Finite Group Databasebeta

← #22597974

Subgroups — #22597974

Order 9088.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D71C211yesmaximal, characteristic, complemented
2C2×#14556134C211yesmaximal, characteristic, complemented
3#17986572C211yesmaximal, characteristic, complemented
4#17986565C211yesmaximal, characteristic
5#17986248C211yesmaximal, characteristic, complemented
6#17986247C211yesmaximal, characteristic, complemented
7#17986244C211yesmaximal, characteristic, complemented
8#17986243C211yesmaximal, characteristic, complemented
9#17986229C211yesmaximal, characteristic, complemented
10#17986228C211yesmaximal, characteristic, complemented
11#17986225C211yesmaximal, characteristic, complemented
12#17986034C211yesmaximal, characteristic, complemented
13#17986006C211yesmaximal, characteristic, complemented
14#17986002C211yesmaximal, characteristic, complemented
15#465×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C2×Q8×D71C2211yescharacteristic
18C2×D4×D71C2211yescharacteristic
19C2×#11765854C2211yescharacteristic
20C2×#11765853C2211yescharacteristic
21C2×#11765852C2211yescharacteristic
22SD16×D71—24no—
23#14556134—24no—
24C2×#11765849C2211yescharacteristic, complemented
25C2×#11765848C2211yescharacteristic
26C2×#11765848C2211yescharacteristic, complemented
27C2×C8×D71C2211yescharacteristic
28C2×C8×D71C2211yescharacteristic, complemented
29#14556127—21no—
30#14556124—21no—
31#14556114—21no—
32#14556114—21no—
33#14556113—21no—
34#14556110—21no—
35#14556108C2211yescharacteristic, complemented
36#14556107C2211yescharacteristic, complemented
37#14556104C2211yescharacteristic, complemented
38#14556103C2211yescharacteristic
39#14556102C2211yescharacteristic
40#14556099C2211yescharacteristic
41#14556098—21no—
42#14556096—21no—
43#14556086C2211yescharacteristic
44#14556086C2211yescharacteristic
45#14556061C2211yescharacteristic, complemented
46#14556061C2211yescharacteristic, complemented
47#14556060C2211yescharacteristic, complemented
48#14556053C2211yescharacteristic, complemented
49#14556052C2211yescharacteristic, complemented
50#14556051C2211yescharacteristic, complemented
51#14556050C2211yescharacteristic
52#14556049C2211yescharacteristic
53#14556048C2211yescharacteristic
54C2×SD16×C71C2211yescharacteristic
55C22×C8×C71C2211yescharacteristic, complemented
56C22⋊Q8×C71C2211yescharacteristic, complemented
57#121×C71C2211yescharacteristic
58#106×C71C2211yescharacteristic, complemented
59#103×C71C2211yescharacteristic, complemented
60#102×C71C2211yescharacteristic
61C23×D71—21no—
62#11765865—22no—
63#11765865—22no—
64C2×Dic142—21no—
65C2×Dic142D411yescharacteristic
66C2×Dic142C2311yescharacteristic
67Q8×D71—22no—
68Q8×D71—22no—
69C2×D284—21no—
70C2×D284D411yescharacteristic
71C2×D284C2311yescharacteristic
72C2×C22⋊D71—21no—
73C2×C22⋊D71—21no—
74C2×C22⋊D71—21no—
75C2×C22⋊D71D411yescharacteristic
76C2×C22⋊D71D411yescharacteristic
77D4×D71—22no—
78D4×D71—22no—
79C2×C4×D71—21no—
80C2×C4×D71C2311yescharacteristic
81C2×C4×D71C2311yescharacteristic
82#11765854—22no—
83#11765853—22no—
84#11765852—22no—
85#11765849—22no—
86#11765848—22no—
87#11765848—22no—
88C2×C71⋊C8C2311yescharacteristic
89C2×C71⋊C8C2311yescharacteristic
90C8×D71—22no—
91C8×D71D414yescomplemented
92C4⋊Dic71—21no—
93C4⋊Dic71C2311yescharacteristic
94C4⋊Dic71C2311yescharacteristic
95Dic71⋊C4—21no—
96Dic71⋊C4—21no—
97C22⋊Dic71—21no—
98C22⋊Dic71—21no—
99(C2×D71)⋊C4—21no—
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4—21no—
102(C2×D71)⋊C4—21no—
103(C2×D71)⋊C4—21no—
104(C2×D71)⋊C4—21no—
105C4×Dic71—21no—
106C4×Dic71—21no—
107C2×Q8×C71C2311yescharacteristic
108C2×D4×C71—21no—
109C2×D4×C71C2311yescharacteristic
110C22×C4×C71C2311yescharacteristic
111SD16×C71—22no—
112C2×C8×C71—22no—
113C2×C8×C71C2311yescharacteristic
114C2×C8×C71C2311yescharacteristic
115C4⋊C4×C71—21no—
116C4⋊C4×C71C2311yescharacteristic
117C4⋊C4×C71C2311yescharacteristic
118C22⋊C4×C71—21no—
119C22⋊C4×C71—21no—
120C23×C71—21no—
121C23×C71C2×D411yescharacteristic
122C22×D71—21no—
123C22×D71—42no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71C2×D411yescharacteristic
127C22×D71C2×D411yescharacteristic
128Q8×C71—21no—
129Q8×C71—21no—
130D4×C71—21no—
131D4×C71—21no—
132D4×C71—81no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2411yescharacteristic
140C22⋊D71—22no—
141C22⋊D71—22no—
142C22⋊D71—42no—
143C22⋊D71—81no—
144C22⋊D71—81no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71C2×D411yescharacteristic
149C2×Dic71C2×D411yescharacteristic
150D284—22no—
151D284—21no—
152D284—21no—
153D284—81no—
154C4×D71—21no—
155C4×D71—21no—
156C4×D71—42no—
157C4×D71C2×D412yes—
158C4×D71#4112yes—
159Dic142—22no—
160Dic142—21no—
161Dic142—21no—
162Dic142—42no—
163C8×C71—21no—
164C8×C71C2×D412yes—
165C71⋊C8—21no—
166C71⋊C8C2×D412yes—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—41no—
171C22×C71—41no—
172C22×C71—41no—
173C22×C71C22×D411yescharacteristic
174C2×D71—21no—
175C2×D71—21no—
176C2×D71—21no—
177C2×D71—21no—
178C2×D71—21no—
179C2×D71—41no—
180C2×D71—41no—
181C2×D71—41no—
182C2×D71—84no—
183C2×D71#12112yes—
184C4×C71—21no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71C22×D411yescharacteristic
189C4×C71C2×#4111yescharacteristic
190Dic71—21no—
191Dic71—41no—
192Dic71—41no—
193Dic71—41no—
194Dic71#12112yes—
195C2×C71—41no—
196C2×C71—81no—
197C2×C71C2×#12111yescharacteristic
198C2×C71#57711yescharacteristic
199C2×C71#57711yescharacteristic
200D71—22no—
201D71—41no—
202D71—81no—
203#3057—711nomaximal, Sylow
204C71#305711yescharacteristic, complemented, Sylow
205C22×SD16—711no—
206C2×#131—711no—
207#536—711no—
208#535—711no—
209#468—711no—
210#468—711no—
211#468—711no—
212#468—711no—
213#465—711no—
214#465—711no—
215#465—711no—
216#465D7111yescharacteristic, complemented
217#427—711no—
218#419—711no—
219#418—711no—
220C2×#41—711no—
221C22×Q8—711no—
222C22×D4—711no—
223C2×SD16—711no—
224C2×SD16—711no—
225C2×SD16—711no—
226C2×SD16—1424no—
227C2×SD16C2×D7111yescharacteristic
228#131—1424no—
229C2×M16—711no—
230C2×M16—711no—
231C2×M16—711no—
232C22×C8—711no—
233C22×C8—711no—
234C22×C8C2×D7111yescharacteristic, complemented
235C4⋊Q8—1421no—
236#127—1421no—
237#124—1421no—
238#124—1421no—
239#123—1421no—
240#123—1421no—
241C22⋊Q8—711no—
242C22⋊Q8—711no—
243C22⋊Q8—711no—
244C22⋊Q8C2×D7111yescharacteristic, complemented
245#121—711no—
246#121—711no—
247#121—711no—
248#121C2×D7111yescharacteristic
249#120—1421no—
250#120—1421no—
251#117—711no—
252#117—711no—
253#107—711no—
254#107—711no—
255#106—711no—
256#106C2×D7111yescharacteristic, complemented
257#103—711no—
258#103—711no—
259#103—711no—
260#103C2×D7111yescharacteristic, complemented
261#102—711no—
262#102—711no—
263#102—711no—
264#102C2×D7111yescharacteristic
265C24—1421no—
266#41—1422no—
267#41—1422no—
268C2×Q8—711no—
269C2×Q8—711no—
270C2×Q8—1421no—
271C2×Q8—1422no—
272C2×Q8—1422no—
273C2×Q8C22×D7111yescharacteristic
274C2×D4—21no—
275C2×D4—711no—
276C2×D4—711no—
277C2×D4—711no—
278C2×D4—711no—
279C2×D4—1421no—
280C2×D4—1422no—
281C2×D4—1422no—
282C2×D4—1421no—
283C2×D4—1421no—
284C2×D4—1421no—
285C2×D4C22×D7111yescharacteristic
286C22×C4—711no—
287C22×C4—711no—
288C22×C4—1421no—
289C22×C4C22×D7111yescharacteristic
290SD16—22no—
291SD16—1422no—
292SD16—1422no—
293SD16—1422no—
294M16—1422no—
295M16—1422no—
296M16—1422no—
297C2×C8—22no—
298C2×C8—711no—
299C2×C8—714no—
300C2×C8—711no—
301C2×C8—1422no—
302C2×C8C22×D7111yescharacteristic
303C2×C8C22×D7111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—711no—
306C4⋊C4—711no—
307C4⋊C4—1421no—
308C4⋊C4—1421no—
309C4⋊C4—1421no—
310C4⋊C4C22×D7111yescharacteristic
311C4⋊C4C22×D7111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—1421no—
315C22⋊C4—1421no—
316C22⋊C4—1421no—
317C22⋊C4—1421no—
318C22⋊C4—1421no—
319C22⋊C4—1421no—
320C22⋊C4—1421no—
321C22⋊C4—1421no—
322C42—1421no—
323C42—1421no—
324C23—21no—
325C23—2842no—
326C23—5681no—
327C23—5681no—
328C23—711no—
329C23—711no—
330C23—1421no—
331C23D4×D7111yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—2842no—
335Q8—1422no—
336Q8—1421no—
337Q8—1421no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—2842no—
342D4—5681no—
343D4—5681no—
344D4—5681no—
345D4—1422no—
346D4—1422no—
347D4—1422no—
348D4—1421no—
349D4—1421no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2842no—
356C2×C4—711no—
357C2×C4—712no—
358C2×C4—712no—
359C2×C4—711no—
360C2×C4—1421no—
361C2×C4—1421no—
362C2×C4—1421no—
363C2×C4—1421no—
364C2×C4—1421no—
365C2×C4D4×D7111yescharacteristic
366C2×C4C23×D7111yescharacteristic
367C8—21no—
368C8—712no—
369C8—1421no—
370C8D4×D7112yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—2841no—
378C22—2841no—
379C22—2841no—
380C22—5684no—
381C22—712no—
382C22—1421no—
383C22—1421no—
384C22—1421no—
385C22—1421no—
386C22—1421no—
387C22C2×D4×D7111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—2841no—
393C4—2841no—
394C4—2841no—
395C4—712no—
396C4—1421no—
397C4C2×D4×D7111yescharacteristic
398C4#41×D7111yescharacteristic
399C2—41no—
400C2—81no—
401C2—2841no—
402C2—5681no—
403C2—1422no—
404C2#121×D7111yescharacteristic
405C2#1798680511yescharacteristic
406C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#465×C71C2
socleC22×C71C22×D4
hypercenter#465D71
hypocenterC71#3057
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D71C2semidirect
2. C2×#14556134C2semidirect
3. #17986572C2semidirect
5. #17986248C2semidirect
6. #17986247C2semidirect
7. #17986244C2semidirect
8. #17986243C2semidirect
9. #17986229C2semidirect
10. #17986228C2semidirect
11. #17986225C2semidirect
12. #17986034C2semidirect
13. #17986006C2semidirect
14. #17986002C2semidirect
15. #465×C71C2semidirect
24. C2×#11765849C22semidirect
26. C2×#11765848C22semidirect
28. C2×C8×D71C22semidirect
35. #14556108C22semidirect
36. #14556107C22semidirect
37. #14556104C22semidirect
45. #14556061C22semidirect
46. #14556061C22semidirect
47. #14556060C22semidirect
48. #14556053C22semidirect
49. #14556052C22semidirect
50. #14556051C22semidirect
55. C22×C8×C71C22semidirect
56. C22⋊Q8×C71C22semidirect
58. #106×C71C22semidirect
59. #103×C71C22semidirect
91. C8×D71D4semidirect
204. C71#3057semidirect
216. #465D71semidirect
234. C22×C8C2×D71semidirect
244. C22⋊Q8C2×D71semidirect
256. #106C2×D71semidirect
260. #103C2×D71semidirect