Finite Group Databasebeta

← #22598789

Subgroups — #22598789

Order 9088.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556155C212yesmaximal, complemented
2C2×#14556152C212yesmaximal, complemented
3#17986548C211yesmaximal, characteristic, complemented
4#127×D71C211yesmaximal, characteristic, complemented
5#17986389C211yesmaximal, characteristic, complemented
6#17986388C211yesmaximal, characteristic, complemented
7#17986386C211yesmaximal, characteristic, complemented
8#17986374C212yesmaximal, complemented
9#17986370C212yesmaximal, complemented
10#17985951C211yesmaximal, characteristic, complemented
11#496×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×#11765864C2211yescharacteristic
14C2×D4×D71—22no—
15C2×D4×D71C2212yes—
16#14556155—28no—
17#14556152—28no—
18C2×#11765854C2212yescomplemented
19C2×#11765853C2212yescomplemented
20C2×#11765852C2212yescomplemented
21C2×D568C2212yescomplemented
22C2×#11765850C2212yescomplemented
23C2×#11765850C2212yescomplemented
24C2×#11765848C2212yes—
25#14556126C2211yescharacteristic, complemented
26#14556125C2211yescharacteristic
27#14556124—22no—
28#14556124C2212yes—
29#14556124C2211yescharacteristic, complemented
30#14556116C2211yescharacteristic, complemented
31#14556101—22no—
32(C22⋊D71)⋊C4—22no—
33C42×D71C2211yescharacteristic
34#14556039C2211yescharacteristic, complemented
35C8×Dic71C2212yescomplemented
36C2×SD16×C71C2212yescomplemented
37C2×D8×C71C2212yescomplemented
38#127×C71C2211yescharacteristic
39#124×C71C2211yescharacteristic, complemented
40#97×C71C2211yescharacteristic, complemented
41C23×D71—22no—
42#11765866—24no—
43#11765864—24no—
44C2×Dic142C2311yescharacteristic
45C2×D284—21no—
46C2×D284—22no—
47C2×D284C2312yes—
48C2×D284C2311yescharacteristic
49C2×C22⋊D71—21no—
50C2×C22⋊D71—24no—
51C2×C22⋊D71—22no—
52C2×C22⋊D71—22no—
53D4×D71—24no—
54D4×D71—24no—
55D4×D71—48no—
56C22×Dic71—21no—
57C2×C4×D71—21no—
58C2×C4×D71D412yes—
59C2×C4×D71C2311yescharacteristic
60#11765854—24no—
61#11765853—24no—
62#11765852—24no—
63D568—24no—
64#11765850—24no—
65#11765850—24no—
66#11765848D418yescomplemented
67C2×C71⋊C8C2312yes—
68Dic71⋊C4—22no—
69(C2×D71)⋊C4—22no—
70C4×Dic71D412yes—
71C4×Dic71C2311yescharacteristic
72C2×Q8×C71C2311yescharacteristic
73C2×D4×C71—22no—
74C2×D4×C71C2312yes—
75C2×D4×C71C2311yescharacteristic
76SD16×C71—24no—
77D8×C71—24no—
78C2×C8×C71C2312yes—
79C22⋊C4×C71—22no—
80C42×C71C2311yescharacteristic
81C23×C71—21no—
82C23×C71—22no—
83C22×D71—21no—
84C22×D71—22no—
85C22×D71—48no—
86C22×D71—44no—
87C22×D71C2×D411yescharacteristic
88Q8×C71—22no—
89D4×C71—22no—
90D4×C71—22no—
91D4×C71—22no—
92D4×C71—44no—
93C2×C4×C71—21no—
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2411yescharacteristic
96C22⋊D71—42no—
97C22⋊D71—44no—
98C22⋊D71—48no—
99C22⋊D71—44no—
100C2×Dic71—21no—
101C2×Dic71—42no—
102C2×Dic71C2×D411yescharacteristic
103C2×Dic71C2×D412yes—
104D284—22no—
105D284—22no—
106D284—22no—
107D284—44no—
108D284—42no—
109C4×D71—24no—
110C4×D71—42no—
111C4×D71C2×D414yes—
112Dic142—22no—
113C8×C71C2×D414yescomplemented
114C71⋊C8C2×D414yescomplemented
115C22×C71—41no—
116C22×C71—42no—
117C22×C71—44no—
118C22×C71—42no—
119C22×C71C22×D411yescharacteristic
120C2×D71—22no—
121C2×D71—41no—
122C2×D71—42no—
123C2×D71—44no—
124C2×D71—42no—
125C2×D71—88no—
126C2×D71#12712yes—
127C4×C71—22no—
128C4×C71—41no—
129C4×C71C22×D412yes—
130Dic71—41no—
131Dic71#12712yes—
132Dic71AGL(1,ℤ/8)14yescomplemented
133C2×C71—82no—
134C2×C71—81no—
135C2×C71C2×#12711yescharacteristic
136C2×C71C2×AGL(1,ℤ/8)12yes—
137D71—22no—
138D71—81no—
139D71—82no—
140#3151—711nomaximal, Sylow
141C71#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—712no—
143C2×AGL(1,ℤ/8)—712no—
144#532—711no—
145C2×#127—711no—
146#496—711no—
147#496—711no—
148#496—711no—
149#496D7111yescharacteristic, complemented
150#493—712no—
151#492—712no—
152C4×M16—711no—
153C2×#41—711no—
154C2×#41—711no—
155C22×D4—712no—
156C22×D4—1422no—
157AGL(1,ℤ/8)—1428no—
158AGL(1,ℤ/8)—1428no—
159C2×SD16—712no—
160C2×SD16—712no—
161C2×SD16—712no—
162C2×SD16C2×D7112yescomplemented
163C2×D8—712no—
164C2×D8—712no—
165C2×D8—712no—
166C2×D8C2×D7112yescomplemented
167C2×M16—712no—
168C4⋊Q8—711no—
169#127—712no—
170#127—711no—
171#127—711no—
172#127—1422no—
173#127C2×D7111yescharacteristic
174#124—711no—
175#124C2×D7111yescharacteristic, complemented
176#121—1422no—
177C4×D4—1422no—
178C2×C42—711no—
179#97—711no—
180#97C2×D7111yescharacteristic, complemented
181C4×C8—712no—
182C24—1422no—
183#41—1424no—
184#41—1424no—
185C2×Q8—711no—
186C2×Q8C22×D7111yescharacteristic
187C2×D4—22no—
188C2×D4—2848no—
189C2×D4—711no—
190C2×D4—712no—
191C2×D4—1421no—
192C2×D4—1424no—
193C2×D4—1424no—
194C2×D4—1422no—
195C2×D4—1424no—
196C2×D4—1422no—
197C2×D4—1421no—
198C2×D4—1422no—
199C2×D4C22×D7111yescharacteristic
200C2×D4C22×D7112yes—
201C22×C4—711no—
202C22×C4—712no—
203C22×C4—1421no—
204C22×C4—1421no—
205SD16—24no—
206SD16—1424no—
207SD16—1424no—
208SD16—1424no—
209D8—24no—
210D8—1424no—
211D8—1424no—
212D8—1424no—
213M16—718no—
214C2×C8—712no—
215C2×C8C22×D7112yes—
216C4⋊C4—1422no—
217C22⋊C4—22no—
218C22⋊C4—1422no—
219C42—712no—
220C42—711no—
221C42C22×D7111yescharacteristic
222C23—21no—
223C23—22no—
224C23—2844no—
225C23—2848no—
226C23—711no—
227C23—1421no—
228C23—1422no—
229Q8—22no—
230Q8—1422no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—2844no—
236D4—2842no—
237D4—2842no—
238D4—2848no—
239D4—2844no—
240D4—2844no—
241D4—1422no—
242D4—1422no—
243D4—1422no—
244C2×C4—21no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—712no—
248C2×C4—711no—
249C2×C4—714no—
250C2×C4—1424no—
251C2×C4—1421no—
252C2×C4D4×D7112yes—
253C2×C4C23×D7111yescharacteristic
254C8—714no—
255C8D4×D7114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2844no—
261C22—2842no—
262C22—2841no—
263C22—2842no—
264C22—5688no—
265C22—712no—
266C22—1422no—
267C22C2×D4×D7111yescharacteristic
268C4—22no—
269C4—41no—
270C4—2841no—
271C4—714no—
272C4—712no—
273C4C2×D4×D7112yes—
274C2—82no—
275C2—81no—
276C2—5682no—
277C2—5681no—
278C2—1422no—
279C2#127×D7111yescharacteristic
280C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#496×C71C2
socleC22×C71C22×D4
hypercenter#496D71
hypocenterC71#3151
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#14556155C2semidirect
2. C2×#14556152C2semidirect
3. #17986548C2semidirect
4. #127×D71C2semidirect
5. #17986389C2semidirect
6. #17986388C2semidirect
7. #17986386C2semidirect
8. #17986374C2semidirect
9. #17986370C2semidirect
10. #17985951C2semidirect
11. #496×C71C2semidirect
18. C2×#11765854C22semidirect
19. C2×#11765853C22semidirect
20. C2×#11765852C22semidirect
21. C2×D568C22semidirect
22. C2×#11765850C22semidirect
23. C2×#11765850C22semidirect
25. #14556126C22semidirect
29. #14556124C22semidirect
30. #14556116C22semidirect
34. #14556039C22semidirect
35. C8×Dic71C22semidirect
36. C2×SD16×C71C22semidirect
37. C2×D8×C71C22semidirect
39. #124×C71C22semidirect
40. #97×C71C22semidirect
66. #11765848D4semidirect
113. C8×C71C2×D4semidirect
114. C71⋊C8C2×D4semidirect
132. Dic71AGL(1,ℤ/8)semidirect
141. C71#3151semidirect
149. #496D71semidirect
162. C2×SD16C2×D71semidirect
166. C2×D8C2×D71semidirect
175. #124C2×D71semidirect
180. #97C2×D71semidirect
255. C8D4×D71semidirect