Finite Group Databasebeta

← #2483×D71 #32166372

Subgroups — #2483×D71 #32166372

Order 18176.

4382 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C4×D71C212yesmaximal
2#22592621C211yesmaximal, characteristic, complemented
3#22592620C214yesmaximal, complemented
4#22592619C214yesmaximal, complemented
5#22592618C212yesmaximal, complemented
6#22592617C214yesmaximal, complemented
7#397×D71C214yesmaximal, complemented
8#385×D71C214yesmaximal, complemented
9#381×D71C214yesmaximal, complemented
10#379×D71C211yesmaximal, characteristic
11C71×#2483C211yesmaximal, characteristic, complemented
12C25×D71C2211yescharacteristic
13C23×C4×D71C2212yes—
14C23×C4×D71C2212yes—
15C2×C4⋊C4×D71C2214yescomplemented
16C22×C22⋊Dic71C2212yes—
17C22×(C2×D71)⋊C4C2212yes—
18C22×(C2×D71)⋊C4C2212yes—
19C2×C22⋊C4×D71—216no—
20C2×C22⋊C4×D71C2214yes—
21C2×C22⋊C4×D71C2214yes—
22C2×C22⋊C4×D71C2214yes—
23C2×C22⋊C4×D71C2214yes—
24C2×C22⋊C4×D71C2214yes—
25C2×C22⋊C4×D71C2212yes—
26#17985928C2218yescomplemented
27#17985927C2218yescomplemented
28#17985926C2214yescomplemented
29#17985925C2214yescomplemented
30#17985924C2214yescomplemented
31#17985874C2214yescomplemented
32#17985873C2218yescomplemented
33#17985872C2214yescomplemented
34#17985871C2214yescomplemented
35#17985870C2218yescomplemented
36#17985862C2214yescomplemented
37#17985861C2218yescomplemented
38#17985861C2218yescomplemented
39#17985860C2214yescomplemented
40#17985860C2214yescomplemented
41#17985855C2211yescharacteristic
42#17985854C2212yes—
43#17985854C2214yes—
44#95×D71C2218yescomplemented
45#95×D71C2214yescomplemented
46C22×C22⋊C4×C71C2212yes—
47#397×C71C2214yescomplemented
48#385×C71C2214yescomplemented
49#381×C71C2214yescomplemented
50#379×C71C2211yescharacteristic
51C24×D71—216no—
52C24×D71—44no—
53C24×D71D418yes—
54C24×D71C2311yescharacteristic
55C24×D71C2311yescharacteristic
56C24×D71C2312yes—
57C24×D71C2312yes—
58C23×Dic71C2312yes—
59C23×Dic71C2312yes—
60C22×C4×D71—28no—
61C22×C4×D71—28no—
62C22×C4×D71—216no—
63C22×C4×D71—48no—
64C22×C4×D71C2314yes—
65C22×C4×D71C2314yes—
66C22×C4×D71C2314yes—
67C22×C4×D71C2314yes—
68C22×C4×D71C2314yes—
69C2×C4⋊Dic71C2314yescomplemented
70C2×Dic71⋊C4C2318yescomplemented
71C4⋊C4×D71—48no—
72C2×C22⋊Dic71—28no—
73C2×C22⋊Dic71C2314yes—
74C2×C22⋊Dic71C2314yes—
75C2×C22⋊Dic71C2314yes—
76C2×C22⋊Dic71C2312yes—
77C2×(C2×D71)⋊C4—28no—
78C2×(C2×D71)⋊C4—28no—
79C2×(C2×D71)⋊C4C2318yes—
80C2×(C2×D71)⋊C4C2314yes—
81C2×(C2×D71)⋊C4C2314yes—
82C2×(C2×D71)⋊C4C2314yes—
83C2×(C2×D71)⋊C4C2314yes—
84C2×(C2×D71)⋊C4C2314yes—
85C22⋊C4×D71—216no—
86C22⋊C4×D71—216no—
87C22⋊C4×D71—216no—
88C22⋊C4×D71—216no—
89C22⋊C4×D71—216no—
90C22⋊C4×D71—44no—
91C71⋊#95C2314yescomplemented
92C71⋊#95C2318yescomplemented
93C71⋊#95C2318yescomplemented
94C71⋊#95C2318yescomplemented
95C71⋊#95C2318yescomplemented
96C25×C71C2311yescharacteristic
97C23×C4×C71C2312yes—
98C23×C4×C71C2312yes—
99C2×C4⋊C4×C71C2314yescomplemented
100C2×C22⋊C4×C71—28no—
101C2×C22⋊C4×C71C2314yes—
102C2×C22⋊C4×C71C2314yes—
103C2×C22⋊C4×C71C2314yes—
104C2×C22⋊C4×C71C2312yes—
105#95×C71C2314yescomplemented
106#95×C71C2318yescomplemented
107C23×D71—28no—
108C23×D71—28no—
109C23×D71—28no—
110C23×D71—264no—
111C23×D71—28no—
112C23×D71—28no—
113C23×D71—28no—
114C23×D71—24no—
115C23×D71—416no—
116C23×D71—42no—
117C23×D71—44no—
118C23×D71—464no—
119C23×D71C2×D414yes—
120C23×D71C2×D414yes—
121C23×D71C2×D418yes—
122C23×D71C2×D418yes—
123C23×D71#41116yes—
124C23×D71C2411yescharacteristic
125C23×D71C2412yes—
126C23×D71C2411yescharacteristic
127C22×Dic71—24no—
128C22×Dic71—24no—
129C22×Dic71—28no—
130C22×Dic71—44no—
131C22×Dic71C2414yes—
132C22×Dic71C2414yes—
133C22×Dic71C2414yes—
134C2×C4×D71—28no—
135C2×C4×D71—28no—
136C2×C4×D71—28no—
137C2×C4×D71—28no—
138C2×C4×D71—216no—
139C2×C4×D71—216no—
140C2×C4×D71—216no—
141C2×C4×D71—216no—
142C2×C4×D71—28no—
143C2×C4×D71—48no—
144C2×C4×D71—432no—
145C2×C4×D71—432no—
146C4⋊Dic71—44no—
147Dic71⋊C4—48no—
148C22⋊Dic71—28no—
149C22⋊Dic71—28no—
150C22⋊Dic71—28no—
151C22⋊Dic71—42no—
152(C2×D71)⋊C4—28no—
153(C2×D71)⋊C4—28no—
154(C2×D71)⋊C4—28no—
155(C2×D71)⋊C4—28no—
156(C2×D71)⋊C4—216no—
157(C2×D71)⋊C4—44no—
158C24×C71—28no—
159C24×C71—42no—
160C24×C71C2×D414yes—
161C24×C71C2411yescharacteristic
162C24×C71C2412yes—
163C22×C4×C71—24no—
164C22×C4×C71—24no—
165C22×C4×C71—28no—
166C22×C4×C71—44no—
167C22×C4×C71C2414yes—
168C22×C4×C71C2414yes—
169C22×C4×C71C2414yes—
170C4⋊C4×C71—44no—
171C22⋊C4×C71—28no—
172C22⋊C4×C71—28no—
173C22⋊C4×C71—28no—
174C22⋊C4×C71—42no—
175C23×C71—24no—
176C23×C71—24no—
177C23×C71—216no—
178C23×C71—24no—
179C23×C71—22no—
180C23×C71—44no—
181C23×C71—416no—
182C23×C71C22×D412yes—
183C23×C71C22×D414yes—
184C23×C71C2×#4118yes—
185C23×C71C2511yescharacteristic
186C22×D71—24no—
187C22×D71—24no—
188C22×D71—216no—
189C22×D71—216no—
190C22×D71—264no—
191C22×D71—216no—
192C22×D71—24no—
193C22×D71—22no—
194C22×D71—216no—
195C22×D71—216no—
196C22×D71—216no—
197C22×D71—28no—
198C22×D71—44no—
199C22×D71—48no—
200C22×D71—416no—
201C22×D71—4128no—
202C22×D71—432no—
203C22×D71—416no—
204C22×D71#123132yes—
205C22×D71C22×D412yes—
206C22×D71C22×D414yes—
207C22×D71C22×D412yes—
208C22×D71C2×#4118yes—
209C22×D71C2×#4118yes—
210C22×D71H(5,𝔽2)14yes—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—28no—
214C2×C4×C71—28no—
215C2×C4×C71—24no—
216C2×C4×C71—28no—
217C2×C4×C71—48no—
218C2×C4×C71—48no—
219C2×Dic71—24no—
220C2×Dic71—24no—
221C2×Dic71—28no—
222C2×Dic71—28no—
223C2×Dic71—24no—
224C2×Dic71—28no—
225C2×Dic71—48no—
226C2×Dic71—48no—
227C4×D71—416no—
228C4×D71—416no—
229C4×D71—416no—
230C4×D71—416no—
231C4×D71—416no—
232C22×C71—24no—
233C22×C71—24no—
234C22×C71—216no—
235C22×C71—24no—
236C22×C71—22no—
237C22×C71—416no—
238C22×C71—44no—
239C22×C71C2×#12318yes—
240C22×C71C23×D411yescharacteristic
241C22×C71C22×#4114yes—
242C22×C71C2×H(5,𝔽2)12yes—
243C2×D71—24no—
244C2×D71—24no—
245C2×D71—216no—
246C2×D71—264no—
247C2×D71—24no—
248C2×D71—22no—
249C2×D71—416no—
250C2×D71—432no—
251C2×D71—44no—
252C2×D71—416no—
253C2×D71C2×#12318yes—
254C2×D71C2×#12318yes—
255C2×D71#53414yes—
256C2×D71#551116yes—
257C4×C71—44no—
258C4×C71—44no—
259C4×C71—44no—
260Dic71—44no—
261Dic71—44no—
262Dic71—44no—
263C2×#2483—711nomaximal, Sylow
264C2×C71—28no—
265C2×C71—42no—
266C2×C71C22×#12312yes—
267C2×C71C2×#53411yescharacteristic
268C2×C71C2×#55114yes—
269D71—28no—
270D71—42no—
271D71#248318yesdirect product factor
272C23×C22⋊C4—712no—
273#2483—714no—
274#2483—714no—
275#2483—714no—
276#2483—711no—
277#2483—712no—
278#2483D7111yescharacteristic, direct product factor
279C2×#397—714no—
280C2×#385—714no—
281C2×#381—714no—
282C2×#379—711no—
283C71C2×#248311yescharacteristic, complemented, Sylow
284C26—711no—
285C24×C4—712no—
286C24×C4—712no—
287C22×C4⋊C4—714no—
288C22×C22⋊C4—712no—
289C22×C22⋊C4—712no—
290C22×C22⋊C4—712no—
291C22×C22⋊C4—712no—
292C22×C22⋊C4—714no—
293C22×C22⋊C4—714no—
294C22×C22⋊C4—714no—
295C22×C22⋊C4—714no—
296C22×C22⋊C4—714no—
297C22×C22⋊C4—14216no—
298C22×C22⋊C4C2×D7112yes—
299#397—714no—
300#397—714no—
301#397—718no—
302#397—714no—
303#397—718no—
304#397C2×D7114yescomplemented
305#385—714no—
306#385—718no—
307#385—714no—
308#385—714no—
309#385—718no—
310#385C2×D7114yescomplemented
311#381—714no—
312#381—714no—
313#381—714no—
314#381—718no—
315#381—718no—
316#381C2×D7114yescomplemented
317#379—711no—
318#379—712no—
319#379—714no—
320#379C2×D7111yescharacteristic
321C2×#95—714no—
322C2×#95—718no—
323C25—2844no—
324C25—718no—
325C25—711no—
326C25—712no—
327C25—712no—
328C25—711no—
329C25—14216no—
330C25C22×D7111yescharacteristic
331C23×C4—2848no—
332C23×C4—712no—
333C23×C4—714no—
334C23×C4—712no—
335C23×C4—714no—
336C23×C4—714no—
337C23×C4—714no—
338C23×C4—714no—
339C23×C4—1428no—
340C23×C4—1428no—
341C23×C4—14216no—
342C23×C4C22×D7112yes—
343C23×C4C22×D7112yes—
344C2×C4⋊C4—2848no—
345C2×C4⋊C4—718no—
346C2×C4⋊C4—714no—
347C2×C4⋊C4C22×D7114yescomplemented
348C2×C22⋊C4—28no—
349C2×C22⋊C4—2844no—
350C2×C22⋊C4—712no—
351C2×C22⋊C4—714no—
352C2×C22⋊C4—714no—
353C2×C22⋊C4—714no—
354C2×C22⋊C4—714no—
355C2×C22⋊C4—714no—
356C2×C22⋊C4—714no—
357C2×C22⋊C4—714no—
358C2×C22⋊C4—714no—
359C2×C22⋊C4—718no—
360C2×C22⋊C4—14216no—
361C2×C22⋊C4—14216no—
362C2×C22⋊C4—14216no—
363C2×C22⋊C4—14216no—
364C2×C22⋊C4—14216no—
365C2×C22⋊C4—1428no—
366C2×C22⋊C4—1428no—
367C2×C22⋊C4—1428no—
368C2×C22⋊C4C22×D7114yes—
369C2×C22⋊C4C22×D7114yes—
370C2×C22⋊C4C22×D7114yes—
371C2×C22⋊C4C22×D7112yes—
372#95—714no—
373#95—718no—
374#95—718no—
375#95—718no—
376#95—718no—
377#95C22×D7114yescomplemented
378#95C22×D7118yescomplemented
379C24—28no—
380C24—42no—
381C24—2844no—
382C24—2842no—
383C24—28464no—
384C24—28416no—
385C24—714no—
386C24—718no—
387C24—711no—
388C24—718no—
389C24—7116no—
390C24—712no—
391C24—714no—
392C24—711no—
393C24—1424no—
394C24—1428no—
395C24—1428no—
396C24—1428no—
397C24—14264no—
398C24—1428no—
399C24—1428no—
400C24—1428no—
401C24D4×D7114yes—
402C24C23×D7111yescharacteristic
403C24C23×D7112yes—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—44no—
408C22×C4—2848no—
409C22×C4—28432no—
410C22×C4—28432no—
411C22×C4—2844no—
412C22×C4—714no—
413C22×C4—714no—
414C22×C4—714no—
415C22×C4—1428no—
416C22×C4—1428no—
417C22×C4—1428no—
418C22×C4—1428no—
419C22×C4—1428no—
420C22×C4—14216no—
421C22×C4—14216no—
422C22×C4—14216no—
423C22×C4—14216no—
424C22×C4—1424no—
425C22×C4—1424no—
426C22×C4—1428no—
427C22×C4C23×D7114yes—
428C22×C4C23×D7114yes—
429C22×C4C23×D7114yes—
430C4⋊C4—44no—
431C4⋊C4—2848no—
432C4⋊C4—2844no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—42no—
437C22⋊C4—2844no—
438C22⋊C4—2842no—
439C22⋊C4—14216no—
440C22⋊C4—1428no—
441C22⋊C4—1428no—
442C22⋊C4—1428no—
443C22⋊C4—1428no—
444C22⋊C4—1428no—
445C22⋊C4—1428no—
446C22⋊C4—1428no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—22no—
451C23—216no—
452C23—44no—
453C23—416no—
454C23—28432no—
455C23—2848no—
456C23—2844no—
457C23—28416no—
458C23—284128no—
459C23—28416no—
460C23—712no—
461C23—718no—
462C23—714no—
463C23—714no—
464C23—718no—
465C23—7132no—
466C23—712no—
467C23—1428no—
468C23—14216no—
469C23—14216no—
470C23—14216no—
471C23—14216no—
472C23—14264no—
473C23—14216no—
474C23—1424no—
475C23—1422no—
476C23—14216no—
477C23—1424no—
478C23—1424no—
479C23C2×D4×D7112yes—
480C23C2×D4×D7114yes—
481C23#41×D7118yes—
482C23C24×D7111yescharacteristic
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—28416no—
492C2×C4—28416no—
493C2×C4—28416no—
494C2×C4—28416no—
495C2×C4—28416no—
496C2×C4—2848no—
497C2×C4—2848no—
498C2×C4—1424no—
499C2×C4—1424no—
500C2×C4—1428no—
501C2×C4—1428no—
502C2×C4—1424no—
503C2×C4—1428no—
504C22—24no—
505C22—216no—
506C22—24no—
507C22—24no—
508C22—22no—
509C22—416no—
510C22—44no—
511C22—28432no—
512C22—28416no—
513C22—28416no—
514C22—2844no—
515C22—718no—
516C22—7116no—
517C22—714no—
518C22—718no—
519C22—14264no—
520C22—1424no—
521C22—1422no—
522C22—1424no—
523C22—14216no—
524C22—1424no—
525C22#123×D7118yes—
526C22C22×D4×D7111yescharacteristic
527C22C2×#41×D7114yes—
528C22H(5,𝔽2)×D7112yes—
529C4—44no—
530C4—44no—
531C4—44no—
532C4—2844no—
533C4—2844no—
534C4—2844no—
535C2—28no—
536C2—42no—
537C2—2842no—
538C2—718no—
539C2—1428no—
540C2C2×#123×D7112yes—
541C2#534×D7111yescharacteristic
542C2#551×D7114yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2483C2
prime radicalC25×D71C22
hypercenter#2483D71
hypocenterC71C2×#2483
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2483

Complements

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

NH ≅ G/Ntype
2. #22592621C2semidirect
3. #22592620C2semidirect
4. #22592619C2semidirect
5. #22592618C2semidirect
6. #22592617C2semidirect
7. #397×D71C2semidirect
8. #385×D71C2semidirect
9. #381×D71C2semidirect
11. C71×#2483C2semidirect
15. C2×C4⋊C4×D71C22semidirect
26. #17985928C22semidirect
27. #17985927C22semidirect
28. #17985926C22semidirect
29. #17985925C22semidirect
30. #17985924C22semidirect
31. #17985874C22semidirect
32. #17985873C22semidirect
33. #17985872C22semidirect
34. #17985871C22semidirect
35. #17985870C22semidirect
36. #17985862C22semidirect
37. #17985861C22semidirect
38. #17985861C22semidirect
39. #17985860C22semidirect
40. #17985860C22semidirect
44. #95×D71C22semidirect
45. #95×D71C22semidirect
47. #397×C71C22semidirect
48. #385×C71C22semidirect
49. #381×C71C22semidirect
69. C2×C4⋊Dic71C23semidirect
70. C2×Dic71⋊C4C23semidirect
91. C71⋊#95C23semidirect
92. C71⋊#95C23semidirect
93. C71⋊#95C23semidirect
94. C71⋊#95C23semidirect
95. C71⋊#95C23semidirect
99. C2×C4⋊C4×C71C23semidirect
105. #95×C71C23semidirect
106. #95×C71C23semidirect
271. D71#2483direct (N × H)
278. #2483D71direct (N × H)
283. C71C2×#2483semidirect
304. #397C2×D71semidirect
310. #385C2×D71semidirect
316. #381C2×D71semidirect
347. C2×C4⋊C4C22×D71semidirect
377. #95C22×D71semidirect
378. #95C22×D71semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C22⋊C4×D712
#225926212
#225926202
#225926192
#225926182
#225926172
#397×D712
#385×D712
#381×D712
#379×D712
C71×#24832
C2×#248371