Finite Group Databasebeta

← #22598725

Subgroups — #22598725

Order 9088.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986554C211yesmaximal, characteristic, complemented
2#124×D71C211yesmaximal, characteristic
3#17986361C211yesmaximal, characteristic, complemented
4#17986359C211yesmaximal, characteristic, complemented
5#17986358C211yesmaximal, characteristic, complemented
6#17986355C212yesmaximal, complemented
7#17986352C212yesmaximal, complemented
8#17986006C212yesmaximal, complemented
9#17986002C212yesmaximal
10#17985951C211yesmaximal, characteristic, complemented
11#489×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765848C2212yes—
15#14556128C2211yescharacteristic, complemented
16#14556119C2212yescomplemented
17#14556117C2211yescharacteristic
18#14556116C2211yescharacteristic
19#14556115C2211yescharacteristic
20#14556114—21no—
21#14556114—21no—
22#14556108—21no—
23#14556104—21no—
24Q8×Dic71—21no—
25#14556093—21no—
26#14556086C2212yes—
27C22⋊C4×D71—21no—
28C22⋊C4×D71—21no—
29C42×D71C2211yescharacteristic
30#14556053C2212yescomplemented
31#14556052C2212yescomplemented
32#14556051C2212yescomplemented
33#14556050C2212yes—
34#14556049C2212yes—
35#14556048C2212yes—
36#14556039C2211yescharacteristic, complemented
37C8×Dic71C2212yescomplemented
38C4⋊Q8×C71C2211yescharacteristic, complemented
39#124×C71C2211yescharacteristic
40#103×C71C2212yescomplemented
41#102×C71C2212yes—
42#97×C71C2211yescharacteristic, complemented
43C23×D71—21no—
44C2×Dic142—21no—
45C2×Dic142—21no—
46C2×Dic142C2311yescharacteristic
47Q8×D71—24no—
48C2×D284C2311yescharacteristic
49C2×C22⋊D71—21no—
50D4×D71—24no—
51C2×C4×D71—21no—
52C2×C4×D71D411yescharacteristic
53C2×C4×D71D411yescharacteristic
54C2×C4×D71C2311yescharacteristic
55#11765848—24no—
56C2×C71⋊C8C2312yes—
57C4⋊Dic71—21no—
58C4⋊Dic71—22no—
59C4⋊Dic71C2312yes—
60Dic71⋊C4—22no—
61C22⋊Dic71—21no—
62C22⋊Dic71—21no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—21no—
68C4×Dic71—22no—
69C4×Dic71D411yescharacteristic
70C4×Dic71D411yescharacteristic
71C4×Dic71C2311yescharacteristic
72C2×Q8×C71—21no—
73C2×Q8×C71C2311yescharacteristic
74C2×D4×C71C2311yescharacteristic
75C2×C8×C71C2312yes—
76C4⋊C4×C71—21no—
77C4⋊C4×C71C2312yes—
78C22⋊C4×C71—21no—
79C22⋊C4×C71—21no—
80C42×C71C2311yescharacteristic
81C23×C71—21no—
82C22×D71—21no—
83C22×D71—42no—
84C22×D71—42no—
85C22×D71—42no—
86C22×D71C2×D411yescharacteristic
87Q8×C71—22no—
88Q8×C71—42no—
89D4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71C2×D411yescharacteristic
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2411yescharacteristic
95C22⋊D71—42no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71C2×D411yescharacteristic
99C2×Dic71C2×D411yescharacteristic
100C2×Dic71C2×D411yescharacteristic
101D284—22no—
102C4×D71—41no—
103C4×D71—41no—
104C4×D71—42no—
105C4×D71#4114yes—
106Dic142—22no—
107Dic142—42no—
108Dic142—42no—
109C8×C71—22no—
110C71⋊C8—22no—
111C22×C71—41no—
112C22×C71—41no—
113C22×C71—41no—
114C22×C71C22×D411yescharacteristic
115C2×D71—21no—
116C2×D71—21no—
117C2×D71—41no—
118C2×D71—41no—
119C2×D71—41no—
120C2×D71—84no—
121C2×D71#12412yes—
122C4×C71—21no—
123C4×C71—21no—
124C4×C71—41no—
125C4×C71—42no—
126C4×C71C2×#4112yes—
127Dic71—21no—
128Dic71—21no—
129Dic71—41no—
130Dic71—42no—
131Dic71#12412yes—
132C2×C71—81no—
133C2×C71C2×#12411yescharacteristic
134C2×C71#57511yescharacteristic
135C2×C71#57511yescharacteristic
136D71—22no—
137D71—81no—
138#3140—711nomaximal, Sylow
139C71#314011yescharacteristic, complemented, Sylow
140#533—711no—
141C2×#124—711no—
142#489—711no—
143#489—711no—
144#489—711no—
145#489D7111yescharacteristic, complemented
146#488—712no—
147#488—712no—
148#419—712no—
149#418—712no—
150C4×M16—711no—
151C22×Q8—711no—
152C22×D4—711no—
153C2×M16—712no—
154C4⋊Q8—711no—
155C4⋊Q8C2×D7111yescharacteristic, complemented
156#125—712no—
157#124—711no—
158#124—711no—
159#124—711no—
160#124—1421no—
161#124—1421no—
162#124C2×D7111yescharacteristic
163C22⋊Q8—1421no—
164C22⋊Q8—1421no—
165C4×Q8—1421no—
166C4×Q8—1421no—
167#117—712no—
168C2×C22⋊C4—1421no—
169C2×C22⋊C4—1421no—
170C2×C42—711no—
171#103—712no—
172#103—712no—
173#103—712no—
174#103C2×D7112yescomplemented
175#102—712no—
176#102—712no—
177#102—712no—
178#102C2×D7112yes—
179#97—711no—
180#97C2×D7111yescharacteristic, complemented
181C4×C8—712no—
182C24—1421no—
183C2×Q8—21no—
184C2×Q8—711no—
185C2×Q8—1421no—
186C2×Q8—1424no—
187C2×Q8—1421no—
188C2×Q8C22×D7111yescharacteristic
189C2×D4—711no—
190C2×D4—1424no—
191C2×D4—1421no—
192C2×D4C22×D7111yescharacteristic
193C22×C4—711no—
194C22×C4—711no—
195C22×C4—711no—
196C22×C4—1421no—
197M16—1424no—
198C2×C8—712no—
199C2×C8C22×D7112yes—
200C4⋊C4—21no—
201C4⋊C4—712no—
202C4⋊C4—1422no—
203C4⋊C4—1422no—
204C4⋊C4—1421no—
205C4⋊C4C22×D7112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1421no—
209C22⋊C4—1421no—
210C22⋊C4—1422no—
211C22⋊C4—1421no—
212C22⋊C4—1421no—
213C22⋊C4—1421no—
214C22⋊C4—1421no—
215C42—711no—
216C42—711no—
217C42—711no—
218C42—1422no—
219C42C22×D7111yescharacteristic
220C23—21no—
221C23—2842no—
222C23—2842no—
223C23—2842no—
224C23—711no—
225C23—1421no—
226Q8—22no—
227Q8—42no—
228Q8—2842no—
229Q8—2842no—
230Q8—1422no—
231D4—22no—
232D4—2842no—
233D4—1422no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—2841no—
237C2×C4—2841no—
238C2×C4—2842no—
239C2×C4—711no—
240C2×C4—714no—
241C2×C4—711no—
242C2×C4—711no—
243C2×C4—1422no—
244C2×C4—1421no—
245C2×C4D4×D7111yescharacteristic
246C2×C4D4×D7111yescharacteristic
247C2×C4C23×D7111yescharacteristic
248C8—22no—
249C8—1422no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—2841no—
254C22—2841no—
255C22—2841no—
256C22—5684no—
257C22—712no—
258C22—1421no—
259C22—1421no—
260C22C2×D4×D7111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—2841no—
266C4—2842no—
267C4—712no—
268C4—1421no—
269C4—1421no—
270C4#41×D7112yes—
271C2—81no—
272C2—5681no—
273C2—1422no—
274C2#124×D7111yescharacteristic
275C2#1798679511yescharacteristic
276C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#489×C71C2
socleC22×C71C22×D4
hypercenter#489D71
hypocenterC71#3140
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986361C2semidirect
4. #17986359C2semidirect
5. #17986358C2semidirect
6. #17986355C2semidirect
7. #17986352C2semidirect
8. #17986006C2semidirect
10. #17985951C2semidirect
11. #489×C71C2semidirect
15. #14556128C22semidirect
16. #14556119C22semidirect
30. #14556053C22semidirect
31. #14556052C22semidirect
32. #14556051C22semidirect
36. #14556039C22semidirect
37. C8×Dic71C22semidirect
38. C4⋊Q8×C71C22semidirect
40. #103×C71C22semidirect
42. #97×C71C22semidirect
139. C71#3140semidirect
145. #489D71semidirect
155. C4⋊Q8C2×D71semidirect
174. #103C2×D71semidirect
180. #97C2×D71semidirect