Finite Group Databasebeta

← #32170920

Subgroups — #32170920

Order 18176.

1986 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596690C212yesmaximal, complemented
2#22596650C211yesmaximal, characteristic, complemented
3#22594151C214yesmaximal
4#22594009C214yesmaximal, complemented
5#22594004C214yesmaximal, complemented
6#22592055C212yesmaximal, complemented
7#22592049C212yesmaximal, complemented
8#22592034C212yesmaximal, complemented
9#22592029C212yesmaximal, complemented
10#22591071C214yesmaximal, complemented
11#22590722C212yesmaximal
12C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
13C71×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D71C2211yescharacteristic
15C2×#14556127C2212yescomplemented
16C2×#14556120C2211yescharacteristic, complemented
17C2×#14556105C2214yes—
18C2×#14556086C2212yes—
19C2×#14556086C2212yes—
20C22×(C2×D71)⋊C4C2212yes—
21C22×C4×Dic71C2211yescharacteristic
22#17985934C2214yes—
23#17985934C2214yes—
24#17985923C2214yes—
25#17985923C2214yes—
26#17985921C2214yes—
27#17985921C2214yes—
28#17985912C2214yes—
29#17985912C2214yes—
30#17985904C2214yescomplemented
31#17985903C2214yescomplemented
32#17985903C2214yescomplemented
33#17985898C2212yescomplemented
34#17985897C2212yescomplemented
35#17985896C2214yescomplemented
36#17985895C2212yescomplemented
37#17985895C2214yescomplemented
38#17985894C2214yescomplemented
39#17985884C2214yescomplemented
40#17985883C2214yescomplemented
41#17985881C2214yescomplemented
42#17985881C2214yescomplemented
43#17985880C2214yescomplemented
44#17985872C2214yes—
45#17985868C2212yescomplemented
46#17985865C2211yescharacteristic, complemented
47#17985861C2214yes—
48#17985859C2212yes—
49#17985857C2214yes—
50#17985856C2212yes—
51#17985856C2212yes—
52#17985854C2214yes—
53C22⋊C4×Dic71C2214yes—
54C4×(C2×D71)⋊C4C2212yes—
55C4×(C2×D71)⋊C4C2212yes—
56C4×(C2×D71)⋊C4C2212yes—
57C4×(C2×D71)⋊C4C2212yes—
58#17985846C2214yes—
59C2×C71⋊#95C2212yes—
60C2×#125×C71C2211yescharacteristic, complemented
61C22×C42×C71C2211yescharacteristic
62#389×C71C2212yescomplemented
63#387×C71C2214yescomplemented
64#385×C71C2214yes—
65#382×C71C2211yescharacteristic, complemented
66#380×C71C2212yes—
67C24×D71C2311yescharacteristic
68C22×Dic142C2314yes—
69C23×Dic71C2311yescharacteristic
70C23×Dic71C2312yes—
71C22×C4×D71—42no—
72C22×C4×D71—42no—
73C22×C4×D71Q818yescomplemented
74C22×C4×D71C2312yes—
75C22×C4×D71C2312yes—
76#14556127—44no—
77#14556120—42no—
78#14556105—216no—
79#14556086—44no—
80#14556086—44no—
81C2×C4⋊Dic71C2312yes—
82C2×C4⋊Dic71C2312yes—
83C2×C4⋊Dic71C2314yes—
84C2×Dic71⋊C4C2314yes—
85C2×Dic71⋊C4C2314yes—
86C2×Dic71⋊C4C2314yes—
87C2×Dic71⋊C4C2314yes—
88C2×Dic71⋊C4C2314yes—
89C2×Dic71⋊C4C2314yes—
90C2×C22⋊Dic71C2314yes—
91C2×(C2×D71)⋊C4—28no—
92C2×(C2×D71)⋊C4C2312yes—
93C2×(C2×D71)⋊C4C2312yes—
94C2×(C2×D71)⋊C4C2312yes—
95C2×(C2×D71)⋊C4C2312yes—
96C2×(C2×D71)⋊C4C2314yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×(C2×D71)⋊C4C2314yes—
99C2×(C2×D71)⋊C4C2314yes—
100C2×C4×Dic71—42no—
101C2×C4×Dic71C2312yes—
102C2×C4×Dic71C2314yes—
103C2×C4×Dic71C2314yes—
104C2×C4×Dic71C2311yescharacteristic
105C2×C4×Dic71C2311yescharacteristic
106C71⋊#95C2312yes—
107C71⋊#95C2312yes—
108C71⋊#95C2312yes—
109C71⋊#95C2312yes—
110C71⋊#95C2314yes—
111C71⋊#95C2314yes—
112C71⋊#95C2314yes—
113C71⋊#95C2314yes—
114C71⋊#95C2314yes—
115C71⋊#95C2314yes—
116C71⋊#95C2314yes—
117C23×C4×C71C2311yescharacteristic
118C23×C4×C71C2312yes—
119#125×C71—42no—
120C2×C4⋊C4×C71C2312yes—
121C2×C4⋊C4×C71C2312yes—
122C2×C4⋊C4×C71C2314yes—
123C2×C4⋊C4×C71C2314yes—
124C2×C22⋊C4×C71C2314yes—
125C2×C42×C71—42no—
126C2×C42×C71C2311yescharacteristic
127C2×C42×C71C2311yescharacteristic
128C2×C42×C71C2312yes—
129#95×C71C2314yes—
130#95×C71C2314yes—
131#95×C71C2314yes—
132C23×D71—24no—
133C23×D71—41no—
134C23×D71—41no—
135C23×D71—42no—
136C23×D71C2×Q814yes—
137C23×D71C2412yes—
138C2×Dic142—28no—
139C2×Dic142—44no—
140C2×Dic142—44no—
141C22×Dic71—41no—
142C22×Dic71—41no—
143C22×Dic71—42no—
144C22×Dic71—42no—
145C22×Dic71C2×Q814yes—
146C22×Dic71#4118yes—
147C22×Dic71C2414yes—
148C22×Dic71C2414yes—
149C22×Dic71C2412yes—
150C22×Dic71C2412yes—
151C22×Dic71C2412yes—
152C2×C4×D71—24no—
153C2×C4×D71—24no—
154C2×C4×D71—416no—
155C2×C4×D71—42no—
156C2×C4×D71—42no—
157C2×C4×D71C2×Q818yes—
158C2×C4×D71C2×Q818yes—
159C4⋊Dic71—24no—
160C4⋊Dic71—24no—
161C4⋊Dic71—28no—
162Dic71⋊C4—28no—
163Dic71⋊C4—28no—
164Dic71⋊C4—44no—
165Dic71⋊C4—44no—
166Dic71⋊C4—44no—
167Dic71⋊C4—44no—
168C22⋊Dic71—28no—
169(C2×D71)⋊C4—24no—
170(C2×D71)⋊C4—24no—
171(C2×D71)⋊C4—24no—
172(C2×D71)⋊C4—24no—
173(C2×D71)⋊C4—44no—
174(C2×D71)⋊C4—44no—
175(C2×D71)⋊C4—44no—
176(C2×D71)⋊C4—44no—
177C4×Dic71—24no—
178C4×Dic71—41no—
179C4×Dic71—41no—
180C4×Dic71—44no—
181C4×Dic71—44no—
182C24×C71C2411yescharacteristic
183C22×C4×C71—22no—
184C22×C4×C71—22no—
185C22×C4×C71—24no—
186C22×C4×C71—42no—
187C22×C4×C71—42no—
188C22×C4×C71C2×Q814yes—
189C22×C4×C71C2414yes—
190C22×C4×C71C2412yes—
191C22×C4×C71C2412yes—
192C22×C4×C71C2414yes—
193C22×C4×C71C2411yescharacteristic
194C22×C4×C71C2411yescharacteristic
195C4⋊C4×C71—24no—
196C4⋊C4×C71—24no—
197C4⋊C4×C71—44no—
198C4⋊C4×C71—44no—
199C22⋊C4×C71—28no—
200C42×C71—22no—
201C42×C71—22no—
202C42×C71—42no—
203C23×C71—21no—
204C23×C71—21no—
205C23×C71—22no—
206C23×C71C22×Q812yes—
207C23×C71C2×#4114yes—
208C23×C71C2511yescharacteristic
209C22×D71—28no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71—44no—
217C22×D71—44no—
218C22×D71—48no—
219C22×D71—48no—
220C22×D71C22×Q814yes—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—28no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—22no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—22no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—22no—
236C2×C4×C71—48no—
237C2×C4×C71C22×Q812yes—
238C2×C4×C71C22×Q812yes—
239C2×C4×C71H(5,𝔽2)12yes—
240C2×C4×C71H(5,𝔽2)12yes—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—24no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—44no—
254C2×Dic71—48no—
255C2×Dic71C22×Q814yes—
256C2×Dic71C2×#4114yes—
257C2×Dic71C2×#4114yes—
258C4×D71—216no—
259C4×D71—48no—
260Dic142—88no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—24no—
264C22×C71—24no—
265C22×C71#53314yes—
266C22×C71C23×Q811yescharacteristic
267C22×C71C22×#4112yes—
268C22×C71C2×H(5,𝔽2)11yescharacteristic
269C22×C71C2×H(5,𝔽2)11yescharacteristic
270C22×C71#58512yes—
271C2×D71—24no—
272C2×D71—24no—
273C2×D71—24no—
274C2×D71—416no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C2×D71—42no—
279C4×C71—22no—
280C4×C71—44no—
281C4×C71—44no—
282C4×C71—42no—
283C4×C71—42no—
284C4×C71#54314yes—
285Dic71—24no—
286Dic71—24no—
287Dic71—24no—
288Dic71—44no—
289Dic71—44no—
290#35033—711nomaximal, Sylow
291C2×C71—24no—
292C2×C71C2×#53311yescharacteristic
293C2×C71C2×#54311yescharacteristic
294C2×C71#345811yescharacteristic
295C2×C71#350512yes—
296C2×C71#350612yes—
297D71—44no—
298#2857—712no—
299#2851—711no—
300#2851D7111yescharacteristic, complemented
301#2626—714no—
302#2608—714no—
303#2607—714no—
304#2441—712no—
305#2441—712no—
306#2439—712no—
307#2439—712no—
308#2329—714no—
309#2280—712no—
310C2×C4×C22⋊C4—711no—
311C71#3503311yescharacteristic, complemented, Sylow
312C24×C4—711no—
313C2×C4⋊Q8—712no—
314C2×#125—711no—
315C2×#125C2×D7111yescharacteristic, complemented
316C2×C22⋊Q8—714no—
317C2×#117—712no—
318C2×#117—712no—
319C22×C22⋊C4—712no—
320C22×C42—711no—
321C22×C42C2×D7111yescharacteristic
322#399—714no—
323#399—714no—
324#396—714no—
325#396—714no—
326#396—714no—
327#396—714no—
328#393—714no—
329#393—714no—
330#391—714no—
331#391—714no—
332#391—714no—
333#389—714no—
334#389—714no—
335#389—712no—
336#389—712no—
337#389—712no—
338#389—714no—
339#389C2×D7112yescomplemented
340#387—714no—
341#387—714no—
342#387—714no—
343#387—714no—
344#387—714no—
345#387C2×D7114yescomplemented
346#385—714no—
347#385C2×D7114yes—
348#384—712no—
349#382—711no—
350#382C2×D7111yescharacteristic, complemented
351#381—714no—
352#380—712no—
353#380—712no—
354#380—712no—
355#380—714no—
356#380C2×D7112yes—
357#379—714no—
358C4×C22⋊C4—714no—
359C4×C22⋊C4—712no—
360C4×C22⋊C4—712no—
361C4×C22⋊C4—712no—
362C4×C22⋊C4—712no—
363C4×C22⋊C4—714no—
364C2×#95—712no—
365C25—711no—
366C22×Q8—714no—
367C23×C4—2842no—
368C23×C4—2842no—
369C23×C4—712no—
370C23×C4—711no—
371C23×C4—718no—
372C23×C4—712no—
373C23×C4—712no—
374C23×C4C22×D7111yescharacteristic
375C23×C4C22×D7112yes—
376C4⋊Q8—2844no—
377#125—42no—
378#125—2842no—
379C22⋊Q8—14216no—
380#117—2844no—
381#117—2844no—
382C2×C4⋊C4—714no—
383C2×C4⋊C4—714no—
384C2×C4⋊C4—714no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—712no—
387C2×C4⋊C4—712no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—714no—
390C2×C4⋊C4—714no—
391C2×C4⋊C4C22×D7112yes—
392C2×C4⋊C4C22×D7112yes—
393C2×C4⋊C4C22×D7114yes—
394C2×C4⋊C4C22×D7114yes—
395C2×C22⋊C4—714no—
396C2×C22⋊C4—714no—
397C2×C22⋊C4—714no—
398C2×C22⋊C4—714no—
399C2×C22⋊C4—714no—
400C2×C22⋊C4—712no—
401C2×C22⋊C4—712no—
402C2×C22⋊C4—712no—
403C2×C22⋊C4—712no—
404C2×C22⋊C4—1428no—
405C2×C22⋊C4C22×D7114yes—
406C2×C42—42no—
407C2×C42—2842no—
408C2×C42—714no—
409C2×C42—714no—
410C2×C42—711no—
411C2×C42—711no—
412C2×C42—712no—
413C2×C42C22×D7111yescharacteristic
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7112yes—
416#95—712no—
417#95—712no—
418#95—712no—
419#95—712no—
420#95—714no—
421#95—714no—
422#95—714no—
423#95—714no—
424#95—714no—
425#95—714no—
426#95—714no—
427#95C22×D7114yes—
428#95C22×D7114yes—
429#95C22×D7114yes—
430C24—2841no—
431C24—2841no—
432C24—2842no—
433C24—714no—
434C24—712no—
435C24—1424no—
436C24C23×D7111yescharacteristic
437C2×Q8—2844no—
438C2×Q8—2844no—
439C2×Q8—1428no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—2842no—
446C22×C4—2842no—
447C22×C4—28416no—
448C22×C4—2842no—
449C22×C4—2841no—
450C22×C4—2841no—
451C22×C4—2842no—
452C22×C4—714no—
453C22×C4—718no—
454C22×C4—718no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—712no—
458C22×C4—718no—
459C22×C4—714no—
460C22×C4—714no—
461C22×C4—1424no—
462C22×C4—1424no—
463C22×C4Q8×D7114yes—
464C22×C4C23×D7112yes—
465C22×C4C23×D7112yes—
466C22×C4C23×D7114yes—
467C22×C4C23×D7114yes—
468C22×C4C23×D7111yescharacteristic
469C22×C4C23×D7111yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—2844no—
475C4⋊C4—2844no—
476C4⋊C4—2844no—
477C4⋊C4—2844no—
478C4⋊C4—1428no—
479C4⋊C4—1428no—
480C4⋊C4—1428no—
481C4⋊C4—1424no—
482C4⋊C4—1424no—
483C22⋊C4—28no—
484C22⋊C4—2844no—
485C22⋊C4—2844no—
486C22⋊C4—2844no—
487C22⋊C4—2844no—
488C22⋊C4—1424no—
489C22⋊C4—1424no—
490C22⋊C4—1424no—
491C22⋊C4—1424no—
492C22⋊C4—1428no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—2841no—
497C42—2841no—
498C42—2844no—
499C42—2844no—
500C42—1424no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—2848no—
505C23—2848no—
506C23—2844no—
507C23—2844no—
508C23—714no—
509C23—1422no—
510C23—1422no—
511C23—1422no—
512C23—1422no—
513C23—1422no—
514C23—1422no—
515C23—1428no—
516C23C2×Q8×D7112yes—
517C23#41×D7114yes—
518C23C24×D7111yescharacteristic
519Q8—5688no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—2848no—
537C2×C4—2844no—
538C2×C4—2848no—
539C2×C4—714no—
540C2×C4—714no—
541C2×C4—714no—
542C2×C4—1424no—
543C2×C4—1424no—
544C2×C4—1424no—
545C2×C4—1424no—
546C2×C4—1424no—
547C2×C4—1424no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4—14216no—
555C2×C4C2×Q8×D7112yes—
556C2×C4C2×Q8×D7112yes—
557C2×C4#1455617312yes—
558C2×C4#1455617412yes—
559C22—22no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—28416no—
564C22—2842no—
565C22—2842no—
566C22—2842no—
567C22—2842no—
568C22—1424no—
569C22—1424no—
570C22—1424no—
571C22#1798655514yes—
572C22C22×Q8×D7111yescharacteristic
573C22C2×#41×D7112yes—
574C22C2×#1455617311yescharacteristic
575C22C2×#1455617411yescharacteristic
576C22#1798683512yes—
577C4—22no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—2844no—
583C4—2844no—
584C4—1424no—
585C4—1424no—
586C4—1424no—
587C4#1798662414yes—
588C2—24no—
589C2—2844no—
590C2C2×#1798655511yescharacteristic
591C2C2×#1798662411yescharacteristic
592C2#2260174311yescharacteristic
593C2#2260217512yes—
594C2#2260218312yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2851C2
prime radicalC24×D71C23
hypercenter#2851D71
hypocenterC71#35033
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #22596690C2semidirect
2. #22596650C2semidirect
4. #22594009C2semidirect
5. #22594004C2semidirect
6. #22592055C2semidirect
7. #22592049C2semidirect
8. #22592034C2semidirect
9. #22592029C2semidirect
10. #22591071C2semidirect
13. C71×#2851C2semidirect
15. C2×#14556127C22semidirect
16. C2×#14556120C22semidirect
30. #17985904C22semidirect
31. #17985903C22semidirect
32. #17985903C22semidirect
33. #17985898C22semidirect
34. #17985897C22semidirect
35. #17985896C22semidirect
36. #17985895C22semidirect
37. #17985895C22semidirect
38. #17985894C22semidirect
39. #17985884C22semidirect
40. #17985883C22semidirect
41. #17985881C22semidirect
42. #17985881C22semidirect
43. #17985880C22semidirect
45. #17985868C22semidirect
46. #17985865C22semidirect
60. C2×#125×C71C22semidirect
62. #389×C71C22semidirect
63. #387×C71C22semidirect
65. #382×C71C22semidirect
73. C22×C4×D71Q8semidirect
300. #2851D71semidirect
311. C71#35033semidirect
315. C2×#125C2×D71semidirect
339. #389C2×D71semidirect
345. #387C2×D71semidirect
350. #382C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225966902
#225966502
#225941512
#225940092
#225940042
#225920552
#225920492
#225920342
#225920292
#225910712
#225907222
C2×C4×(C2×D71)⋊C42
C71×#28512
#3503371