Finite Group Databasebeta

← #22599307

Subgroups — #22599307

Order 9088.

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

normal subgroups only

This group has 410 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#17986533C211yesmaximal, characteristic, complemented
2C2×#14556116C211yesmaximal, characteristic, complemented
3#17986383C212yesmaximal, complemented
4#17986354C212yesmaximal, complemented
5#17986222C212yesmaximal, complemented
6#17986215C212yesmaximal, complemented
7#17986157C211yesmaximal, characteristic, complemented
8#17986152C211yesmaximal, characteristic, complemented
9#17986140C211yesmaximal, characteristic, complemented
10#17986135C211yesmaximal, characteristic
11#431×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C2×Dic284C2212yescomplemented
15C2×#11765854C2212yes—
16C2×#11765854C2212yescomplemented
17C2×D568C2212yes—
18#14556122—21no—
19#14556120C2211yescharacteristic, complemented
20#14556116—21no—
21#14556116—21no—
22#14556116C2212yes—
23#14556116C2212yes—
24#14556116C2211yescharacteristic, complemented
25#14556111—21no—
26#14556107C2211yescharacteristic
27#14556102C2211yescharacteristic
28C4×Dic142C2211yescharacteristic, complemented
29C4×D284C2211yescharacteristic, complemented
30C4×C22⋊D71—21no—
31#14556085—21no—
32C2×(C2×D71)⋊C4—21no—
33C2×(C2×D71)⋊C4—21no—
34#14556053C2212yescomplemented
35#14556053C2212yescomplemented
36#14556050C2212yes—
37#14556050C2212yes—
38C2×C42×C71C2211yescharacteristic
39C4⋊C8×C71C2212yescomplemented
40C22⋊C8×C71C2211yescharacteristic
41C22⋊C8×C71C2211yescharacteristic
42C4×C8×C71C2212yescomplemented
43C23×D71—21no—
44C2×Dic142—21no—
45C2×Dic142—42no—
46C2×Dic142C2312yes—
47C2×Dic142C2311yescharacteristic
48C2×D284—21no—
49C2×D284—42no—
50C2×D284C2312yes—
51C2×D284C2311yescharacteristic
52C2×C22⋊D71—21no—
53C22×Dic71—21no—
54C2×C4×D71—21no—
55Dic284—42no—
56#11765854—42no—
57#11765854—42no—
58D568—42no—
59C4⋊Dic71C2311yescharacteristic
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72(C2×D71)⋊C4—21no—
73(C2×D71)⋊C4—21no—
74C4×Dic71—21no—
75C22×C4×C71D411yescharacteristic
76C22×C4×C71D411yescharacteristic
77C22×C4×C71C2311yescharacteristic
78C2×C8×C71C2312yes—
79C2×C8×C71C2312yes—
80C42×C71D411yescharacteristic
81C42×C71D411yescharacteristic
82C42×C71C2311yescharacteristic
83C42×C71C2311yescharacteristic
84C23×C71C2×D411yescharacteristic
85C22×D71—21no—
86C22×D71—22no—
87C22×D71—42no—
88C22×D71—42no—
89C22×D71—42no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—41no—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71C2×D411yescharacteristic
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C2×Dic71—21no—
101C2×Dic71—21no—
102C2×Dic71—22no—
103C2×Dic71—21no—
104C2×Dic71—42no—
105D284—22no—
106D284—22no—
107D284—22no—
108D284—42no—
109C4×D71—42no—
110Dic142—22no—
111Dic142—22no—
112Dic142—22no—
113Dic142—42no—
114C8×C71—22no—
115C8×C71—22no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71#13512yes—
119C22×C71C22×D411yescharacteristic
120C2×D71—42no—
121C2×D71—41no—
122C2×D71—41no—
123C2×D71—42no—
124C2×D71—41no—
125C2×D71—42no—
126C2×D71—84no—
127C4×C71—21no—
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—21no—
131C4×C71—21no—
132C4×C71H(5,𝔽2)12yes—
133Dic71—42no—
134Dic71—41no—
135Dic71—41no—
136Dic71—41no—
137C2×C71—22no—
138C2×C71#53511yescharacteristic
139C2×C71C2×#13511yescharacteristic
140C2×C71#57511yescharacteristic
141D71—81no—
142D71—82no—
143#3211—711nomaximal, Sylow
144C71#321111yescharacteristic, complemented, Sylow
145#529—711no—
146C2×#124—711no—
147#495—712no—
148#488—712no—
149#464—712no—
150#463—712no—
151#451—711no—
152#450—711no—
153#448—711no—
154#447—711no—
155#431D7111yescharacteristic, complemented
156C22×Q8—711no—
157C22×D4—711no—
158C2×Q16—712no—
159C2×SD16—712no—
160C2×SD16—712no—
161C2×D8—712no—
162#126—1421no—
163#125—711no—
164#124—712no—
165#124—712no—
166#124—711no—
167#124—1421no—
168#124—1421no—
169#123—1421no—
170C22⋊Q8—711no—
171#121—711no—
172C4×Q8—711no—
173C4×D4—711no—
174C4×D4—1421no—
175#117—1421no—
176C2×C22⋊C4—1421no—
177C2×C22⋊C4—1421no—
178C2×C42C2×D7111yescharacteristic
179C4⋊C8C2×D7112yescomplemented
180#103—712no—
181#103—712no—
182#102—712no—
183#102—712no—
184C22⋊C8C2×D7111yescharacteristic
185C22⋊C8C2×D7111yescharacteristic
186C4×C8C2×D7112yescomplemented
187C24—1421no—
188C2×Q8—2842no—
189C2×Q8—712no—
190C2×Q8—711no—
191C2×Q8—1421no—
192C2×D4—2842no—
193C2×D4—712no—
194C2×D4—711no—
195C2×D4—1421no—
196C2×D4—1421no—
197C22×C4—1421no—
198C22×C4—1421no—
199C22×C4D28411yescharacteristic
200C22×C4D28411yescharacteristic
201C22×C4C22×D7111yescharacteristic
202Q16—2842no—
203SD16—2842no—
204SD16—2842no—
205D8—2842no—
206C2×C8C22×D7112yes—
207C2×C8C22×D7112yes—
208C4⋊C4—711no—
209C4⋊C4—711no—
210C4⋊C4—1421no—
211C4⋊C4—1421no—
212C4⋊C4—1421no—
213C4⋊C4—1421no—
214C22⋊C4—1421no—
215C22⋊C4—1422no—
216C22⋊C4—1422no—
217C22⋊C4—1422no—
218C22⋊C4—1422no—
219C22⋊C4—1421no—
220C22⋊C4—1421no—
221C22⋊C4—1421no—
222C22⋊C4—1421no—
223C42—1421no—
224C42D28411yescharacteristic
225C42D28411yescharacteristic
226C42C22×D7111yescharacteristic
227C42C22×D7111yescharacteristic
228C23—2842no—
229C23—2842no—
230C23—2842no—
231C23—1421no—
232C23—1422no—
233C23C2×D28411yescharacteristic
234Q8—2842no—
235Q8—1422no—
236Q8—1422no—
237Q8—1422no—
238D4—2842no—
239D4—2842no—
240D4—1422no—
241D4—1422no—
242D4—1422no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—2842no—
247C2×C4—2842no—
248C2×C4—1421no—
249C2×C4—1421no—
250C2×C4—1422no—
251C2×C4—1421no—
252C2×C4C2×D28411yescharacteristic
253C2×C4C2×D28411yescharacteristic
254C2×C4C2×D28411yescharacteristic
255C2×C4C2×D28411yescharacteristic
256C2×C4C2×D28411yescharacteristic
257C2×C4C23×D7111yescharacteristic
258C8—22no—
259C8—22no—
260C22—21no—
261C22—21no—
262C22—2842no—
263C22—2841no—
264C22—2841no—
265C22—2842no—
266C22—2841no—
267C22—2842no—
268C22—5684no—
269C22#1455615012yes—
270C22C22×D28411yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—2842no—
277C4—2841no—
278C4—2841no—
279C4—2841no—
280C4#1455617412yes—
281C2—22no—
282C2—5682no—
283C2—5681no—
284C2#1798656811yescharacteristic
285C2C2×#1455615011yescharacteristic
286C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#431×C71C2
socleC22×C71C22×D4
hypercenter#431D71
hypocenterC71#3211
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #17986533C2semidirect
2. C2×#14556116C2semidirect
3. #17986383C2semidirect
4. #17986354C2semidirect
5. #17986222C2semidirect
6. #17986215C2semidirect
7. #17986157C2semidirect
8. #17986152C2semidirect
9. #17986140C2semidirect
11. #431×C71C2semidirect
14. C2×Dic284C22semidirect
16. C2×#11765854C22semidirect
19. #14556120C22semidirect
24. #14556116C22semidirect
28. C4×Dic142C22semidirect
29. C4×D284C22semidirect
34. #14556053C22semidirect
35. #14556053C22semidirect
39. C4⋊C8×C71C22semidirect
42. C4×C8×C71C22semidirect
144. C71#3211semidirect
155. #431D71semidirect
179. C4⋊C8C2×D71semidirect
186. C4×C8C2×D71semidirect