Finite Group Databasebeta

← #22598290

Subgroups — #22598290

Order 9088.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14556108C211yesmaximal, characteristic
2#117×D71C211yesmaximal, characteristic
3#17986343C212yesmaximal, complemented
4#17986342C212yesmaximal, complemented
5#17986336C212yesmaximal, complemented
6#17986331C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic, complemented
8#17986009C211yesmaximal, characteristic, complemented
9#17986007C211yesmaximal, characteristic, complemented
10#17985962C211yesmaximal, characteristic
11#428×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C2×#11765849C2211yescharacteristic
15#14556108—21no—
16#14556108—21no—
17#14556108C2212yes—
18#14556108C2212yes—
19#14556087C2211yescharacteristic
20#14556086C2211yescharacteristic
21#14556086C2211yescharacteristic
22#14556085—21no—
23#14556084—21no—
24C2×C4⋊Dic71—21no—
25C4⋊C4×D71C2211yescharacteristic
26C4⋊C4×D71C2211yescharacteristic
27C2×(C2×D71)⋊C4—21no—
28C22⋊C4×D71—21no—
29C42×D71—21no—
30#14556061C2212yescomplemented
31#14556059C2212yescomplemented
32#14556053C2212yescomplemented
33#14556053C2212yescomplemented
34#14556051C2212yescomplemented
35#14556051C2212yescomplemented
36(C2×D71)⋊C8C2212yes—
37(C2×D71)⋊C8C2212yes—
38C2×M16×C71C2211yescharacteristic
39#117×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#107×C71C2212yescomplemented
42#106×C71C2212yescomplemented
43C23×D71D411yescharacteristic
44C2×Dic142—22no—
45C2×Dic142—22no—
46C2×Dic142—21no—
47C2×Dic142C2312yes—
48C22×Dic71—21no—
49C22×Dic71D411yescharacteristic
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—21no—
53C2×C4×D71—21no—
54C2×C4×D71D411yescharacteristic
55C2×C4×D71D411yescharacteristic
56C2×C4×D71C2311yescharacteristic
57C2×C4×D71C2311yescharacteristic
58#11765849—22no—
59C2×C71⋊C8C2312yes—
60C4⋊Dic71—22no—
61C4⋊Dic71—22no—
62C4⋊Dic71C2311yescharacteristic
63C4⋊Dic71C2311yescharacteristic
64Dic71⋊C4—21no—
65Dic71⋊C4—21no—
66C22⋊Dic71—21no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—21no—
69(C2×D71)⋊C4—22no—
70(C2×D71)⋊C4—22no—
71C4×Dic71—21no—
72C4×Dic71—21no—
73C4×Dic71—21no—
74C22×C4×C71—21no—
75C22×C4×C71C2311yescharacteristic
76M16×C71—22no—
77C2×C8×C71C2312yes—
78C4⋊C4×C71—21no—
79C4⋊C4×C71C2311yescharacteristic
80C4⋊C4×C71C2312yes—
81C4⋊C4×C71C2311yescharacteristic
82C22⋊C4×C71—21no—
83C42×C71—21no—
84C23×C71C2×D411yescharacteristic
85C22×D71—22no—
86C22×D71—22no—
87C22×D71—41no—
88C22×D71C2×D411yescharacteristic
89C22×D71C2×D411yescharacteristic
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—42no—
94C2×C4×C71C2×D411yescharacteristic
95C2×C4×C71#4112yes—
96C2×C4×C71#4112yes—
97C2×C4×C71C2411yescharacteristic
98C2×Dic71—22no—
99C2×Dic71—21no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—42no—
103C2×Dic71C2×D411yescharacteristic
104C2×Dic71C2×D411yescharacteristic
105C4×D71—22no—
106C4×D71—22no—
107C4×D71—21no—
108C4×D71—21no—
109C4×D71—21no—
110C4×D71—21no—
111C4×D71—42no—
112C4×D71—42no—
113Dic142—22no—
114Dic142—22no—
115Dic142—42no—
116C8×C71—22no—
117C71⋊C8—22no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71#12312yes—
121C22×C71C22×D411yescharacteristic
122C2×D71—21no—
123C2×D71—21no—
124C2×D71—21no—
125C2×D71—21no—
126C2×D71—21no—
127C2×D71—44no—
128C2×D71#13712yes—
129C4×C71—41no—
130C4×C71—41no—
131C4×C71—42no—
132C4×C71#12312yes—
133C4×C71C2×#4111yescharacteristic
134C4×C71C2×#4111yescharacteristic
135Dic71—21no—
136Dic71—21no—
137Dic71—42no—
138Dic71—41no—
139Dic71—41no—
140C2×C71—22no—
141C2×C71C2×#12311yescharacteristic
142C2×C71C2×#13711yescharacteristic
143C2×C71#57511yescharacteristic
144D71—22no—
145D71—41no—
146#3095—711nomaximal, Sylow
147C71#309511yescharacteristic, complemented, Sylow
148C2×C22⋊Q8—711no—
149C2×#117—711no—
150#485—712no—
151#485—712no—
152#484—712no—
153#483—712no—
154#428—711no—
155#428D7111yescharacteristic, complemented
156#419—711no—
157#419—711no—
158#407—711no—
159C22×Q8—711no—
160C23×C4—711no—
161C2×M16—711no—
162C2×M16C2×D7111yescharacteristic
163C22⋊Q8—712no—
164C22⋊Q8—712no—
165C22⋊Q8—1421no—
166C22⋊Q8—1421no—
167#117—711no—
168#117—711no—
169#117—711no—
170#117—1421no—
171#117—1421no—
172#117C2×D7111yescharacteristic
173C2×C4⋊C4—711no—
174C2×C4⋊C4—711no—
175C2×C4⋊C4—1421no—
176C2×C4⋊C4C2×D7111yescharacteristic
177C2×C22⋊C4—1421no—
178C2×C22⋊C4—1421no—
179C2×C42—1421no—
180#107—712no—
181#107C2×D7112yescomplemented
182#106—712no—
183#106C2×D7112yescomplemented
184#103—712no—
185#103—712no—
186#103—712no—
187#103—712no—
188C22⋊C8—712no—
189C22⋊C8—712no—
190C24—711no—
191C2×Q8—712no—
192C2×Q8—1422no—
193C2×Q8—1422no—
194C2×Q8—1421no—
195C22×C4—21no—
196C22×C4—711no—
197C22×C4—711no—
198C22×C4—711no—
199C22×C4—711no—
200C22×C4—711no—
201C22×C4—1421no—
202C22×C4—1421no—
203C22×C4—1421no—
204C22×C4—1422no—
205C22×C4—1422no—
206C22×C4C22×D7111yescharacteristic
207M16—22no—
208M16—1422no—
209C2×C8—712no—
210C2×C8C22×D7112yes—
211C4⋊C4—21no—
212C4⋊C4—711no—
213C4⋊C4—711no—
214C4⋊C4—1421no—
215C4⋊C4—1421no—
216C4⋊C4—1422no—
217C4⋊C4—1422no—
218C4⋊C4C22×D7111yescharacteristic
219C4⋊C4C22×D7111yescharacteristic
220C4⋊C4C22×D7112yes—
221C22⋊C4—21no—
222C22⋊C4—1421no—
223C22⋊C4—1421no—
224C22⋊C4—1422no—
225C22⋊C4—1422no—
226C22⋊C4—1421no—
227C42—21no—
228C42—1421no—
229C42—1421no—
230C42—1421no—
231C23—2841no—
232C23—711no—
233C23—711no—
234C23—1422no—
235C23—1422no—
236C23D4×D7111yescharacteristic
237Q8—2842no—
238Q8—1422no—
239Q8—1422no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—711no—
248C2×C4—711no—
249C2×C4—1421no—
250C2×C4—1421no—
251C2×C4—1422no—
252C2×C4—1422no—
253C2×C4—1421no—
254C2×C4—1421no—
255C2×C4—1421no—
256C2×C4—1421no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4D4×D7111yescharacteristic
260C2×C4#1176586512yes—
261C2×C4#1176586612yes—
262C2×C4C23×D7111yescharacteristic
263C8—22no—
264C8—1422no—
265C22—21no—
266C22—21no—
267C22—2844no—
268C22—712no—
269C22—1421no—
270C22—1421no—
271C22—1421no—
272C22—1421no—
273C22—1421no—
274C22#1455611312yes—
275C22C2×D4×D7111yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2842no—
280C4—2841no—
281C4—2841no—
282C4—1421no—
283C4—1421no—
284C4#1455611312yes—
285C4C2×#1176586511yescharacteristic
286C4C2×#1176586611yescharacteristic
287C2—22no—
288C2—2841no—
289C2—1422no—
290C2C2×#1455611311yescharacteristic
291C2#137×D7111yescharacteristic
292C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3095
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #17986343C2semidirect
4. #17986342C2semidirect
5. #17986336C2semidirect
6. #17986331C2semidirect
7. #17986039C2semidirect
8. #17986009C2semidirect
9. #17986007C2semidirect
11. #428×C71C2semidirect
30. #14556061C22semidirect
31. #14556059C22semidirect
32. #14556053C22semidirect
33. #14556053C22semidirect
34. #14556051C22semidirect
35. #14556051C22semidirect
41. #107×C71C22semidirect
42. #106×C71C22semidirect
147. C71#3095semidirect
155. #428D71semidirect
181. #107C2×D71semidirect
183. #106C2×D71semidirect