Finite Group Databasebeta

← AGL(1,ℤ/16)×D71 #32150889

Subgroups — AGL(1,ℤ/16)×D71 #32150889

Order 18176.

710 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#509×D71C211yesmaximal, characteristic
2#444×D71C211yesmaximal, characteristic, complemented
3#442×D71C211yesmaximal, characteristic, complemented
4#22590241C211yesmaximal, characteristic, complemented
5#22590240C211yesmaximal, characteristic, complemented
6#22590239C211yesmaximal, characteristic, complemented
7#22590238C211yesmaximal, characteristic, complemented
8#22590237C211yesmaximal, characteristic, complemented
9#22590236C211yesmaximal, characteristic, complemented
10#22590235C211yesmaximal, characteristic, complemented
11#365×D71C211yesmaximal, characteristic, complemented
12#361×D71C211yesmaximal, characteristic, complemented
13#360×D71C211yesmaximal, characteristic, complemented
14#347×D71C211yesmaximal, characteristic, complemented
15C71×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#135×D71C2211yescharacteristic
17C2×D8×D71C2211yescharacteristic
18#131×D71—21no—
19C4×D4×D71—21no—
20#17986437C2211yescharacteristic
21#17986436C412yescomplemented
22#17986435C2211yescharacteristic
23#17986434C2211yescharacteristic
24#17986433C412yescomplemented
25SD32×D71C412yescomplemented
26D16×D71C412yescomplemented
27M32×D71C2211yescharacteristic
28M32×D71C2211yescharacteristic
29#17986122C2211yescharacteristic, complemented
30#17986121C2211yescharacteristic, complemented
31#17986120C2211yescharacteristic, complemented
32#17986119—21no—
33#17986118—21no—
34#17986112C2211yescharacteristic, complemented
35#17986111C2211yescharacteristic, complemented
36#17986110C2211yescharacteristic, complemented
37#17986109—21no—
38#17986108—21no—
39#108×D71C2211yescharacteristic
40#106×D71C2211yescharacteristic
41#104×D71—21no—
42#102×D71—21no—
43#97×D71C2211yescharacteristic
44#17985823C2211yescharacteristic, complemented
45#17985822C2211yescharacteristic, complemented
46#17985821C2211yescharacteristic, complemented
47#17985813C2211yescharacteristic, complemented
48#17985812C2211yescharacteristic, complemented
49#17985811C2211yescharacteristic, complemented
50#17985810C2211yescharacteristic, complemented
51#17985809C2211yescharacteristic, complemented
52#17985808C2211yescharacteristic, complemented
53#17985777C2211yescharacteristic, complemented
54#17985776C2211yescharacteristic, complemented
55#17985776C2211yescharacteristic, complemented
56#509×C71C2211yescharacteristic
57#444×C71C2211yescharacteristic, complemented
58#442×C71C2211yescharacteristic, complemented
59#365×C71C2211yescharacteristic, complemented
60#361×C71C2211yescharacteristic, complemented
61#360×C71C2211yescharacteristic, complemented
62#347×C71C2211yescharacteristic, complemented
63#41×D71—21no—
64C2×D4×D71—21no—
65C22×C4×D71—41no—
66#14556151C2×C411yescharacteristic, complemented
67#14556150C2311yescharacteristic
68#14556149—21no—
69#14556148—21no—
70#14556147C2×C411yescharacteristic, complemented
71Q16×D71C2×C411yescharacteristic, complemented
72SD16×D71—21no—
73C2×D568C2311yescharacteristic
74C2×#11765850—21no—
75D8×D71—21no—
76D8×D71—21no—
77D8×D71C2×C412yes—
78D8×D71C2×C412yes—
79D8×D71C2×C411yescharacteristic, complemented
80#14556136—21no—
81#14556136—41no—
82#14556135—21no—
83#14556134—21no—
84#14556134—41no—
85M16×D71—21no—
86M16×D71—41no—
87C2×C8×D71—41no—
88C2×C8×D71D411yescharacteristic
89C2×C8×D71C2311yescharacteristic
90C4×D284—21no—
91D4×Dic71—21no—
92#14556090—21no—
93C4×C22⋊D71—41no—
94(C22⋊D71)⋊C4—41no—
95C4⋊C4×D71—21no—
96C22⋊C4×D71—41no—
97C42×D71D411yescharacteristic
98#14556073C2×C412yescomplemented
99#14556072C2×C412yescomplemented
100#14556071C2×C412yescomplemented
101D1136C2×C412yescomplemented
102#14556069C2×C412yescomplemented
103#14556069C2×C412yescomplemented
104#14556068C2311yescharacteristic
105#14556068C2311yescharacteristic
106#14556067—21no—
107#14556067C2×C412yescomplemented
108C16×D71—21no—
109C16×D71C2×C412yescomplemented
110#14556064C2311yescharacteristic
111#14556063—21no—
112#14556060C2311yescharacteristic
113#14556059—21no—
114#14556056—21no—
115#14556055—21no—
116#14556054—21no—
117#14556050—21no—
118#14556049—21no—
119#14556048—21no—
120#14556039D411yescharacteristic
121#14556038D411yescharacteristic
122#14556038C2311yescharacteristic
123#135×C71C2311yescharacteristic
124C2×D8×C71C2311yescharacteristic
125#131×C71—21no—
126C4×D4×C71—21no—
127SD32×C71C2×C412yescomplemented
128D16×C71C2×C412yescomplemented
129M32×C71C2311yescharacteristic
130M32×C71C2311yescharacteristic
131#108×C71C2311yescharacteristic
132#106×C71C2311yescharacteristic
133#104×C71—21no—
134#102×C71—21no—
135#97×C71C2311yescharacteristic
136C23×D71—41no—
137#11765866—21no—
138#11765865—21no—
139#11765865—41no—
140#11765864—21no—
141#11765864—41no—
142Q8×D71—21no—
143C2×D284—21no—
144C2×C22⋊D71—41no—
145D4×D71—22no—
146D4×D71—22no—
147D4×D71—21no—
148D4×D71—41no—
149D4×D71—41no—
150D4×D71—41no—
151C22×Dic71—41no—
152C2×C4×D71—21no—
153C2×C4×D71—41no—
154C2×C4×D71—41no—
155C2×C4×D71—41no—
156C2×C4×D71—41no—
157C2×C4×D71—84no—
158C2×C4×D71C2×D411yescharacteristic
159Dic284C22×C411yescharacteristic, complemented
160#11765855—21no—
161#11765854—21no—
162#11765853—21no—
163#11765852—21no—
164D568—21no—
165D568C22×C412yes—
166D568C22×C411yescharacteristic, complemented
167#11765850—22no—
168#11765850—22no—
169#11765850—21no—
170#11765849—21no—
171#11765849—41no—
172#11765848—41no—
173#11765848—41no—
174#11765848—41no—
175C2×C71⋊C8—41no—
176C2×C71⋊C8C2×D411yescharacteristic
177C2×C71⋊C8C2×D411yescharacteristic
178C8×D71—21no—
179C8×D71—21no—
180C8×D71—41no—
181C8×D71—41no—
182C8×D71—41no—
183C8×D71C22×C411yescharacteristic
184C8×D71C22×C411yescharacteristic
185C8×D71#4111yescharacteristic
186C8×D71#4111yescharacteristic
187C4⋊Dic71—21no—
188Dic71⋊C4—41no—
189C22⋊Dic71—41no—
190(C2×D71)⋊C4—41no—
191(C2×D71)⋊C4—41no—
192C4×Dic71—21no—
193C4×Dic71C2×D411yescharacteristic
194C71⋊C16—21no—
195C71⋊C16C22×C412yescomplemented
196#41×C71—21no—
197C2×D4×C71—21no—
198C22×C4×C71—41no—
199Q16×C71C22×C411yescharacteristic, complemented
200SD16×C71—21no—
201D8×C71—21no—
202D8×C71C22×C412yes—
203D8×C71C22×C411yescharacteristic, complemented
204M16×C71—21no—
205M16×C71—41no—
206C2×C8×C71—41no—
207C2×C8×C71C2×D411yescharacteristic
208C2×C8×C71C2411yescharacteristic
209C4⋊C4×C71—21no—
210C22⋊C4×C71—41no—
211C42×C71C2×D411yescharacteristic
212C16×C71—21no—
213C16×C71C22×C412yescomplemented
214C23×C71—41no—
215C22×D71—41no—
216C22×D71—41no—
217C22×D71—42no—
218C22×D71—42no—
219C22×D71—82no—
220C22×D71—82no—
221C22×D71AGL(1,ℤ/8)11yescharacteristic
222Q8×C71—21no—
223D4×C71—22no—
224D4×C71—21no—
225D4×C71—41no—
226D4×C71—41no—
227C2×C4×C71—21no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—82no—
231C2×C4×C71C22×D411yescharacteristic
232C22⋊D71—41no—
233C22⋊D71—41no—
234C22⋊D71—42no—
235C22⋊D71—42no—
236C22⋊D71—41no—
237C2×Dic71—21no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—82no—
241C2×Dic71AGL(1,ℤ/8)11yescharacteristic
242D284—22no—
243D284—21no—
244D284—41no—
245D284—41no—
246C4×D71—42no—
247C4×D71—41no—
248C4×D71—41no—
249C4×D71—41no—
250C4×D71—82no—
251C4×D71—82no—
252C4×D71—82no—
253C4×D71C4×D411yescharacteristic
254C4×D71C4×D411yescharacteristic
255C4×D71AGL(1,ℤ/8)11yescharacteristic
256C4×D71AGL(1,ℤ/8)11yescharacteristic
257Dic142—21no—
258Dic142—41no—
259C8×C71—21no—
260C8×C71—41no—
261C8×C71C23×C411yescharacteristic
262C8×C71C2×#4111yescharacteristic
263C71⋊C8—21no—
264C71⋊C8—41no—
265C71⋊C8C4×D411yescharacteristic
266C71⋊C8C4×D411yescharacteristic
267C22×C71—41no—
268C22×C71—42no—
269C22×C71—81no—
270C22×C71—81no—
271C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
272C2×D71—22no—
273C2×D71—41no—
274C2×D71—42no—
275C2×D71—84no—
276C2×D71—81no—
277C2×D71—81no—
278C2×D71—82no—
279C2×D71—84no—
280C2×D71#44211yescharacteristic
281C2×D71#44211yescharacteristic
282C4×C71—41no—
283C4×C71—41no—
284C4×C71—81no—
285C4×C71C2×C4×D411yescharacteristic
286C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
287Dic71—41no—
288Dic71—41no—
289Dic71—81no—
290Dic71#44211yescharacteristic
291Dic71#44211yescharacteristic
292C2×AGL(1,ℤ/16)—711nomaximal, Sylow
293C2×C71—21no—
294C2×C71—82no—
295C2×C71—81no—
296C2×C71C2×#44211yescharacteristic
297D71—21no—
298D71—81no—
299D71—82no—
300D71AGL(1,ℤ/16)12yesdirect product factor
301C2×#509—711no—
302C2×#444—711no—
303C2×#442—711no—
304AGL(1,ℤ/16)—711no—
305AGL(1,ℤ/16)—711no—
306AGL(1,ℤ/16)—711no—
307AGL(1,ℤ/16)—711no—
308AGL(1,ℤ/16)—711no—
309AGL(1,ℤ/16)—711no—
310AGL(1,ℤ/16)—711no—
311AGL(1,ℤ/16)D7111yescharacteristic, direct product factor
312C2×#365—711no—
313C2×#361—711no—
314C2×#360—711no—
315C2×#347—711no—
316C71C2×AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
317C2×#135—711no—
318C22×D8—711no—
319C2×#131—1421no—
320C2×C4×D4—1421no—
321#509—711no—
322#509—711no—
323#509—712no—
324#509—712no—
325#509—711no—
326#509C2×D7111yescharacteristic
327C2×SD32—712no—
328C2×D16—712no—
329C2×M32—711no—
330C2×M32—711no—
331#444—711no—
332#444—711no—
333#444—711no—
334#444—1421no—
335#444—1421no—
336#444C2×D7111yescharacteristic, complemented
337#442—711no—
338#442—711no—
339#442—711no—
340#442—1421no—
341#442—1421no—
342#442C2×D7111yescharacteristic, complemented
343C2×#108—711no—
344C2×#106—711no—
345C2×#104—1421no—
346C2×#102—1421no—
347C2×#97—711no—
348#365—711no—
349#365—711no—
350#365—711no—
351#365C2×D7111yescharacteristic, complemented
352#361—711no—
353#361—711no—
354#361—711no—
355#361C2×D7111yescharacteristic, complemented
356#360—711no—
357#360—711no—
358#360—711no—
359#360C2×D7111yescharacteristic, complemented
360#347—711no—
361#347—711no—
362#347—711no—
363#347C2×D7111yescharacteristic, complemented
364C2×#41—1421no—
365C22×D4—1421no—
366C23×C4—2841no—
367#135—711no—
368#135—711no—
369#135—711no—
370#135—1421no—
371#135—1421no—
372#135C22×D7111yescharacteristic
373C2×Q16—711no—
374C2×SD16—1421no—
375C2×D8—712no—
376C2×D8—712no—
377C2×D8—711no—
378C2×D8—711no—
379C2×D8—1421no—
380C2×D8—1421no—
381C2×D8—1421no—
382C2×D8C22×D7111yescharacteristic
383#131—21no—
384#131—2841no—
385#131—2841no—
386#131—1421no—
387#131—1421no—
388#131—1421no—
389C2×M16—2841no—
390C2×M16—1421no—
391C22×C8—2841no—
392C22×C8—711no—
393C22×C8—711no—
394C4×D4—21no—
395C4×D4—2841no—
396C4×D4—2841no—
397C4×D4—1421no—
398C4×D4—1421no—
399C4×D4—1421no—
400C2×C4⋊C4—1421no—
401C2×C22⋊C4—2841no—
402C2×C42—711no—
403SD32—712no—
404SD32—712no—
405SD32—712no—
406SD32C4×D7112yescomplemented
407D16—712no—
408D16—712no—
409D16—712no—
410D16C4×D7112yescomplemented
411M32—711no—
412M32—712no—
413M32—711no—
414M32—1421no—
415M32C22×D7111yescharacteristic
416M32C22×D7111yescharacteristic
417C2×C16—712no—
418C2×C16—1421no—
419#108—711no—
420#108—1421no—
421#108C22×D7111yescharacteristic
422#106—711no—
423#106—1421no—
424#106C22×D7111yescharacteristic
425#104—21no—
426#104—1421no—
427#104—1421no—
428#104—1421no—
429#102—21no—
430#102—1421no—
431#102—1421no—
432#102—1421no—
433#97—711no—
434#97—711no—
435#97—711no—
436#97C22×D7111yescharacteristic
437C24—2841no—
438#41—21no—
439#41—2841no—
440#41—2841no—
441#41—1421no—
442#41—1421no—
443#41—1421no—
444C2×Q8—1421no—
445C2×D4—21no—
446C2×D4—2841no—
447C2×D4—2841no—
448C2×D4—2841no—
449C2×D4—2841no—
450C2×D4—1422no—
451C2×D4—1422no—
452C2×D4—1421no—
453C2×D4—1421no—
454C22×C4—41no—
455C22×C4—2841no—
456C22×C4—2841no—
457C22×C4—2841no—
458C22×C4—2841no—
459C22×C4—2841no—
460C22×C4—5684no—
461C22×C4—711no—
462C22×C4—1421no—
463Q16—711no—
464Q16—1421no—
465Q16C2×C4×D7111yescharacteristic, complemented
466SD16—21no—
467SD16—1421no—
468SD16—1421no—
469SD16—1421no—
470D8—21no—
471D8—712no—
472D8—711no—
473D8—1422no—
474D8—1422no—
475D8—1421no—
476D8—1421no—
477D8C2×C4×D7112yes—
478D8C2×C4×D7111yescharacteristic, complemented
479M16—21no—
480M16—41no—
481M16—2841no—
482M16—2841no—
483M16—2841no—
484M16—2841no—
485M16—1421no—
486C2×C8—41no—
487C2×C8—2841no—
488C2×C8—2841no—
489C2×C8—2841no—
490C2×C8—2841no—
491C2×C8—711no—
492C2×C8—711no—
493C2×C8—711no—
494C2×C8—711no—
495C2×C8—711no—
496C2×C8—711no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8D4×D7111yescharacteristic
500C2×C8C23×D7111yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—2841no—
503C4⋊C4—1421no—
504C22⋊C4—41no—
505C22⋊C4—2841no—
506C22⋊C4—2841no—
507C22⋊C4—2841no—
508C42—711no—
509C42—1421no—
510C42D4×D7111yescharacteristic
511C16—21no—
512C16—712no—
513C16—1421no—
514C16C2×C4×D7112yescomplemented
515C23—41no—
516C23—2841no—
517C23—2842no—
518C23—2842no—
519C23—2841no—
520C23—5682no—
521C23—5682no—
522C23—711no—
523Q8—21no—
524Q8—2841no—
525Q8—1421no—
526D4—22no—
527D4—21no—
528D4—41no—
529D4—41no—
530D4—2841no—
531D4—2841no—
532D4—2841no—
533D4—2842no—
534D4—2842no—
535D4—2841no—
536D4—2841no—
537D4—1422no—
538D4—1421no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—82no—
543C2×C4—2841no—
544C2×C4—2842no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—5682no—
550C2×C4—5682no—
551C2×C4—5682no—
552C2×C4—5682no—
553C2×C4—711no—
554C2×C4—711no—
555C2×C4—711no—
556C2×C4—711no—
557C2×C4—711no—
558C2×C4—1421no—
559C2×C4C2×D4×D7111yescharacteristic
560C8—21no—
561C8—41no—
562C8—2841no—
563C8—711no—
564C8—711no—
565C8—1421no—
566C8C22×C4×D7111yescharacteristic
567C8#41×D7111yescharacteristic
568C22—41no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—2841no—
573C22—2842no—
574C22—5681no—
575C22—5681no—
576C22—5684no—
577C22—5682no—
578C22—5684no—
579C22—711no—
580C22—711no—
581C22—1422no—
582C22AGL(1,ℤ/8)×D7111yescharacteristic
583C4—41no—
584C4—41no—
585C4—81no—
586C4—2841no—
587C4—2841no—
588C4—5681no—
589C4—711no—
590C4—711no—
591C4C4×D4×D7111yescharacteristic
592C4AGL(1,ℤ/8)×D7111yescharacteristic
593C2—21no—
594C2—82no—
595C2—81no—
596C2—5682no—
597C2—5681no—
598C2—712no—
599C2—1421no—
600C2#442×D7111yescharacteristic
subgroupG/H
centerC2#442×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×AGL(1,ℤ/16)C2
socleC2×C71C2×#442
prime radical#509×D71C2
cosocleC2×C8×C71C24
hypercenterAGL(1,ℤ/16)D71
hypocenterC71C2×AGL(1,ℤ/16)
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ AGL(1,ℤ/16)

Complements

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

NH ≅ G/Ntype
2. #444×D71C2semidirect
3. #442×D71C2semidirect
4. #22590241C2semidirect
5. #22590240C2semidirect
6. #22590239C2semidirect
7. #22590238C2semidirect
8. #22590237C2semidirect
9. #22590236C2semidirect
10. #22590235C2semidirect
11. #365×D71C2semidirect
12. #361×D71C2semidirect
13. #360×D71C2semidirect
14. #347×D71C2semidirect
15. C71×AGL(1,ℤ/16)C2semidirect
21. #17986436C4semidirect
24. #17986433C4semidirect
25. SD32×D71C4semidirect
26. D16×D71C4semidirect
29. #17986122C22semidirect
30. #17986121C22semidirect
31. #17986120C22semidirect
34. #17986112C22semidirect
35. #17986111C22semidirect
36. #17986110C22semidirect
44. #17985823C22semidirect
45. #17985822C22semidirect
46. #17985821C22semidirect
47. #17985813C22semidirect
48. #17985812C22semidirect
49. #17985811C22semidirect
50. #17985810C22semidirect
51. #17985809C22semidirect
52. #17985808C22semidirect
53. #17985777C22semidirect
54. #17985776C22semidirect
55. #17985776C22semidirect
57. #444×C71C22semidirect
58. #442×C71C22semidirect
59. #365×C71C22semidirect
60. #361×C71C22semidirect
61. #360×C71C22semidirect
62. #347×C71C22semidirect
66. #14556151C2×C4semidirect
70. #14556147C2×C4semidirect
71. Q16×D71C2×C4semidirect
79. D8×D71C2×C4semidirect
98. #14556073C2×C4semidirect
99. #14556072C2×C4semidirect
100. #14556071C2×C4semidirect
101. D1136C2×C4semidirect
102. #14556069C2×C4semidirect
103. #14556069C2×C4semidirect
107. #14556067C2×C4semidirect
109. C16×D71C2×C4semidirect
127. SD32×C71C2×C4semidirect
128. D16×C71C2×C4semidirect
159. Dic284C22×C4semidirect
166. D568C22×C4semidirect
195. C71⋊C16C22×C4semidirect
199. Q16×C71C22×C4semidirect
203. D8×C71C22×C4semidirect
213. C16×C71C22×C4semidirect
300. D71AGL(1,ℤ/16)direct (N × H)
311. AGL(1,ℤ/16)D71direct (N × H)
316. C71C2×AGL(1,ℤ/16)semidirect
336. #444C2×D71semidirect
342. #442C2×D71semidirect
351. #365C2×D71semidirect
355. #361C2×D71semidirect
359. #360C2×D71semidirect
363. #347C2×D71semidirect
406. SD32C4×D71semidirect
410. D16C4×D71semidirect
465. Q16C2×C4×D71semidirect
478. D8C2×C4×D71semidirect
514. C16C2×C4×D71semidirect

Maximal subgroups

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

subgroupindex
#509×D712
#444×D712
#442×D712
#225902412
#225902402
#225902392
#225902382
#225902372
#225902362
#225902352
#365×D712
#361×D712
#360×D712
#347×D712
C71×AGL(1,ℤ/16)2
C2×AGL(1,ℤ/16)71