Finite Group Databasebeta

← #22599963

Subgroups — #22599963

Order 9088.

556 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556152C212yesmaximal, complemented
2#17986669C211yesmaximal, characteristic, complemented
3D4×D284C211yesmaximal, characteristic, complemented
4#17986375C211yesmaximal, characteristic, complemented
5#17986251C212yesmaximal, complemented
6#17986200C211yesmaximal, characteristic, complemented
7#17986145C212yesmaximal, complemented
8#17986131C212yesmaximal, complemented
9C4×#11765853C211yesmaximal, characteristic, complemented
10#17986076C211yesmaximal, characteristic, complemented
11#459×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2212yescomplemented
13C22×D284—21no—
14C2×D4×D71C2212yes—
15#14556152—28no—
16C2×#11765854C2212yescomplemented
17C2×#11765853C2211yescharacteristic
18C2×#11765850C2212yescomplemented
19C2×#11765850C2212yescomplemented
20C2×#11765848C2212yescomplemented
21#14556125C2211yescharacteristic
22#14556124—21no—
23#14556103C2212yes—
24#14556102—21no—
25#14556100—22no—
26#14556099C2212yescomplemented
27#14556098—22no—
28C4×Dic142C2211yescharacteristic, complemented
29C4×D284C2211yescharacteristic
30D4×Dic71—21no—
31#14556059C2211yescharacteristic, complemented
32#14556051C2211yescharacteristic, complemented
33#14556050C2212yescomplemented
34#14556049C2211yescharacteristic
35(C2×D71)⋊C8C2212yescomplemented
36C4×C71⋊C8C2211yescharacteristic, complemented
37C2×D8×C71C2212yes—
38#127×C71C2211yescharacteristic, complemented
39C4×D4×C71C2211yescharacteristic
40C4⋊C8×C71C2211yescharacteristic, complemented
41#102×C71C2212yescomplemented
42C23×D71—22no—
43#11765864—24no—
44#11765864—42no—
45C2×Dic142C2311yescharacteristic
46C2×D284—22no—
47C2×D284—21no—
48C2×D284—21no—
49C2×D284—44no—
50C2×D284D411yescharacteristic
51C2×D284C2312yes—
52C2×C22⋊D71—22no—
53C2×C22⋊D71—22no—
54C2×C22⋊D71—22no—
55D4×D71—24no—
56D4×D71—42no—
57C22×Dic71—22no—
58C2×C4×D71C2312yes—
59#11765854—24no—
60#11765853D414yescomplemented
61#11765850—24no—
62#11765850—42no—
63#11765848—24no—
64C2×C71⋊C8C2311yescharacteristic
65C2×C71⋊C8C2311yescharacteristic
66C4⋊Dic71D411yescharacteristic
67C4⋊Dic71C2311yescharacteristic
68Dic71⋊C4—21no—
69C22⋊Dic71—22no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4D412yes—
72C4×Dic71—21no—
73C2×D4×C71—21no—
74C2×D4×C71—21no—
75C2×D4×C71C2312yes—
76C2×D4×C71C2311yescharacteristic
77C22×C4×C71—21no—
78D8×C71—24no—
79C2×C8×C71C2312yes—
80C4⋊C4×C71C2311yescharacteristic
81C22⋊C4×C71—21no—
82C42×C71C2311yescharacteristic
83C23×C71—21no—
84C23×C71—22no—
85C22×D71—22no—
86C22×D71—44no—
87C22×D71—44no—
88C22×D71—82no—
89C22×D71C2×D412yes—
90D4×C71—22no—
91D4×C71—21no—
92D4×C71—22no—
93D4×C71—42no—
94D4×C71—81no—
95D4×C71C2×D412yes—
96C2×C4×C71—21no—
97C2×C4×C71—42no—
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2411yescharacteristic
101C22⋊D71—44no—
102C22⋊D71—44no—
103C22⋊D71—82no—
104C2×Dic71—21no—
105C2×Dic71—21no—
106C2×Dic71—44no—
107C2×Dic71C2×D412yes—
108D284—22no—
109D284—22no—
110D284—22no—
111D284—44no—
112D284—42no—
113D284—81no—
114C4×D71—22no—
115C4×D71—22no—
116Dic142—21no—
117Dic142C2×D412yescomplemented
118C8×C71—22no—
119C71⋊C8—21no—
120C71⋊C8C2×D412yescomplemented
121C22×C71—22no—
122C22×C71—41no—
123C22×C71—41no—
124C22×C71—42no—
125C22×C71—42no—
126C22×C71—42no—
127C22×C71C22×D411yescharacteristic
128C2×D71—22no—
129C2×D71—22no—
130C2×D71—22no—
131C2×D71—42no—
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—88no—
135C4×C71—21no—
136C4×C71—41no—
137C4×C71AGL(1,ℤ/8)12yescomplemented
138C4×C71C22×D411yescharacteristic
139C4×C71H(5,𝔽2)11yescharacteristic
140Dic71—22no—
141Dic71—22no—
142Dic71—41no—
143C2×C71—42no—
144C2×C71—82no—
145C2×C71D4211yescharacteristic
146C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C71#57611yescharacteristic
148D71—42no—
149D71—82no—
150#3274—711nomaximal, Sylow
151C71#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—712no—
153#550—711no—
154D42—711no—
155#493—711no—
156#469—712no—
157#460—711no—
158#459D7111yescharacteristic, complemented
159#449—712no—
160#447—712no—
161C4×SD16—711no—
162#436—711no—
163C2×#41—712no—
164C22×D4—712no—
165C22×D4—1421no—
166AGL(1,ℤ/8)—1428no—
167C2×SD16—711no—
168C2×SD16—712no—
169C2×D8—712no—
170C2×D8—712no—
171C2×D8C2×D7112yes—
172C2×M16—712no—
173#127—711no—
174#127—1421no—
175#127C2×D7111yescharacteristic, complemented
176#121—712no—
177#121—712no—
178#121—1422no—
179#121—1421no—
180#120—1422no—
181C4×Q8—711no—
182C4×D4—711no—
183C4×D4—1421no—
184C4×D4C2×D7111yescharacteristic
185#106—711no—
186C4⋊C8C2×D7111yescharacteristic, complemented
187#103—711no—
188#102—711no—
189#102—712no—
190#102C2×D7112yescomplemented
191C22⋊C8—712no—
192C4×C8—711no—
193C24—1422no—
194#41—2842no—
195#41—1424no—
196C2×Q8—711no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—2844no—
200C2×D4—2842no—
201C2×D4—711no—
202C2×D4—712no—
203C2×D4—1422no—
204C2×D4—1422no—
205C2×D4—1422no—
206C2×D4—1424no—
207C2×D4—1422no—
208C2×D4—1421no—
209C2×D4—1421no—
210C2×D4C22×D7112yes—
211C2×D4C22×D7111yescharacteristic
212C22×C4—21no—
213C22×C4—712no—
214C22×C4—1422no—
215SD16—714no—
216SD16—1424no—
217D8—24no—
218D8—2842no—
219D8—1424no—
220M16—1424no—
221C2×C8—711no—
222C2×C8—711no—
223C2×C8C22×D7112yes—
224C4⋊C4—711no—
225C4⋊C4—711no—
226C4⋊C4—1421no—
227C4⋊C4C22×D7111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—712no—
230C22⋊C4—1422no—
231C22⋊C4—1422no—
232C42—1421no—
233C42C22×D7111yescharacteristic
234C23—21no—
235C23—22no—
236C23—2844no—
237C23—2844no—
238C23—5682no—
239C23—712no—
240C23—1422no—
241Q8—712no—
242Q8—1421no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—42no—
247D4—81no—
248D4—2844no—
249D4—2844no—
250D4—2844no—
251D4—2842no—
252D4—5681no—
253D4—5682no—
254D4—1422no—
255D4—1422no—
256D4—1422no—
257D4D4×D7112yes—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—2844no—
261C2×C4—712no—
262C2×C4—1421no—
263C2×C4—1421no—
264C2×C4—1422no—
265C2×C4—1422no—
266C2×C4D4×D7111yescharacteristic
267C2×C4D4×D7111yescharacteristic
268C2×C4C23×D7111yescharacteristic
269C8—22no—
270C8—712no—
271C8—1421no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—2842no—
279C22—2842no—
280C22—2842no—
281C22—5688no—
282C22—1422no—
283C22—1422no—
284C22—1422no—
285C22C2×D4×D7111yescharacteristic
286C4—21no—
287C4—41no—
288C4—2841no—
289C4—1422no—
290C4—1422no—
291C4#1455615212yescomplemented
292C4C2×D4×D7111yescharacteristic
293C4#1455617411yescharacteristic
294C2—42no—
295C2—82no—
296C2—2842no—
297C2—5682no—
298C2#1798663311yescharacteristic
299C2C2×#1455615211yescharacteristic
300C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#459×C71C2
socleC22×C71C22×D4
hypercenter#459D71
hypocenterC71#3274
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#14556152C2semidirect
2. #17986669C2semidirect
3. D4×D284C2semidirect
4. #17986375C2semidirect
5. #17986251C2semidirect
6. #17986200C2semidirect
7. #17986145C2semidirect
8. #17986131C2semidirect
9. C4×#11765853C2semidirect
10. #17986076C2semidirect
11. #459×C71C2semidirect
12. C2×#11765864C22semidirect
16. C2×#11765854C22semidirect
18. C2×#11765850C22semidirect
19. C2×#11765850C22semidirect
20. C2×#11765848C22semidirect
26. #14556099C22semidirect
28. C4×Dic142C22semidirect
31. #14556059C22semidirect
32. #14556051C22semidirect
33. #14556050C22semidirect
35. (C2×D71)⋊C8C22semidirect
36. C4×C71⋊C8C22semidirect
38. #127×C71C22semidirect
40. C4⋊C8×C71C22semidirect
41. #102×C71C22semidirect
60. #11765853D4semidirect
117. Dic142C2×D4semidirect
120. C71⋊C8C2×D4semidirect
137. C4×C71AGL(1,ℤ/8)semidirect
151. C71#3274semidirect
158. #459D71semidirect
175. #127C2×D71semidirect
186. C4⋊C8C2×D71semidirect
190. #102C2×D71semidirect
291. C4#14556152semidirect