Finite Group Databasebeta

← #22598285

Subgroups — #22598285

Order 9088.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D71C211yesmaximal, characteristic
2C2×#14556086C211yesmaximal, characteristic
3#17986342C212yesmaximal, complemented
4#17986338C212yesmaximal, complemented
5#17986334C211yesmaximal, characteristic, complemented
6#17986330C211yesmaximal, characteristic, complemented
7#17986327C211yesmaximal, characteristic, complemented
8#17986040C212yesmaximal, complemented
9#17986006C212yesmaximal, complemented
10#17985961C211yesmaximal, characteristic
11#484×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C2×#11765848C2212yes—
15#14556108C2211yescharacteristic
16#14556107C2211yescharacteristic
17#14556106—21no—
18#14556105—21no—
19#14556104C2211yescharacteristic
20#14556086—21no—
21#14556086—21no—
22#14556086C2212yes—
23#14556086C2212yes—
24C2×C4⋊Dic71C2211yescharacteristic
25C4⋊C4×D71—21no—
26C4⋊C4×D71C2211yescharacteristic
27C2×(C2×D71)⋊C4—21no—
28C22⋊C4×D71—21no—
29C2×C4×Dic71—21no—
30#14556061C2212yescomplemented
31#14556060C2212yescomplemented
32#14556059C2212yescomplemented
33#14556053C2212yescomplemented
34#14556052C2212yescomplemented
35#14556051C2212yescomplemented
36C22⋊(C71⋊C8)C2211yescharacteristic
37(C2×D71)⋊C8C2212yes—
38C22⋊Q8×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#107×C71C2212yescomplemented
41#103×C71C2212yescomplemented
42C22⋊C8×C71C2211yescharacteristic
43C23×D71D411yescharacteristic
44C2×Dic142—21no—
45C2×Dic142C2311yescharacteristic
46Q8×D71—24no—
47C22×Dic71—21no—
48C22×Dic71D411yescharacteristic
49C2×C4×D71—24no—
50C2×C4×D71—21no—
51C2×C4×D71—21no—
52C2×C4×D71D411yescharacteristic
53C2×C4×D71D411yescharacteristic
54C2×C4×D71C2311yescharacteristic
55C2×C4×D71C2311yescharacteristic
56#11765848—24no—
57C2×C71⋊C8C2312yes—
58C4⋊Dic71—21no—
59C4⋊Dic71—21no—
60C4⋊Dic71C2312yes—
61C4⋊Dic71C2311yescharacteristic
62Dic71⋊C4—21no—
63Dic71⋊C4—21no—
64Dic71⋊C4—21no—
65C22⋊Dic71—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—21no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—22no—
70C4×Dic71—22no—
71C4×Dic71—22no—
72C2×Q8×C71C2311yescharacteristic
73C22×C4×C71—21no—
74C22×C4×C71C2311yescharacteristic
75C2×C8×C71C2312yes—
76C4⋊C4×C71—21no—
77C4⋊C4×C71—21no—
78C4⋊C4×C71C2312yes—
79C4⋊C4×C71C2311yescharacteristic
80C22⋊C4×C71—21no—
81C23×C71C2×D411yescharacteristic
82C22×D71—22no—
83C22×D71—22no—
84C22×D71—41no—
85C22×D71C2×D411yescharacteristic
86C22×D71C2×D411yescharacteristic
87Q8×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—22no—
92C2×C4×C71—42no—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2411yescharacteristic
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—42no—
100C2×Dic71C2×D411yescharacteristic
101C2×Dic71C2×D411yescharacteristic
102C4×D71—22no—
103C4×D71—22no—
104C4×D71—22no—
105C4×D71—42no—
106C4×D71—42no—
107C4×D71#4114yes—
108Dic142—22no—
109Dic142—42no—
110C8×C71—22no—
111C71⋊C8—22no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71#13712yes—
115C22×C71C22×D411yescharacteristic
116C2×D71—21no—
117C2×D71—21no—
118C2×D71—21no—
119C2×D71—21no—
120C2×D71—21no—
121C2×D71—44no—
122C2×D71#12312yes—
123C4×C71—21no—
124C4×C71—41no—
125C4×C71—41no—
126C4×C71—42no—
127C4×C71C2×#4112yes—
128Dic71—21no—
129Dic71—41no—
130Dic71—42no—
131Dic71—41no—
132Dic71#12312yes—
133C2×C71—22no—
134C2×C71C2×#12311yescharacteristic
135C2×C71C2×#13711yescharacteristic
136C2×C71#57511yescharacteristic
137D71—22no—
138D71—41no—
139#3095—711nomaximal, Sylow
140C71#309511yescharacteristic, complemented, Sylow
141C2×C22⋊Q8—711no—
142C2×#117—711no—
143#485—712no—
144#485—712no—
145#484—711no—
146#484D7111yescharacteristic, complemented
147#483—711no—
148#483—711no—
149#428—712no—
150#419—712no—
151#407—711no—
152C22×Q8—711no—
153C23×C4—711no—
154C2×M16—712no—
155C22⋊Q8—711no—
156C22⋊Q8—711no—
157C22⋊Q8—711no—
158C22⋊Q8—1421no—
159C22⋊Q8—1421no—
160C22⋊Q8C2×D7111yescharacteristic
161#117—712no—
162#117—712no—
163#117—1421no—
164#117—1421no—
165C2×C4⋊C4—711no—
166C2×C4⋊C4—711no—
167C2×C4⋊C4—1421no—
168C2×C4⋊C4C2×D7111yescharacteristic
169C2×C22⋊C4—1421no—
170C2×C22⋊C4—1421no—
171C2×C42—1421no—
172#107—712no—
173#107C2×D7112yescomplemented
174#106—712no—
175#106—712no—
176#103—712no—
177#103—712no—
178#103—712no—
179#103C2×D7112yescomplemented
180C22⋊C8—711no—
181C22⋊C8—712no—
182C22⋊C8C2×D7111yescharacteristic
183C24—711no—
184C2×Q8—711no—
185C2×Q8—1421no—
186C2×Q8—1424no—
187C2×Q8C22×D7111yescharacteristic
188C22×C4—21no—
189C22×C4—711no—
190C22×C4—711no—
191C22×C4—711no—
192C22×C4—711no—
193C22×C4—711no—
194C22×C4—1421no—
195C22×C4—1421no—
196C22×C4—1424no—
197C22×C4—1421no—
198C22×C4C22×D7111yescharacteristic
199M16—1424no—
200C2×C8—712no—
201C2×C8C22×D7112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—712no—
205C4⋊C4—711no—
206C4⋊C4—1421no—
207C4⋊C4—1421no—
208C4⋊C4—1421no—
209C4⋊C4—1421no—
210C4⋊C4—1421no—
211C4⋊C4C22×D7112yes—
212C4⋊C4C22×D7111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—1422no—
215C22⋊C4—1422no—
216C22⋊C4—1421no—
217C22⋊C4—1421no—
218C22⋊C4—1421no—
219C42—1422no—
220C42—1422no—
221C23—2841no—
222C23—711no—
223C23—711no—
224C23—1422no—
225C23—1422no—
226C23D4×D7111yescharacteristic
227Q8—22no—
228Q8—2842no—
229Q8—1422no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2842no—
236C2×C4—2842no—
237C2×C4—2842no—
238C2×C4—711no—
239C2×C4—714no—
240C2×C4—711no—
241C2×C4—1422no—
242C2×C4—1422no—
243C2×C4—1422no—
244C2×C4—1421no—
245C2×C4—1421no—
246C2×C4—1422no—
247C2×C4—1422no—
248C2×C4D4×D7111yescharacteristic
249C2×C4C23×D7111yescharacteristic
250C8—22no—
251C8—1422no—
252C22—21no—
253C22—21no—
254C22—2844no—
255C22—712no—
256C22—1421no—
257C22—1421no—
258C22—1421no—
259C22—1421no—
260C22—1421no—
261C22#1455616012yes—
262C22C2×D4×D7111yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—2841no—
268C4—2842no—
269C4—2841no—
270C4—712no—
271C4—1421no—
272C4#41×D7112yes—
273C2—22no—
274C2—2841no—
275C2—1422no—
276C2#123×D7111yescharacteristic
277C2C2×#1455616011yescharacteristic
278C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#484×C71C2
socleC22×C71C22×D4
hypercenter#484D71
hypocenterC71#3095
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #17986342C2semidirect
4. #17986338C2semidirect
5. #17986334C2semidirect
6. #17986330C2semidirect
7. #17986327C2semidirect
8. #17986040C2semidirect
9. #17986006C2semidirect
11. #484×C71C2semidirect
30. #14556061C22semidirect
31. #14556060C22semidirect
32. #14556059C22semidirect
33. #14556053C22semidirect
34. #14556052C22semidirect
35. #14556051C22semidirect
40. #107×C71C22semidirect
41. #103×C71C22semidirect
140. C71#3095semidirect
146. #484D71semidirect
173. #107C2×D71semidirect
179. #103C2×D71semidirect