Finite Group Databasebeta

← D61×#1719 #29667765

Subgroups — D61×#1719 #29667765

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D61C211yesmaximal, characteristic, complemented
2#492×D61C211yesmaximal, characteristic
3#487×D61C211yesmaximal, characteristic, complemented
4#487×D61C211yesmaximal, characteristic, complemented
5#486×D61C211yesmaximal, characteristic
6#486×D61C211yesmaximal, characteristic
7#21445475C211yesmaximal, characteristic, complemented
8#21445474C211yesmaximal, characteristic, complemented
9#21445473C211yesmaximal, characteristic, complemented
10#21445472C211yesmaximal, characteristic, complemented
11#21445471C211yesmaximal, characteristic, complemented
12#21445470C211yesmaximal, characteristic, complemented
13#21445469C211yesmaximal, characteristic, complemented
14#322×D61C211yesmaximal, characteristic, complemented
15C61×#1719C211yesmaximal, characteristic, complemented
16C2×SD16×D61—22no—
17C4⋊Q8×D61C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21#16840167C2211yescharacteristic, complemented
22#16840166—22no—
23#16840165C2211yescharacteristic, complemented
24#16840165C2211yescharacteristic, complemented
25#16840136C2211yescharacteristic
26#16840135C2211yescharacteristic
27#16840134C2211yescharacteristic
28#16840133—22no—
29#16840114C2211yescharacteristic, complemented
30#16840114C2211yescharacteristic, complemented
31#16840113C2211yescharacteristic, complemented
32#16840113C2211yescharacteristic, complemented
33#16840112C2211yescharacteristic, complemented
34#16840112C2211yescharacteristic, complemented
35#16840111—22no—
36#16840111—22no—
37#16840110C2211yescharacteristic
38#16840110C2211yescharacteristic
39#16840109C2211yescharacteristic
40#16840109C2211yescharacteristic
41#16840108C2211yescharacteristic
42#16840108C2211yescharacteristic
43#16840107—22no—
44#16840107—22no—
45#107×D61—22no—
46#103×D61—22no—
47#103×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×#500C2211yescharacteristic, complemented
57C61×#492C2211yescharacteristic
58C61×#487C2211yescharacteristic, complemented
59C61×#487C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#322C2211yescharacteristic, complemented
63C2×Q8×D61—21no—
64C2×Q8×D61—21no—
65C2×Q8×D61—21no—
66C2×D4×D61—21no—
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×#11758190—22no—
70C2×#11758200—22no—
71C2×#11758201—22no—
72SD16×D61—48no—
73C2×C8×D61D412yes—
74C2×C8×D61D412yes—
75C2×C8×D61D412yes—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519776—21no—
82#14519776—21no—
83#14519776—21no—
84#14519776—21no—
85#14519775—21no—
86#14519775—21no—
87#14519775—21no—
88#14519774C2311yescharacteristic
89#14519773—21no—
90#14519773—21no—
91#14519773—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C42×D61C2311yescharacteristic
99#14519711—22no—
100#14519710—22no—
101#14519710—22no—
102#14519702—22no—
103#14519702—22no—
104#14519701—22no—
105#14519701—22no—
106#14519700—22no—
107#14519700—22no—
108#14519699—22no—
109#14519699—22no—
110#14519698—22no—
111#14519698—22no—
112#14519697—22no—
113#14519697—22no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C8×Dic61D412yes—
118C8×Dic61D412yes—
119C8×Dic61D412yes—
120C2×SD16×C61—22no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#127×C61C2311yescharacteristic
125#107×C61—22no—
126#103×C61—22no—
127#103×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C4×C8×C61C2311yescharacteristic
133C23×D61—41no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×C22⋊D61—41no—
141C2×C22⋊D61—41no—
142C2×C22⋊D61—41no—
143Q8×D61—44no—
144Q8×D61—44no—
145Q8×D61—44no—
146D4×D61—44no—
147D4×D61—44no—
148D4×D61—44no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122—41no—
163C2×Dic122—41no—
164SD16×C61—44no—
165C2×C8×C61C2×D412yes—
166C2×C8×C61C2×D412yes—
167C2×C8×C61C2×D412yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C42×C61C2411yescharacteristic
175#11758201—44no—
176#11758200—44no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183#11758196—41no—
184#11758196—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C4×Dic61C2×D411yescharacteristic
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C2×#92418C2×D412yes—
195#11758190—44no—
196C8×D61—24no—
197C8×D61—24no—
198C8×D61—24no—
199C23×C61—41no—
200C22×D61—41no—
201C22×D61—82no—
202C22×D61—82no—
203C22×D61—82no—
204C22×D61#12711yescharacteristic
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61—42no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—42no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C22⋊D61—82no—
218C22⋊D61—82no—
219C22⋊D61—82no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61#12711yescharacteristic
224C2×Dic61#12711yescharacteristic
225C2×Dic61#12711yescharacteristic
226D244—42no—
227D244—42no—
228D244—42no—
229C4×D61—82no—
230C4×D61—82no—
231C4×D61—82no—
232C4×D61#13514yes—
233C4×D61AGL(1,ℤ/8)14yes—
234C4×D61#13714yes—
235Dic122—42no—
236Dic122—42no—
237Dic122—42no—
238Dic122—82no—
239Dic122—82no—
240Dic122—82no—
241C8×C61—22no—
242C8×C61—22no—
243C8×C61—22no—
244#92418—22no—
245#92418—22no—
246#92418—22no—
247C2×#1719—611nomaximal, Sylow
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C2×#12711yescharacteristic
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—164no—
256C2×D61#49512yes—
257C2×D61#49612yes—
258C2×D61#49712yes—
259C4×C61—81no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C2×#13512yes—
263C4×C61C2×AGL(1,ℤ/8)12yes—
264C4×C61C2×#13712yes—
265Dic61—81no—
266Dic61—81no—
267Dic61—81no—
268Dic61#49512yes—
269Dic61#49612yes—
270Dic61#49712yes—
271C2×#500—611no—
272C2×#492—611no—
273C2×#487—611no—
274C2×#487—611no—
275C2×#486—611no—
276C2×#486—611no—
277#1719—611no—
278#1719—611no—
279#1719—611no—
280#1719—611no—
281#1719—611no—
282#1719—611no—
283#1719—611no—
284#1719D6111yescharacteristic, direct product factor
285C2×#322—611no—
286C2×C61—161no—
287C2×C61C2×#49511yescharacteristic
288C2×C61C2×#49611yescharacteristic
289C2×C61C2×#49711yescharacteristic
290D61—161no—
291D61#171914yesdirect product factor
292C22×SD16—1222no—
293C2×C4⋊Q8—611no—
294C2×C4⋊Q8—611no—
295C2×C4⋊Q8—611no—
296C2×#127—611no—
297#500—611no—
298#500—611no—
299#500—611no—
300#500—1222no—
301#500C2×D6111yescharacteristic, complemented
302#492—611no—
303#492—611no—
304#492—611no—
305#492—1222no—
306#492C2×D6111yescharacteristic
307#487—611no—
308#487—611no—
309#487—611no—
310#487—611no—
311#487—611no—
312#487—611no—
313#487—1222no—
314#487—1222no—
315#487C2×D6111yescharacteristic, complemented
316#487C2×D6111yescharacteristic, complemented
317#486—611no—
318#486—611no—
319#486—611no—
320#486—611no—
321#486—611no—
322#486—611no—
323#486—1222no—
324#486—1222no—
325#486C2×D6111yescharacteristic
326#486C2×D6111yescharacteristic
327C2×#107—1222no—
328C2×#103—1222no—
329C2×#103—1222no—
330C2×#102—1222no—
331C2×#102—1222no—
332C2×C4×C8—611no—
333C2×C4×C8—611no—
334C2×C4×C8—611no—
335#322—611no—
336#322—611no—
337#322—611no—
338#322C2×D6111yescharacteristic, complemented
339C61C2×#171911yescharacteristic, complemented, Sylow
340C22×Q8—1221no—
341C22×Q8—1221no—
342C22×Q8—1221no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—1221no—
346C2×SD16—22no—
347C2×SD16—1222no—
348C2×SD16—1222no—
349C2×SD16—1222no—
350C2×SD16—2448no—
351C22×C8—612no—
352C22×C8—612no—
353C22×C8—612no—
354C4⋊Q8—611no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362C4⋊Q8—1221no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8C22×D6111yescharacteristic
367C4⋊Q8C22×D6111yescharacteristic
368C4⋊Q8C22×D6111yescharacteristic
369#127—611no—
370#127—1221no—
371#127—1221no—
372#127—1221no—
373#127C22×D6111yescharacteristic
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C42—611no—
381#107—22no—
382#107—1222no—
383#107—1222no—
384#107—1222no—
385#103—22no—
386#103—22no—
387#103—1222no—
388#103—1222no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#103—1222no—
393#102—22no—
394#102—22no—
395#102—1222no—
396#102—1222no—
397#102—1222no—
398#102—1222no—
399#102—1222no—
400#102—1222no—
401C4×C8—612no—
402C4×C8—612no—
403C4×C8—612no—
404C4×C8—611no—
405C4×C8—611no—
406C4×C8—611no—
407C4×C8C22×D6111yescharacteristic
408C4×C8C22×D6111yescharacteristic
409C4×C8C22×D6111yescharacteristic
410C24—2441no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—1221no—
415C2×Q8—1221no—
416C2×Q8—1221no—
417C2×Q8—2441no—
418C2×Q8—2441no—
419C2×Q8—2441no—
420C2×Q8—2444no—
421C2×Q8—2444no—
422C2×Q8—2444no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—1221no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—2444no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2441no—
433C2×D4—2441no—
434C2×D4—2441no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2441no—
441SD16—44no—
442SD16—2444no—
443SD16—2444no—
444SD16—2444no—
445C2×C8—612no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—1224no—
449C2×C8—1224no—
450C2×C8—1224no—
451C2×C8D4×D6112yes—
452C2×C8D4×D6112yes—
453C2×C8D4×D6112yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C42—611no—
473C42—611no—
474C42—611no—
475C42C23×D6111yescharacteristic
476C23—41no—
477C23—611no—
478C23—4882no—
479C23—4882no—
480C23—4882no—
481C23—2441no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—4882no—
486Q8—4882no—
487Q8—4882no—
488Q8—2442no—
489Q8—2442no—
490Q8—2442no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—4882no—
495D4—4882no—
496D4—4882no—
497D4—2442no—
498D4—2442no—
499D4—2442no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—614no—
507C2×C4—614no—
508C2×C4—614no—
509C2×C4—4882no—
510C2×C4—4882no—
511C2×C4—4882no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C8—22no—
519C8—22no—
520C8—22no—
521C8—1222no—
522C8—1222no—
523C8—1222no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—612no—
528C22—612no—
529C22—612no—
530C22—9764no—
531C22—4881no—
532C22—4881no—
533C22—4881no—
534C22#127×D6111yescharacteristic
535C4—81no—
536C4—81no—
537C4—81no—
538C4—612no—
539C4—612no—
540C4—612no—
541C4—4881no—
542C4—4881no—
543C4—4881no—
544C4#135×D6112yes—
545C4AGL(1,ℤ/8)×D6112yes—
546C4#137×D6112yes—
547C2—161no—
548C2—614no—
549C2—9761no—
550C2#495×D6111yescharacteristic
551C2#496×D6111yescharacteristic
552C2#497×D6111yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1719C2
socleC22×C61C2×#127
prime radical#127×D61C22
hypercenter#1719D61
hypocenterC61C2×#1719
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1719

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
3. #487×D61C2semidirect
4. #487×D61C2semidirect
7. #21445475C2semidirect
8. #21445474C2semidirect
9. #21445473C2semidirect
10. #21445472C2semidirect
11. #21445471C2semidirect
12. #21445470C2semidirect
13. #21445469C2semidirect
14. #322×D61C2semidirect
15. C61×#1719C2semidirect
21. #16840167C22semidirect
23. #16840165C22semidirect
24. #16840165C22semidirect
29. #16840114C22semidirect
30. #16840114C22semidirect
31. #16840113C22semidirect
32. #16840113C22semidirect
33. #16840112C22semidirect
34. #16840112C22semidirect
53. #16839484C22semidirect
54. #16839484C22semidirect
55. #16839483C22semidirect
56. C61×#500C22semidirect
58. C61×#487C22semidirect
59. C61×#487C22semidirect
62. C61×#322C22semidirect
284. #1719D61direct (N × H)
291. D61#1719direct (N × H)
301. #500C2×D61semidirect
315. #487C2×D61semidirect
316. #487C2×D61semidirect
338. #322C2×D61semidirect
339. C61C2×#1719semidirect

Maximal subgroups

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

subgroupindex
#500×D612
#492×D612
#487×D612
#486×D612
#214454752
#214454742
#214454732
#214454722
#214454712
#214454702
#214454692
#322×D612
C61×#17192
C2×#171961