Finite Group Databasebeta

← D61×AGL(1,ℤ/16) #29833450

Subgroups — D61×AGL(1,ℤ/16) #29833450

Order 15616.

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×D61C211yesmaximal, characteristic
2#444×D61C211yesmaximal, characteristic, complemented
3#442×D61C211yesmaximal, characteristic, complemented
4#21447720C211yesmaximal, characteristic, complemented
5#21447719C211yesmaximal, characteristic, complemented
6#21447718C211yesmaximal, characteristic, complemented
7#21447717C211yesmaximal, characteristic, complemented
8#21447716C211yesmaximal, characteristic, complemented
9#21447715C211yesmaximal, characteristic, complemented
10#21447714C211yesmaximal, characteristic, complemented
11#365×D61C211yesmaximal, characteristic, complemented
12#361×D61C211yesmaximal, characteristic, complemented
13#360×D61C211yesmaximal, characteristic, complemented
14#347×D61C211yesmaximal, characteristic, complemented
15C61×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#135×D61C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#131×D61—21no—
19C4×D4×D61—21no—
20#16840200C2211yescharacteristic
21#16840199C412yescomplemented
22#16840198C2211yescharacteristic
23#16840197C2211yescharacteristic
24#16840196C412yescomplemented
25SD32×D61C412yescomplemented
26D16×D61C412yescomplemented
27M32×D61C2211yescharacteristic
28M32×D61C2211yescharacteristic
29#16839885C2211yescharacteristic, complemented
30#16839884C2211yescharacteristic, complemented
31#16839883C2211yescharacteristic, complemented
32#16839882—21no—
33#16839881—21no—
34#16839875C2211yescharacteristic, complemented
35#16839874C2211yescharacteristic, complemented
36#16839873C2211yescharacteristic, complemented
37#16839872—21no—
38#16839871—21no—
39#108×D61C2211yescharacteristic
40#106×D61C2211yescharacteristic
41#104×D61—21no—
42#102×D61—21no—
43#97×D61C2211yescharacteristic
44#16839586C2211yescharacteristic, complemented
45#16839585C2211yescharacteristic, complemented
46#16839584C2211yescharacteristic, complemented
47#16839576C2211yescharacteristic, complemented
48#16839575C2211yescharacteristic, complemented
49#16839574C2211yescharacteristic, complemented
50#16839573C2211yescharacteristic, complemented
51#16839572C2211yescharacteristic, complemented
52#16839571C2211yescharacteristic, complemented
53#16839540C2211yescharacteristic, complemented
54#16839539C2211yescharacteristic, complemented
55#16839539C2211yescharacteristic, complemented
56C61×#509C2211yescharacteristic
57C61×#444C2211yescharacteristic, complemented
58C61×#442C2211yescharacteristic, complemented
59C61×#365C2211yescharacteristic, complemented
60C61×#361C2211yescharacteristic, complemented
61C61×#360C2211yescharacteristic, complemented
62C61×#347C2211yescharacteristic, complemented
63#41×D61—21no—
64C2×D4×D61—21no—
65C22×C4×D61—41no—
66#14519800C2×C411yescharacteristic, complemented
67#14519799C2311yescharacteristic
68#14519798—21no—
69#14519797—21no—
70#14519796C2×C411yescharacteristic, complemented
71Q16×D61C2×C411yescharacteristic, complemented
72SD16×D61—21no—
73C2×D488C2311yescharacteristic
74C2×#11758199—21no—
75D8×D61—21no—
76D8×D61—21no—
77D8×D61C2×C412yes—
78D8×D61C2×C412yes—
79D8×D61C2×C411yescharacteristic, complemented
80#14519785—21no—
81#14519785—41no—
82#14519784—21no—
83#14519783—21no—
84#14519783—41no—
85M16×D61—21no—
86M16×D61—41no—
87C2×C8×D61—41no—
88C2×C8×D61D411yescharacteristic
89C2×C8×D61C2311yescharacteristic
90C4×D244—21no—
91D4×Dic61—21no—
92#14519739—21no—
93C4×C22⋊D61—41no—
94#14519737—41no—
95C4⋊C4×D61—21no—
96C22⋊C4×D61—41no—
97C42×D61D411yescharacteristic
98#14519722C2×C412yescomplemented
99#14519721C2×C412yescomplemented
100#14519720C2×C412yescomplemented
101D976C2×C412yescomplemented
102#14519718C2×C412yescomplemented
103#14519718C2×C412yescomplemented
104#14519717C2311yescharacteristic
105#14519717C2311yescharacteristic
106#14519716—21no—
107#14519716C2×C412yescomplemented
108C16×D61—21no—
109C16×D61C2×C412yescomplemented
110#14519713C2311yescharacteristic
111#14519712—21no—
112#14519709C2311yescharacteristic
113#14519708—21no—
114#14519705—21no—
115#14519704—21no—
116#14519703—21no—
117#14519699—21no—
118#14519698—21no—
119#14519697—21no—
120#14519688D411yescharacteristic
121#14519687D411yescharacteristic
122#14519687C2311yescharacteristic
123#135×C61C2311yescharacteristic
124C2×D8×C61C2311yescharacteristic
125#131×C61—21no—
126C4×D4×C61—21no—
127SD32×C61C2×C412yescomplemented
128D16×C61C2×C412yescomplemented
129M32×C61C2311yescharacteristic
130M32×C61C2311yescharacteristic
131#108×C61C2311yescharacteristic
132#106×C61C2311yescharacteristic
133#104×C61—21no—
134#102×C61—21no—
135#97×C61C2311yescharacteristic
136C23×D61—41no—
137#41×C61—21no—
138C2×D4×C61—21no—
139C22×C4×C61—41no—
140C2×C22⋊D61—41no—
141C22×Dic61—41no—
142#11758226—21no—
143Q8×D61—21no—
144#11758224—21no—
145#11758224—41no—
146D4×D61—22no—
147D4×D61—22no—
148D4×D61—21no—
149D4×D61—41no—
150D4×D61—41no—
151D4×D61—41no—
152#11758222—21no—
153#11758222—41no—
154C2×D244—21no—
155C2×C4×D61—21no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—84no—
161C2×C4×D61C2×D411yescharacteristic
162Q16×C61C22×C411yescharacteristic, complemented
163SD16×C61—21no—
164D8×C61—21no—
165D8×C61C22×C412yes—
166D8×C61C22×C411yescharacteristic, complemented
167M16×C61—21no—
168M16×C61—41no—
169C2×C8×C61—41no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2411yescharacteristic
172C4⋊C4×C61—21no—
173C22⋊C4×C61—41no—
174C42×C61C2×D411yescharacteristic
175C22⋊Dic61—41no—
176#11758202—21no—
177#11758201—21no—
178#11758200—21no—
179#11758199—22no—
180#11758199—22no—
181#11758199—21no—
182#11758198—41no—
183#11758198—41no—
184C4⋊Dic61—21no—
185#11758196—41no—
186C4×Dic61—21no—
187C4×Dic61C2×D411yescharacteristic
188#11758194—21no—
189#11758194—41no—
190C2×#92418—41no—
191C2×#92418C2×D411yescharacteristic
192C2×#92418C2×D411yescharacteristic
193Dic244C22×C411yescharacteristic, complemented
194D488—21no—
195D488C22×C412yes—
196D488C22×C411yescharacteristic, complemented
197#11758190—21no—
198#11758189—41no—
199#11758189—41no—
200#11758189—41no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—41no—
204C8×D61—41no—
205C8×D61—41no—
206C8×D61C22×C411yescharacteristic
207C8×D61C22×C411yescharacteristic
208C8×D61#4111yescharacteristic
209C8×D61#4111yescharacteristic
210C16×C61—21no—
211C16×C61C22×C412yescomplemented
212#11758185—21no—
213#11758185C22×C412yescomplemented
214C23×C61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61AGL(1,ℤ/8)11yescharacteristic
222Q8×C61—21no—
223D4×C61—22no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61C22×D411yescharacteristic
232C22⋊D61—41no—
233C22⋊D61—41no—
234C22⋊D61—41no—
235C22⋊D61—42no—
236C22⋊D61—42no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241C2×Dic61AGL(1,ℤ/8)11yescharacteristic
242D244—21no—
243D244—22no—
244D244—41no—
245D244—41no—
246C4×D61—42no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—82no—
251C4×D61—82no—
252C4×D61—82no—
253C4×D61C4×D411yescharacteristic
254C4×D61C4×D411yescharacteristic
255C4×D61AGL(1,ℤ/8)11yescharacteristic
256C4×D61AGL(1,ℤ/8)11yescharacteristic
257Dic122—21no—
258Dic122—41no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61C23×C411yescharacteristic
262C8×C61C2×#4111yescharacteristic
263#92418—21no—
264#92418—41no—
265#92418C4×D411yescharacteristic
266#92418C4×D411yescharacteristic
267C2×AGL(1,ℤ/16)—611nomaximal, Sylow
268C22×C61—41no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C2×D61—22no—
274C2×D61—41no—
275C2×D61—42no—
276C2×D61—84no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—82no—
280C2×D61—84no—
281C2×D61#44211yescharacteristic
282C2×D61#44211yescharacteristic
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—81no—
286C4×C61C2×C4×D411yescharacteristic
287C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61#44211yescharacteristic
292Dic61#44211yescharacteristic
293C2×#509—611no—
294C2×#444—611no—
295C2×#442—611no—
296AGL(1,ℤ/16)—611no—
297AGL(1,ℤ/16)—611no—
298AGL(1,ℤ/16)—611no—
299AGL(1,ℤ/16)—611no—
300AGL(1,ℤ/16)—611no—
301AGL(1,ℤ/16)—611no—
302AGL(1,ℤ/16)—611no—
303AGL(1,ℤ/16)D6111yescharacteristic, direct product factor
304C2×#365—611no—
305C2×#361—611no—
306C2×#360—611no—
307C2×#347—611no—
308C2×C61—21no—
309C2×C61—82no—
310C2×C61—81no—
311C2×C61C2×#44211yescharacteristic
312D61—21no—
313D61—81no—
314D61—82no—
315D61AGL(1,ℤ/16)12yesdirect product factor
316C2×#135—611no—
317C22×D8—611no—
318C2×#131—1221no—
319C2×C4×D4—1221no—
320#509—611no—
321#509—611no—
322#509—612no—
323#509—612no—
324#509—611no—
325#509C2×D6111yescharacteristic
326C2×SD32—612no—
327C2×D16—612no—
328C2×M32—611no—
329C2×M32—611no—
330#444—611no—
331#444—611no—
332#444—611no—
333#444—1221no—
334#444—1221no—
335#444C2×D6111yescharacteristic, complemented
336#442—611no—
337#442—611no—
338#442—611no—
339#442—1221no—
340#442—1221no—
341#442C2×D6111yescharacteristic, complemented
342C2×#108—611no—
343C2×#106—611no—
344C2×#104—1221no—
345C2×#102—1221no—
346C2×#97—611no—
347#365—611no—
348#365—611no—
349#365—611no—
350#365C2×D6111yescharacteristic, complemented
351#361—611no—
352#361—611no—
353#361—611no—
354#361C2×D6111yescharacteristic, complemented
355#360—611no—
356#360—611no—
357#360—611no—
358#360C2×D6111yescharacteristic, complemented
359#347—611no—
360#347—611no—
361#347—611no—
362#347C2×D6111yescharacteristic, complemented
363C61C2×AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
364C2×#41—1221no—
365C22×D4—1221no—
366C23×C4—2441no—
367#135—611no—
368#135—611no—
369#135—611no—
370#135—1221no—
371#135—1221no—
372#135C22×D6111yescharacteristic
373C2×Q16—611no—
374C2×SD16—1221no—
375C2×D8—612no—
376C2×D8—612no—
377C2×D8—611no—
378C2×D8—611no—
379C2×D8—1221no—
380C2×D8—1221no—
381C2×D8—1221no—
382C2×D8C22×D6111yescharacteristic
383#131—21no—
384#131—1221no—
385#131—1221no—
386#131—1221no—
387#131—2441no—
388#131—2441no—
389C2×M16—1221no—
390C2×M16—2441no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—2441no—
394C4×D4—21no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—2441no—
399C4×D4—2441no—
400C2×C4⋊C4—1221no—
401C2×C22⋊C4—2441no—
402C2×C42—611no—
403SD32—612no—
404SD32—612no—
405SD32—612no—
406SD32C4×D6112yescomplemented
407D16—612no—
408D16—612no—
409D16—612no—
410D16C4×D6112yescomplemented
411M32—611no—
412M32—612no—
413M32—611no—
414M32—1221no—
415M32C22×D6111yescharacteristic
416M32C22×D6111yescharacteristic
417C2×C16—612no—
418C2×C16—1221no—
419#108—611no—
420#108—1221no—
421#108C22×D6111yescharacteristic
422#106—611no—
423#106—1221no—
424#106C22×D6111yescharacteristic
425#104—21no—
426#104—1221no—
427#104—1221no—
428#104—1221no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#97—611no—
434#97—611no—
435#97—611no—
436#97C22×D6111yescharacteristic
437C24—2441no—
438#41—21no—
439#41—1221no—
440#41—1221no—
441#41—1221no—
442#41—2441no—
443#41—2441no—
444C2×Q8—1221no—
445C2×D4—21no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2441no—
454C22×C4—41no—
455C22×C4—611no—
456C22×C4—1221no—
457C22×C4—4884no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463Q16—611no—
464Q16—1221no—
465Q16C2×C4×D6111yescharacteristic, complemented
466SD16—21no—
467SD16—1221no—
468SD16—1221no—
469SD16—1221no—
470D8—21no—
471D8—611no—
472D8—612no—
473D8—1222no—
474D8—1222no—
475D8—1221no—
476D8—1221no—
477D8C2×C4×D6112yes—
478D8C2×C4×D6111yescharacteristic, complemented
479M16—21no—
480M16—41no—
481M16—1221no—
482M16—2441no—
483M16—2441no—
484M16—2441no—
485M16—2441no—
486C2×C8—41no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2441no—
496C2×C8—2441no—
497C2×C8—2441no—
498C2×C8—2441no—
499C2×C8D4×D6111yescharacteristic
500C2×C8C23×D6111yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—1221no—
503C4⋊C4—2441no—
504C22⋊C4—41no—
505C22⋊C4—2441no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—611no—
509C42—1221no—
510C42D4×D6111yescharacteristic
511C16—21no—
512C16—612no—
513C16—1221no—
514C16C2×C4×D6112yescomplemented
515C23—41no—
516C23—611no—
517C23—4882no—
518C23—4882no—
519C23—2441no—
520C23—2442no—
521C23—2442no—
522C23—2441no—
523Q8—21no—
524Q8—1221no—
525Q8—2441no—
526D4—22no—
527D4—21no—
528D4—41no—
529D4—41no—
530D4—1222no—
531D4—1221no—
532D4—2441no—
533D4—2442no—
534D4—2442no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—82no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—1221no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C2×D4×D6111yescharacteristic
560C8—21no—
561C8—41no—
562C8—611no—
563C8—611no—
564C8—1221no—
565C8—2441no—
566C8C22×C4×D6111yescharacteristic
567C8#41×D6111yescharacteristic
568C22—41no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—611no—
573C22—611no—
574C22—1222no—
575C22—4881no—
576C22—4881no—
577C22—4884no—
578C22—4882no—
579C22—4884no—
580C22—2441no—
581C22—2442no—
582C22AGL(1,ℤ/8)×D6111yescharacteristic
583C4—41no—
584C4—41no—
585C4—81no—
586C4—611no—
587C4—611no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C4×D4×D6111yescharacteristic
592C4AGL(1,ℤ/8)×D6111yescharacteristic
593C2—21no—
594C2—82no—
595C2—81no—
596C2—612no—
597C2—1221no—
598C2—4882no—
599C2—4881no—
600C2#442×D6111yescharacteristic
subgroupG/H
centerC2#442×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×AGL(1,ℤ/16)C2
socleC2×C61C2×#442
prime radical#509×D61C2
cosocleC2×C8×C61C24
hypercenterAGL(1,ℤ/16)D61
hypocenterC61C2×AGL(1,ℤ/16)
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
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×D61C2semidirect
3. #442×D61C2semidirect
4. #21447720C2semidirect
5. #21447719C2semidirect
6. #21447718C2semidirect
7. #21447717C2semidirect
8. #21447716C2semidirect
9. #21447715C2semidirect
10. #21447714C2semidirect
11. #365×D61C2semidirect
12. #361×D61C2semidirect
13. #360×D61C2semidirect
14. #347×D61C2semidirect
15. C61×AGL(1,ℤ/16)C2semidirect
21. #16840199C4semidirect
24. #16840196C4semidirect
25. SD32×D61C4semidirect
26. D16×D61C4semidirect
29. #16839885C22semidirect
30. #16839884C22semidirect
31. #16839883C22semidirect
34. #16839875C22semidirect
35. #16839874C22semidirect
36. #16839873C22semidirect
44. #16839586C22semidirect
45. #16839585C22semidirect
46. #16839584C22semidirect
47. #16839576C22semidirect
48. #16839575C22semidirect
49. #16839574C22semidirect
50. #16839573C22semidirect
51. #16839572C22semidirect
52. #16839571C22semidirect
53. #16839540C22semidirect
54. #16839539C22semidirect
55. #16839539C22semidirect
57. C61×#444C22semidirect
58. C61×#442C22semidirect
59. C61×#365C22semidirect
60. C61×#361C22semidirect
61. C61×#360C22semidirect
62. C61×#347C22semidirect
66. #14519800C2×C4semidirect
70. #14519796C2×C4semidirect
71. Q16×D61C2×C4semidirect
79. D8×D61C2×C4semidirect
98. #14519722C2×C4semidirect
99. #14519721C2×C4semidirect
100. #14519720C2×C4semidirect
101. D976C2×C4semidirect
102. #14519718C2×C4semidirect
103. #14519718C2×C4semidirect
107. #14519716C2×C4semidirect
109. C16×D61C2×C4semidirect
127. SD32×C61C2×C4semidirect
128. D16×C61C2×C4semidirect
162. Q16×C61C22×C4semidirect
166. D8×C61C22×C4semidirect
193. Dic244C22×C4semidirect
196. D488C22×C4semidirect
211. C16×C61C22×C4semidirect
213. #11758185C22×C4semidirect
303. AGL(1,ℤ/16)D61direct (N × H)
315. D61AGL(1,ℤ/16)direct (N × H)
335. #444C2×D61semidirect
341. #442C2×D61semidirect
350. #365C2×D61semidirect
354. #361C2×D61semidirect
358. #360C2×D61semidirect
362. #347C2×D61semidirect
363. C61C2×AGL(1,ℤ/16)semidirect
406. SD32C4×D61semidirect
410. D16C4×D61semidirect
465. Q16C2×C4×D61semidirect
478. D8C2×C4×D61semidirect
514. C16C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
#509×D612
#444×D612
#442×D612
#214477202
#214477192
#214477182
#214477172
#214477162
#214477152
#214477142
#365×D612
#361×D612
#360×D612
#347×D612
C61×AGL(1,ℤ/16)2
C2×AGL(1,ℤ/16)61