Finite Group Databasebeta

← #20252865

Subgroups — #20252865

Order 6016.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13094051C212yesmaximal, complemented
2#127×D47C211yesmaximal, characteristic, complemented
3C4×#10625354C211yesmaximal, characteristic, complemented
4#15689323C211yesmaximal, characteristic, complemented
5#15689321C212yesmaximal, complemented
6#15689319C211yesmaximal, characteristic, complemented
7#15689317C212yesmaximal, complemented
8#15689315C211yesmaximal, characteristic, complemented
9#15689145C211yesmaximal, characteristic, complemented
10#15689126C212yesmaximal, complemented
11C47×#460C211yesmaximal, characteristic, complemented
12C2×#10625354C2211yescharacteristic
13C2×D4×D47—21no—
14C2×D4×D47—21no—
15C2×D4×D47C2212yes—
16#13094051—28no—
17C2×#10625336C2212yescomplemented
18C2×D376C2212yescomplemented
19C2×#10625334C2212yescomplemented
20C2×#10625324C2212yes—
21#13094021C2211yescharacteristic
22#13094020—21no—
23#13094020—21no—
24#13094020C2212yes—
25Q8×Dic47C2211yescharacteristic, complemented
26C4×D188—21no—
27C4×D188C2211yescharacteristic, complemented
28#13093986—21no—
29#13093986C2211yescharacteristic, complemented
30#13093982—21no—
31#13093982C2211yescharacteristic
32C42×D47—21no—
33C42×D47C2211yescharacteristic
34C4⋊(C47⋊C8)C2211yescharacteristic, complemented
35Dic47⋊C8C2212yescomplemented
36#13093946C2212yescomplemented
37#13093945C2212yescomplemented
38#13093944C2212yescomplemented
39C2×SD16×C47C2212yescomplemented
40#127×C47C2211yescharacteristic
41C4×Q8×C47C2211yescharacteristic, complemented
42C4⋊C8×C47C2211yescharacteristic, complemented
43#102×C47C2212yescomplemented
44C23×D47—22no—
45C2×Q8×C47C2311yescharacteristic
46C2×D4×C47—21no—
47C2×D4×C47—21no—
48C2×D4×C47C2312yes—
49C2×C22⋊D47—22no—
50C2×C22⋊D47—22no—
51C2×C22⋊D47—22no—
52#10625354—22no—
53#10625354D414yescomplemented
54D4×D47—24no—
55D4×D47—24no—
56D4×D47—44no—
57D4×D47—44no—
58C2×D188—21no—
59C2×D188—21no—
60C2×D188—21no—
61C2×D188C2312yes—
62C2×D188C2311yescharacteristic
63C2×C4×D47—21no—
64C2×C4×D47—21no—
65C2×C4×D47—21no—
66C2×C4×D47D411yescharacteristic
67C2×C4×D47D411yescharacteristic
68C2×C4×D47C2311yescharacteristic
69SD16×C47—24no—
70C2×C8×C47C2312yes—
71C4⋊C4×C47—21no—
72C4⋊C4×C47C2311yescharacteristic
73C42×C47—21no—
74C42×C47C2311yescharacteristic
75#10625336—24no—
76#10625334—24no—
77(C2×D47)⋊C4—21no—
78(C2×D47)⋊C4—21no—
79(C2×D47)⋊C4—21no—
80C4⋊Dic47—21no—
81C4⋊Dic47C2311yescharacteristic
82C4×Dic47—21no—
83C4×Dic47—21no—
84C4×Dic47D411yescharacteristic
85C4×Dic47D411yescharacteristic
86C4×Dic47C2311yescharacteristic
87C2×C47⋊C8C2312yes—
88D376—24no—
89#10625324—24no—
90C23×C47—22no—
91C22×D47—21no—
92C22×D47—22no—
93C22×D47—44no—
94C22×D47—44no—
95C22×D47—44no—
96C22×D47C2×D411yescharacteristic
97Q8×C47—21no—
98Q8×C47C2×D412yescomplemented
99D4×C47—22no—
100D4×C47—22no—
101D4×C47—42no—
102D4×C47—42no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47C2×D411yescharacteristic
106C2×C4×C47C2×D411yescharacteristic
107C2×C4×C47C2411yescharacteristic
108C22⋊D47—44no—
109C22⋊D47—44no—
110C22⋊D47—44no—
111C2×Dic47—21no—
112C2×Dic47C2×D411yescharacteristic
113C2×Dic47C2×D411yescharacteristic
114C2×Dic47C2×D411yescharacteristic
115D188—24no—
116D188—22no—
117D188—21no—
118D188—22no—
119D188—42no—
120D188—42no—
121D188C2×D412yescomplemented
122C4×D47—22no—
123C4×D47—24no—
124C4×D47—22no—
125C4×D47—42no—
126C4×D47—42no—
127C4×D47C2×D412yes—
128C4×D47#4112yes—
129C8×C47—22no—
130C47⋊C8—22no—
131C22×C47—42no—
132C22×C47—42no—
133C22×C47—42no—
134C22×C47C22×D411yescharacteristic
135C2×D47—21no—
136C2×D47—21no—
137C2×D47—22no—
138C2×D47—41no—
139C2×D47—41no—
140C2×D47—42no—
141C2×D47—42no—
142C2×D47—42no—
143C2×D47—88no—
144C2×D47#12112yes—
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—41no—
148C4×C47AGL(1,ℤ/8)12yescomplemented
149C4×C47C22×D411yescharacteristic
150C4×C47C2×#4111yescharacteristic
151Dic47—21no—
152Dic47—41no—
153Dic47#12112yes—
154Dic47AGL(1,ℤ/8)12yescomplemented
155#3038—471nomaximal, Sylow
156C2×C47—82no—
157C2×C47C2×#12111yescharacteristic
158C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C47C2×AGL(1,ℤ/8)11yescharacteristic
160D47—22no—
161D47—42no—
162D47—82no—
163C2×AGL(1,ℤ/8)—472no—
164C2×#127—471no—
165C4×#41—471no—
166#460—472no—
167#460—471no—
168#460D4711yescharacteristic, complemented
169#459—471no—
170#459—471no—
171#459—472no—
172#423—471no—
173#418—472no—
174C47#303811yescharacteristic, complemented, Sylow
175C2×#41—471no—
176C22×D4—472no—
177C22×D4—941no—
178C22×D4—941no—
179AGL(1,ℤ/8)—948no—
180C2×SD16—472no—
181C2×SD16C2×D4712yescomplemented
182C2×D8—472no—
183C2×D8—472no—
184C2×M16—472no—
185#127—472no—
186#127—471no—
187#127—941no—
188#127—941no—
189#127C2×D4711yescharacteristic
190C4×Q8—471no—
191C4×Q8C2×D4711yescharacteristic, complemented
192C4×D4—471no—
193C4×D4—471no—
194C4×D4—941no—
195C4×D4—941no—
196#117—471no—
197#117—941no—
198C2×C42—471no—
199C2×C42—941no—
200C4⋊C8—472no—
201C4⋊C8—471no—
202C4⋊C8C2×D4711yescharacteristic, complemented
203#102—472no—
204#102—472no—
205#102—472no—
206#102C2×D4712yescomplemented
207C24—942no—
208#41—474no—
209#41—942no—
210C2×Q8C22×D4711yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—471no—
214C2×D4—472no—
215C2×D4—941no—
216C2×D4—944no—
217C2×D4—944no—
218C2×D4—942no—
219C2×D4—942no—
220C2×D4—942no—
221C2×D4—941no—
222C2×D4—941no—
223C2×D4—1884no—
224C2×D4—1884no—
225C2×D4C22×D4712yes—
226C22×C4—471no—
227C22×C4—471no—
228C22×C4—471no—
229C22×C4—941no—
230C22×C4—941no—
231C22×C4—941no—
232SD16—24no—
233SD16—944no—
234D8—944no—
235D8—944no—
236M16—944no—
237C2×C8—472no—
238C2×C8C22×D4712yes—
239C4⋊C4—21no—
240C4⋊C4—471no—
241C4⋊C4—941no—
242C4⋊C4C22×D4711yescharacteristic
243C22⋊C4—941no—
244C22⋊C4—941no—
245C22⋊C4—941no—
246C42—21no—
247C42—471no—
248C42—471no—
249C42—471no—
250C42—941no—
251C42—941no—
252C42C22×D4711yescharacteristic
253C23—22no—
254C23—471no—
255C23—941no—
256C23—942no—
257C23—1884no—
258C23—1884no—
259C23—1884no—
260Q8—21no—
261Q8D4×D4712yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—472no—
267D4—944no—
268D4—942no—
269D4—941no—
270D4—942no—
271D4—1882no—
272D4—1884no—
273D4—1884no—
274D4—1884no—
275D4—1882no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—471no—
279C2×C4—471no—
280C2×C4—472no—
281C2×C4—472no—
282C2×C4—471no—
283C2×C4—942no—
284C2×C4—944no—
285C2×C4—942no—
286C2×C4—941no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4D4×D4711yescharacteristic
290C2×C4D4×D4711yescharacteristic
291C2×C4C23×D4711yescharacteristic
292C8—22no—
293C8—942no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—472no—
298C22—941no—
299C22—941no—
300C22—942no—
301C22—3768no—
302C22—1882no—
303C22—1882no—
304C22—1882no—
305C22—1881no—
306C22—1881no—
307C22C2×D4×D4711yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—472no—
312C4—472no—
313C4—941no—
314C4—1881no—
315C4#1309405112yescomplemented
316C4C2×D4×D4711yescharacteristic
317C4#41×D4711yescharacteristic
318C2—82no—
319C2—942no—
320C2—3762no—
321C2—1882no—
322C2#121×D4711yescharacteristic
323C2AGL(1,ℤ/8)×D4711yescharacteristic
324C2C2×#1309405111yescharacteristic
subgroupG/H
centerC22C2×D4×D47
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC47×#460C2
socleC22×C47C22×D4
hypercenter#460D47
hypocenterC47#3038
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#13094051C2semidirect
2. #127×D47C2semidirect
3. C4×#10625354C2semidirect
4. #15689323C2semidirect
5. #15689321C2semidirect
6. #15689319C2semidirect
7. #15689317C2semidirect
8. #15689315C2semidirect
9. #15689145C2semidirect
10. #15689126C2semidirect
11. C47×#460C2semidirect
17. C2×#10625336C22semidirect
18. C2×D376C22semidirect
19. C2×#10625334C22semidirect
25. Q8×Dic47C22semidirect
27. C4×D188C22semidirect
29. #13093986C22semidirect
34. C4⋊(C47⋊C8)C22semidirect
35. Dic47⋊C8C22semidirect
36. #13093946C22semidirect
37. #13093945C22semidirect
38. #13093944C22semidirect
39. C2×SD16×C47C22semidirect
41. C4×Q8×C47C22semidirect
42. C4⋊C8×C47C22semidirect
43. #102×C47C22semidirect
53. #10625354D4semidirect
98. Q8×C47C2×D4semidirect
121. D188C2×D4semidirect
148. C4×C47AGL(1,ℤ/8)semidirect
154. Dic47AGL(1,ℤ/8)semidirect
168. #460D47semidirect
174. C47#3038semidirect
181. C2×SD16C2×D47semidirect
191. C4×Q8C2×D47semidirect
202. C4⋊C8C2×D47semidirect
206. #102C2×D47semidirect
261. Q8D4×D47semidirect
315. C4#13094051semidirect