Finite Group Databasebeta

← #32170643

Subgroups — #32170643

Order 18176.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22596650C211yesmaximal, characteristic, complemented
2#22596648C212yesmaximal, complemented
3#22594312C214yesmaximal, complemented
4#22593563C214yesmaximal, complemented
5#22593562C214yesmaximal, complemented
6#22592067C212yesmaximal, complemented
7#22592066C212yesmaximal, complemented
8#22592062C212yesmaximal, complemented
9#22592060C212yesmaximal, complemented
10#22590951C214yesmaximal
11#380×D71C212yesmaximal
12C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
13C71×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D71C2211yescharacteristic
15C2×#14556120C2211yescharacteristic, complemented
16C2×#14556119C2212yescomplemented
17C2×C4⋊C4×D71C2212yes—
18C2×C4⋊C4×D71C2212yes—
19C22×(C2×D71)⋊C4C2212yes—
20C2×C22⋊C4×D71C2214yes—
21C22×C4×Dic71C2211yescharacteristic
22#17985940C2214yescomplemented
23#17985940C2214yescomplemented
24#17985939C2214yescomplemented
25#17985937C2214yescomplemented
26#17985933C2214yescomplemented
27#17985932C2214yescomplemented
28#17985930C2214yescomplemented
29#17985929C2214yescomplemented
30#17985898C2212yescomplemented
31#17985897C2212yescomplemented
32#17985895C2212yescomplemented
33#17985886C2214yescomplemented
34#17985883C2214yescomplemented
35#17985882C2214yescomplemented
36#17985874C2214yes—
37#17985873C2214yes—
38#17985873C2214yes—
39#17985872C2214yes—
40#17985871C2214yes—
41#17985870C2214yes—
42#17985870C2214yes—
43#17985865C2211yescharacteristic, complemented
44#17985864C2212yescomplemented
45#17985859C2212yes—
46#17985858C2214yes—
47#17985858C2214yes—
48#17985858C2214yes—
49#17985857C2214yes—
50#17985856C2214yes—
51#17985856C2214yes—
52#17985856C2212yes—
53#17985856C2212yes—
54C4×(C2×D71)⋊C4C2212yes—
55C4×(C2×D71)⋊C4C2212yes—
56C4×(C2×D71)⋊C4C2212yes—
57C4×(C2×D71)⋊C4C2212yes—
58C2×C71⋊#95C2212yes—
59#95×D71C2214yes—
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
68C23×Dic71C2312yes—
69C23×Dic71C2311yescharacteristic
70C22×C4×D71—42no—
71C22×C4×D71—42no—
72C22×C4×D71Q818yes—
73C22×C4×D71C2312yes—
74C22×C4×D71C2314yes—
75C22×C4×D71C2314yes—
76C22×C4×D71C2312yes—
77#14556120—42no—
78#14556119—44no—
79C2×C4⋊Dic71C2312yes—
80C2×C4⋊Dic71C2312yes—
81C2×C4⋊Dic71C2314yes—
82C2×C4⋊Dic71C2314yes—
83C2×Dic71⋊C4C2314yes—
84C2×Dic71⋊C4C2314yes—
85C2×Dic71⋊C4C2314yes—
86C2×Dic71⋊C4C2314yes—
87C2×Dic71⋊C4C2314yes—
88C2×Dic71⋊C4C2314yes—
89C4⋊C4×D71—44no—
90C4⋊C4×D71—44no—
91C2×C22⋊Dic71C2314yes—
92C2×(C2×D71)⋊C4—28no—
93C2×(C2×D71)⋊C4C2312yes—
94C2×(C2×D71)⋊C4C2312yes—
95C2×(C2×D71)⋊C4C2312yes—
96C2×(C2×D71)⋊C4C2312yes—
97C2×(C2×D71)⋊C4C2314yes—
98C2×(C2×D71)⋊C4C2314yes—
99C22⋊C4×D71—216no—
100C2×C4×Dic71—42no—
101C2×C4×Dic71C2312yes—
102C2×C4×Dic71C2311yescharacteristic
103C2×C4×Dic71C2311yescharacteristic
104C71⋊#95C2314yes—
105C71⋊#95C2314yes—
106C71⋊#95C2314yes—
107C71⋊#95C2312yes—
108C71⋊#95C2312yes—
109C71⋊#95C2312yes—
110C71⋊#95C2312yes—
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—41no—
133C23×D71—41no—
134C23×D71—42no—
135C23×D71C2×Q814yes—
136C23×D71#4118yes—
137C23×D71C2412yes—
138C22×Dic71—24no—
139C22×Dic71—41no—
140C22×Dic71—41no—
141C22×Dic71—42no—
142C22×Dic71—42no—
143C22×Dic71C2×Q814yes—
144C22×Dic71C2414yes—
145C22×Dic71C2414yes—
146C22×Dic71C2412yes—
147C22×Dic71C2412yes—
148C22×Dic71C2412yes—
149C2×C4×D71—24no—
150C2×C4×D71—24no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—416no—
154C2×C4×D71—42no—
155C2×C4×D71—42no—
156C2×C4×D71—44no—
157C2×C4×D71—44no—
158C2×C4×D71—44no—
159C2×C4×D71—44no—
160C2×C4×D71C2×Q818yes—
161C2×C4×D71C2×Q818yes—
162C4⋊Dic71—24no—
163C4⋊Dic71—24no—
164C4⋊Dic71—44no—
165C4⋊Dic71—44no—
166Dic71⋊C4—44no—
167Dic71⋊C4—44no—
168Dic71⋊C4—44no—
169Dic71⋊C4—44no—
170Dic71⋊C4—44no—
171Dic71⋊C4—44no—
172C22⋊Dic71—28no—
173(C2×D71)⋊C4—28no—
174(C2×D71)⋊C4—28no—
175(C2×D71)⋊C4—24no—
176(C2×D71)⋊C4—24no—
177(C2×D71)⋊C4—24no—
178(C2×D71)⋊C4—24no—
179C4×Dic71—24no—
180C4×Dic71—41no—
181C4×Dic71—41no—
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×C71C2414yes—
191C22×C4×C71C2412yes—
192C22×C4×C71C2412yes—
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—22no—
210C22×D71—22no—
211C22×D71—22no—
212C22×D71—22no—
213C22×D71—44no—
214C22×D71—44no—
215C22×D71—48no—
216C22×D71—48no—
217C22×D71#125116yescomplemented
218C22×D71C22×Q814yes—
219C22×D71C2×#4114yes—
220C22×D71C2×#4114yes—
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—24no—
246C2×Dic71—24no—
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—22no—
253C2×Dic71—24no—
254C2×Dic71—22no—
255C2×Dic71—44no—
256C2×Dic71—48no—
257C2×Dic71C22×Q814yes—
258C4×D71—216no—
259C4×D71—48no—
260C4×D71—88no—
261C4×D71—88no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—24no—
265C22×C71—24no—
266C22×C71C2×#12514yes—
267C22×C71C23×Q811yescharacteristic
268C22×C71C22×#4112yes—
269C22×C71C2×H(5,𝔽2)11yescharacteristic
270C22×C71C2×H(5,𝔽2)11yescharacteristic
271C22×C71#58512yes—
272C2×D71—24no—
273C2×D71—24no—
274C2×D71—24no—
275C2×D71—24no—
276C2×D71—24no—
277C2×D71—24no—
278C2×D71—416no—
279C2×D71C2×#12518yes—
280C4×C71—22no—
281C4×C71—44no—
282C4×C71—44no—
283C4×C71—42no—
284C4×C71—42no—
285C4×C71#54314yes—
286Dic71—24no—
287Dic71—44no—
288Dic71—44no—
289Dic71—42no—
290Dic71—42no—
291#35020—711nomaximal, Sylow
292C2×C71—24no—
293C2×C71C22×#12511yescharacteristic
294C2×C71C2×#54311yescharacteristic
295C2×C71#345811yescharacteristic
296C2×C71#350512yes—
297C2×C71#350512yes—
298D71—28no—
299#2851—711no—
300#2851—712no—
301#2851D7111yescharacteristic, complemented
302#2646—714no—
303#2569—714no—
304#2569—714no—
305#2442—712no—
306#2442—712no—
307#2442—712no—
308#2442—712no—
309#2315—714no—
310C2×#380—712no—
311C2×C4×C22⋊C4—711no—
312C71#3502011yescharacteristic, complemented, Sylow
313C24×C4—711no—
314C2×#125—712no—
315C2×#125—711no—
316C2×#125C2×D7111yescharacteristic, complemented
317C22×C4⋊C4—712no—
318C22×C4⋊C4—712no—
319C22×C22⋊C4—712no—
320C22×C22⋊C4—714no—
321C22×C42—711no—
322C22×C42C2×D7111yescharacteristic
323#400—714no—
324#400—714no—
325#400—714no—
326#400—714no—
327#398—714no—
328#398—714no—
329#398—714no—
330#398—714no—
331#389—712no—
332#389—712no—
333#389—712no—
334#389C2×D7112yescomplemented
335#387—714no—
336#387—714no—
337#387—714no—
338#387C2×D7114yescomplemented
339#385—714no—
340#385—714no—
341#385—714no—
342#385—714no—
343#385—714no—
344#385—714no—
345#385—714no—
346#385C2×D7114yes—
347#382—712no—
348#382—711no—
349#382C2×D7111yescharacteristic, complemented
350#380—712no—
351#380—714no—
352#380—714no—
353#380—712no—
354#380—712no—
355#380—714no—
356#380—714no—
357#380—714no—
358#380—714no—
359#380C2×D7112yes—
360C4×C22⋊C4—712no—
361C4×C22⋊C4—712no—
362C4×C22⋊C4—712no—
363C4×C22⋊C4—712no—
364C2×#95—712no—
365C2×#95—714no—
366C25—711no—
367C23×C4—2842no—
368C23×C4—2842no—
369C23×C4—711no—
370C23×C4—718no—
371C23×C4—712no—
372C23×C4—712no—
373C23×C4—714no—
374C23×C4—714no—
375C23×C4—712no—
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7112yes—
378#125—42no—
379#125—2842no—
380#125—2844no—
381C2×C4⋊C4—2844no—
382C2×C4⋊C4—2844no—
383C2×C4⋊C4—714no—
384C2×C4⋊C4—714no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—714no—
387C2×C4⋊C4—714no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—714no—
390C2×C4⋊C4—714no—
391C2×C4⋊C4—712no—
392C2×C4⋊C4—712no—
393C2×C4⋊C4C22×D7112yes—
394C2×C4⋊C4C22×D7112yes—
395C2×C4⋊C4C22×D7114yes—
396C2×C4⋊C4C22×D7114yes—
397C2×C22⋊C4—714no—
398C2×C22⋊C4—712no—
399C2×C22⋊C4—712no—
400C2×C22⋊C4—712no—
401C2×C22⋊C4—712no—
402C2×C22⋊C4—714no—
403C2×C22⋊C4—714no—
404C2×C22⋊C4—14216no—
405C2×C22⋊C4—1428no—
406C2×C22⋊C4C22×D7114yes—
407C2×C42—42no—
408C2×C42—2842no—
409C2×C42—711no—
410C2×C42—711no—
411C2×C42—712no—
412C2×C42C22×D7111yescharacteristic
413C2×C42C22×D7111yescharacteristic
414C2×C42C22×D7112yes—
415#95—712no—
416#95—712no—
417#95—712no—
418#95—712no—
419#95—714no—
420#95—714no—
421#95—714no—
422#95—714no—
423#95—714no—
424#95—714no—
425#95—714no—
426#95—714no—
427#95—714no—
428#95C22×D7114yes—
429#95C22×D7114yes—
430#95C22×D7114yes—
431C24—2841no—
432C24—2841no—
433C24—2842no—
434C24—714no—
435C24—718no—
436C24—712no—
437C24C23×D7111yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—2842no—
444C22×C4—2842no—
445C22×C4—28416no—
446C22×C4—2844no—
447C22×C4—2844no—
448C22×C4—2844no—
449C22×C4—2844no—
450C22×C4—2842no—
451C22×C4—2841no—
452C22×C4—2841no—
453C22×C4—2842no—
454C22×C4—714no—
455C22×C4—718no—
456C22×C4—712no—
457C22×C4—718no—
458C22×C4—712no—
459C22×C4—712no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—1428no—
463C22×C4—1428no—
464C22×C4—1424no—
465C22×C4—1424no—
466C22×C4—1424no—
467C22×C4Q8×D7114yes—
468C22×C4C23×D7112yes—
469C22×C4C23×D7112yes—
470C22×C4C23×D7114yes—
471C22×C4C23×D7114yes—
472C22×C4C23×D7111yescharacteristic
473C22×C4C23×D7111yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—2844no—
479C4⋊C4—2844no—
480C4⋊C4—2844no—
481C4⋊C4—2844no—
482C4⋊C4—2844no—
483C4⋊C4—2844no—
484C4⋊C4—2844no—
485C4⋊C4—2844no—
486C4⋊C4—1424no—
487C4⋊C4—1424no—
488C22⋊C4—28no—
489C22⋊C4—1428no—
490C22⋊C4—1428no—
491C22⋊C4—1424no—
492C22⋊C4—1424no—
493C22⋊C4—1424no—
494C22⋊C4—1424no—
495C22⋊C4—1428no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—2841no—
500C42—2841no—
501C42—1424no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—2844no—
506C23—2844no—
507C23—2848no—
508C23—2848no—
509C23—7116no—
510C23—714no—
511C23—714no—
512C23—714no—
513C23—1422no—
514C23—1422no—
515C23—1422no—
516C23—1422no—
517C23C2×Q8×D7112yes—
518C23#41×D7114yes—
519C23C24×D7111yescharacteristic
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—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—2848no—
537C2×C4—2848no—
538C2×C4—2844no—
539C2×C4—5688no—
540C2×C4—5688no—
541C2×C4—714no—
542C2×C4—1424no—
543C2×C4—1422no—
544C2×C4—1422no—
545C2×C4—1424no—
546C2×C4—1424no—
547C2×C4—1422no—
548C2×C4—1422no—
549C2×C4—1424no—
550C2×C4—1424no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4—1424no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—14216no—
557C2×C4C2×Q8×D7112yes—
558C2×C4C2×Q8×D7112yes—
559C2×C4#1455617312yes—
560C2×C4#1455617412yes—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—28416no—
566C22—718no—
567C22—1424no—
568C22—1424no—
569C22—1424no—
570C22—1424no—
571C22—1424no—
572C22—1424no—
573C22#125×D7114yes—
574C22C22×Q8×D7111yescharacteristic
575C22C2×#41×D7112yes—
576C22C2×#1455617311yescharacteristic
577C22C2×#1455617411yescharacteristic
578C22#1798683512yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—2844no—
585C4—2842no—
586C4—2842no—
587C4—2844no—
588C4—1424no—
589C4#1798662414yes—
590C2—24no—
591C2—1428no—
592C2C2×#125×D7111yescharacteristic
593C2C2×#1798662411yescharacteristic
594C2#2260174311yescharacteristic
595C2#2260216812yes—
596C2#2260217512yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2851C2
prime radicalC24×D71C23
hypercenter#2851D71
hypocenterC71#35020
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #22596650C2semidirect
2. #22596648C2semidirect
3. #22594312C2semidirect
4. #22593563C2semidirect
5. #22593562C2semidirect
6. #22592067C2semidirect
7. #22592066C2semidirect
8. #22592062C2semidirect
9. #22592060C2semidirect
13. C71×#2851C2semidirect
15. C2×#14556120C22semidirect
16. C2×#14556119C22semidirect
22. #17985940C22semidirect
23. #17985940C22semidirect
24. #17985939C22semidirect
25. #17985937C22semidirect
26. #17985933C22semidirect
27. #17985932C22semidirect
28. #17985930C22semidirect
29. #17985929C22semidirect
30. #17985898C22semidirect
31. #17985897C22semidirect
32. #17985895C22semidirect
33. #17985886C22semidirect
34. #17985883C22semidirect
35. #17985882C22semidirect
43. #17985865C22semidirect
44. #17985864C22semidirect
60. C2×#125×C71C22semidirect
62. #389×C71C22semidirect
63. #387×C71C22semidirect
65. #382×C71C22semidirect
217. C22×D71#125semidirect
301. #2851D71semidirect
312. C71#35020semidirect
316. C2×#125C2×D71semidirect
334. #389C2×D71semidirect
338. #387C2×D71semidirect
349. #382C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225966502
#225966482
#225943122
#225935632
#225935622
#225920672
#225920662
#225920622
#225920602
#225909512
#380×D712
C2×C4×(C2×D71)⋊C42
C71×#28512
#3502071