—
299C4—1881no—
300C4—1881no—
301C4#1309406912yes—
302C2—22no—
303C2—82no—
304C2—942no—
305C2—942no—
306C2#1568968311yescharacteristic
307C2C2×SD16×D4711yescharacteristic
308C2#1568991911yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#450C2
socleC22×C47C22×D4
hypercenter#450D47
hypocenterC47#3187
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D47C2semidirect
3. #15689454C2semidirect
4. #15689451C2semidirect
5. #15689440C2semidirect
6. #15689351C2semidirect
7. #15689349C2semidirect
8. #15689274C2semidirect
9. #15689260C2semidirect
10. C22⋊C8×D47C2semidirect
11. C47×#450C2semidirect
12. C2×Q8×D47C22semidirect
15. C2×#10625335C22semidirect
16. C2×C8×D47C22semidirect
17. #13094004C22semidirect
21. #13094000C22semidirect
28. C4⋊C4×D47C22semidirect
31. #13093956C22semidirect
32. #13093955C22semidirect
33. #13093949C22semidirect
34. #13093948C22semidirect
35. #13093945C22semidirect
37. (C2×D47)⋊C8C22semidirect
39. C2×SD16×C47C22semidirect
41. #102×C47C22semidirect
92. C22×D47SD16semidirect
163. #450D47semidirect
166. C47#3187semidirect
172. C2×SD16C2×D47semidirect
194. #102C2×D47semidirect