Finite Group Databasebeta

← #22599156

Subgroups — #22599156

Order 9088.

466 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986510C211yesmaximal, characteristic, complemented
2C2×#14556102C211yesmaximal, characteristic, complemented
3#17986330C211yesmaximal, characteristic, complemented
4#17986324C212yesmaximal, complemented
5#17986313C211yesmaximal, characteristic, complemented
6#17986252C212yesmaximal, complemented
7#17986245C212yesmaximal, complemented
8#17986152C211yesmaximal, characteristic, complemented
9#17986146C212yesmaximal, complemented
10#17986135C211yesmaximal, characteristic, complemented
11#407×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic, complemented
13C22×D284C2211yescharacteristic
14C22×C22⋊D71—21no—
15C2×#11765854C2212yescomplemented
16C2×D568C2212yescomplemented
17#14556111—21no—
18#14556107C2211yescharacteristic
19#14556107C2211yescharacteristic, complemented
20#14556102—21no—
21#14556102—21no—
22#14556102C2211yescharacteristic
23#14556102C2212yes—
24#14556102C2212yes—
25#14556102C2211yescharacteristic, complemented
26#14556097—21no—
27C4×C22⋊D71—21no—
28C4×C22⋊D71—21no—
29#14556087C2211yescharacteristic, complemented
30C2×C4⋊Dic71C2211yescharacteristic
31C2×(C2×D71)⋊C4—21no—
32#14556062C2212yescomplemented
33#14556060C2212yescomplemented
34#14556053C2212yescomplemented
35#14556050C2212yescomplemented
36#14556050C2212yescomplemented
37#14556050C2212yescomplemented
38C23×C4×C71C2211yescharacteristic
39C2×M16×C71C2212yescomplemented
40C22⋊C8×C71C2211yescharacteristic
41C22⋊C8×C71C2211yescharacteristic
42C22⋊C8×C71C2212yescomplemented
43C23×D71—21no—
44#11765865—42no—
45C2×Dic142C2311yescharacteristic
46C2×D284—21no—
47C2×D284—42no—
48C2×D284C2312yes—
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—22no—
51C2×C22⋊D71—22no—
52C2×C22⋊D71—21no—
53C2×C22⋊D71—21no—
54C2×C22⋊D71—21no—
55C2×C22⋊D71—44no—
56C22×Dic71—21no—
57C2×C4×D71—21no—
58#11765854—42no—
59D568—42no—
60C4⋊Dic71—21no—
61C4⋊Dic71C2311yescharacteristic
62C4⋊Dic71C2311yescharacteristic
63C4⋊Dic71C2312yes—
64Dic71⋊C4—21no—
65Dic71⋊C4—21no—
66C22⋊Dic71—21no—
67C22⋊Dic71—21no—
68C22⋊Dic71—21no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—22no—
71(C2×D71)⋊C4—21no—
72(C2×D71)⋊C4—21no—
73C4×Dic71—21no—
74C24×C71D411yescharacteristic
75C22×C4×C71—42no—
76C22×C4×C71D411yescharacteristic
77C22×C4×C71D411yescharacteristic
78C22×C4×C71D411yescharacteristic
79C22×C4×C71C2311yescharacteristic
80C22×C4×C71C2311yescharacteristic
81C22×C4×C71C2311yescharacteristic
82M16×C71—42no—
83C2×C8×C71C2312yes—
84C2×C8×C71C2312yes—
85C23×C71—22no—
86C23×C71—41no—
87C23×C71—41no—
88C23×C71C2×D411yescharacteristic
89C23×C71C2×D411yescharacteristic
90C23×C71C2×D411yescharacteristic
91C22×D71—21no—
92C22×D71—22no—
93C22×D71—42no—
94C22×D71—42no—
95C22×D71—42no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D411yescharacteristic
105C2×C4×C71C2411yescharacteristic
106C22⋊D71—42no—
107C22⋊D71—44no—
108C22⋊D71—44no—
109C22⋊D71—42no—
110C22⋊D71—42no—
111C2×Dic71—21no—
112C2×Dic71—21no—
113C2×Dic71—22no—
114C2×Dic71—21no—
115C2×Dic71—42no—
116D284—22no—
117D284—22no—
118D284—22no—
119D284—42no—
120C4×D71—42no—
121Dic142—22no—
122C8×C71—22no—
123C8×C71—22no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—21no—
132C22×C71—44no—
133C22×C71AGL(1,ℤ/8)12yes—
134C22×C71C22×D411yescharacteristic
135C2×D71—41no—
136C2×D71—42no—
137C2×D71—42no—
138C2×D71—41no—
139C2×D71—41no—
140C2×D71—42no—
141C2×D71—84no—
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—21no—
145C4×C71H(5,𝔽2)12yes—
146Dic71—41no—
147Dic71—42no—
148Dic71—41no—
149Dic71—41no—
150C2×C71—22no—
151C2×C71—41no—
152C2×C71—41no—
153C2×C71#53411yescharacteristic
154C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
155C2×C71#57511yescharacteristic
156D71—81no—
157D71—82no—
158#3194—711nomaximal, Sylow
159C71#319411yescharacteristic, complemented, Sylow
160#525—711no—
161C2×#121—711no—
162#483—711no—
163#482—712no—
164#480—711no—
165#469—712no—
166#468—712no—
167#450—711no—
168#449—712no—
169#447—711no—
170#407D7111yescharacteristic, complemented
171C2×#41—711no—
172C22×D4—711no—
173C22×D4—1421no—
174C23×C4C2×D7111yescharacteristic
175C2×SD16—712no—
176C2×D8—712no—
177C2×M16C2×D7112yescomplemented
178#123—1421no—
179C22⋊Q8—711no—
180C22⋊Q8—711no—
181#121—712no—
182#121—712no—
183#121—711no—
184#121—711no—
185#121—1421no—
186#121—1421no—
187#120—1421no—
188C4×D4—1421no—
189C4×D4—1421no—
190#117—711no—
191C2×C4⋊C4—711no—
192C2×C22⋊C4—1421no—
193#107—712no—
194#106—712no—
195#103—712no—
196#102—712no—
197#102—712no—
198#102—712no—
199C22⋊C8C2×D7111yescharacteristic
200C22⋊C8C2×D7111yescharacteristic
201C22⋊C8C2×D7112yescomplemented
202C24—1421no—
203C24D28411yescharacteristic
204#41—2842no—
205C2×Q8—711no—
206C2×D4—2842no—
207C2×D4—2844no—
208C2×D4—711no—
209C2×D4—712no—
210C2×D4—1421no—
211C2×D4—1421no—
212C2×D4—1421no—
213C2×D4—1422no—
214C2×D4—1422no—
215C2×D4—1421no—
216C22×C4—42no—
217C22×C4—1421no—
218C22×C4—1421no—
219C22×C4D28411yescharacteristic
220C22×C4D28411yescharacteristic
221C22×C4D28411yescharacteristic
222C22×C4C22×D7111yescharacteristic
223C22×C4C22×D7111yescharacteristic
224C22×C4C22×D7111yescharacteristic
225SD16—2842no—
226D8—2842no—
227M16—42no—
228C2×C8C22×D7112yes—
229C2×C8C22×D7112yes—
230C4⋊C4—711no—
231C4⋊C4—711no—
232C4⋊C4—712no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4—1421no—
236C22⋊C4—1421no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1422no—
240C22⋊C4—1422no—
241C22⋊C4—1421no—
242C22⋊C4—1421no—
243C42—1421no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—2842no—
248C23—2842no—
249C23—2842no—
250C23—1421no—
251C23—1422no—
252C23C2×D28411yescharacteristic
253C23C2×D28411yescharacteristic
254C23C2×D28411yescharacteristic
255Q8—1422no—
256D4—2842no—
257D4—2844no—
258D4—2844no—
259D4—2842no—
260D4—2842no—
261D4—2842no—
262D4—1422no—
263D4—1422no—
264D4—1422no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—2842no—
272C2×C4—2842no—
273C2×C4—1421no—
274C2×C4—1421no—
275C2×C4—1422no—
276C2×C4—1421no—
277C2×C4C2×D28411yescharacteristic
278C2×C4C2×D28411yescharacteristic
279C2×C4C2×D28411yescharacteristic
280C2×C4C23×D7111yescharacteristic
281C8—22no—
282C8—22no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—2841no—
293C22—2842no—
294C22—2842no—
295C22—2841no—
296C22—2841no—
297C22—2842no—
298C22—5684no—
299C22#1455615612yes—
300C22C22×D28411yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—2841no—
305C4—2842no—
306C4—2841no—
307C4—2841no—
308C4#1455617312yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—5682no—
313C2—5681no—
314C2#1798656111yescharacteristic
315C2C2×#1455615611yescharacteristic
316C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3194
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #17986510C2semidirect
2. C2×#14556102C2semidirect
3. #17986330C2semidirect
4. #17986324C2semidirect
5. #17986313C2semidirect
6. #17986252C2semidirect
7. #17986245C2semidirect
8. #17986152C2semidirect
9. #17986146C2semidirect
10. #17986135C2semidirect
11. #407×C71C2semidirect
12. C2×#11765865C22semidirect
15. C2×#11765854C22semidirect
16. C2×D568C22semidirect
19. #14556107C22semidirect
25. #14556102C22semidirect
29. #14556087C22semidirect
32. #14556062C22semidirect
33. #14556060C22semidirect
34. #14556053C22semidirect
35. #14556050C22semidirect
36. #14556050C22semidirect
37. #14556050C22semidirect
39. C2×M16×C71C22semidirect
42. C22⋊C8×C71C22semidirect
159. C71#3194semidirect
170. #407D71semidirect
177. C2×M16C2×D71semidirect
201. C22⋊C8C2×D71semidirect