Finite Group Databasebeta

← D61×#1717 #29667749

Subgroups — D61×#1717 #29667749

Order 15616.

966 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#493×D61C211yesmaximal, characteristic
2#492×D61C211yesmaximal, characteristic, complemented
3#487×D61C211yesmaximal, characteristic, complemented
4#486×D61C211yesmaximal, characteristic, complemented
5#486×D61C211yesmaximal, characteristic, complemented
6#486×D61C211yesmaximal, characteristic, complemented
7#21445461C211yesmaximal, characteristic, complemented
8#21445460C211yesmaximal, characteristic, complemented
9#21445459C211yesmaximal, characteristic, complemented
10#21445458C211yesmaximal, characteristic, complemented
11#21445457C211yesmaximal, characteristic, complemented
12#21445456C211yesmaximal, characteristic, complemented
13#21445455C211yesmaximal, characteristic, complemented
14#322×D61C211yesmaximal, characteristic, complemented
15C61×#1717C211yesmaximal, characteristic, complemented
16C2×SD16×D61—22no—
17C2×D8×D61—22no—
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21#127×D61C2211yescharacteristic
22#16840139C2211yescharacteristic
23#16840138C2211yescharacteristic
24#16840138C2211yescharacteristic
25#16840137—22no—
26#16840136C2211yescharacteristic, complemented
27#16840135C2211yescharacteristic, complemented
28#16840134C2211yescharacteristic, complemented
29#16840133—22no—
30#16840114C2211yescharacteristic, complemented
31#16840113C2211yescharacteristic, complemented
32#16840112C2211yescharacteristic, complemented
33#16840111—22no—
34#16840110C2211yescharacteristic, complemented
35#16840110C2211yescharacteristic, complemented
36#16840110C2211yescharacteristic, complemented
37#16840109C2211yescharacteristic, complemented
38#16840109C2211yescharacteristic, complemented
39#16840109C2211yescharacteristic, complemented
40#16840108C2211yescharacteristic, complemented
41#16840108C2211yescharacteristic, complemented
42#16840108C2211yescharacteristic, complemented
43#16840107—22no—
44#16840107—22no—
45#16840107—22no—
46#103×D61—22no—
47#102×D61—22no—
48#102×D61—22no—
49#102×D61—22no—
50C4×C8×D61C2211yescharacteristic
51C4×C8×D61C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic, complemented
54#16839484C2211yescharacteristic, complemented
55#16839483C2211yescharacteristic, complemented
56C61×#493C2211yescharacteristic
57C61×#492C2211yescharacteristic, complemented
58C61×#487C2211yescharacteristic, complemented
59C61×#486C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic, complemented
62C61×#322C2211yescharacteristic, complemented
63C2×Q8×D61—21no—
64C2×Q8×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—21no—
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×#11758190—22no—
72C2×#11758200—22no—
73C2×#11758201—22no—
74SD16×D61—48no—
75C2×D488—22no—
76C2×#11758199—22no—
77C2×#11758199—22no—
78D8×D61—48no—
79C2×C8×D61D412yes—
80C2×C8×D61D412yes—
81C2×C8×D61D412yes—
82#14519777C2311yescharacteristic
83#14519777C2311yescharacteristic
84#14519776—21no—
85#14519776—21no—
86#14519776—21no—
87#14519776—21no—
88#14519775—21no—
89#14519775—21no—
90#14519774C2311yescharacteristic
91#14519774C2311yescharacteristic
92#14519773—21no—
93#14519773—21no—
94#14519773—21no—
95#14519773—21no—
96#14519773—21no—
97#14519773—21no—
98C4⋊C4×D61—21no—
99C4⋊C4×D61—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C42×D61C2311yescharacteristic
103#14519702—22no—
104#14519701—22no—
105#14519700—22no—
106#14519699—22no—
107#14519699—22no—
108#14519699—22no—
109#14519698—22no—
110#14519698—22no—
111#14519698—22no—
112#14519697—22no—
113#14519697—22no—
114#14519697—22no—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C8×Dic61D412yes—
120C8×Dic61D412yes—
121C2×SD16×C61—22no—
122C2×D8×C61—22no—
123C4⋊Q8×C61C2311yescharacteristic
124C4⋊Q8×C61C2311yescharacteristic
125#127×C61C2311yescharacteristic
126#127×C61C2311yescharacteristic
127#103×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130#102×C61—22no—
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C4×C8×C61C2311yescharacteristic
134C23×D61—41no—
135C23×D61—41no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150Q8×D61—44no—
151Q8×D61—44no—
152D4×D61—44no—
153D4×D61—44no—
154D4×D61—44no—
155D4×D61—44no—
156D4×D61—44no—
157D4×D61—44no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122—41no—
173SD16×C61—44no—
174D8×C61—44no—
175C2×C8×C61C2×D412yes—
176C2×C8×C61C2×D412yes—
177C2×C8×C61C2×D412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C42×C61C2411yescharacteristic
183#11758201—44no—
184#11758200—44no—
185#11758199—44no—
186#11758199—44no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—41no—
192#11758196—41no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C2×#92418C2×D412yes—
199C2×#92418C2×D412yes—
200C2×#92418C2×D412yes—
201D488—44no—
202#11758190—44no—
203C8×D61—24no—
204C8×D61—24no—
205C8×D61—24no—
206C23×C61—41no—
207C23×C61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—82no—
211C22×D61—82no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61#12711yescharacteristic
217Q8×C61—42no—
218Q8×C61—42no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—42no—
224D4×C61—42no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61#12711yescharacteristic
239C2×Dic61#12711yescharacteristic
240C2×Dic61#12711yescharacteristic
241D244—42no—
242D244—42no—
243D244—42no—
244D244—42no—
245D244—42no—
246D244—42no—
247C4×D61—82no—
248C4×D61—82no—
249C4×D61#13514yes—
250C4×D61AGL(1,ℤ/8)14yes—
251C4×D61AGL(1,ℤ/8)14yes—
252Dic122—42no—
253Dic122—42no—
254Dic122—82no—
255Dic122—82no—
256C8×C61—22no—
257C8×C61—22no—
258C8×C61—22no—
259#92418—22no—
260#92418—22no—
261#92418—22no—
262C2×#1717—611nomaximal, Sylow
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12711yescharacteristic
270C2×D61—81no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—164no—
277C2×D61—164no—
278C2×D61#49512yes—
279C2×D61#49612yes—
280C2×D61#49612yes—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61C2×#13512yes—
284C4×C61C2×AGL(1,ℤ/8)12yes—
285C4×C61C2×AGL(1,ℤ/8)12yes—
286Dic61—81no—
287Dic61—81no—
288Dic61#49512yes—
289Dic61#49612yes—
290Dic61#49612yes—
291C2×#493—611no—
292C2×#492—611no—
293C2×#487—611no—
294C2×#486—611no—
295C2×#486—611no—
296C2×#486—611no—
297#1717—611no—
298#1717—611no—
299#1717—611no—
300#1717—611no—
301#1717—611no—
302#1717—611no—
303#1717—611no—
304#1717D6111yescharacteristic, direct product factor
305C2×#322—611no—
306C2×C61—161no—
307C2×C61—161no—
308C2×C61C2×#49511yescharacteristic
309C2×C61C2×#49611yescharacteristic
310C2×C61C2×#49611yescharacteristic
311D61—161no—
312D61—161no—
313D61#171714yesdirect product factor
314C22×SD16—1222no—
315C22×D8—1222no—
316C2×C4⋊Q8—611no—
317C2×C4⋊Q8—611no—
318C2×#127—611no—
319C2×#127—611no—
320#493—611no—
321#493—611no—
322#493—611no—
323#493—1222no—
324#493C2×D6111yescharacteristic
325#492—611no—
326#492—611no—
327#492—611no—
328#492—1222no—
329#492C2×D6111yescharacteristic, complemented
330#487—611no—
331#487—611no—
332#487—611no—
333#487—1222no—
334#487C2×D6111yescharacteristic, complemented
335#486—611no—
336#486—611no—
337#486—611no—
338#486—611no—
339#486—611no—
340#486—611no—
341#486—611no—
342#486—611no—
343#486—611no—
344#486—1222no—
345#486—1222no—
346#486—1222no—
347#486C2×D6111yescharacteristic, complemented
348#486C2×D6111yescharacteristic, complemented
349#486C2×D6111yescharacteristic, complemented
350C2×#103—1222no—
351C2×#102—1222no—
352C2×#102—1222no—
353C2×#102—1222no—
354C2×C4×C8—611no—
355C2×C4×C8—611no—
356C2×C4×C8—611no—
357#322—611no—
358#322—611no—
359#322—611no—
360#322C2×D6111yescharacteristic, complemented
361C61C2×#171711yescharacteristic, complemented, Sylow
362C22×Q8—1221no—
363C22×Q8—1221no—
364C22×D4—1221no—
365C22×D4—1221no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4—1221no—
370C2×SD16—22no—
371C2×SD16—1222no—
372C2×SD16—1222no—
373C2×SD16—1222no—
374C2×SD16—2448no—
375C2×D8—22no—
376C2×D8—1222no—
377C2×D8—1222no—
378C2×D8—1222no—
379C2×D8—2448no—
380C22×C8—612no—
381C22×C8—612no—
382C22×C8—612no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—1221no—
390C4⋊Q8—1221no—
391C4⋊Q8C22×D6111yescharacteristic
392C4⋊Q8C22×D6111yescharacteristic
393#127—611no—
394#127—611no—
395#127—1221no—
396#127—1221no—
397#127—1221no—
398#127—1221no—
399#127—1221no—
400#127—1221no—
401#127C22×D6111yescharacteristic
402#127C22×D6111yescharacteristic
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C42—611no—
408#103—22no—
409#103—1222no—
410#103—1222no—
411#103—1222no—
412#102—22no—
413#102—22no—
414#102—22no—
415#102—1222no—
416#102—1222no—
417#102—1222no—
418#102—1222no—
419#102—1222no—
420#102—1222no—
421#102—1222no—
422#102—1222no—
423#102—1222no—
424C4×C8—612no—
425C4×C8—612no—
426C4×C8—612no—
427C4×C8—611no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8C22×D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433C24—2441no—
434C24—2441no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—2441no—
440C2×Q8—2441no—
441C2×Q8—2444no—
442C2×Q8—2444no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—1221no—
450C2×D4—1221no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—2444no—
456C2×D4—2444no—
457C2×D4—2444no—
458C2×D4—2444no—
459C2×D4—2444no—
460C2×D4—2444no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2441no—
464C2×D4—2441no—
465C2×D4—2441no—
466C2×D4—2441no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—2441no—
471C22×C4—2441no—
472SD16—44no—
473SD16—2444no—
474SD16—2444no—
475SD16—2444no—
476D8—44no—
477D8—2444no—
478D8—2444no—
479D8—2444no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—612no—
483C2×C8—1224no—
484C2×C8—1224no—
485C2×C8—1224no—
486C2×C8D4×D6112yes—
487C2×C8D4×D6112yes—
488C2×C8D4×D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42C23×D6111yescharacteristic
505C23—41no—
506C23—41no—
507C23—611no—
508C23—4882no—
509C23—4882no—
510C23—4882no—
511C23—4882no—
512C23—4882no—
513C23—4882no—
514C23—2441no—
515C23—2441no—
516Q8—42no—
517Q8—42no—
518Q8—4882no—
519Q8—4882no—
520Q8—2442no—
521Q8—2442no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—4882no—
529D4—4882no—
530D4—4882no—
531D4—4882no—
532D4—4882no—
533D4—4882no—
534D4—2442no—
535D4—2442no—
536D4—2442no—
537D4—2442no—
538D4—2442no—
539D4—2442no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—614no—
546C2×C4—614no—
547C2×C4—614no—
548C2×C4—4882no—
549C2×C4—4882no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C8—22no—
556C8—22no—
557C8—22no—
558C8—1222no—
559C8—1222no—
560C8—1222no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—612no—
568C22—612no—
569C22—612no—
570C22—9764no—
571C22—9764no—
572C22—4881no—
573C22—4881no—
574C22—4881no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22#127×D6111yescharacteristic
579C4—81no—
580C4—81no—
581C4—612no—
582C4—612no—
583C4—612no—
584C4—4881no—
585C4—4881no—
586C4#135×D6112yes—
587C4AGL(1,ℤ/8)×D6112yes—
588C4AGL(1,ℤ/8)×D6112yes—
589C2—161no—
590C2—161no—
591C2—614no—
592C2—9761no—
593C2—9761no—
594C2#495×D6111yescharacteristic
595C2#496×D6111yescharacteristic
596C2#496×D6111yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1717C2
socleC22×C61C2×#127
prime radical#493×D61C2
hypercenter#1717D61
hypocenterC61C2×#1717
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1717

Complements

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

NH ≅ G/Ntype
2. #492×D61C2semidirect
3. #487×D61C2semidirect
4. #486×D61C2semidirect
5. #486×D61C2semidirect
6. #486×D61C2semidirect
7. #21445461C2semidirect
8. #21445460C2semidirect
9. #21445459C2semidirect
10. #21445458C2semidirect
11. #21445457C2semidirect
12. #21445456C2semidirect
13. #21445455C2semidirect
14. #322×D61C2semidirect
15. C61×#1717C2semidirect
26. #16840136C22semidirect
27. #16840135C22semidirect
28. #16840134C22semidirect
30. #16840114C22semidirect
31. #16840113C22semidirect
32. #16840112C22semidirect
34. #16840110C22semidirect
35. #16840110C22semidirect
36. #16840110C22semidirect
37. #16840109C22semidirect
38. #16840109C22semidirect
39. #16840109C22semidirect
40. #16840108C22semidirect
41. #16840108C22semidirect
42. #16840108C22semidirect
53. #16839484C22semidirect
54. #16839484C22semidirect
55. #16839483C22semidirect
57. C61×#492C22semidirect
58. C61×#487C22semidirect
59. C61×#486C22semidirect
60. C61×#486C22semidirect
61. C61×#486C22semidirect
62. C61×#322C22semidirect
304. #1717D61direct (N × H)
313. D61#1717direct (N × H)
329. #492C2×D61semidirect
334. #487C2×D61semidirect
347. #486C2×D61semidirect
348. #486C2×D61semidirect
349. #486C2×D61semidirect
360. #322C2×D61semidirect
361. C61C2×#1717semidirect

Maximal subgroups

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

subgroupindex
#493×D612
#492×D612
#487×D612
#486×D612
#214454612
#214454602
#214454592
#214454582
#214454572
#214454562
#214454552
#322×D612
C61×#17172
C2×#171761