Finite Group Databasebeta

← #20254166

Subgroups — #20254166

Order 6016.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#15689633C211yesmaximal, characteristic
2C2×#13094000C211yesmaximal, characteristic, complemented
3#15689462C212yesmaximal, complemented
4#15689454C211yesmaximal, characteristic, complemented
5#15689434C211yesmaximal, characteristic, complemented
6#15689363C212yesmaximal, complemented
7#15689349C212yesmaximal, complemented
8#15689282C212yesmaximal, complemented
9#15689279C211yesmaximal, characteristic, complemented
10C22⋊C8×D47C211yesmaximal, characteristic, complemented
11C47×#448C211yesmaximal, characteristic, complemented
12C2×#10625352C2211yescharacteristic
13C22×C4×D47C2211yescharacteristic
14C2×#10625337C2212yescomplemented
15C2×C8×D47C2212yes—
16#13094006—21no—
17#13094004C2211yescharacteristic
18#13094003C2211yescharacteristic
19#13094000—21no—
20#13094000—21no—
21#13094000C2212yes—
22#13094000C2212yes—
23#13093995C2211yescharacteristic
24#13093992—21no—
25D4×Dic47—21no—
26(C22⋊D47)⋊C4—21no—
27#13093982C2211yescharacteristic
28C2×C4⋊Dic47C2211yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×(C2×D47)⋊C4—21no—
31#13093957C2212yescomplemented
32#13093956C2212yescomplemented
33#13093949C2212yescomplemented
34#13093948C2212yescomplemented
35#13093945C2212yescomplemented
36C22⋊(C47⋊C8)C2211yescharacteristic
37(C2×D47)⋊C8C2212yes—
38C22×Q8×C47C2211yescharacteristic
39C2×SD16×C47C2212yescomplemented
40#121×C47C2211yescharacteristic
41#103×C47C2212yescomplemented
42C22⋊C8×C47C2211yescharacteristic
43C23×D47D411yescharacteristic
44C2×Q8×C47—21no—
45C2×Q8×C47—42no—
46C2×Q8×C47C2312yes—
47C2×D4×C47—21no—
48C2×D4×C47C2311yescharacteristic
49C22×C4×C47—21no—
50C22×C4×C47C2311yescharacteristic
51C2×C22⋊D47—21no—
52C2×C22⋊D47—21no—
53C22×Dic47—21no—
54C22×Dic47—21no—
55C22×Dic47D411yescharacteristic
56#10625352—42no—
57C2×C4×D47—42no—
58C2×C4×D47D411yescharacteristic
59C2×C4×D47D411yescharacteristic
60C2×C4×D47C2311yescharacteristic
61C2×C4×D47C2311yescharacteristic
62C2×Dic94C2311yescharacteristic
63SD16×C47—42no—
64C2×C8×C47C2312yes—
65C4⋊C4×C47C2311yescharacteristic
66C22⋊C4×C47—21no—
67C22⋊Dic47—21no—
68C22⋊Dic47—21no—
69#10625337—42no—
70(C2×D47)⋊C4—22no—
71(C2×D47)⋊C4—22no—
72(C2×D47)⋊C4—21no—
73(C2×D47)⋊C4—21no—
74C4⋊Dic47—21no—
75C4⋊Dic47—21no—
76C4⋊Dic47C2312yes—
77C4⋊Dic47C2311yescharacteristic
78Dic47⋊C4—21no—
79Dic47⋊C4—22no—
80Dic47⋊C4—22no—
81C4×Dic47—21no—
82C2×C47⋊C8C2312yes—
83C8×D47—42no—
84C23×C47—21no—
85C23×C47C2×D411yescharacteristic
86C22×D47—22no—
87C22×D47—22no—
88C22×D47—41no—
89C22×D47C2×D411yescharacteristic
90C22×D47C2×D411yescharacteristic
91Q8×C47—22no—
92Q8×C47—22no—
93Q8×C47—42no—
94D4×C47—22no—
95D4×C47—42no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—42no—
100C2×C4×C47C2×D411yescharacteristic
101C2×C4×C47C2411yescharacteristic
102C22⋊D47—42no—
103C22⋊D47—42no—
104C2×Dic47—21no—
105C2×Dic47—22no—
106C2×Dic47—21no—
107C2×Dic47—41no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47C2×D411yescharacteristic
111C2×Dic47C2×D411yescharacteristic
112C4×D47—22no—
113C4×D47—22no—
114C4×D47—41no—
115C4×D47—41no—
116Dic94—22no—
117C8×C47—22no—
118C47⋊C8—22no—
119C22×C47—21no—
120C22×C47—21no—
121C22×C47—41no—
122C22×C47—41no—
123C22×C47—41no—
124C22×C47#13512yes—
125C22×C47C22×D411yescharacteristic
126C2×D47—21no—
127C2×D47—21no—
128C2×D47—21no—
129C2×D47—21no—
130C2×D47—21no—
131C2×D47—44no—
132C2×D47#13712yes—
133C4×C47—21no—
134C4×C47—42no—
135C4×C47—41no—
136C4×C47H(5,𝔽2)12yes—
137Dic47—21no—
138Dic47—21no—
139Dic47—41no—
140Dic47—42no—
141Dic47—41no—
142#3190—471nomaximal, Sylow
143C2×C47—22no—
144C2×C47—81no—
145C2×C47#53411yescharacteristic
146C2×C47C2×#13511yescharacteristic
147C2×C47C2×#13711yescharacteristic
148D47—22no—
149D47—41no—
150#525—471no—
151C2×C22⋊Q8—471no—
152#485—472no—
153#483—471no—
154#480—471no—
155#467—472no—
156#465—472no—
157#452—472no—
158#451—471no—
159#448D4711yescharacteristic, complemented
160C2×C22⋊C8—471no—
161C47#319011yescharacteristic, complemented, Sylow
162C2×#41—471no—
163C22×Q8C2×D4711yescharacteristic
164C23×C4—471no—
165C2×Q16—472no—
166C2×SD16C2×D4712yescomplemented
167C22×C8—472no—
168#123—941no—
169C22⋊Q8—472no—
170C22⋊Q8—472no—
171C22⋊Q8—471no—
172C22⋊Q8—471no—
173C22⋊Q8—941no—
174C22⋊Q8—941no—
175#121—471no—
176#121C2×D4711yescharacteristic
177#120—941no—
178C4×D4—941no—
179C4×D4—941no—
180#117—471no—
181C2×C4⋊C4—471no—
182C2×C4⋊C4—941no—
183C2×C22⋊C4—941no—
184#107—472no—
185#106—472no—
186#103—472no—
187#103—472no—
188#103C2×D4712yescomplemented
189#102—472no—
190C22⋊C8—471no—
191C22⋊C8—472no—
192C22⋊C8C2×D4711yescharacteristic
193C24—471no—
194#41—1882no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—471no—
198C2×Q8C22×D4712yes—
199C2×D4—21no—
200C2×D4—941no—
201C2×D4—941no—
202C2×D4C22×D4711yescharacteristic
203C22×C4—21no—
204C22×C4—471no—
205C22×C4—471no—
206C22×C4—471no—
207C22×C4—471no—
208C22×C4—471no—
209C22×C4—941no—
210C22×C4—941no—
211C22×C4—1882no—
212C22×C4C22×D4711yescharacteristic
213Q16—1882no—
214SD16—42no—
215C2×C8—472no—
216C2×C8—1882no—
217C2×C8C22×D4712yes—
218C4⋊C4—472no—
219C4⋊C4—471no—
220C4⋊C4—942no—
221C4⋊C4—942no—
222C4⋊C4—941no—
223C4⋊C4—941no—
224C4⋊C4—941no—
225C4⋊C4C22×D4711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—942no—
228C22⋊C4—942no—
229C22⋊C4—941no—
230C22⋊C4—941no—
231C22⋊C4—941no—
232C22⋊C4—941no—
233C42—941no—
234C23—21no—
235C23—471no—
236C23—471no—
237C23—942no—
238C23—942no—
239C23—1881no—
240C23D4×D4711yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—942no—
245D4—22no—
246D4—42no—
247D4—1882no—
248D4—1882no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—471no—
254C2×C4—471no—
255C2×C4—942no—
256C2×C4—942no—
257C2×C4—941no—
258C2×C4—942no—
259C2×C4—941no—
260C2×C4—1881no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4D4×D4711yescharacteristic
266C2×C4C23×D4711yescharacteristic
267C8—22no—
268C8—942no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—472no—
275C22—941no—
276C22—941no—
277C22—941no—
278C22—941no—
279C22—941no—
280C22—1884no—
281C22#1309404512yes—
282C22C2×D4×D4711yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—941no—
287C4—941no—
288C4—1882no—
289C4—1881no—
290C4—1881no—
291C4#1309406912yes—
292C2—22no—
293C2—81no—
294C2—942no—
295C2—1881no—
296C2#1568968311yescharacteristic
297C2C2×#1309404511yescharacteristic
298C2#137×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#448C2
socleC22×C47C22×D4
hypercenter#448D47
hypocenterC47#3190
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×#13094000C2semidirect
3. #15689462C2semidirect
4. #15689454C2semidirect
5. #15689434C2semidirect
6. #15689363C2semidirect
7. #15689349C2semidirect
8. #15689282C2semidirect
9. #15689279C2semidirect
10. C22⋊C8×D47C2semidirect
11. C47×#448C2semidirect
14. C2×#10625337C22semidirect
31. #13093957C22semidirect
32. #13093956C22semidirect
33. #13093949C22semidirect
34. #13093948C22semidirect
35. #13093945C22semidirect
39. C2×SD16×C47C22semidirect
41. #103×C47C22semidirect
159. #448D47semidirect
161. C47#3190semidirect
166. C2×SD16C2×D47semidirect
188. #103C2×D47semidirect