Finite Group Databasebeta

← #22598788

Subgroups — #22598788

Order 9088.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14556156C211yesmaximal, characteristic, complemented
2C2×#14556154C211yesmaximal, characteristic, complemented
3C2×#14556153C211yesmaximal, characteristic, complemented
4#17986545C211yesmaximal, characteristic, complemented
5C2×#14556124C211yesmaximal, characteristic, complemented
6#17986387C212yesmaximal, complemented
7#17986385C212yesmaximal, complemented
8#17986374C212yesmaximal, complemented
9#17986370C212yesmaximal, complemented
10M16×Dic71C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C22⋊D71—22no—
15#14556156—24no—
16#14556154—24no—
17#14556153—24no—
18C2×#11765854C2212yescomplemented
19C2×#11765853C2212yescomplemented
20C2×#11765852C2212yescomplemented
21C2×D568C2212yescomplemented
22C2×#11765850C2212yescomplemented
23C2×#11765850C2212yescomplemented
24C2×#11765849C2211yescharacteristic
25#14556126C2211yescharacteristic, complemented
26#14556124—22no—
27#14556124C2212yes—
28#14556124C2212yes—
29#14556124C2211yescharacteristic, complemented
30#14556117C2211yescharacteristic, complemented
31#14556115C2211yescharacteristic, complemented
32#14556100—22no—
33C4×C22⋊D71—22no—
34C2×C4×Dic71C2211yescharacteristic
35#14556038C2212yescomplemented
36C8×Dic71C2212yescomplemented
37C2×#41×C71C2211yescharacteristic
38C22×D4×C71C2211yescharacteristic
39AGL(1,ℤ/8)×C71—24no—
40C2×SD16×C71C2212yescomplemented
41C2×D8×C71C2212yescomplemented
42C2×M16×C71C2211yescharacteristic
43C23×D71—21no—
44#11765865—22no—
45#11765865—22no—
46C2×Dic142C2311yescharacteristic
47C2×D284—22no—
48C2×D284—22no—
49C2×D284—21no—
50C2×D284C2312yes—
51C2×D284C2311yescharacteristic
52C2×C22⋊D71—21no—
53C2×C22⋊D71—24no—
54C2×C22⋊D71—24no—
55C2×C22⋊D71—22no—
56C2×C22⋊D71—48no—
57C22×Dic71D412yes—
58C2×C4×D71—21no—
59#11765854—24no—
60#11765853—24no—
61#11765852—24no—
62D568—24no—
63#11765850—24no—
64#11765850—24no—
65#11765849D414yescomplemented
66C2×C71⋊C8C2312yes—
67Dic71⋊C4—22no—
68C22⋊Dic71—22no—
69(C2×D71)⋊C4—22no—
70C4×Dic71D412yes—
71C4×Dic71C2311yescharacteristic
72C4×Dic71C2311yescharacteristic
73C24×C71—21no—
74#41×C71—22no—
75#41×C71—22no—
76C2×Q8×C71C2311yescharacteristic
77C2×D4×C71—21no—
78C2×D4×C71—22no—
79C2×D4×C71—22no—
80C2×D4×C71—21no—
81C2×D4×C71C2311yescharacteristic
82C2×D4×C71C2312yes—
83C22×C4×C71—21no—
84C22×C4×C71C2311yescharacteristic
85SD16×C71—24no—
86D8×C71—24no—
87M16×C71D414yescomplemented
88C2×C8×C71C2312yes—
89C23×C71—21no—
90C23×C71—22no—
91C23×C71—44no—
92C23×C71—42no—
93C23×C71C2×D411yescharacteristic
94C22×D71—21no—
95C22×D71—22no—
96C22×D71—44no—
97C22×D71—42no—
98Q8×C71—21no—
99Q8×C71—21no—
100D4×C71—21no—
101D4×C71—22no—
102D4×C71—22no—
103D4×C71—21no—
104D4×C71—42no—
105D4×C71—42no—
106C2×C4×C71—21no—
107C2×C4×C71—42no—
108C2×C4×C71C2×D411yescharacteristic
109C2×C4×C71C2×D412yes—
110C2×C4×C71C2×D412yes—
111C2×C4×C71C2411yescharacteristic
112C22⋊D71—48no—
113C22⋊D71—42no—
114C22⋊D71—44no—
115C22⋊D71—48no—
116C2×Dic71—24no—
117C2×Dic71—21no—
118C2×Dic71C2×D412yes—
119C2×Dic71C2×D412yes—
120D284—21no—
121D284—22no—
122D284—21no—
123D284—22no—
124D284—42no—
125C4×D71—42no—
126Dic142—21no—
127Dic142—21no—
128C8×C71C2×D414yescomplemented
129C71⋊C8C2×D414yescomplemented
130C22×C71—22no—
131C22×C71—41no—
132C22×C71—42no—
133C22×C71—44no—
134C22×C71—42no—
135C22×C71—84no—
136C22×C71#12712yes—
137C22×C71C22×D411yescharacteristic
138C2×D71—41no—
139C2×D71—42no—
140C2×D71—44no—
141C2×D71—42no—
142C2×D71—84no—
143C4×C71—41no—
144C4×C71#12712yes—
145C4×C71C22×D411yescharacteristic
146C4×C71C22×D411yescharacteristic
147Dic71—22no—
148Dic71—41no—
149Dic71AGL(1,ℤ/8)14yescomplemented
150C2×C71—22no—
151C2×C71—81no—
152C2×C71—82no—
153C2×C71C2×#12711yescharacteristic
154C2×C71C2×AGL(1,ℤ/8)12yes—
155D71—81no—
156D71—82no—
157#3151—711nomaximal, Sylow
158C71#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—711no—
160C2×AGL(1,ℤ/8)—711no—
161C2×AGL(1,ℤ/8)—711no—
162C2×AGL(1,ℤ/8)D7111yescharacteristic, complemented
163#532—711no—
164C2×#127—711no—
165#496—712no—
166#496—712no—
167#493—712no—
168#492—712no—
169C4×M16—711no—
170C2×#41—711no—
171C2×#41C2×D7111yescharacteristic
172C22×D4—711no—
173C22×D4—1422no—
174C22×D4C2×D7111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—1424no—
177AGL(1,ℤ/8)—1424no—
178AGL(1,ℤ/8)—1424no—
179C2×SD16—712no—
180C2×SD16—712no—
181C2×SD16—712no—
182C2×SD16C2×D7112yescomplemented
183C2×D8—712no—
184C2×D8—712no—
185C2×D8—712no—
186C2×D8C2×D7112yescomplemented
187C2×M16—711no—
188C2×M16C2×D7111yescharacteristic
189C4⋊Q8—711no—
190#127—712no—
191#127—712no—
192#127—711no—
193#127—1422no—
194#124—711no—
195#124—711no—
196#121—1422no—
197C4×D4—1422no—
198C2×C42—711no—
199#97—712no—
200C4×C8—712no—
201C24—21no—
202C24—1421no—
203#41—22no—
204#41—22no—
205#41—1422no—
206#41—1422no—
207C2×Q8—711no—
208C2×Q8C22×D7111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2848no—
214C2×D4—711no—
215C2×D4—712no—
216C2×D4—1422no—
217C2×D4—1424no—
218C2×D4—1424no—
219C2×D4—1421no—
220C2×D4—1421no—
221C2×D4—1422no—
222C2×D4—1422no—
223C2×D4C22×D7111yescharacteristic
224C2×D4C22×D7112yes—
225C22×C4—21no—
226C22×C4—712no—
227C22×C4—1421no—
228C22×C4C22×D7111yescharacteristic
229SD16—24no—
230SD16—1424no—
231SD16—1424no—
232SD16—1424no—
233D8—24no—
234D8—1424no—
235D8—1424no—
236D8—1424no—
237M16—714no—
238M16C22⋊D7114yescomplemented
239C2×C8—712no—
240C2×C8C22×D7112yes—
241C4⋊C4—1422no—
242C22⋊C4—1422no—
243C22⋊C4—1422no—
244C42—712no—
245C42—711no—
246C42—711no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—2844no—
252C23—2842no—
253C23—1421no—
254C23—1422no—
255C23D4×D7111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—1421no—
259Q8—1421no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2842no—
267D4—2848no—
268D4—2844no—
269D4—2848no—
270D4—2842no—
271D4—1421no—
272D4—1422no—
273D4—1421no—
274D4—1422no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2842no—
278C2×C4—712no—
279C2×C4—712no—
280C2×C4—1421no—
281C2×C4—1424no—
282C2×C4D4×D7112yes—
283C2×C4D4×D7111yescharacteristic
284C2×C4C2×C22⋊D7112yes—
285C2×C4C23×D7111yescharacteristic
286C8—714no—
287C8C2×C22⋊D7114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—2844no—
295C22—2842no—
296C22—2841no—
297C22—2842no—
298C22—5684no—
299C22#1455612412yes—
300C22C2×D4×D7111yescharacteristic
301C4—41no—
302C4—2841no—
303C4—714no—
304C4—1422no—
305C4#1455612412yes—
306C4C2×D4×D7111yescharacteristic
307C4C22×C22⋊D7111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—5682no—
312C2—5681no—
313C2C2×#1455612411yescharacteristic
314C2AGL(1,ℤ/8)×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×AGL(1,ℤ/8)×C71C2
socleC22×C71C22×D4
hypercenterC2×AGL(1,ℤ/8)D71
hypocenterC71#3151
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#14556156C2semidirect
2. C2×#14556154C2semidirect
3. C2×#14556153C2semidirect
4. #17986545C2semidirect
5. C2×#14556124C2semidirect
6. #17986387C2semidirect
7. #17986385C2semidirect
8. #17986374C2semidirect
9. #17986370C2semidirect
10. M16×Dic71C2semidirect
11. C2×AGL(1,ℤ/8)×C71C2semidirect
18. C2×#11765854C22semidirect
19. C2×#11765853C22semidirect
20. C2×#11765852C22semidirect
21. C2×D568C22semidirect
22. C2×#11765850C22semidirect
23. C2×#11765850C22semidirect
25. #14556126C22semidirect
29. #14556124C22semidirect
30. #14556117C22semidirect
31. #14556115C22semidirect
35. #14556038C22semidirect
36. C8×Dic71C22semidirect
40. C2×SD16×C71C22semidirect
41. C2×D8×C71C22semidirect
65. #11765849D4semidirect
87. M16×C71D4semidirect
128. C8×C71C2×D4semidirect
129. C71⋊C8C2×D4semidirect
149. Dic71AGL(1,ℤ/8)semidirect
158. C71#3151semidirect
162. C2×AGL(1,ℤ/8)D71semidirect
182. C2×SD16C2×D71semidirect
186. C2×D8C2×D71semidirect
238. M16C22⋊D71semidirect
287. C8C2×C22⋊D71semidirect