Finite Group Databasebeta

← #22599172

Subgroups — #22599172

Order 9088.

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

normal subgroups only

This group has 414 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
2C2×#14556107C211yesmaximal, characteristic, complemented
3#17986341C212yesmaximal, complemented
4#17986335C211yesmaximal, characteristic, complemented
5#17986318C211yesmaximal, characteristic, complemented
6#17986257C212yesmaximal, complemented
7#17986245C212yesmaximal, complemented
8#17986163C212yesmaximal, complemented
9#17986157C211yesmaximal, characteristic, complemented
10#17986140C211yesmaximal, characteristic, complemented
11#407×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C2×Dic284C2212yescomplemented
15C2×#11765854C2212yescomplemented
16#14556111—21no—
17#14556107—21no—
18#14556107—21no—
19#14556107C2211yescharacteristic
20#14556107C2212yes—
21#14556107C2212yes—
22#14556107C2211yescharacteristic
23#14556102C2211yescharacteristic
24#14556102C2211yescharacteristic
25#14556097—21no—
26C4×C22⋊D71—21no—
27C4×C22⋊D71—21no—
28#14556087C2211yescharacteristic
29C2×C4⋊Dic71C2211yescharacteristic
30C2×Dic71⋊C4—21no—
31C2×C22⋊Dic71—21no—
32#14556062C2212yescomplemented
33#14556060C2212yescomplemented
34#14556053C2212yescomplemented
35#14556053C2212yescomplemented
36#14556053C2212yescomplemented
37#14556050C2212yescomplemented
38C23×C4×C71C2211yescharacteristic
39C2×M16×C71C2212yes—
40C22⋊C8×C71C2211yescharacteristic
41C22⋊C8×C71C2211yescharacteristic
42C22⋊C8×C71C2212yes—
43#11765865—42no—
44C2×Dic142—21no—
45C2×Dic142—42no—
46C2×Dic142C2312yes—
47C2×Dic142C2311yescharacteristic
48C2×D284C2311yescharacteristic
49C2×C22⋊D71—21no—
50C2×C22⋊D71—21no—
51C2×C22⋊D71—21no—
52C22×Dic71—21no—
53C22×Dic71—21no—
54C2×C4×D71—21no—
55Dic284—42no—
56#11765854—42no—
57C4⋊Dic71—21no—
58C4⋊Dic71C2312yes—
59C4⋊Dic71C2311yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—22no—
62Dic71⋊C4—22no—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65C22⋊Dic71—22no—
66C22⋊Dic71—22no—
67C22⋊Dic71—21no—
68C22⋊Dic71—21no—
69C22⋊Dic71—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73C24×C71D411yescharacteristic
74C22×C4×C71—42no—
75C22×C4×C71D411yescharacteristic
76C22×C4×C71D411yescharacteristic
77C22×C4×C71D411yescharacteristic
78C22×C4×C71C2311yescharacteristic
79C22×C4×C71C2311yescharacteristic
80C22×C4×C71C2311yescharacteristic
81M16×C71—42no—
82C2×C8×C71C2312yes—
83C2×C8×C71C2312yes—
84C23×C71—22no—
85C23×C71—41no—
86C23×C71—41no—
87C23×C71C2×D411yescharacteristic
88C23×C71C2×D411yescharacteristic
89C23×C71C2×D411yescharacteristic
90C22×D71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2×D411yescharacteristic
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2411yescharacteristic
101C22⋊D71—42no—
102C22⋊D71—42no—
103C22⋊D71—42no—
104C2×Dic71—21no—
105C2×Dic71—21no—
106C2×Dic71—22no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111D284—22no—
112C4×D71—42no—
113Dic142—22no—
114Dic142—22no—
115Dic142—22no—
116Dic142—42no—
117C8×C71—22no—
118C8×C71—22no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—44no—
128C22×C71#13712yes—
129C22×C71C22×D411yescharacteristic
130C2×D71—41no—
131C2×D71—41no—
132C2×D71—41no—
133C4×C71—21no—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71H(5,𝔽2)12yes—
137Dic71—42no—
138Dic71—41no—
139Dic71—41no—
140Dic71—42no—
141Dic71—41no—
142C2×C71—22no—
143C2×C71—41no—
144C2×C71—41no—
145C2×C71#53411yescharacteristic
146C2×C71C2×#13711yescharacteristic
147C2×C71#57511yescharacteristic
148D71—81no—
149#3196—711nomaximal, Sylow
150C71#319611yescharacteristic, complemented, Sylow
151#525—711no—
152C2×C22⋊Q8—711no—
153#485—712no—
154#484—711no—
155#481—711no—
156#470—712no—
157#468—712no—
158#452—712no—
159#451—711no—
160#448—711no—
161#407D7111yescharacteristic, complemented
162C2×#41—711no—
163C22×Q8—711no—
164C23×C4C2×D7111yescharacteristic
165C2×Q16—712no—
166C2×SD16—712no—
167C2×M16C2×D7112yes—
168#123—1421no—
169C22⋊Q8—712no—
170C22⋊Q8—712no—
171C22⋊Q8—711no—
172C22⋊Q8—711no—
173C22⋊Q8—1421no—
174C22⋊Q8—1421no—
175#121—711no—
176#121—711no—
177#120—1421no—
178C4×D4—1421no—
179C4×D4—1421no—
180#117—711no—
181C2×C4⋊C4—711no—
182C2×C4⋊C4—1421no—
183C2×C22⋊C4—1421no—
184#107—712no—
185#106—712no—
186#103—712no—
187#103—712no—
188#103—712no—
189#102—712no—
190C22⋊C8C2×D7111yescharacteristic
191C22⋊C8C2×D7111yescharacteristic
192C22⋊C8C2×D7112yes—
193C24D28411yescharacteristic
194#41—2842no—
195C2×Q8—2842no—
196C2×Q8—712no—
197C2×Q8—711no—
198C2×Q8—1421no—
199C2×D4—711no—
200C2×D4—1421no—
201C2×D4—1421no—
202C2×D4—1421no—
203C22×C4—42no—
204C22×C4—1421no—
205C22×C4—1421no—
206C22×C4—1421no—
207C22×C4D28411yescharacteristic
208C22×C4D28411yescharacteristic
209C22×C4D28411yescharacteristic
210C22×C4C22×D7111yescharacteristic
211C22×C4C22×D7111yescharacteristic
212C22×C4C22×D7111yescharacteristic
213Q16—2842no—
214SD16—2842no—
215M16—42no—
216C2×C8C22×D7112yes—
217C2×C8C22×D7112yes—
218C4⋊C4—712no—
219C4⋊C4—711no—
220C4⋊C4—711no—
221C4⋊C4—1422no—
222C4⋊C4—1422no—
223C4⋊C4—1421no—
224C4⋊C4—1421no—
225C4⋊C4—1421no—
226C22⋊C4—1422no—
227C22⋊C4—1422no—
228C22⋊C4—1421no—
229C22⋊C4—1421no—
230C22⋊C4—1421no—
231C22⋊C4—1421no—
232C22⋊C4—1421no—
233C42—1421no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—1421no—
238C23C2×D28411yescharacteristic
239C23C2×D28411yescharacteristic
240C23C2×D28411yescharacteristic
241Q8—2842no—
242Q8—1422no—
243Q8—1422no—
244Q8—1422no—
245D4—2842no—
246D4—2842no—
247D4—2842no—
248D4—1422no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—2842no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—1421no—
259C2×C4—1421no—
260C2×C4—1422no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4C2×D28411yescharacteristic
264C2×C4C2×D28411yescharacteristic
265C2×C4C2×D28411yescharacteristic
266C2×C4C23×D7111yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—2841no—
279C22—2841no—
280C22—2841no—
281C22#1455616112yes—
282C22C22×D28411yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—2841no—
287C4—2842no—
288C4—2841no—
289C4—2842no—
290C4—2841no—
291C4#1455617312yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—5681no—
296C2#1798656111yescharacteristic
297C2C2×#1455616111yescharacteristic
298C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3196
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
2. C2×#14556107C2semidirect
3. #17986341C2semidirect
4. #17986335C2semidirect
5. #17986318C2semidirect
6. #17986257C2semidirect
7. #17986245C2semidirect
8. #17986163C2semidirect
9. #17986157C2semidirect
10. #17986140C2semidirect
11. #407×C71C2semidirect
14. C2×Dic284C22semidirect
15. C2×#11765854C22semidirect
32. #14556062C22semidirect
33. #14556060C22semidirect
34. #14556053C22semidirect
35. #14556053C22semidirect
36. #14556053C22semidirect
37. #14556050C22semidirect
150. C71#3196semidirect
161. #407D71semidirect