Finite Group Databasebeta

← #22599124

Subgroups — #22599124

Order 9088.

420 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986512C211yesmaximal, characteristic
2C2×#14556104C211yesmaximal, characteristic, complemented
3#17986339C212yesmaximal, complemented
4#17986327C211yesmaximal, characteristic, complemented
5#17986313C211yesmaximal, characteristic, complemented
6#17986237C212yesmaximal, complemented
7#17986231C212yesmaximal, complemented
8#17986160C212yesmaximal, complemented
9#17986138C211yesmaximal, characteristic, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#451×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C2×#11765852C2212yescomplemented
15C2×C8×D71C2212yes—
16#14556113—21no—
17#14556104—21no—
18#14556104—21no—
19#14556104C2212yes—
20#14556104C2212yes—
21#14556104C2211yescharacteristic
22#14556103C2211yescharacteristic
23#14556102C2211yescharacteristic
24#14556098—21no—
25#14556090—21no—
26(C22⋊D71)⋊C4—21no—
27#14556086C2211yescharacteristic
28C2×C4⋊Dic71C2211yescharacteristic
29C2×Dic71⋊C4—21no—
30C2×(C2×D71)⋊C4—21no—
31#14556062C2212yescomplemented
32#14556059C2212yescomplemented
33#14556052C2212yescomplemented
34#14556052C2212yescomplemented
35#14556050C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38C22×Q8×C71C2211yescharacteristic
39C2×Q16×C71C2212yescomplemented
40C22⋊Q8×C71C2211yescharacteristic
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44#11765866—42no—
45C2×D284—21no—
46C2×D284—21no—
47C2×D284C2311yescharacteristic
48C2×C22⋊D71—21no—
49C22×Dic71—21no—
50C22×Dic71D411yescharacteristic
51C2×C4×D71—21no—
52C2×C4×D71—42no—
53C2×C4×D71D411yescharacteristic
54C2×C4×D71D411yescharacteristic
55C2×C4×D71C2311yescharacteristic
56C2×C4×D71C2311yescharacteristic
57#11765852—42no—
58C2×C71⋊C8C2312yes—
59C8×D71—42no—
60C4⋊Dic71—21no—
61C4⋊Dic71C2312yes—
62C4⋊Dic71C2311yescharacteristic
63Dic71⋊C4—22no—
64Dic71⋊C4—22no—
65Dic71⋊C4—21no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—21no—
69(C2×D71)⋊C4—21no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72C4×Dic71—21no—
73C2×Q8×C71—21no—
74C2×Q8×C71—42no—
75C2×Q8×C71C2311yescharacteristic
76C2×Q8×C71C2312yes—
77C22×C4×C71—21no—
78C22×C4×C71C2311yescharacteristic
79Q16×C71—42no—
80C2×C8×C71C2312yes—
81C4⋊C4×C71—21no—
82C4⋊C4×C71C2311yescharacteristic
83C22⋊C4×C71—21no—
84C23×C71C2×D411yescharacteristic
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—21no—
88C22×D71—41no—
89C22×D71C2×D411yescharacteristic
90C22×D71C2×D411yescharacteristic
91Q8×C71—22no—
92Q8×C71—22no—
93Q8×C71—22no—
94Q8×C71—42no—
95C2×C4×C71—22no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—22no—
99C2×C4×C71—42no—
100C2×C4×C71C2×D411yescharacteristic
101C2×C4×C71C2411yescharacteristic
102C22⋊D71—42no—
103C2×Dic71—21no—
104C2×Dic71—22no—
105C2×Dic71—41no—
106C2×Dic71—42no—
107C2×Dic71C2×D411yescharacteristic
108C2×Dic71C2×D411yescharacteristic
109D284—22no—
110D284—42no—
111D284—42no—
112C4×D71—22no—
113C4×D71—22no—
114C4×D71—41no—
115C4×D71—41no—
116C4×D71—42no—
117C8×C71—22no—
118C71⋊C8—22no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71#13512yes—
122C22×C71C22×D411yescharacteristic
123C2×D71—21no—
124C2×D71—21no—
125C2×D71—21no—
126C2×D71—21no—
127C2×D71—21no—
128C2×D71—44no—
129C2×D71—41no—
130C2×D71—41no—
131C2×D71—41no—
132C2×D71#13712yes—
133C4×C71—21no—
134C4×C71—41no—
135C4×C71—42no—
136C4×C71—41no—
137C4×C71H(5,𝔽2)12yes—
138Dic71—21no—
139Dic71—21no—
140Dic71—42no—
141Dic71—41no—
142C2×C71—22no—
143C2×C71#53411yescharacteristic
144C2×C71C2×#13511yescharacteristic
145C2×C71C2×#13711yescharacteristic
146D71—22no—
147D71—41no—
148D71—81no—
149#3190—711nomaximal, Sylow
150C71#319011yescharacteristic, complemented, Sylow
151#525—711no—
152C2×C22⋊Q8—711no—
153#485—712no—
154#483—711no—
155#480—711no—
156#467—712no—
157#465—712no—
158#452—712no—
159#451D7111yescharacteristic, complemented
160#448—711no—
161C2×C22⋊C8—711no—
162C2×#41—711no—
163C22×Q8C2×D7111yescharacteristic
164C23×C4—711no—
165C2×Q16C2×D7112yescomplemented
166C2×SD16—712no—
167C22×C8—712no—
168#123—1421no—
169C22⋊Q8—712no—
170C22⋊Q8—712no—
171C22⋊Q8—711no—
172C22⋊Q8—1421no—
173C22⋊Q8—1421no—
174C22⋊Q8C2×D7111yescharacteristic
175#121—711no—
176#121—711no—
177#120—1421no—
178C4×D4—1421no—
179C4×D4—1421no—
180#117—711no—
181C2×C4⋊C4—711no—
182C2×C4⋊C4—1421no—
183C2×C22⋊C4—1421no—
184#107—712no—
185#106—712no—
186#103—712no—
187#103—712no—
188#103C2×D7112yescomplemented
189#102—712no—
190C22⋊C8—711no—
191C22⋊C8—712no—
192C22⋊C8C2×D7111yescharacteristic
193C24—711no—
194#41—2842no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D7111yescharacteristic
198C2×Q8C22×D7112yes—
199C2×D4—711no—
200C2×D4—1421no—
201C2×D4—1421no—
202C2×D4—1421no—
203C22×C4—21no—
204C22×C4—2842no—
205C22×C4—711no—
206C22×C4—711no—
207C22×C4—711no—
208C22×C4—711no—
209C22×C4—711no—
210C22×C4—1421no—
211C22×C4—1421no—
212C22×C4C22×D7111yescharacteristic
213Q16—42no—
214SD16—2842no—
215C2×C8—2842no—
216C2×C8—712no—
217C2×C8C22×D7112yes—
218C4⋊C4—21no—
219C4⋊C4—712no—
220C4⋊C4—711no—
221C4⋊C4—1422no—
222C4⋊C4—1422no—
223C4⋊C4—1421no—
224C4⋊C4—1421no—
225C4⋊C4C22×D7111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1421no—
228C22⋊C4—1422no—
229C22⋊C4—1422no—
230C22⋊C4—1421no—
231C22⋊C4—1421no—
232C22⋊C4—1421no—
233C42—1421no—
234C23—2841no—
235C23—711no—
236C23—711no—
237C23—1422no—
238C23—1421no—
239C23—1422no—
240C23D4×D7111yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—2842no—
246D4—2842no—
247D4—2842no—
248D4—1422no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—2841no—
255C2×C4—2841no—
256C2×C4—2842no—
257C2×C4—2842no—
258C2×C4—2841no—
259C2×C4—711no—
260C2×C4—711no—
261C2×C4—1421no—
262C2×C4—1422no—
263C2×C4—1422no—
264C2×C4—1422no—
265C2×C4D4×D7111yescharacteristic
266C2×C4C23×D7111yescharacteristic
267C8—22no—
268C8—1422no—
269C22—21no—
270C22—21no—
271C22—2844no—
272C22—2841no—
273C22—2841no—
274C22—2841no—
275C22—712no—
276C22—1421no—
277C22—1421no—
278C22—1421no—
279C22—1421no—
280C22—1421no—
281C22#1455615112yes—
282C22C2×D4×D7111yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—2842no—
288C4—2841no—
289C4—1421no—
290C4—1421no—
291C4#1455617312yes—
292C2—22no—
293C2—2841no—
294C2—5681no—
295C2—1422no—
296C2#1798655911yescharacteristic
297C2C2×#1455615111yescharacteristic
298C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#451×C71C2
socleC22×C71C22×D4
hypercenter#451D71
hypocenterC71#3190
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#14556104C2semidirect
3. #17986339C2semidirect
4. #17986327C2semidirect
5. #17986313C2semidirect
6. #17986237C2semidirect
7. #17986231C2semidirect
8. #17986160C2semidirect
9. #17986138C2semidirect
10. C22⋊C8×D71C2semidirect
11. #451×C71C2semidirect
14. C2×#11765852C22semidirect
31. #14556062C22semidirect
32. #14556059C22semidirect
33. #14556052C22semidirect
34. #14556052C22semidirect
35. #14556050C22semidirect
39. C2×Q16×C71C22semidirect
41. #103×C71C22semidirect
150. C71#3190semidirect
159. #451D71semidirect
165. C2×Q16C2×D71semidirect
188. #103C2×D71semidirect