Finite Group Databasebeta

← #22598807

Subgroups — #22598807

Order 9088.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14556159C211yesmaximal, characteristic, complemented
2C2×#14556156C211yesmaximal, characteristic, complemented
3C2×#14556153C211yesmaximal, characteristic, complemented
4#17986545C211yesmaximal, characteristic, complemented
5C2×#14556117C211yesmaximal, characteristic, complemented
6#17986392C212yesmaximal, complemented
7#17986385C212yesmaximal, complemented
8#17986384C212yesmaximal, complemented
9#17986380C212yesmaximal, complemented
10M16×Dic71C211yesmaximal, characteristic, complemented
11C2×#137×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×D284C2211yescharacteristic
14#14556159—24no—
15#14556156—24no—
16#14556153—24no—
17C2×#11765855C2212yescomplemented
18C2×#11765854C2212yes—
19C2×#11765852C2212yes—
20C2×#11765852C2212yescomplemented
21C2×D568C2212yes—
22C2×#11765850C2212yes—
23C2×#11765849C2211yescharacteristic
24#14556126C2211yescharacteristic, complemented
25#14556124C2211yescharacteristic, complemented
26#14556117—22no—
27#14556117C2212yes—
28#14556117C2212yes—
29#14556117C2211yescharacteristic, complemented
30#14556115C2211yescharacteristic, complemented
31#14556100—22no—
32C4×C22⋊D71—22no—
33C2×(C2×D71)⋊C4—22no—
34C2×C4×Dic71C2211yescharacteristic
35#14556038C2212yescomplemented
36C8×Dic71C2212yescomplemented
37C2×#41×C71C2211yescharacteristic
38C22×Q8×C71C2211yescharacteristic
39#137×C71—24no—
40C2×Q16×C71C2212yescomplemented
41C2×SD16×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—22no—
54C22×Dic71D412yes—
55C2×C4×D71—21no—
56#11765855—24no—
57#11765854—24no—
58#11765852—24no—
59#11765852—24no—
60D568—24no—
61#11765850—24no—
62#11765849D414yescomplemented
63C2×C71⋊C8C2312yes—
64Dic71⋊C4—22no—
65C22⋊Dic71—22no—
66(C2×D71)⋊C4—24no—
67(C2×D71)⋊C4—24no—
68(C2×D71)⋊C4—22no—
69C4×Dic71D412yes—
70C4×Dic71C2311yescharacteristic
71C4×Dic71C2311yescharacteristic
72#41×C71—22no—
73#41×C71—22no—
74C2×Q8×C71—21no—
75C2×Q8×C71—22no—
76C2×Q8×C71—22no—
77C2×Q8×C71C2312yes—
78C2×Q8×C71C2311yescharacteristic
79C2×D4×C71—21no—
80C2×D4×C71C2311yescharacteristic
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71C2311yescharacteristic
84Q16×C71—24no—
85SD16×C71—24no—
86M16×C71D414yescomplemented
87C2×C8×C71C2312yes—
88C23×C71—21no—
89C23×C71C2×D411yescharacteristic
90C22×D71—22no—
91C22×D71—21no—
92C22×D71—44no—
93C22×D71—42no—
94Q8×C71—22no—
95Q8×C71—21no—
96Q8×C71—22no—
97Q8×C71—21no—
98Q8×C71—42no—
99D4×C71—21no—
100D4×C71—21no—
101D4×C71—42no—
102C2×C4×C71—21no—
103C2×C4×C71—22no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71C2×D411yescharacteristic
107C2×C4×C71C2×D412yes—
108C2×C4×C71C2×D412yes—
109C2×C4×C71C2411yescharacteristic
110C22⋊D71—44no—
111C22⋊D71—42no—
112C2×Dic71—21no—
113C2×Dic71—42no—
114C2×Dic71C2×D412yes—
115C2×Dic71C2×D412yes—
116D284—22no—
117D284—21no—
118D284—22no—
119D284—21no—
120D284—42no—
121C4×D71—42no—
122Dic142—21no—
123Dic142—21no—
124C8×C71C2×D414yes—
125C71⋊C8C2×D414yes—
126C22×C71—22no—
127C22×C71—41no—
128C22×C71—42no—
129C22×C71#12712yes—
130C22×C71C22×D411yescharacteristic
131C2×D71—41no—
132C2×D71—42no—
133C2×D71—42no—
134C2×D71—44no—
135C2×D71—84no—
136C4×C71—41no—
137C4×C71—42no—
138C4×C71#12712yes—
139C4×C71C22×D411yescharacteristic
140C4×C71C22×D411yescharacteristic
141Dic71—22no—
142Dic71—22no—
143Dic71—41no—
144C2×C71—22no—
145C2×C71—81no—
146C2×C71C2×#12711yescharacteristic
147C2×C71#57512yes—
148D71—81no—
149D71—82no—
150#3153—711nomaximal, Sylow
151C71#315311yescharacteristic, complemented, Sylow
152C2×#137—711no—
153C2×#137D7111yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—711no—
155C2×AGL(1,ℤ/8)—711no—
156#532—711no—
157C2×#124—711no—
158#497—712no—
159#496—712no—
160#495—712no—
161#495—712no—
162C4×M16—711no—
163C2×#41—711no—
164C2×#41C2×D7111yescharacteristic
165C22×Q8C2×D7111yescharacteristic
166C22×D4—711no—
167#137—24no—
168#137—1424no—
169AGL(1,ℤ/8)—1424no—
170AGL(1,ℤ/8)—1424no—
171C2×Q16—712no—
172C2×Q16C2×D7112yescomplemented
173C2×SD16—712no—
174C2×SD16—712no—
175C2×SD16—712no—
176C2×SD16C2×D7112yescomplemented
177C2×D8—712no—
178C2×D8—712no—
179C2×M16—711no—
180C2×M16C2×D7111yescharacteristic
181C4⋊Q8—711no—
182#127—711no—
183#124—712no—
184#124—712no—
185#124—711no—
186#124—711no—
187#124—1422no—
188#121—1422no—
189C4×D4—1422no—
190C2×C22⋊C4—1422no—
191C2×C42—711no—
192#97—712no—
193C4×C8—712no—
194C24—1421no—
195#41—22no—
196#41—22no—
197#41—1422no—
198#41—1422no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—711no—
203C2×Q8C22×D7112yes—
204C2×Q8C22×D7111yescharacteristic
205C2×D4—21no—
206C2×D4—712no—
207C2×D4—711no—
208C2×D4—1422no—
209C2×D4—1421no—
210C2×D4—1421no—
211C2×D4—1422no—
212C2×D4—1422no—
213C2×D4C22×D7111yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—712no—
217C22×C4—1421no—
218C22×C4C22×D7111yescharacteristic
219Q16—24no—
220Q16—1424no—
221SD16—24no—
222SD16—1424no—
223SD16—1424no—
224SD16—1424no—
225D8—1424no—
226D8—1424no—
227M16—714no—
228M16C22⋊D7114yescomplemented
229C2×C8—712no—
230C2×C8C22×D7112yes—
231C4⋊C4—1422no—
232C22⋊C4—1424no—
233C22⋊C4—1424no—
234C22⋊C4—1422no—
235C22⋊C4—1422no—
236C42—712no—
237C42—711no—
238C42—711no—
239C23—21no—
240C23—2844no—
241C23—2842no—
242C23—1422no—
243C23—1421no—
244C23D4×D7111yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—1421no—
251Q8—1421no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—2842no—
256D4—2844no—
257D4—2842no—
258D4—1422no—
259D4—1421no—
260D4—1421no—
261D4—1422no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—2842no—
267C2×C4—2842no—
268C2×C4—712no—
269C2×C4—712no—
270C2×C4—1421no—
271C2×C4D4×D7112yes—
272C2×C4D4×D7111yescharacteristic
273C2×C4C2×C22⋊D7112yes—
274C2×C4C23×D7111yescharacteristic
275C8—714no—
276C8C2×C22⋊D7114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—2841no—
281C22—2842no—
282C22—2842no—
283C22—2844no—
284C22—5684no—
285C22#1455612412yes—
286C22C2×D4×D7111yescharacteristic
287C4—41no—
288C4—42no—
289C4—2841no—
290C4—1422no—
291C4—1422no—
292C4#1455612412yes—
293C4C2×D4×D7111yescharacteristic
294C4C22×C22⋊D7111yescharacteristic
295C2—22no—
296C2—81no—
297C2—5682no—
298C2—5681no—
299C2C2×#1455612411yescharacteristic
300C2#1798679812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#137×C71C2
socleC22×C71C22×D4
hypercenterC2×#137D71
hypocenterC71#3153
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#14556159C2semidirect
2. C2×#14556156C2semidirect
3. C2×#14556153C2semidirect
4. #17986545C2semidirect
5. C2×#14556117C2semidirect
6. #17986392C2semidirect
7. #17986385C2semidirect
8. #17986384C2semidirect
9. #17986380C2semidirect
10. M16×Dic71C2semidirect
11. C2×#137×C71C2semidirect
17. C2×#11765855C22semidirect
20. C2×#11765852C22semidirect
24. #14556126C22semidirect
25. #14556124C22semidirect
29. #14556117C22semidirect
30. #14556115C22semidirect
35. #14556038C22semidirect
36. C8×Dic71C22semidirect
40. C2×Q16×C71C22semidirect
41. C2×SD16×C71C22semidirect
62. #11765849D4semidirect
86. M16×C71D4semidirect
151. C71#3153semidirect
153. C2×#137D71semidirect
172. C2×Q16C2×D71semidirect
176. C2×SD16C2×D71semidirect
228. M16C22⋊D71semidirect