Finite Group Databasebeta

← #20246525

Subgroups — #20246525

Order 6016.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13093992C211yesmaximal, characteristic
2#15689023C212yesmaximal, complemented
3#15689014C212yesmaximal, complemented
4#15689013C212yesmaximal, complemented
5#15689011C212yesmaximal, complemented
6#15689003C211yesmaximal, characteristic, complemented
7#15689000C212yesmaximal, complemented
8#15688978C212yesmaximal, complemented
9C47×#390C211yesmaximal, characteristic, complemented
10C24×D47C2211yescharacteristic
11C22×C22⋊D47C2212yes—
12C23×Dic47C2211yescharacteristic, complemented
13#13093992C418yescomplemented
14C2×Dic47⋊C4C2212yescomplemented
15C2×C22⋊Dic47C2212yes—
16C2×C22⋊Dic47C2211yescharacteristic
17C2×C22⋊Dic47C2211yescharacteristic, complemented
18C2×(C2×D47)⋊C4C2212yes—
19C2×(C2×D47)⋊C4C2212yes—
20C2×(C2×D47)⋊C4C2212yes—
21C2×(C2×D47)⋊C4C2212yes—
22C2×(C2×D47)⋊C4C2212yes—
23C2×C4×Dic47C2212yescomplemented
24C47⋊#95C2212yescomplemented
25C47⋊#95C2212yescomplemented
26C47⋊#95C2212yescomplemented
27C22×D4×C47C2211yescharacteristic
28C2×C4⋊C4×C47C2211yescharacteristic, complemented
29C2×C22⋊C4×C47C2212yes—
30C2×C22⋊C4×C47C2212yescomplemented
31C2×C42×C47C2211yescharacteristic, complemented
32C24×C47C2311yescharacteristic
33C24×C47C2311yescharacteristic
34C23×D47—41no—
35C23×D47—41no—
36C23×D47—42no—
37C23×D47—42no—
38C23×D47C2×C414yescomplemented
39C23×D47C2312yes—
40C2×D4×C47—41no—
41C2×D4×C47—41no—
42C2×D4×C47C2×C414yes—
43C22×C4×C47C2312yes—
44C22×C4×C47C2312yes—
45C22×C4×C47C2311yescharacteristic
46C2×C22⋊D47—42no—
47C2×C22⋊D47—42no—
48C2×C22⋊D47C2×C418yes—
49C22×Dic47—41no—
50C22×Dic47—42no—
51C22×Dic47C2312yes—
52C22×Dic47C2312yes—
53C22×Dic47C2312yes—
54C4⋊C4×C47—41no—
55C22⋊C4×C47—42no—
56C22⋊C4×C47—42no—
57C42×C47—41no—
58C22⋊Dic47—42no—
59C22⋊Dic47—41no—
60C22⋊Dic47C2×C414yescomplemented
61(C2×D47)⋊C4—42no—
62(C2×D47)⋊C4—42no—
63(C2×D47)⋊C4—42no—
64(C2×D47)⋊C4—42no—
65(C2×D47)⋊C4C2×C418yescomplemented
66Dic47⋊C4—42no—
67C4×Dic47—42no—
68C23×C47—21no—
69C23×C47—22no—
70C23×C47—22no—
71C23×C47—22no—
72C23×C47—22no—
73C23×C47—21no—
74C23×C47—21no—
75C23×C47—21no—
76C23×C47C22×C412yes—
77C23×C47C22×C412yes—
78C23×C47C2411yescharacteristic
79C22×D47—22no—
80C22×D47—22no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47—22no—
84C22×D47—22no—
85C22×D47—48no—
86C22×D47—44no—
87C22×D47—44no—
88C22×D47—48no—
89C22×D47—44no—
90C22×D47C22×C414yes—
91D4×C47—44no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47C22×C412yes—
101C22⋊D47—48no—
102C2×Dic47—22no—
103C2×Dic47—22no—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—22no—
108C2×Dic47—22no—
109C2×Dic47—22no—
110C2×Dic47—44no—
111C2×Dic47C22×C414yes—
112C22×C47—24no—
113C22×C47—22no—
114C22×C47—22no—
115C22×C47—24no—
116C22×C47—41no—
117C22×C47—42no—
118C22×C47—41no—
119C22×C47—42no—
120C22×C47—41no—
121C22×C47—41no—
122C22×C47C23×C411yescharacteristic
123C22×C47H(5,𝔽2)12yes—
124C22×C47H(5,𝔽2)11yescharacteristic
125C22×C47H(5,𝔽2)11yescharacteristic
126C22×C47H(5,𝔽2)12yes—
127C2×D47—24no—
128C2×D47—24no—
129C2×D47—24no—
130C2×D47—416no—
131C2×D47—42no—
132C2×D47—42no—
133C2×D47—42no—
134C2×D47—42no—
135C4×C47—22no—
136C4×C47—42no—
137C4×C47—42no—
138Dic47—24no—
139Dic47—42no—
140Dic47—42no—
141#2375—471nomaximal, Sylow
142C2×C47—42no—
143C2×C47—42no—
144C2×C47#51812yes—
145C2×C47#51811yescharacteristic
146C2×C47#56011yescharacteristic
147C2×C47#56012yes—
148C2×C47H(3,𝔽4)11yescharacteristic
149D47—44no—
150C2×#120—471no—
151#390—472no—
152#390D4711yescharacteristic, complemented
153#388—472no—
154#388—472no—
155#388—472no—
156#386—471no—
157#386—472no—
158#379—472no—
159C47#237511yescharacteristic, complemented, Sylow
160C25—471no—
161C22×D4—472no—
162C22×D4C2×D4711yescharacteristic
163C23×C4—471no—
164#120—478no—
165C2×C4⋊C4—472no—
166C2×C4⋊C4C2×D4711yescharacteristic, complemented
167C2×C22⋊C4—471no—
168C2×C22⋊C4—472no—
169C2×C22⋊C4—472no—
170C2×C22⋊C4—471no—
171C2×C22⋊C4—472no—
172C2×C22⋊C4—472no—
173C2×C22⋊C4—472no—
174C2×C22⋊C4—472no—
175C2×C22⋊C4C2×D4712yes—
176C2×C22⋊C4C2×D4712yescomplemented
177C2×C42—472no—
178C2×C42C2×D4711yescharacteristic, complemented
179#95—472no—
180#95—472no—
181#95—472no—
182C24—472no—
183C24—474no—
184C24—1882no—
185C24—1881no—
186C24—1881no—
187C24—1882no—
188C24C22×D4711yescharacteristic
189C24C22×D4711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—478no—
193C2×D4—1882no—
194C2×D4—1882no—
195C2×D4C4×D4714yes—
196C22×C4—472no—
197C22×C4—472no—
198C22×C4—472no—
199C22×C4—1881no—
200C22×C4—1882no—
201C22×C4C22×D4712yes—
202C22×C4C22×D4712yes—
203C22×C4C22×D4711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—1882no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—474no—
209C22⋊C4—478no—
210C22⋊C4—1882no—
211C22⋊C4—1882no—
212C22⋊C4—1882no—
213C22⋊C4—1882no—
214C22⋊C4—1881no—
215C22⋊C4—1882no—
216C42—41no—
217C42—1882no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—474no—
227C23—942no—
228C23—942no—
229C23—942no—
230C23—942no—
231C23—942no—
232C23—942no—
233C23—1888no—
234C23—1884no—
235C23—1884no—
236C23—1888no—
237C23—1884no—
238C23C2×C4×D4712yes—
239C23C2×C4×D4712yes—
240C23C23×D4711yescharacteristic
241D4—44no—
242D4—1888no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—474no—
252C2×C4—942no—
253C2×C4—942no—
254C2×C4—942no—
255C2×C4—942no—
256C2×C4—942no—
257C2×C4—942no—
258C2×C4—942no—
259C2×C4—942no—
260C2×C4—1884no—
261C2×C4C2×C4×D4712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—944no—
273C22—944no—
274C22—944no—
275C22—18816no—
276C22—1882no—
277C22—1882no—
278C22—1882no—
279C22—1882no—
280C22C22×C4×D4711yescharacteristic
281C22#1309406912yes—
282C22#1309406911yescharacteristic
283C22#1309406911yescharacteristic
284C22#1309407012yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—944no—
289C4—1882no—
290C4—1882no—
291C2—42no—
292C2—42no—
293C2—1884no—
294C2#1568959512yes—
295C2#1568959611yescharacteristic
296C2#1568986011yescharacteristic
297C2#1568986112yes—
298C2#1568986511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D47
commutatorC22×C47C23×C4
FittingC47×#390C2
socleC23×C47C24
hypercenter#390D47
hypocenterC47#2375
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #15689023C2semidirect
3. #15689014C2semidirect
4. #15689013C2semidirect
5. #15689011C2semidirect
6. #15689003C2semidirect
7. #15689000C2semidirect
8. #15688978C2semidirect
9. C47×#390C2semidirect
12. C23×Dic47C22semidirect
13. #13093992C4semidirect
14. C2×Dic47⋊C4C22semidirect
17. C2×C22⋊Dic47C22semidirect
23. C2×C4×Dic47C22semidirect
24. C47⋊#95C22semidirect
25. C47⋊#95C22semidirect
26. C47⋊#95C22semidirect
28. C2×C4⋊C4×C47C22semidirect
30. C2×C22⋊C4×C47C22semidirect
31. C2×C42×C47C22semidirect
38. C23×D47C2×C4semidirect
60. C22⋊Dic47C2×C4semidirect
65. (C2×D47)⋊C4C2×C4semidirect
152. #390D47semidirect
159. C47#2375semidirect
166. C2×C4⋊C4C2×D47semidirect
176. C2×C22⋊C4C2×D47semidirect
178. C2×C42C2×D47semidirect