Finite Group Databasebeta

← #20254174

Subgroups — #20254174

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#15689636C211yesmaximal, characteristic
2C2×#13094004C211yesmaximal, characteristic, complemented
3#15689467C212yesmaximal, complemented
4#15689436C211yesmaximal, characteristic, complemented
5#15689363C212yesmaximal, complemented
6#15689353C212yesmaximal, complemented
7#15689288C212yesmaximal, complemented
8#15689278C211yesmaximal, characteristic, complemented
9#15689264C211yesmaximal, characteristic, complemented
10C22⋊C8×D47C211yesmaximal, characteristic, complemented
11C47×#483C211yesmaximal, characteristic, complemented
12C2×#10625354C2211yescharacteristic
13C22×Dic94C2211yescharacteristic
14C22×C4×D47C2211yescharacteristic
15C2×#10625337C2212yescomplemented
16C2×#10625325C2212yescomplemented
17C2×C8×D47C2212yes—
18#13094009—21no—
19#13094004—21no—
20#13094004—21no—
21#13094004C2212yes—
22#13094004C2212yes—
23#13094000C2211yescharacteristic
24#13093999C2211yescharacteristic
25#13093998C2211yescharacteristic
26#13093994—21no—
27#13093986—21no—
28(C22⋊D47)⋊C4—21no—
29#13093982C2211yescharacteristic
30C2×C4⋊Dic47—21no—
31C2×(C2×D47)⋊C4—21no—
32#13093957C2212yescomplemented
33#13093949C2212yescomplemented
34#13093947C2212yescomplemented
35#13093944C2212yescomplemented
36C22⋊(C47⋊C8)C2211yescharacteristic
37(C2×D47)⋊C8C2212yes—
38C22⋊Q8×C47C2211yescharacteristic
39C2×C4⋊C4×C47C2211yescharacteristic
40#106×C47C2212yescomplemented
41#103×C47C2212yescomplemented
42C22⋊C8×C47C2211yescharacteristic
43C23×D47D411yescharacteristic
44C2×Q8×C47C2311yescharacteristic
45C22×C4×C47—21no—
46C22×C4×C47C2311yescharacteristic
47C2×C22⋊D47—21no—
48C22×Dic47—21no—
49C22×Dic47D411yescharacteristic
50#10625354—42no—
51C2×D188—21no—
52C2×D188—21no—
53C2×D188C2311yescharacteristic
54C2×C4×D47—21no—
55C2×C4×D47—42no—
56C2×C4×D47D411yescharacteristic
57C2×C4×D47D411yescharacteristic
58C2×C4×D47C2311yescharacteristic
59C2×C4×D47C2311yescharacteristic
60C2×Dic94—21no—
61C2×Dic94—42no—
62C2×Dic94C2312yes—
63C2×C8×C47C2312yes—
64C4⋊C4×C47—21no—
65C4⋊C4×C47—21no—
66C4⋊C4×C47C2312yes—
67C4⋊C4×C47C2311yescharacteristic
68C22⋊C4×C47—21no—
69#10625337—42no—
70(C2×D47)⋊C4—22no—
71(C2×D47)⋊C4—22no—
72(C2×D47)⋊C4—21no—
73(C2×D47)⋊C4—21no—
74(C2×D47)⋊C4—21no—
75(C2×D47)⋊C4—21no—
76C4⋊Dic47—22no—
77C4⋊Dic47—22no—
78C4⋊Dic47C2311yescharacteristic
79Dic47⋊C4—21no—
80C4×Dic47—21no—
81C2×C47⋊C8C2312yes—
82#10625325—42no—
83C8×D47—42no—
84C23×C47C2×D411yescharacteristic
85C22×D47—22no—
86C22×D47—22no—
87C22×D47—21no—
88C22×D47—41no—
89C22×D47C2×D411yescharacteristic
90C22×D47C2×D411yescharacteristic
91Q8×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47—42no—
97C2×C4×C47C2×D411yescharacteristic
98C2×C4×C47C2411yescharacteristic
99C22⋊D47—42no—
100C2×Dic47—21no—
101C2×Dic47—22no—
102C2×Dic47—41no—
103C2×Dic47—42no—
104C2×Dic47C2×D411yescharacteristic
105C2×Dic47C2×D411yescharacteristic
106D188—22no—
107D188—42no—
108D188—42no—
109C4×D47—22no—
110C4×D47—22no—
111C4×D47—41no—
112C4×D47—41no—
113C4×D47—42no—
114Dic94—22no—
115Dic94—22no—
116Dic94—42no—
117C8×C47—22no—
118C47⋊C8—22no—
119C22×C47—21no—
120C22×C47—21no—
121C22×C47#13512yes—
122C22×C47C22×D411yescharacteristic
123C2×D47—21no—
124C2×D47—21no—
125C2×D47—21no—
126C2×D47—21no—
127C2×D47—21no—
128C2×D47—44no—
129C2×D47—41no—
130C2×D47—41no—
131C2×D47—41no—
132C2×D47#13712yes—
133C4×C47—21no—
134C4×C47—41no—
135C4×C47—42no—
136C4×C47—41no—
137C4×C47H(5,𝔽2)12yes—
138Dic47—21no—
139Dic47—21no—
140Dic47—42no—
141Dic47—41no—
142#3190—471nomaximal, Sylow
143C2×C47—22no—
144C2×C47#53411yescharacteristic
145C2×C47C2×#13511yescharacteristic
146C2×C47C2×#13711yescharacteristic
147D47—22no—
148D47—41no—
149D47—81no—
150#525—471no—
151C2×C22⋊Q8—471no—
152#485—472no—
153#483D4711yescharacteristic, complemented
154#480—471no—
155#467—472no—
156#465—472no—
157#452—472no—
158#451—471no—
159#448—471no—
160C2×C22⋊C8—471no—
161C47#319011yescharacteristic, complemented, Sylow
162C2×#41—471no—
163C22×Q8—471no—
164C23×C4—471no—
165C2×Q16—472no—
166C2×SD16—472no—
167C22×C8—472no—
168#123—941no—
169C22⋊Q8—471no—
170C22⋊Q8—472no—
171C22⋊Q8—472no—
172C22⋊Q8—941no—
173C22⋊Q8—941no—
174C22⋊Q8C2×D4711yescharacteristic
175#121—471no—
176#121—471no—
177#120—941no—
178C4×D4—941no—
179C4×D4—941no—
180#117—471no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4C2×D4711yescharacteristic
183C2×C22⋊C4—941no—
184#107—472no—
185#106C2×D4712yescomplemented
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—472no—
196C2×Q8—941no—
197C2×Q8—1882no—
198C2×Q8C22×D4711yescharacteristic
199C2×D4—471no—
200C2×D4—941no—
201C2×D4—941no—
202C2×D4—941no—
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—1882no—
215C2×C8—472no—
216C2×C8—1882no—
217C2×C8C22×D4712yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—471no—
221C4⋊C4—941no—
222C4⋊C4—942no—
223C4⋊C4—942no—
224C4⋊C4C22×D4712yes—
225C4⋊C4C22×D4711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—941no—
228C22⋊C4—942no—
229C22⋊C4—942no—
230C22⋊C4—941no—
231C22⋊C4—941no—
232C22⋊C4—941no—
233C42—941no—
234C23—471no—
235C23—471no—
236C23—942no—
237C23—942no—
238C23—941no—
239C23—1881no—
240C23D4×D4711yescharacteristic
241Q8—22no—
242Q8—942no—
243Q8—942no—
244Q8—1882no—
245D4—942no—
246D4—1882no—
247D4—1882no—
248D4—1882no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—471no—
255C2×C4—471no—
256C2×C4—942no—
257C2×C4—942no—
258C2×C4—941no—
259C2×C4—942no—
260C2×C4—1881no—
261C2×C4—1881no—
262C2×C4—1882no—
263C2×C4—1881no—
264C2×C4—1882no—
265C2×C4D4×D4711yescharacteristic
266C2×C4C23×D4711yescharacteristic
267C8—22no—
268C8—942no—
269C22—21no—
270C22—21no—
271C22—472no—
272C22—941no—
273C22—941no—
274C22—941no—
275C22—941no—
276C22—941no—
277C22—1884no—
278C22—1881no—
279C22—1881no—
280C22—1881no—
281C22#1309404512yes—
282C22C2×D4×D4711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—941no—
288C4—941no—
289C4—1881no—
290C4—1882no—
291C4#1309407012yes—
292C2—22no—
293C2—942no—
294C2—3761no—
295C2—1881no—
296C2#1568968411yescharacteristic
297C2C2×#1309404511yescharacteristic
298C2#137×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#483C2
socleC22×C47C22×D4
hypercenter#483D47
hypocenterC47#3190
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#13094004C2semidirect
3. #15689467C2semidirect
4. #15689436C2semidirect
5. #15689363C2semidirect
6. #15689353C2semidirect
7. #15689288C2semidirect
8. #15689278C2semidirect
9. #15689264C2semidirect
10. C22⋊C8×D47C2semidirect
11. C47×#483C2semidirect
15. C2×#10625337C22semidirect
16. C2×#10625325C22semidirect
32. #13093957C22semidirect
33. #13093949C22semidirect
34. #13093947C22semidirect
35. #13093944C22semidirect
40. #106×C47C22semidirect
41. #103×C47C22semidirect
153. #483D47semidirect
161. C47#3190semidirect
185. #106C2×D47semidirect
188. #103C2×D47semidirect