Finite Group Databasebeta

← #20254327

Subgroups — #20254327

Order 6016.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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#15689675C211yesmaximal, characteristic, complemented
2C2×#13094020C211yesmaximal, characteristic
3#15689494C212yesmaximal, complemented
4#15689468C212yesmaximal, complemented
5#15689322C212yesmaximal, complemented
6#15689321C212yesmaximal, complemented
7#15689276C211yesmaximal, characteristic, complemented
8#15689274C211yesmaximal, characteristic, complemented
9#15689273C211yesmaximal, characteristic, complemented
10#15689173C211yesmaximal, characteristic, complemented
11C47×#450C211yesmaximal, characteristic, complemented
12C22×D188C2211yescharacteristic
13C22×C22⋊D47—21no—
14C22×C22⋊D47—21no—
15C2×#10625325C2212yescomplemented
16C2×#10625335C2212yescomplemented
17C2×#10625336C2212yescomplemented
18#13094023C2211yescharacteristic, complemented
19#13094022C2211yescharacteristic, complemented
20#13094020—21no—
21#13094020—21no—
22#13094020C2212yes—
23#13094020C2212yes—
24#13094014—21no—
25#13094003C2211yescharacteristic
26#13094001—21no—
27Q8×Dic47C2211yescharacteristic, complemented
28#13093989—21no—
29#13093989C2211yescharacteristic, complemented
30#13093980—21no—
31C2×C4×Dic47C2211yescharacteristic
32Dic47⋊C8C2212yescomplemented
33#13093946C2212yescomplemented
34#13093945C2212yescomplemented
35#13093944C2212yescomplemented
36C22⋊(C47⋊C8)C2211yescharacteristic
37C8×Dic47C2212yescomplemented
38C22×D4×C47C2211yescharacteristic
39C2×SD16×C47C2212yescomplemented
40C22⋊Q8×C47C2211yescharacteristic
41#102×C47C2212yescomplemented
42C22⋊C8×C47C2211yescharacteristic
43C24×C47—21no—
44C23×D47—21no—
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—22no—
53C2×C22⋊D47—22no—
54C2×C22⋊D47—44no—
55C2×C22⋊D47—44no—
56C22×Dic47D411yescharacteristic
57C22×Dic47D411yescharacteristic
58C2×D188—21no—
59C2×D188—42no—
60C2×D188C2312yes—
61C2×Dic94—21no—
62C2×Dic94C2311yescharacteristic
63SD16×C47—42no—
64C2×C8×C47C2312yes—
65C4⋊C4×C47—21no—
66C4⋊C4×C47C2311yescharacteristic
67C22⋊C4×C47—21no—
68C22⋊Dic47—21no—
69#10625336—42no—
70#10625335—42no—
71C4⋊Dic47—21no—
72C4⋊Dic47C2311yescharacteristic
73Dic47⋊C4—21no—
74Dic47⋊C4—21no—
75Dic47⋊C4—21no—
76Dic47⋊C4—21no—
77Dic47⋊C4—21no—
78C4×Dic47—21no—
79C4×Dic47—21no—
80C4×Dic47D411yescharacteristic
81C4×Dic47D411yescharacteristic
82C4×Dic47C2311yescharacteristic
83C4×Dic47C2311yescharacteristic
84C2×C47⋊C8C2312yes—
85#10625325—42no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47—42no—
90C23×C47C2×D411yescharacteristic
91C22×D47—22no—
92C22×D47—42no—
93C22×D47—42no—
94C22×D47—42no—
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—
106C22⋊D47—44no—
107C22⋊D47—44no—
108C2×Dic47—22no—
109C2×Dic47—21no—
110C2×Dic47—21no—
111C2×Dic47SD1614yescomplemented
112C2×Dic47C2×D411yescharacteristic
113C2×Dic47C2×D411yescharacteristic
114C2×Dic47C2×D411yescharacteristic
115C2×Dic47C2×D411yescharacteristic
116D188—22no—
117D188—22no—
118D188—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—42no—
132C2×D47—42no—
133C2×D47—42no—
134C2×D47—84no—
135C4×C47—21no—
136C4×C47—41no—
137C4×C47—41no—
138C4×C47H(5,𝔽2)12yes—
139Dic47—21no—
140Dic47—41no—
141Dic47—41no—
142Dic47C2×SD1612yes—
143Dic47C2×SD1612yes—
144Dic47AGL(1,ℤ/8)12yes—
145#3208—471nomaximal, Sylow
146C2×C47—22no—
147C2×C47—82no—
148C2×C47#53511yescharacteristic
149C2×C47C22×SD1611yescharacteristic
150C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
151D47—82no—
152#533—471no—
153C2×#127—471no—
154#492—472no—
155#486—472no—
156#460—472no—
157#460—472no—
158#450—471no—
159#450—471no—
160#450—471no—
161#450D4711yescharacteristic, complemented
162#431—471no—
163C47#320811yescharacteristic, complemented, Sylow
164C22×D4—471no—
165C22×D4—941no—
166C22×D4—941no—
167C22×D4C2×D4711yescharacteristic
168C2×SD16—472no—
169C2×SD16—472no—
170C2×SD16—472no—
171C2×SD16C2×D4712yescomplemented
172C4⋊Q8—471no—
173C4⋊Q8—471no—
174#127—472no—
175#127—472no—
176#127—941no—
177#127—941no—
178#125—941no—
179C22⋊Q8—471no—
180C22⋊Q8—941no—
181C22⋊Q8C2×D4711yescharacteristic
182C4×Q8—471no—
183C4×Q8—471no—
184C4×Q8—941no—
185#117—941no—
186C2×C42—471no—
187C4⋊C8—472no—
188#102—472no—
189#102—472no—
190#102—472no—
191#102C2×D4712yescomplemented
192C22⋊C8—471no—
193C22⋊C8C2×D4711yescharacteristic
194C4×C8—472no—
195C24—21no—
196C24—941no—
197C2×Q8—471no—
198C2×Q8—941no—
199C2×Q8C22×D4711yescharacteristic
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—472no—
203C2×D4—942no—
204C2×D4—942no—
205C2×D4—942no—
206C2×D4—942no—
207C2×D4—941no—
208C2×D4—1882no—
209C2×D4—1884no—
210C2×D4—1884no—
211C2×D4C22×D4712yes—
212C22×C4—471no—
213C22×C4—471no—
214C22×C4C22×D4711yescharacteristic
215SD16—42no—
216SD16—1882no—
217SD16—1882no—
218SD16—1882no—
219C2×C8—472no—
220C2×C8C22×D4712yes—
221C4⋊C4—21no—
222C4⋊C4—471no—
223C4⋊C4—941no—
224C4⋊C4—941no—
225C4⋊C4—941no—
226C4⋊C4—941no—
227C4⋊C4—941no—
228C4⋊C4—941no—
229C4⋊C4C22×D4711yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—941no—
232C42—471no—
233C42—471no—
234C42—471no—
235C42—471no—
236C42—941no—
237C42—941no—
238C23—22no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—942no—
243C23—1882no—
244C23—1882no—
245C23—1882no—
246C23D4×D4711yescharacteristic
247Q8—22no—
248Q8—942no—
249Q8—1882no—
250D4—22no—
251D4—22no—
252D4—42no—
253D4—942no—
254D4—942no—
255D4—1882no—
256D4—1884no—
257D4—1884no—
258D4—1884no—
259D4—1884no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—474no—
264C2×C4—471no—
265C2×C4—471no—
266C2×C4—471no—
267C2×C4—471no—
268C2×C4—942no—
269C2×C4—941no—
270C2×C4—941no—
271C2×C4D4×D4711yescharacteristic
272C2×C4C23×D4711yescharacteristic
273C8—22no—
274C8—942no—
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—3764no—
282C22—1882no—
283C22—1882no—
284C22—1882no—
285C22SD16×D4712yes—
286C22C2×D4×D4711yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—472no—
291C4—472no—
292C4—472no—
293C4—941no—
294C4—1881no—
295C4—1881no—
296C4#1309406912yes—
297C2—22no—
298C2—82no—
299C2—3762no—
300C2#1568968811yescharacteristic
301C2C2×SD16×D4711yescharacteristic
302C2AGL(1,ℤ/8)×D4711yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#450C2
socleC22×C47C22×D4
hypercenter#450D47
hypocenterC47#3208
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. #15689675C2semidirect
3. #15689494C2semidirect
4. #15689468C2semidirect
5. #15689322C2semidirect
6. #15689321C2semidirect
7. #15689276C2semidirect
8. #15689274C2semidirect
9. #15689273C2semidirect
10. #15689173C2semidirect
11. C47×#450C2semidirect
15. C2×#10625325C22semidirect
16. C2×#10625335C22semidirect
17. C2×#10625336C22semidirect
18. #13094023C22semidirect
19. #13094022C22semidirect
27. Q8×Dic47C22semidirect
29. #13093989C22semidirect
32. Dic47⋊C8C22semidirect
33. #13093946C22semidirect
34. #13093945C22semidirect
35. #13093944C22semidirect
37. C8×Dic47C22semidirect
39. C2×SD16×C47C22semidirect
41. #102×C47C22semidirect
111. C2×Dic47SD16semidirect
161. #450D47semidirect
163. C47#3208semidirect
171. C2×SD16C2×D47semidirect
191. #102C2×D47semidirect