Finite Group Databasebeta

← C59×#18689 #28430798

Subgroups — C59×#18689 #28430798

Order 15104.

1006 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3543C211yesmaximal, characteristic
2C59×#3239C211yesmaximal, characteristic, complemented
3C59×#3218C211yesmaximal, characteristic, complemented
4C2×C59×#486C211yesmaximal, characteristic, complemented
5C59×#1642C212yesmaximal, complemented
6C59×#1641C212yesmaximal, complemented
7C59×#1640C212yesmaximal, complemented
8C59×#1639C212yesmaximal, complemented
9C59×#1555C211yesmaximal, characteristic, complemented
10C59×#1494C211yesmaximal, characteristic, complemented
11C59×#1494C211yesmaximal, characteristic, complemented
12C59×#560—22no—
13C59×#559C2211yescharacteristic
14C59×#550C2211yescharacteristic
15C59×#550C2211yescharacteristic
16C59×#546—21no—
17C59×#546—21no—
18D42×C59—21no—
19D42×C59—21no—
20D42×C59C2211yescharacteristic
21C59×#540—21no—
22C59×#540—21no—
23C59×#535—21no—
24C59×#535—21no—
25C59×#534—22no—
26C2×C4⋊Q8×C59C2211yescharacteristic, complemented
27C2×#127×C59C2211yescharacteristic
28C59×#529C2211yescharacteristic
29C59×#529C2211yescharacteristic
30C59×#501—21no—
31C59×#497—21no—
32C59×#491—21no—
33C59×#489—21no—
34C59×#486—22no—
35C59×#486C2212yes—
36C59×#486C2212yescomplemented
37C59×#484—21no—
38C59×#481—21no—
39C59×#475C2212yescomplemented
40C59×#474C2212yescomplemented
41C59×#462C2212yescomplemented
42C59×#461C2212yescomplemented
43C59×#451—21no—
44C59×#448—21no—
45C59×#432C2211yescharacteristic, complemented
46C59×#432C2211yescharacteristic, complemented
47C2×#102×C59—22no—
48C2×C4×C8×C59C2211yescharacteristic, complemented
49C59×#331C2212yes—
50C59×#331C2212yescomplemented
51C59×Q8⋊C8C2212yescomplemented
52C59×Q8⋊C8C2212yescomplemented
53C59×D4⋊C8C2212yescomplemented
54C59×D4⋊C8C2212yescomplemented
55C2×#41×C59—21no—
56C2×#41×C59—21no—
57C22×Q8×C59—21no—
58C22×D4×C59—22no—
59C22×D4×C59—21no—
60C22×D4×C59—21no—
61C22×D4×C59—21no—
62C22×D4×C59—21no—
63C22×D4×C59D412yes—
64C2×Q16×C59—22no—
65C2×SD16×C59—22no—
66C22×C8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59—21no—
70C4⋊Q8×C59C2312yes—
71#127×C59—21no—
72#127×C59—21no—
73#127×C59—21no—
74#127×C59—21no—
75#127×C59D412yes—
76#127×C59C2311yescharacteristic
77#127×C59C2311yescharacteristic
78#126×C59—21no—
79#126×C59—21no—
80#125×C59D411yescharacteristic
81#125×C59D411yescharacteristic
82#124×C59—21no—
83#124×C59—21no—
84#124×C59D411yescharacteristic
85#124×C59D411yescharacteristic
86#123×C59—22no—
87#123×C59—22no—
88#123×C59—22no—
89#123×C59—22no—
90#123×C59—21no—
91#123×C59—21no—
92C22⋊Q8×C59D411yescharacteristic
93C22⋊Q8×C59D411yescharacteristic
94#121×C59—22no—
95#121×C59—22no—
96#121×C59—22no—
97#121×C59—22no—
98#121×C59—22no—
99#121×C59—22no—
100#121×C59—22no—
101#121×C59—22no—
102#121×C59—21no—
103#121×C59—21no—
104#121×C59—21no—
105#121×C59—21no—
106#121×C59D411yescharacteristic
107#121×C59D411yescharacteristic
108#120×C59—22no—
109#120×C59—22no—
110#120×C59—22no—
111#120×C59—22no—
112#120×C59—22no—
113#120×C59—22no—
114C4×Q8×C59C2311yescharacteristic
115C4×Q8×C59C2311yescharacteristic
116C4×D4×C59—21no—
117C4×D4×C59—21no—
118C4×D4×C59—21no—
119C4×D4×C59—21no—
120C4×D4×C59—21no—
121C4×D4×C59—21no—
122C4×D4×C59C2311yescharacteristic
123C4×D4×C59C2311yescharacteristic
124#117×C59—21no—
125#117×C59—21no—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—21no—
128C2×C22⋊C4×C59—22no—
129C2×C42×C59C2311yescharacteristic
130#107×C59—22no—
131#106×C59—22no—
132C4⋊C8×C59C2312yes—
133C4⋊C8×C59C2312yes—
134#103×C59—22no—
135#103×C59—22no—
136#103×C59—22no—
137#102×C59—24no—
138#102×C59—24no—
139#102×C59—22no—
140C22⋊C8×C59—21no—
141C22⋊C8×C59—21no—
142#97×C59—21no—
143#97×C59—21no—
144C4×C8×C59—21no—
145C4×C8×C59C2312yes—
146C24×C59—22no—
147C24×C59—22no—
148#41×C59—82no—
149#41×C59—82no—
150C2×Q8×C59—22no—
151C2×Q8×C59—41no—
152C2×Q8×C59—82no—
153C2×Q8×C59C2×D411yescharacteristic
154C2×Q8×C59C2×D411yescharacteristic
155C2×D4×C59—21no—
156C2×D4×C59—21no—
157C2×D4×C59—22no—
158C2×D4×C59—22no—
159C2×D4×C59—22no—
160C2×D4×C59—22no—
161C2×D4×C59—22no—
162C2×D4×C59—22no—
163C2×D4×C59—22no—
164C2×D4×C59—22no—
165C2×D4×C59—22no—
166C2×D4×C59—21no—
167C2×D4×C59—21no—
168C2×D4×C59—21no—
169C2×D4×C59—21no—
170C2×D4×C59—44no—
171C2×D4×C59—82no—
172C2×D4×C59—82no—
173C2×D4×C59—82no—
174C2×D4×C59—82no—
175C2×D4×C59—84no—
176C2×D4×C59C2×D411yescharacteristic
177C2×D4×C59C2×D411yescharacteristic
178C2×D4×C59C2×D412yes—
179C2×D4×C59C2×D412yes—
180C22×C4×C59—22no—
181C22×C4×C59—22no—
182C22×C4×C59—22no—
183C22×C4×C59—21no—
184C22×C4×C59—21no—
185C22×C4×C59—41no—
186C22×C4×C59C2×D411yescharacteristic
187C22×C4×C59C2×D411yescharacteristic
188C22×C4×C59C2×D411yescharacteristic
189Q16×C59—82no—
190SD16×C59—82no—
191C2×C8×C59—22no—
192C2×C8×C59—22no—
193C2×C8×C59—22no—
194C2×C8×C59—42no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—21no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—21no—
200C4⋊C4×C59—21no—
201C4⋊C4×C59—21no—
202C4⋊C4×C59—21no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59—22no—
205C4⋊C4×C59—41no—
206C4⋊C4×C59—41no—
207C4⋊C4×C59C2×D411yescharacteristic
208C4⋊C4×C59C2×D411yescharacteristic
209C4⋊C4×C59C2×D411yescharacteristic
210C4⋊C4×C59C2×D411yescharacteristic
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—22no—
216C22⋊C4×C59—22no—
217C22⋊C4×C59—21no—
218C22⋊C4×C59—21no—
219C22⋊C4×C59—21no—
220C22⋊C4×C59—21no—
221C22⋊C4×C59—21no—
222C22⋊C4×C59—21no—
223C22⋊C4×C59—21no—
224C22⋊C4×C59—21no—
225C22⋊C4×C59—21no—
226C22⋊C4×C59—21no—
227C42×C59—21no—
228C42×C59—21no—
229C42×C59C2×D411yescharacteristic
230C42×C59C2×D411yescharacteristic
231C42×C59C2×D411yescharacteristic
232C42×C59C2411yescharacteristic
233C23×C59—21no—
234C23×C59—21no—
235C23×C59—22no—
236C23×C59—22no—
237C23×C59—44no—
238C23×C59—82no—
239C23×C59—82no—
240C23×C59—82no—
241C23×C59—82no—
242C23×C59—82no—
243C23×C59#12011yescharacteristic
244Q8×C59—22no—
245Q8×C59—22no—
246Q8×C59—42no—
247Q8×C59—42no—
248Q8×C59—82no—
249D4×C59—22no—
250D4×C59—22no—
251D4×C59—42no—
252D4×C59—42no—
253D4×C59—42no—
254D4×C59—42no—
255D4×C59—42no—
256D4×C59—42no—
257D4×C59—42no—
258D4×C59—42no—
259D4×C59—82no—
260D4×C59—82no—
261D4×C59—82no—
262D4×C59—82no—
263D4×C59—82no—
264D4×C59—82no—
265D4×C59—82no—
266D4×C59—82no—
267D4×C59—82no—
268C2×C4×C59—22no—
269C2×C4×C59—22no—
270C2×C4×C59—22no—
271C2×C4×C59—21no—
272C2×C4×C59—21no—
273C2×C4×C59—21no—
274C2×C4×C59—21no—
275C2×C4×C59—21no—
276C2×C4×C59—21no—
277C2×C4×C59—42no—
278C2×C4×C59—42no—
279C2×C4×C59—42no—
280C2×C4×C59—82no—
281C2×C4×C59—82no—
282C2×C4×C59—82no—
283C2×C4×C59—82no—
284C2×C4×C59#12011yescharacteristic
285C2×C4×C59#12011yescharacteristic
286C2×C4×C59#12011yescharacteristic
287C2×C4×C59C22×D411yescharacteristic
288C2×C4×C59C22×D411yescharacteristic
289C2×C4×C59C22×D411yescharacteristic
290C8×C59—42no—
291C8×C59—42no—
292C8×C59—42no—
293#18689C5911yesmaximal, characteristic, direct product factor, Sylow
294C22×C59—21no—
295C22×C59—21no—
296C22×C59—41no—
297C22×C59—42no—
298C22×C59—41no—
299C22×C59—42no—
300C22×C59—41no—
301C22×C59—42no—
302C22×C59—41no—
303C22×C59—42no—
304C22×C59—42no—
305C22×C59—41no—
306C22×C59—41no—
307C22×C59—42no—
308C22×C59—164no—
309C22×C59—164no—
310C22×C59#45512yes—
311C22×C59C2×#12011yescharacteristic
312C4×C59—21no—
313C4×C59—21no—
314C4×C59—21no—
315C4×C59—21no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—41no—
320C4×C59—41no—
321C4×C59—41no—
322C4×C59—82no—
323C4×C59#57612yes—
324C4×C59#57712yes—
325#3543C2×C5911yescharacteristic
326#3239C2×C5911yescharacteristic, complemented
327#3218C2×C5911yescharacteristic, complemented
328C2×#486C2×C5911yescharacteristic, complemented
329#1642C2×C5912yescomplemented
330#1641C2×C5912yescomplemented
331#1640C2×C5912yescomplemented
332#1639C2×C5912yescomplemented
333#1555C2×C5911yescharacteristic, complemented
334#1494C2×C5911yescharacteristic, complemented
335#1494C2×C5911yescharacteristic, complemented
336C2×C59—22no—
337C2×C59—82no—
338C2×C59—81no—
339C2×C59—81no—
340C2×C59—82no—
341C2×C59#300811yescharacteristic
342C2×C59#300911yescharacteristic
343C2×C59C2×#45511yescharacteristic
344#560—22no—
345#559C22×C5911yescharacteristic
346#550C22×C5911yescharacteristic
347#550C22×C5911yescharacteristic
348#546—21no—
349#546—21no—
350D42—21no—
351D42—21no—
352D42C22×C5911yescharacteristic
353#540—21no—
354#540—21no—
355#535—21no—
356#535—21no—
357#534—22no—
358C2×C4⋊Q8C22×C5911yescharacteristic, complemented
359C2×#127C22×C5911yescharacteristic
360#529C22×C5911yescharacteristic
361#529C22×C5911yescharacteristic
362#501—21no—
363#497—21no—
364#491—21no—
365#489—21no—
366#486—22no—
367#486C22×C5912yes—
368#486C22×C5912yescomplemented
369#484—21no—
370#481—21no—
371#475C22×C5912yescomplemented
372#474C22×C5912yescomplemented
373#462C22×C5912yescomplemented
374#461C22×C5912yescomplemented
375#451—21no—
376#448—21no—
377#432C22×C5911yescharacteristic, complemented
378#432C22×C5911yescharacteristic, complemented
379C2×#102—22no—
380C2×C4×C8C22×C5911yescharacteristic, complemented
381#331C22×C5912yes—
382#331C22×C5912yescomplemented
383Q8⋊C8C22×C5912yescomplemented
384Q8⋊C8C22×C5912yescomplemented
385D4⋊C8C22×C5912yescomplemented
386D4⋊C8C22×C5912yescomplemented
387C59#1868911yescharacteristic, direct product factor, Sylow
388C2×#41—21no—
389C2×#41—21no—
390C22×Q8—21no—
391C22×D4—22no—
392C22×D4—21no—
393C22×D4—21no—
394C22×D4—21no—
395C22×D4—21no—
396C22×D4D4×C5912yes—
397C2×Q16—22no—
398C2×SD16—22no—
399C22×C8—21no—
400C4⋊Q8—21no—
401C4⋊Q8—21no—
402C4⋊Q8—21no—
403C4⋊Q8C23×C5912yes—
404#127—21no—
405#127—21no—
406#127—21no—
407#127—21no—
408#127D4×C5912yes—
409#127C23×C5911yescharacteristic
410#127C23×C5911yescharacteristic
411#126—21no—
412#126—21no—
413#125D4×C5911yescharacteristic
414#125D4×C5911yescharacteristic
415#124—21no—
416#124—21no—
417#124D4×C5911yescharacteristic
418#124D4×C5911yescharacteristic
419#123—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423#123—21no—
424#123—21no—
425C22⋊Q8D4×C5911yescharacteristic
426C22⋊Q8D4×C5911yescharacteristic
427#121—22no—
428#121—22no—
429#121—22no—
430#121—22no—
431#121—22no—
432#121—22no—
433#121—22no—
434#121—22no—
435#121—21no—
436#121—21no—
437#121—21no—
438#121—21no—
439#121D4×C5911yescharacteristic
440#121D4×C5911yescharacteristic
441#120—22no—
442#120—22no—
443#120—22no—
444#120—22no—
445#120—22no—
446#120—22no—
447C4×Q8C23×C5911yescharacteristic
448C4×Q8C23×C5911yescharacteristic
449C4×D4—21no—
450C4×D4—21no—
451C4×D4—21no—
452C4×D4—21no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4C23×C5911yescharacteristic
456C4×D4C23×C5911yescharacteristic
457#117—21no—
458#117—21no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—21no—
461C2×C22⋊C4—22no—
462C2×C42C23×C5911yescharacteristic
463#107—22no—
464#106—22no—
465C4⋊C8C23×C5912yes—
466C4⋊C8C23×C5912yes—
467#103—22no—
468#103—22no—
469#103—22no—
470#102—24no—
471#102—24no—
472#102—22no—
473C22⋊C8—21no—
474C22⋊C8—21no—
475#97—21no—
476#97—21no—
477C4×C8—21no—
478C4×C8C23×C5912yes—
479C24—22no—
480C24—22no—
481#41—82no—
482#41—82no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—82no—
486C2×Q8C2×D4×C5911yescharacteristic
487C2×Q8C2×D4×C5911yescharacteristic
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—44no—
504C2×D4—82no—
505C2×D4—82no—
506C2×D4—82no—
507C2×D4—82no—
508C2×D4—84no—
509C2×D4C2×D4×C5911yescharacteristic
510C2×D4C2×D4×C5911yescharacteristic
511C2×D4C2×D4×C5912yes—
512C2×D4C2×D4×C5912yes—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—41no—
519C22×C4C2×D4×C5911yescharacteristic
520C22×C4C2×D4×C5911yescharacteristic
521C22×C4C2×D4×C5911yescharacteristic
522Q16—82no—
523SD16—82no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—42no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4C2×D4×C5911yescharacteristic
541C4⋊C4C2×D4×C5911yescharacteristic
542C4⋊C4C2×D4×C5911yescharacteristic
543C4⋊C4C2×D4×C5911yescharacteristic
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—21no—
559C22⋊C4—21no—
560C42—21no—
561C42—21no—
562C42C2×D4×C5911yescharacteristic
563C42C2×D4×C5911yescharacteristic
564C42C2×D4×C5911yescharacteristic
565C42C24×C5911yescharacteristic
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—44no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23#120×C5911yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—42no—
580Q8—42no—
581Q8—82no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—82no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—82no—
599D4—82no—
600D4—82no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4#120×C5911yescharacteristic
618C2×C4#120×C5911yescharacteristic
619C2×C4#120×C5911yescharacteristic
620C2×C4C22×D4×C5911yescharacteristic
621C2×C4C22×D4×C5911yescharacteristic
622C2×C4C22×D4×C5911yescharacteristic
623C8—42no—
624C8—42no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—41no—
629C22—42no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—164no—
641C22—164no—
642C22C59×#45512yes—
643C22C2×#120×C5911yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4C59×#57612yes—
656C4C59×#57712yes—
657C2—22no—
658C2—81no—
659C2—81no—
660C2—82no—
661C2—82no—
662C2C59×#300811yescharacteristic
663C2C59×#300911yescharacteristic
664C2C2×C59×#45511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3543C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3239C2semidirect
3. C59×#3218C2semidirect
4. C2×C59×#486C2semidirect
5. C59×#1642C2semidirect
6. C59×#1641C2semidirect
7. C59×#1640C2semidirect
8. C59×#1639C2semidirect
9. C59×#1555C2semidirect
10. C59×#1494C2semidirect
11. C59×#1494C2semidirect
26. C2×C4⋊Q8×C59C22semidirect
36. C59×#486C22semidirect
39. C59×#475C22semidirect
40. C59×#474C22semidirect
41. C59×#462C22semidirect
42. C59×#461C22semidirect
45. C59×#432C22semidirect
46. C59×#432C22semidirect
48. C2×C4×C8×C59C22semidirect
50. C59×#331C22semidirect
51. C59×Q8⋊C8C22semidirect
52. C59×Q8⋊C8C22semidirect
53. C59×D4⋊C8C22semidirect
54. C59×D4⋊C8C22semidirect
293. #18689C59direct (N × H)
326. #3239C2×C59semidirect
327. #3218C2×C59semidirect
328. C2×#486C2×C59semidirect
329. #1642C2×C59semidirect
330. #1641C2×C59semidirect
331. #1640C2×C59semidirect
332. #1639C2×C59semidirect
333. #1555C2×C59semidirect
334. #1494C2×C59semidirect
335. #1494C2×C59semidirect
358. C2×C4⋊Q8C22×C59semidirect
368. #486C22×C59semidirect
371. #475C22×C59semidirect
372. #474C22×C59semidirect
373. #462C22×C59semidirect
374. #461C22×C59semidirect
377. #432C22×C59semidirect
378. #432C22×C59semidirect
380. C2×C4×C8C22×C59semidirect
382. #331C22×C59semidirect
383. Q8⋊C8C22×C59semidirect
384. Q8⋊C8C22×C59semidirect
385. D4⋊C8C22×C59semidirect
386. D4⋊C8C22×C59semidirect
387. C59#18689direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35432
C59×#32392
C59×#32182
C2×C59×#4862
C59×#16422
C59×#16412
C59×#16402
C59×#16392
C59×#15552
C59×#14942
#1868959