Finite Group Databasebeta

← #22598803

Subgroups — #22598803

Order 9088.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#14556160C212yesmaximal, complemented
2C2×#14556155C212yesmaximal, complemented
3#17986550C211yesmaximal, characteristic, complemented
4#124×D71C211yesmaximal, characteristic, complemented
5#17986391C211yesmaximal, characteristic, complemented
6#17986389C211yesmaximal, characteristic, complemented
7#17986388C211yesmaximal, characteristic, complemented
8#17986384C212yesmaximal, complemented
9#17986380C212yesmaximal, complemented
10#17985951C211yesmaximal, characteristic, complemented
11#497×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yes—
13C2×Q8×D71C2211yescharacteristic
14C2×D4×D71C2211yescharacteristic
15#14556160—28no—
16#14556155—28no—
17C2×#11765855C2212yescomplemented
18C2×#11765854C2212yescomplemented
19C2×#11765852C2212yes—
20C2×#11765852C2212yescomplemented
21C2×D568C2212yes—
22C2×#11765850C2212yes—
23C2×#11765848C2212yes—
24#14556125C2211yescharacteristic, complemented
25#14556117C2211yescharacteristic
26#14556117C2212yescomplemented
27#14556116C2211yescharacteristic
28#14556115C2211yescharacteristic
29#14556114—22no—
30#14556103—22no—
31#14556090—22no—
32C22⋊C4×D71—22no—
33C42×D71C2211yescharacteristic
34#14556039C2211yescharacteristic, complemented
35C8×Dic71C2212yescomplemented
36C2×Q16×C71C2212yescomplemented
37C2×SD16×C71C2212yes—
38C4⋊Q8×C71C2211yescharacteristic, complemented
39#124×C71C2211yescharacteristic
40#97×C71C2211yescharacteristic, complemented
41C23×D71—21no—
42#11765866—24no—
43#11765866—24no—
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—24no—
47C2×D284—22no—
48C2×D284—22no—
49C2×D284C2311yescharacteristic
50C2×D284C2312yes—
51C2×C22⋊D71—21no—
52D4×D71—24no—
53C2×C4×D71—21no—
54C2×C4×D71—22no—
55C2×C4×D71D412yes—
56C2×C4×D71C2311yescharacteristic
57#11765855—24no—
58#11765854—24no—
59#11765852—24no—
60#11765852—24no—
61D568—24no—
62#11765850—24no—
63#11765848D418yescomplemented
64C2×C71⋊C8C2312yes—
65C22⋊Dic71—22no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—24no—
69C4×Dic71D412yes—
70C4×Dic71C2311yescharacteristic
71C2×Q8×C71C2311yescharacteristic
72C2×Q8×C71C2312yes—
73C2×D4×C71C2311yescharacteristic
74Q16×C71—24no—
75SD16×C71—24no—
76C2×C8×C71C2312yes—
77C4⋊C4×C71—22no—
78C22⋊C4×C71—22no—
79C42×C71C2311yescharacteristic
80C23×C71—21no—
81C22×D71—21no—
82C22×D71—22no—
83C22×D71—44no—
84C22×D71—42no—
85C22×D71C2×D411yescharacteristic
86Q8×C71—22no—
87Q8×C71—22no—
88Q8×C71—22no—
89D4×C71—22no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71C2×D412yes—
93C2×C4×C71C2411yescharacteristic
94C22⋊D71—42no—
95C2×Dic71—21no—
96C2×Dic71C2×D412yes—
97C2×Dic71C2×D411yescharacteristic
98D284—22no—
99D284—22no—
100D284—22no—
101D284—44no—
102D284—44no—
103C4×D71—42no—
104C4×D71—42no—
105C4×D71—44no—
106C4×D71C2×D414yes—
107Dic142—22no—
108Dic142—42no—
109C8×C71C2×D414yes—
110C71⋊C8C2×D414yes—
111C22×C71—41no—
112C22×C71—42no—
113C22×C71C22×D411yescharacteristic
114C2×D71—22no—
115C2×D71—42no—
116C2×D71—41no—
117C2×D71—44no—
118C2×D71—42no—
119C2×D71—84no—
120C2×D71#12712yes—
121C4×C71—22no—
122C4×C71—41no—
123C4×C71—42no—
124C4×C71C22×D412yes—
125Dic71—22no—
126Dic71—41no—
127Dic71#12712yes—
128C2×C71—81no—
129C2×C71C2×#12711yescharacteristic
130C2×C71#57512yes—
131D71—22no—
132D71—82no—
133D71—81no—
134#3153—711nomaximal, Sylow
135C71#315311yescharacteristic, complemented, Sylow
136C2×#137—712no—
137C2×AGL(1,ℤ/8)—712no—
138#532—711no—
139C2×#124—711no—
140#497—711no—
141#497D7111yescharacteristic, complemented
142#496—711no—
143#496—711no—
144#495—712no—
145#495—712no—
146C4×M16—711no—
147C2×#41—712no—
148C22×Q8—711no—
149C22×D4—711no—
150#137—1428no—
151AGL(1,ℤ/8)—1428no—
152C2×Q16—712no—
153C2×Q16C2×D7112yescomplemented
154C2×SD16—712no—
155C2×SD16—712no—
156C2×SD16—712no—
157C2×SD16C2×D7112yes—
158C2×D8—712no—
159C2×D8—712no—
160C2×M16—712no—
161C4⋊Q8C2×D7111yescharacteristic, complemented
162#127—711no—
163#124—711no—
164#124—712no—
165#124—711no—
166#124—711no—
167#124—1422no—
168#124C2×D7111yescharacteristic
169#121—1422no—
170C4×D4—1422no—
171C2×C22⋊C4—1422no—
172C2×C42—711no—
173#97—711no—
174#97C2×D7111yescharacteristic, complemented
175C4×C8—712no—
176C24—1421no—
177#41—1424no—
178#41—1424no—
179C2×Q8—711no—
180C2×Q8—1421no—
181C2×Q8—1424no—
182C2×Q8C22×D7111yescharacteristic
183C2×Q8C22×D7112yes—
184C2×D4—711no—
185C2×D4—712no—
186C2×D4—1422no—
187C2×D4—1424no—
188C2×D4—1421no—
189C2×D4—1422no—
190C2×D4C22×D7111yescharacteristic
191C22×C4—712no—
192C22×C4—711no—
193C22×C4—1422no—
194C22×C4—1421no—
195Q16—24no—
196Q16—1424no—
197SD16—24no—
198SD16—1424no—
199SD16—1424no—
200SD16—1424no—
201D8—1424no—
202D8—1424no—
203M16—718no—
204C2×C8—712no—
205C2×C8C22×D7112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—1422no—
209C22⋊C4—1424no—
210C22⋊C4—1422no—
211C22⋊C4—1422no—
212C42—712no—
213C42—711no—
214C42C22×D7111yescharacteristic
215C23—21no—
216C23—2842no—
217C23—2844no—
218C23—711no—
219C23—1421no—
220C23—1422no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—2842no—
225Q8—1422no—
226D4—22no—
227D4—2844no—
228D4—2844no—
229D4—2842no—
230D4—1422no—
231D4—1422no—
232D4—1422no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—2842no—
236C2×C4—2842no—
237C2×C4—2844no—
238C2×C4—711no—
239C2×C4—714no—
240C2×C4—712no—
241C2×C4—1421no—
242C2×C4D4×D7112yes—
243C2×C4C23×D7111yescharacteristic
244C8—714no—
245C8D4×D7114yes—
246C22—41no—
247C22—42no—
248C22—2842no—
249C22—2841no—
250C22—2844no—
251C22—2842no—
252C22—5684no—
253C22—712no—
254C22—1422no—
255C22C2×D4×D7111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—2841no—
260C4—712no—
261C4—1422no—
262C4C2×D4×D7112yes—
263C2—81no—
264C2—5681no—
265C2—5682no—
266C2—1422no—
267C2#127×D7111yescharacteristic
268C2#1798679812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#497×C71C2
socleC22×C71C22×D4
hypercenter#497D71
hypocenterC71#3153
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#14556160C2semidirect
2. C2×#14556155C2semidirect
3. #17986550C2semidirect
4. #124×D71C2semidirect
5. #17986391C2semidirect
6. #17986389C2semidirect
7. #17986388C2semidirect
8. #17986384C2semidirect
9. #17986380C2semidirect
10. #17985951C2semidirect
11. #497×C71C2semidirect
17. C2×#11765855C22semidirect
18. C2×#11765854C22semidirect
20. C2×#11765852C22semidirect
24. #14556125C22semidirect
26. #14556117C22semidirect
34. #14556039C22semidirect
35. C8×Dic71C22semidirect
36. C2×Q16×C71C22semidirect
38. C4⋊Q8×C71C22semidirect
40. #97×C71C22semidirect
63. #11765848D4semidirect
135. C71#3153semidirect
141. #497D71semidirect
153. C2×Q16C2×D71semidirect
161. C4⋊Q8C2×D71semidirect
174. #97C2×D71semidirect