Finite Group Databasebeta

← #22599305

Subgroups — #22599305

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
2#17986381C212yesmaximal, complemented
3#17986353C212yesmaximal, complemented
4#17986218C212yesmaximal, complemented
5#17986211C212yesmaximal, complemented
6#17986155C211yesmaximal, characteristic, complemented
7#17986150C211yesmaximal, characteristic, complemented
8#17986138C211yesmaximal, characteristic, complemented
9#17986133C211yesmaximal, characteristic
10#17986052C211yesmaximal, characteristic, complemented
11C2×#124×C71C211yesmaximal, characteristic, complemented
12C2×#11765855C2212yescomplemented
13C2×#11765853C2212yes—
14C2×#11765852C2212yescomplemented
15C2×#11765850C2212yes—
16#14556122—21no—
17#14556120C2211yescharacteristic, complemented
18#14556116C2211yescharacteristic, complemented
19#14556111—21no—
20#14556107C2211yescharacteristic
21#14556102C2211yescharacteristic
22C4×Dic142C2211yescharacteristic, complemented
23C4×D284C2211yescharacteristic, complemented
24C4×C22⋊D71—21no—
25#14556085—21no—
26C4⋊(C71⋊C8)C2212yescomplemented
27#14556052C2212yescomplemented
28#14556052C2212yescomplemented
29#14556049C2212yes—
30#14556049C2212yes—
31C22⋊(C71⋊C8)C2211yescharacteristic
32C22⋊(C71⋊C8)C2211yescharacteristic
33C4×C71⋊C8C2212yescomplemented
34C22×Q8×C71C2211yescharacteristic
35C22×D4×C71C2211yescharacteristic
36#124×C71—21no—
37#124×C71—21no—
38#124×C71C2212yes—
39#124×C71C2212yes—
40C2×C22⋊C4×C71—21no—
41C2×C22⋊C4×C71—21no—
42C2×C42×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C4×D71—21no—
47#11765855—42no—
48#11765853—42no—
49#11765852—42no—
50#11765850—42no—
51C2×C71⋊C8C2312yes—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59C22⋊Dic71—21no—
60(C2×D71)⋊C4—21no—
61(C2×D71)⋊C4—21no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—21no—
64C4×Dic71—21no—
65C24×C71—21no—
66C2×Q8×C71—21no—
67C2×Q8×C71—42no—
68C2×Q8×C71C2312yes—
69C2×D4×C71—21no—
70C2×D4×C71—42no—
71C2×D4×C71C2312yes—
72C22×C4×C71—21no—
73C22×C4×C71D411yescharacteristic
74C22×C4×C71D411yescharacteristic
75C22×C4×C71C2311yescharacteristic
76C22⋊C4×C71—22no—
77C22⋊C4×C71—22no—
78C22⋊C4×C71—22no—
79C22⋊C4×C71—22no—
80C42×C71D411yescharacteristic
81C42×C71D411yescharacteristic
82C42×C71C2311yescharacteristic
83C42×C71C2311yescharacteristic
84C23×C71—22no—
85C23×C71—42no—
86C23×C71—42no—
87C23×C71—42no—
88C23×C71C2×D411yescharacteristic
89C22×D71—21no—
90Q8×C71—22no—
91Q8×C71—22no—
92Q8×C71—42no—
93D4×C71—22no—
94D4×C71—22no—
95D4×C71—42no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—41no—
100C2×C4×C71—42no—
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71C2×D411yescharacteristic
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D411yescharacteristic
105C2×C4×C71C2×D411yescharacteristic
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—42no—
108C2×Dic71—21no—
109C2×Dic71—21no—
110C2×Dic71—21no—
111D284—22no—
112C4×D71—42no—
113Dic142—22no—
114C71⋊C8—22no—
115C71⋊C8—22no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71—42no—
121C22×C71—84no—
122C22×C71#13512yes—
123C22×C71C22×D411yescharacteristic
124C2×D71—41no—
125C2×D71—41no—
126C2×D71—41no—
127C4×C71—21no—
128C4×C71—21no—
129C4×C71—21no—
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—42no—
133C4×C71H(5,𝔽2)12yes—
134Dic71—41no—
135Dic71—41no—
136Dic71—41no—
137C2×C71—22no—
138C2×C71—82no—
139C2×C71#53511yescharacteristic
140C2×C71C2×#13511yescharacteristic
141C2×C71#57511yescharacteristic
142D71—81no—
143#3211—711nomaximal, Sylow
144C71#321111yescharacteristic, complemented, Sylow
145#529—711no—
146C2×#124D7111yescharacteristic, complemented
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#431—711no—
156C22×Q8C2×D7111yescharacteristic
157C22×D4C2×D7111yescharacteristic
158C2×Q16—712no—
159C2×SD16—712no—
160C2×SD16—712no—
161C2×D8—712no—
162#126—1421no—
163#125—711no—
164#124—21no—
165#124—21no—
166#124—711no—
167#124C2×D7112yes—
168#124C2×D7112yes—
169#123—1421no—
170C22⋊Q8—711no—
171#121—711no—
172C4×Q8—711no—
173C4×D4—711no—
174C4×D4—1421no—
175#117—1421no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C42C2×D7111yescharacteristic
179C4⋊C8—712no—
180#103—712no—
181#103—712no—
182#102—712no—
183#102—712no—
184C22⋊C8—711no—
185C22⋊C8—711no—
186C4×C8—712no—
187C24—21no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8—711no—
191C2×Q8C22×D7112yes—
192C2×D4—21no—
193C2×D4—42no—
194C2×D4—711no—
195C2×D4—1421no—
196C2×D4C22×D7112yes—
197C22×C4—21no—
198C22×C4—1421no—
199C22×C4C22⋊D7111yescharacteristic
200C22×C4C22⋊D7111yescharacteristic
201C22×C4C22×D7111yescharacteristic
202Q16—2842no—
203SD16—2842no—
204SD16—2842no—
205D8—2842no—
206C2×C8—712no—
207C2×C8—712no—
208C4⋊C4—711no—
209C4⋊C4—711no—
210C4⋊C4—1421no—
211C4⋊C4—1421no—
212C4⋊C4—1421no—
213C4⋊C4—1421no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1421no—
219C22⋊C4—1421no—
220C22⋊C4—1421no—
221C22⋊C4—1421no—
222C22⋊C4—1421no—
223C42—1421no—
224C42C22⋊D7111yescharacteristic
225C42C22⋊D7111yescharacteristic
226C42C22×D7111yescharacteristic
227C42C22×D7111yescharacteristic
228C23—22no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—1421no—
233C23C2×C22⋊D7111yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—1422no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—2842no—
242D4—1422no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—2842no—
249C2×C4—1421no—
250C2×C4—1421no—
251C2×C4—1421no—
252C2×C4C2×C22⋊D7111yescharacteristic
253C2×C4C2×C22⋊D7111yescharacteristic
254C2×C4C2×C22⋊D7111yescharacteristic
255C2×C4C2×C22⋊D7111yescharacteristic
256C2×C4C2×C22⋊D7111yescharacteristic
257C2×C4C23×D7111yescharacteristic
258C8—1422no—
259C8—1422no—
260C22—21no—
261C22—21no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—2841no—
267C22—2841no—
268C22—2841no—
269C22#1455614812yes—
270C22C22×C22⋊D7111yescharacteristic
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—2841no—
278C4—2841no—
279C4—2841no—
280C4#1455617312yes—
281C2—22no—
282C2—82no—
283C2—5681no—
284C2#1798656611yescharacteristic
285C2C2×#1455614811yescharacteristic
286C2#1798679711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#124×C71C2
socleC22×C71C22×D4
hypercenterC2×#124D71
hypocenterC71#3211
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #17986533C2semidirect
2. #17986381C2semidirect
3. #17986353C2semidirect
4. #17986218C2semidirect
5. #17986211C2semidirect
6. #17986155C2semidirect
7. #17986150C2semidirect
8. #17986138C2semidirect
10. #17986052C2semidirect
11. C2×#124×C71C2semidirect
12. C2×#11765855C22semidirect
14. C2×#11765852C22semidirect
17. #14556120C22semidirect
18. #14556116C22semidirect
22. C4×Dic142C22semidirect
23. C4×D284C22semidirect
26. C4⋊(C71⋊C8)C22semidirect
27. #14556052C22semidirect
28. #14556052C22semidirect
33. C4×C71⋊C8C22semidirect
144. C71#3211semidirect
146. C2×#124D71semidirect