Finite Group Databasebeta

← #20253021

Subgroups — #20253021

Order 6016.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#13094054C211yesmaximal, characteristic, complemented
2C2×M16×D47C211yesmaximal, characteristic, complemented
3#15689630C211yesmaximal, characteristic
4C2×#13094004C211yesmaximal, characteristic, complemented
5#15689380C212yesmaximal, complemented
6#15689368C212yesmaximal, complemented
7#15689363C212yesmaximal, complemented
8#15689351C212yesmaximal, complemented
9#15689163C211yesmaximal, characteristic, complemented
10#15689133C211yesmaximal, characteristic, complemented
11C47×#417C211yesmaximal, characteristic, complemented
12C22×Dic94C2211yescharacteristic
13C22×C4×D47C2211yescharacteristic
14#13094054—24no—
15C2×#10625337C2212yescomplemented
16C2×#10625335C2212yescomplemented
17C2×#10625329C2211yescharacteristic
18C2×#10625324C2212yes—
19M16×D47—24no—
20C2×C8×D47C2212yes—
21#13094005—21no—
22#13094004—21no—
23#13094004—21no—
24#13094004C2212yes—
25#13094004C2212yes—
26#13094000C2211yescharacteristic
27#13094000C2211yescharacteristic
28#13093995C2211yescharacteristic
29#13093995C2211yescharacteristic
30#13093992—21no—
31C4×C22⋊D47—21no—
32C4×C22⋊D47—21no—
33#13093983C2211yescharacteristic
34C2×C4⋊Dic47—21no—
35C2×(C2×D47)⋊C4—21no—
36#13093957C2212yescomplemented
37#13093955C2212yescomplemented
38#13093949C2212yescomplemented
39#13093949C2212yescomplemented
40C2×#41×C47C2211yescharacteristic
41C2×M16×C47C2211yescharacteristic
42C2×C4⋊C4×C47C2211yescharacteristic
43#103×C47C2212yescomplemented
44#102×C47C2212yescomplemented
45C23×D47D411yescharacteristic
46#41×C47—22no—
47#41×C47—22no—
48C2×Q8×C47C2311yescharacteristic
49C2×D4×C47—21no—
50C2×D4×C47C2311yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47C2311yescharacteristic
54C2×C22⋊D47—21no—
55C2×C22⋊D47—21no—
56C22×Dic47—21no—
57C22×Dic47D411yescharacteristic
58C2×C4×D47—22no—
59C2×C4×D47—22no—
60C2×C4×D47D411yescharacteristic
61C2×C4×D47D411yescharacteristic
62C2×C4×D47C2311yescharacteristic
63C2×C4×D47C2311yescharacteristic
64C2×Dic94—22no—
65C2×Dic94—22no—
66C2×Dic94—21no—
67C2×Dic94C2312yes—
68M16×C47—22no—
69C2×C8×C47C2312yes—
70C4⋊C4×C47—21no—
71C4⋊C4×C47C2312yes—
72C22⋊Dic47—21no—
73C22⋊Dic47—21no—
74C22⋊Dic47—21no—
75#10625337—24no—
76#10625335—24no—
77(C2×D47)⋊C4—22no—
78(C2×D47)⋊C4—22no—
79(C2×D47)⋊C4—21no—
80(C2×D47)⋊C4—21no—
81C4⋊Dic47—22no—
82C4⋊Dic47—22no—
83C4⋊Dic47C2311yescharacteristic
84C4⋊Dic47C2311yescharacteristic
85Dic47⋊C4—21no—
86Dic47⋊C4—21no—
87C4×Dic47—21no—
88#10625329D414yescomplemented
89C2×C47⋊C8C2312yes—
90#10625324—24no—
91C8×D47—24no—
92C23×C47—21no—
93C23×C47C2×D411yescharacteristic
94C22×D47—22no—
95C22×D47—22no—
96C22×D47—41no—
97C22×D47C2×D411yescharacteristic
98C22×D47C2×D411yescharacteristic
99Q8×C47—21no—
100Q8×C47—21no—
101D4×C47—21no—
102D4×C47—21no—
103D4×C47—42no—
104C2×C4×C47—21no—
105C2×C4×C47—22no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47C2×D411yescharacteristic
109C2×C4×C47C2×D412yes—
110C2×C4×C47#4112yes—
111C2×C4×C47C2411yescharacteristic
112C22⋊D47—42no—
113C22⋊D47—42no—
114C2×Dic47—22no—
115C2×Dic47—21no—
116C2×Dic47—21no—
117C2×Dic47—22no—
118C2×Dic47—42no—
119C2×Dic47C2×D411yescharacteristic
120C2×Dic47C2×D411yescharacteristic
121C4×D47—22no—
122C4×D47—22no—
123C4×D47—21no—
124C4×D47—21no—
125C4×D47—21no—
126C4×D47—21no—
127Dic94—22no—
128Dic94—22no—
129Dic94—42no—
130C8×C47—22no—
131C47⋊C8C2×D414yes—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47#12112yes—
138C22×C47C22×D411yescharacteristic
139C2×D47—21no—
140C2×D47—21no—
141C2×D47—21no—
142C2×D47—21no—
143C2×D47—21no—
144C2×D47—44no—
145C2×D47#13712yescomplemented
146C4×C47—41no—
147C4×C47—42no—
148C4×C47#12112yes—
149C4×C47C22×D411yescharacteristic
150C4×C47C2×#4111yescharacteristic
151Dic47—21no—
152Dic47—21no—
153Dic47—42no—
154Dic47—41no—
155Dic47—41no—
156#3056—471nomaximal, Sylow
157C2×C47—22no—
158C2×C47—81no—
159C2×C47C2×#12111yescharacteristic
160C2×C47C2×#13711yescharacteristic
161C2×C47#57511yescharacteristic
162D47—22no—
163D47—41no—
164C2×#137—471no—
165C22×M16—471no—
166#525—471no—
167C2×C22⋊Q8—471no—
168#470—472no—
169#468—472no—
170#467—472no—
171#465—472no—
172#428—471no—
173#419—471no—
174#417D4711yescharacteristic, complemented
175C47#305611yescharacteristic, complemented, Sylow
176C2×#41C2×D4711yescharacteristic
177C22×Q8—471no—
178C23×C4—471no—
179#137—944no—
180C2×Q16—472no—
181C2×SD16—472no—
182C2×M16—471no—
183C2×M16—472no—
184C2×M16—944no—
185C2×M16C2×D4711yescharacteristic
186C22×C8—472no—
187#123—941no—
188C22⋊Q8—471no—
189C22⋊Q8—471no—
190C22⋊Q8—472no—
191C22⋊Q8—472no—
192C22⋊Q8—941no—
193C22⋊Q8—941no—
194#121—471no—
195#121—471no—
196#120—941no—
197C4×D4—941no—
198C4×D4—941no—
199#117—471no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4C2×D4711yescharacteristic
202C2×C22⋊C4—941no—
203#107—472no—
204#106—472no—
205#103—472no—
206#103—472no—
207#103C2×D4712yescomplemented
208#102C2×D4712yescomplemented
209C24—471no—
210#41—22no—
211#41—22no—
212C2×Q8—472no—
213C2×Q8—941no—
214C2×Q8—942no—
215C2×Q8—942no—
216C2×Q8C22×D4711yescharacteristic
217C2×D4—21no—
218C2×D4—941no—
219C2×D4—941no—
220C2×D4C22×D4711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—471no—
224C22×C4—471no—
225C22×C4—471no—
226C22×C4—471no—
227C22×C4—471no—
228C22×C4—942no—
229C22×C4—942no—
230C22×C4—941no—
231C22×C4C22×D4711yescharacteristic
232Q16—944no—
233SD16—944no—
234M16—22no—
235M16—474no—
236M16—944no—
237C2×C8—472no—
238C2×C8—944no—
239C2×C8C22×D4712yes—
240C4⋊C4—21no—
241C4⋊C4—471no—
242C4⋊C4—471no—
243C4⋊C4—941no—
244C4⋊C4—941no—
245C4⋊C4—942no—
246C4⋊C4—942no—
247C4⋊C4C22×D4712yes—
248C22⋊C4—942no—
249C22⋊C4—942no—
250C22⋊C4—941no—
251C22⋊C4—941no—
252C22⋊C4—941no—
253C22⋊C4—941no—
254C22⋊C4—941no—
255C42—941no—
256C23—21no—
257C23—471no—
258C23—471no—
259C23—942no—
260C23—942no—
261C23—1881no—
262C23D4×D4711yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—942no—
266Q8—942no—
267Q8—1882no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—1882no—
272D4—1882no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—471no—
278C2×C4—471no—
279C2×C4—942no—
280C2×C4—942no—
281C2×C4—942no—
282C2×C4—941no—
283C2×C4—941no—
284C2×C4—941no—
285C2×C4—941no—
286C2×C4—941no—
287C2×C4—941no—
288C2×C4—942no—
289C2×C4—1882no—
290C2×C4#1062535012yes—
291C2×C4D4×D4711yescharacteristic
292C2×C4D4×D4712yes—
293C2×C4C23×D4711yescharacteristic
294C8—22no—
295C8—474no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—472no—
302C22—941no—
303C22—941no—
304C22—941no—
305C22—941no—
306C22—941no—
307C22—1884no—
308C22#1309399712yes—
309C22C2×D4×D4711yescharacteristic
310C4—41no—
311C4—42no—
312C4—941no—
313C4—941no—
314C4—1882no—
315C4—1881no—
316C4—1881no—
317C4#1309399712yes—
318C4C2×D4×D4711yescharacteristic
319C4C2×#1062535011yescharacteristic
320C2—22no—
321C2—81no—
322C2—942no—
323C2—1881no—
324C2C2×#1309399711yescharacteristic
325C2#137×D4711yescharacteristic
326C2#1568991911yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#417C2
socleC22×C47C22×D4
hypercenter#417D47
hypocenterC47#3056
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#13094054C2semidirect
2. C2×M16×D47C2semidirect
4. C2×#13094004C2semidirect
5. #15689380C2semidirect
6. #15689368C2semidirect
7. #15689363C2semidirect
8. #15689351C2semidirect
9. #15689163C2semidirect
10. #15689133C2semidirect
11. C47×#417C2semidirect
15. C2×#10625337C22semidirect
16. C2×#10625335C22semidirect
36. #13093957C22semidirect
37. #13093955C22semidirect
38. #13093949C22semidirect
39. #13093949C22semidirect
43. #103×C47C22semidirect
44. #102×C47C22semidirect
88. #10625329D4semidirect
145. C2×D47#137semidirect
174. #417D47semidirect
175. C47#3056semidirect
207. #103C2×D47semidirect
208. #102C2×D47semidirect