Finite Group Databasebeta

← #22599126

Subgroups — #22599126

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×#14556108C211yesmaximal, characteristic, complemented
3#17986343C212yesmaximal, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986239C212yesmaximal, complemented
6#17986229C212yesmaximal, complemented
7#17986164C212yesmaximal, complemented
8#17986154C211yesmaximal, characteristic, complemented
9#17986140C211yesmaximal, characteristic, complemented
10C22⋊C8×D71C211yesmaximal, characteristic, complemented
11#483×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C2×#11765855C2212yescomplemented
16C2×#11765854C2212yescomplemented
17C2×C8×D71C2212yes—
18#14556113—21no—
19#14556108—21no—
20#14556108—21no—
21#14556108C2212yes—
22#14556108C2212yes—
23#14556104C2211yescharacteristic
24#14556103C2211yescharacteristic
25#14556102C2211yescharacteristic
26#14556098—21no—
27#14556090—21no—
28(C22⋊D71)⋊C4—21no—
29#14556086C2211yescharacteristic
30C2×C4⋊Dic71—21no—
31C2×(C2×D71)⋊C4—21no—
32#14556061C2212yescomplemented
33#14556053C2212yescomplemented
34#14556051C2212yescomplemented
35#14556048C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38C22⋊Q8×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#106×C71C2212yescomplemented
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44#11765866—42no—
45C2×Dic142—21no—
46C2×Dic142—42no—
47C2×Dic142C2312yes—
48C2×D284—21no—
49C2×D284—21no—
50C2×D284C2311yescharacteristic
51C2×C22⋊D71—21no—
52C22×Dic71—21no—
53C22×Dic71D411yescharacteristic
54C2×C4×D71—21no—
55C2×C4×D71—42no—
56C2×C4×D71D411yescharacteristic
57C2×C4×D71D411yescharacteristic
58C2×C4×D71C2311yescharacteristic
59C2×C4×D71C2311yescharacteristic
60#11765855—42no—
61#11765854—42no—
62C2×C71⋊C8C2312yes—
63C8×D71—42no—
64C4⋊Dic71—22no—
65C4⋊Dic71—22no—
66C4⋊Dic71C2311yescharacteristic
67Dic71⋊C4—21no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—21no—
71(C2×D71)⋊C4—21no—
72(C2×D71)⋊C4—21no—
73(C2×D71)⋊C4—21no—
74C4×Dic71—21no—
75C2×Q8×C71C2311yescharacteristic
76C22×C4×C71—21no—
77C22×C4×C71C2311yescharacteristic
78C2×C8×C71C2312yes—
79C4⋊C4×C71—21no—
80C4⋊C4×C71—21no—
81C4⋊C4×C71C2312yes—
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—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—42no—
97C2×C4×C71C2×D411yescharacteristic
98C2×C4×C71C2411yescharacteristic
99C22⋊D71—42no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—41no—
103C2×Dic71—42no—
104C2×Dic71C2×D411yescharacteristic
105C2×Dic71C2×D411yescharacteristic
106D284—22no—
107D284—42no—
108D284—42no—
109C4×D71—22no—
110C4×D71—22no—
111C4×D71—41no—
112C4×D71—41no—
113C4×D71—42no—
114Dic142—22no—
115Dic142—22no—
116Dic142—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#483D7111yescharacteristic, complemented
155#480—711no—
156#467—712no—
157#465—712no—
158#452—712no—
159#451—711no—
160#448—711no—
161C2×C22⋊C8—711no—
162C2×#41—711no—
163C22×Q8—711no—
164C23×C4—711no—
165C2×Q16—712no—
166C2×SD16—712no—
167C22×C8—712no—
168#123—1421no—
169C22⋊Q8—711no—
170C22⋊Q8—712no—
171C22⋊Q8—712no—
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—1421no—
182C2×C4⋊C4C2×D7111yescharacteristic
183C2×C22⋊C4—1421no—
184#107—712no—
185#106C2×D7112yescomplemented
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—2842no—
196C2×Q8—712no—
197C2×Q8—1421no—
198C2×Q8C22×D7111yescharacteristic
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—2842no—
214SD16—2842no—
215C2×C8—2842no—
216C2×C8—712no—
217C2×C8C22×D7112yes—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—711no—
221C4⋊C4—1421no—
222C4⋊C4—1422no—
223C4⋊C4—1422no—
224C4⋊C4C22×D7112yes—
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—2842no—
243Q8—1422no—
244Q8—1422no—
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#1455614912yes—
282C22C2×D4×D7111yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—2842no—
288C4—2841no—
289C4—1421no—
290C4—1421no—
291C4#1455617412yes—
292C2—22no—
293C2—2841no—
294C2—5681no—
295C2—1422no—
296C2#1798656011yescharacteristic
297C2C2×#1455614911yescharacteristic
298C2#137×D7111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#483×C71C2
socleC22×C71C22×D4
hypercenter#483D71
hypocenterC71#3190
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. C2×#14556108C2semidirect
3. #17986343C2semidirect
4. #17986312C2semidirect
5. #17986239C2semidirect
6. #17986229C2semidirect
7. #17986164C2semidirect
8. #17986154C2semidirect
9. #17986140C2semidirect
10. C22⋊C8×D71C2semidirect
11. #483×C71C2semidirect
15. C2×#11765855C22semidirect
16. C2×#11765854C22semidirect
32. #14556061C22semidirect
33. #14556053C22semidirect
34. #14556051C22semidirect
35. #14556048C22semidirect
40. #106×C71C22semidirect
41. #103×C71C22semidirect
150. C71#3190semidirect
154. #483D71semidirect
185. #106C2×D71semidirect
188. #103C2×D71semidirect