Finite Group Databasebeta

← #22598261

Subgroups — #22598261

Order 9088.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#17986506C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#17986333C212yesmaximal, complemented
4#17986328C212yesmaximal, complemented
5#17986316C212yesmaximal, complemented
6#17986311C212yesmaximal, complemented
7#17986041C211yesmaximal, characteristic, complemented
8#17986039C211yesmaximal, characteristic, complemented
9#17986000C211yesmaximal, characteristic, complemented
10#17985962C211yesmaximal, characteristic, complemented
11#417×C71C211yesmaximal, characteristic, complemented
12C22×C4×D71—21no—
13C22×C4×D71C2211yescharacteristic
14C2×#11765849C2211yescharacteristic
15#14556109—21no—
16#14556104C2211yescharacteristic
17#14556104C2211yescharacteristic
18#14556099C2211yescharacteristic
19#14556099C2211yescharacteristic
20#14556096—21no—
21C4×C22⋊D71—21no—
22C4×C22⋊D71—21no—
23#14556087C2211yescharacteristic
24C2×C4⋊Dic71C2211yescharacteristic
25C2×Dic71⋊C4—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71—21no—
28C4⋊C4×D71C2212yes—
29C4⋊C4×D71C2212yes—
30#14556062C2212yescomplemented
31#14556061C2212yescomplemented
32#14556060C2212yescomplemented
33#14556059C2212yescomplemented
34#14556052C2212yescomplemented
35#14556049C2212yescomplemented
36(C2×D71)⋊C8C2212yes—
37(C2×D71)⋊C8C2212yes—
38C2×#41×C71C2211yescharacteristic
39C2×M16×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#103×C71C2212yescomplemented
42#102×C71C2212yescomplemented
43C23×D71D411yescharacteristic
44C2×C22⋊D71—21no—
45C2×C22⋊D71—21no—
46C22×Dic71—21no—
47C22×Dic71D411yescharacteristic
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—44no—
53C2×C4×D71D411yescharacteristic
54C2×C4×D71D411yescharacteristic
55C2×C4×D71C2311yescharacteristic
56C2×C4×D71C2311yescharacteristic
57#11765849—22no—
58C2×C71⋊C8C2312yes—
59C4⋊Dic71—21no—
60C4⋊Dic71C2312yes—
61C4⋊Dic71C2311yescharacteristic
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—22no—
64Dic71⋊C4—22no—
65Dic71⋊C4—21no—
66Dic71⋊C4—21no—
67C22⋊Dic71—21no—
68C22⋊Dic71—21no—
69C22⋊Dic71—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73#41×C71—22no—
74#41×C71—22no—
75C2×Q8×C71C2311yescharacteristic
76C2×D4×C71—21no—
77C2×D4×C71C2311yescharacteristic
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71C2311yescharacteristic
81M16×C71—22no—
82C2×C8×C71C2312yes—
83C4⋊C4×C71—21no—
84C4⋊C4×C71C2312yes—
85C23×C71—21no—
86C23×C71C2×D411yescharacteristic
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—22no—
90C22×D71C2×D411yescharacteristic
91C22×D71C2×D411yescharacteristic
92Q8×C71—21no—
93Q8×C71—21no—
94D4×C71—21no—
95D4×C71—21no—
96D4×C71—42no—
97C2×C4×C71—22no—
98C2×C4×C71—21no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71C2×D411yescharacteristic
102C2×C4×C71#4112yes—
103C2×C4×C71#4112yes—
104C2×C4×C71C2411yescharacteristic
105C22⋊D71—42no—
106C22⋊D71—42no—
107C2×Dic71—22no—
108C2×Dic71—21no—
109C2×Dic71—21no—
110C2×Dic71—22no—
111C2×Dic71—42no—
112C2×Dic71C2×D411yescharacteristic
113C2×Dic71C2×D411yescharacteristic
114C4×D71—22no—
115C4×D71—22no—
116C4×D71—21no—
117C4×D71—21no—
118C4×D71—21no—
119C4×D71—21no—
120C4×D71—44no—
121C4×D71—44no—
122C8×C71—22no—
123C71⋊C8—22no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—41no—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71#12312yes—
130C22×C71C22×D411yescharacteristic
131C2×D71—28no—
132C2×D71—21no—
133C2×D71—21no—
134C2×D71—21no—
135C2×D71—21no—
136C2×D71AGL(1,ℤ/8)12yes—
137C2×D71#13712yes—
138C4×C71—41no—
139C4×C71—42no—
140C4×C71#12312yes—
141C4×C71C2×#4111yescharacteristic
142C4×C71C2×#4111yescharacteristic
143Dic71—21no—
144Dic71—21no—
145Dic71—42no—
146Dic71—41no—
147Dic71—41no—
148C2×C71—22no—
149C2×C71—81no—
150C2×C71C2×#12311yescharacteristic
151C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C71C2×#13711yescharacteristic
153D71—22no—
154D71—22no—
155#3092—711nomaximal, Sylow
156C71#309211yescharacteristic, complemented, Sylow
157#525—711no—
158C22×C4⋊C4—711no—
159#484—712no—
160#483—712no—
161#481—712no—
162#480—712no—
163#428—711no—
164#428—711no—
165#417—711no—
166#417D7111yescharacteristic, complemented
167#407—711no—
168C2×#41C2×D7111yescharacteristic
169C23×C4—711no—
170C23×C4—1421no—
171C2×M16—711no—
172C2×M16C2×D7111yescharacteristic
173#123—1421no—
174C22⋊Q8—711no—
175C22⋊Q8—711no—
176#121—711no—
177#121—711no—
178#120—1421no—
179C4×D4—1421no—
180C4×D4—1421no—
181#117—711no—
182C2×C4⋊C4—711no—
183C2×C4⋊C4—712no—
184C2×C4⋊C4—712no—
185C2×C4⋊C4—1421no—
186C2×C4⋊C4—1421no—
187C2×C4⋊C4—1421no—
188C2×C4⋊C4C2×D7111yescharacteristic
189#107—712no—
190#107—712no—
191#106—712no—
192#106—712no—
193#103—712no—
194#103C2×D7112yescomplemented
195#102—712no—
196#102C2×D7112yescomplemented
197C22⋊C8—712no—
198C22⋊C8—712no—
199C24—711no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D7111yescharacteristic
203C2×D4—21no—
204C2×D4—1421no—
205C2×D4—1421no—
206C2×D4C22×D7111yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2844no—
210C22×C4—711no—
211C22×C4—711no—
212C22×C4—711no—
213C22×C4—711no—
214C22×C4—711no—
215C22×C4—1422no—
216C22×C4—1422no—
217C22×C4—1421no—
218C22×C4—1422no—
219C22×C4—1422no—
220C22×C4C22×D7111yescharacteristic
221M16—22no—
222M16—1422no—
223C2×C8—712no—
224C2×C8C22×D7112yes—
225C4⋊C4—21no—
226C4⋊C4—712no—
227C4⋊C4—711no—
228C4⋊C4—711no—
229C4⋊C4—1422no—
230C4⋊C4—1422no—
231C4⋊C4—1421no—
232C4⋊C4—1421no—
233C4⋊C4—1421no—
234C4⋊C4C22×D7112yes—
235C22⋊C4—1421no—
236C22⋊C4—1421no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C42—1421no—
241C23—21no—
242C23—711no—
243C23—711no—
244C23—1422no—
245C23—1422no—
246C23—1422no—
247C23D4×D7111yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—2842no—
254D4—2842no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—2844no—
260C2×C4—2844no—
261C2×C4—2842no—
262C2×C4—711no—
263C2×C4—711no—
264C2×C4—1421no—
265C2×C4—1421no—
266C2×C4—1422no—
267C2×C4—1422no—
268C2×C4—1421no—
269C2×C4—1421no—
270C2×C4—1421no—
271C2×C4—1421no—
272C2×C4—1422no—
273C2×C4—1422no—
274C2×C4D4×D7111yescharacteristic
275C2×C4#1176586412yes—
276C2×C4#1176586512yes—
277C2×C4C23×D7111yescharacteristic
278C8—22no—
279C8—1422no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—712no—
286C22—712no—
287C22—1428no—
288C22—1421no—
289C22—1421no—
290C22—1421no—
291C22—1421no—
292C22#1455611012yes—
293C22C2×D4×D7111yescharacteristic
294C4—41no—
295C4—42no—
296C4—2842no—
297C4—2841no—
298C4—2841no—
299C4—1421no—
300C4—1421no—
301C4#1455611012yes—
302C4C2×#1176586411yescharacteristic
303C4C2×#1176586511yescharacteristic
304C2—22no—
305C2—81no—
306C2—1422no—
307C2—1422no—
308C2C2×#1455611011yescharacteristic
309C2AGL(1,ℤ/8)×D7111yescharacteristic
310C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#417×C71C2
socleC22×C71C22×D4
hypercenter#417D71
hypocenterC71#3092
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D71C2semidirect
3. #17986333C2semidirect
4. #17986328C2semidirect
5. #17986316C2semidirect
6. #17986311C2semidirect
7. #17986041C2semidirect
8. #17986039C2semidirect
9. #17986000C2semidirect
10. #17985962C2semidirect
11. #417×C71C2semidirect
30. #14556062C22semidirect
31. #14556061C22semidirect
32. #14556060C22semidirect
33. #14556059C22semidirect
34. #14556052C22semidirect
35. #14556049C22semidirect
41. #103×C71C22semidirect
42. #102×C71C22semidirect
156. C71#3092semidirect
166. #417D71semidirect
194. #103C2×D71semidirect
196. #102C2×D71semidirect