Finite Group Databasebeta

← #22597971

Subgroups — #22597971

Order 9088.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14556161C211yesmaximal, characteristic, complemented
2C2×M16×D71C211yesmaximal, characteristic, complemented
3#17986511C211yesmaximal, characteristic
4C2×#14556108C211yesmaximal, characteristic, complemented
5#17986258C212yesmaximal, complemented
6#17986246C212yesmaximal, complemented
7#17986241C212yesmaximal, complemented
8#17986229C212yesmaximal, complemented
9#17986007C211yesmaximal, characteristic, complemented
10#17985998C211yesmaximal, characteristic, complemented
11#428×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15#14556161—24no—
16C2×Dic284C2212yescomplemented
17C2×#11765854C2212yescomplemented
18C2×#11765849C2211yescharacteristic
19C2×#11765848C2212yes—
20M16×D71—24no—
21C2×C8×D71C2212yes—
22#14556112—21no—
23#14556108—21no—
24#14556108—21no—
25#14556108C2212yes—
26#14556108C2211yescharacteristic
27#14556108C2212yes—
28#14556108C2211yescharacteristic
29#14556103C2211yescharacteristic
30#14556103C2211yescharacteristic
31#14556098—21no—
32C4×D284—21no—
33C4×D284—21no—
34C2×C4⋊Dic71—21no—
35C2×(C2×D71)⋊C4—21no—
36#14556051C2212yescomplemented
37#14556051C2212yescomplemented
38#14556051C2212yescomplemented
39#14556048C2212yescomplemented
40C2×M16×C71C2211yescharacteristic
41#117×C71C2211yescharacteristic
42C2×C4⋊C4×C71C2211yescharacteristic
43#107×C71C2212yescomplemented
44#106×C71C2212yescomplemented
45C23×D71D411yescharacteristic
46#11765865—22no—
47#11765865—22no—
48C2×Dic142—22no—
49C2×Dic142—22no—
50C2×Dic142—21no—
51C2×Dic142C2312yes—
52C2×Dic142C2311yescharacteristic
53C2×D284—21no—
54C2×D284—21no—
55C2×D284C2311yescharacteristic
56C2×C22⋊D71—21no—
57C22×Dic71—21no—
58C22×Dic71D411yescharacteristic
59C2×C4×D71—22no—
60C2×C4×D71—22no—
61C2×C4×D71—21no—
62C2×C4×D71D411yescharacteristic
63C2×C4×D71D411yescharacteristic
64C2×C4×D71C2311yescharacteristic
65C2×C4×D71C2311yescharacteristic
66Dic284—24no—
67#11765854—24no—
68#11765849—22no—
69#11765848—24no—
70C2×C71⋊C8C2312yes—
71C8×D71—24no—
72C4⋊Dic71—22no—
73C4⋊Dic71—22no—
74C4⋊Dic71—21no—
75C4⋊Dic71—21no—
76(C2×D71)⋊C4—22no—
77(C2×D71)⋊C4—22no—
78(C2×D71)⋊C4—21no—
79(C2×D71)⋊C4—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82C22×C4×C71—21no—
83C22×C4×C71C2311yescharacteristic
84M16×C71D414yescomplemented
85C2×C8×C71C2312yes—
86C4⋊C4×C71—21no—
87C4⋊C4×C71C2311yescharacteristic
88C4⋊C4×C71C2312yes—
89C4⋊C4×C71C2311yescharacteristic
90C22⋊C4×C71—21no—
91C42×C71—21no—
92C23×C71C2×D411yescharacteristic
93C22×D71—22no—
94C22×D71—22no—
95C22×D71—21no—
96C22×D71—41no—
97C22×D71C2×D411yescharacteristic
98C22×D71C2×D411yescharacteristic
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—22no—
102C2×C4×C71—42no—
103C2×C4×C71C2×D411yescharacteristic
104C2×C4×C71C2×D412yes—
105C2×C4×C71#4112yes—
106C2×C4×C71C2411yescharacteristic
107C22⋊D71—42no—
108C2×Dic71—22no—
109C2×Dic71—21no—
110C2×Dic71—22no—
111C2×Dic71—42no—
112C2×Dic71C2×D411yescharacteristic
113C2×Dic71C2×D411yescharacteristic
114D284—21no—
115D284—21no—
116D284—42no—
117D284—42no—
118C4×D71—22no—
119C4×D71—22no—
120C4×D71—21no—
121C4×D71—21no—
122C4×D71—21no—
123C4×D71—21no—
124C4×D71—42no—
125Dic142—22no—
126Dic142—21no—
127Dic142—22no—
128Dic142—21no—
129Dic142—42no—
130C8×C71C2×D414yes—
131C71⋊C8—22no—
132C22×C71—21no—
133C22×C71—21no—
134C22×C71#12112yes—
135C22×C71C22×D411yescharacteristic
136C2×D71—21no—
137C2×D71—21no—
138C2×D71—21no—
139C2×D71—21no—
140C2×D71—21no—
141C2×D71—44no—
142C2×D71—41no—
143C2×D71—41no—
144C2×D71—41no—
145C2×D71#13712yescomplemented
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—42no—
149C4×C71#12112yes—
150C4×C71C22×D411yescharacteristic
151C4×C71C2×#4111yescharacteristic
152Dic71—21no—
153Dic71—21no—
154Dic71—42no—
155Dic71—41no—
156C2×C71—22no—
157C2×C71C2×#12111yescharacteristic
158C2×C71C2×#13711yescharacteristic
159C2×C71#57511yescharacteristic
160D71—22no—
161D71—41no—
162D71—81no—
163#3056—711nomaximal, Sylow
164C71#305611yescharacteristic, complemented, Sylow
165C2×#137—711no—
166C22×M16—711no—
167#525—711no—
168C2×C22⋊Q8—711no—
169#470—712no—
170#468—712no—
171#467—712no—
172#465—712no—
173#428D7111yescharacteristic, complemented
174#419—711no—
175#417—711no—
176C2×#41—711no—
177C22×Q8—711no—
178C23×C4—711no—
179#137—1424no—
180C2×Q16—712no—
181C2×SD16—712no—
182C2×M16—711no—
183C2×M16—712no—
184C2×M16—1424no—
185C2×M16C2×D7111yescharacteristic
186C22×C8—712no—
187#123—1421no—
188C22⋊Q8—712no—
189C22⋊Q8—712no—
190C22⋊Q8—711no—
191C22⋊Q8—711no—
192C22⋊Q8—1421no—
193C22⋊Q8—1421no—
194#121—711no—
195#121—711no—
196#120—1421no—
197C4×D4—1421no—
198C4×D4—1421no—
199#117C2×D7111yescharacteristic
200C2×C4⋊C4—1421no—
201C2×C4⋊C4C2×D7111yescharacteristic
202C2×C22⋊C4—1421no—
203#107C2×D7112yescomplemented
204#106C2×D7112yescomplemented
205#103—712no—
206#103—712no—
207#103—712no—
208#102—712no—
209C24—711no—
210#41—1422no—
211#41—1422no—
212C2×Q8—712no—
213C2×Q8—711no—
214C2×Q8—1422no—
215C2×Q8—1422no—
216C2×Q8—1421no—
217C2×D4—711no—
218C2×D4—1421no—
219C2×D4—1421no—
220C2×D4—1421no—
221C22×C4—21no—
222C22×C4—711no—
223C22×C4—711no—
224C22×C4—711no—
225C22×C4—711no—
226C22×C4—711no—
227C22×C4—1421no—
228C22×C4—1421no—
229C22×C4—1422no—
230C22×C4—1422no—
231C22×C4C22×D7111yescharacteristic
232Q16—1424no—
233SD16—1424no—
234M16—1422no—
235M16—1424no—
236M16D28414yescomplemented
237C2×C8—712no—
238C2×C8—1424no—
239C2×C8C22×D7112yes—
240C4⋊C4—21no—
241C4⋊C4—1422no—
242C4⋊C4—1422no—
243C4⋊C4—1421no—
244C4⋊C4—1421no—
245C4⋊C4C22×D7112yes—
246C4⋊C4C22×D7111yescharacteristic
247C4⋊C4C22×D7111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—1421no—
250C22⋊C4—1421no—
251C22⋊C4—1422no—
252C22⋊C4—1422no—
253C22⋊C4—1421no—
254C22⋊C4—1421no—
255C42—21no—
256C23—2841no—
257C23—711no—
258C23—711no—
259C23—1422no—
260C23—1421no—
261C23—1422no—
262C23D4×D7111yescharacteristic
263Q8—2842no—
264Q8—1422no—
265Q8—1421no—
266Q8—1422no—
267Q8—1421no—
268D4—2842no—
269D4—2842no—
270D4—2842no—
271D4—1421no—
272D4—1421no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—2842no—
278C2×C4—2842no—
279C2×C4—711no—
280C2×C4—711no—
281C2×C4—1421no—
282C2×C4—1422no—
283C2×C4—1422no—
284C2×C4—1421no—
285C2×C4—1421no—
286C2×C4—1421no—
287C2×C4—1421no—
288C2×C4—1422no—
289C2×C4—1422no—
290C2×C4D4×D7111yescharacteristic
291C2×C4C2×D28412yes—
292C2×C4#1176586612yes—
293C2×C4C23×D7111yescharacteristic
294C8—1422no—
295C8C2×D28414yes—
296C22—21no—
297C22—21no—
298C22—2844no—
299C22—2841no—
300C22—2841no—
301C22—2841no—
302C22—712no—
303C22—1421no—
304C22—1421no—
305C22—1421no—
306C22—1421no—
307C22—1421no—
308C22#1455610312yes—
309C22C2×D4×D7111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—2841no—
314C4—2842no—
315C4—1421no—
316C4—1421no—
317C4#1455610312yes—
318C4C22×D28411yescharacteristic
319C4C2×#1176586611yescharacteristic
320C2—22no—
321C2—2841no—
322C2—5681no—
323C2—1422no—
324C2C2×#1455610311yescharacteristic
325C2#137×D7111yescharacteristic
326C2#1798679511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3056
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
1. C2×#14556161C2semidirect
2. C2×M16×D71C2semidirect
4. C2×#14556108C2semidirect
5. #17986258C2semidirect
6. #17986246C2semidirect
7. #17986241C2semidirect
8. #17986229C2semidirect
9. #17986007C2semidirect
10. #17985998C2semidirect
11. #428×C71C2semidirect
16. C2×Dic284C22semidirect
17. C2×#11765854C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
38. #14556051C22semidirect
39. #14556048C22semidirect
43. #107×C71C22semidirect
44. #106×C71C22semidirect
84. M16×C71D4semidirect
145. C2×D71#137semidirect
164. C71#3056semidirect
173. #428D71semidirect
203. #107C2×D71semidirect
204. #106C2×D71semidirect
236. M16D284semidirect