Finite Group Databasebeta

← C59×#18586 #28430695

Subgroups — C59×#18586 #28430695

Order 15104.

964 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3520C211yesmaximal, characteristic
2C59×#3239C211yesmaximal, characteristic, complemented
3C59×#3165C211yesmaximal, characteristic, complemented
4C2×C59×#461C211yesmaximal, characteristic, complemented
5C59×#1654C212yesmaximal, complemented
6C59×#1651C212yesmaximal, complemented
7C59×#1650C212yesmaximal, complemented
8C59×#1647C212yesmaximal, complemented
9C59×#1552C211yesmaximal, characteristic, complemented
10C59×#1490C211yesmaximal, characteristic, complemented
11C59×#1483C211yesmaximal, characteristic, complemented
12C22×SD16×C59—21no—
13C59×#556C2211yescharacteristic
14C59×#554—21no—
15C59×#552—21no—
16C59×#551—22no—
17D4×Q8×C59C2211yescharacteristic
18C59×#548—21no—
19C59×#547C2211yescharacteristic
20C59×#547C2211yescharacteristic
21C59×#546—21no—
22C59×#542—21no—
23C59×#539—22no—
24C59×#533C2211yescharacteristic
25C2×C4⋊Q8×C59C2211yescharacteristic
26C59×#529C2211yescharacteristic
27C59×#525—21no—
28C2×C22⋊Q8×C59—21no—
29C59×#520—21no—
30C2×C4×D4×C59C2211yescharacteristic
31C59×#501—22no—
32C59×#501—21no—
33C59×#500C2212yescomplemented
34C59×#498C2212yescomplemented
35C59×#489—21no—
36C59×#484—21no—
37C59×#481—21no—
38C59×#475C2212yescomplemented
39C59×#474C2212yescomplemented
40C59×#461—21no—
41C59×#461—21no—
42C59×#461C2212yes—
43C59×#461C2212yes—
44C59×#432C2211yescharacteristic
45C59×#428—22no—
46C2×C4⋊C8×C59C2211yescharacteristic
47C2×#103×C59—21no—
48C4×M16×C59C2211yescharacteristic
49C59×#332C2212yescomplemented
50C59×#332C2212yescomplemented
51C59×Q8⋊C8C2212yescomplemented
52C59×D4⋊C8C2212yes—
53C59×D4⋊C8C2212yes—
54C59×D4⋊C8C2212yescomplemented
55C2×#41×C59—21no—
56C2×#41×C59—21no—
57C22×Q8×C59—21no—
58C22×Q8×C59—21no—
59C22×D4×C59D411yescharacteristic
60C23×C4×C59—21no—
61C2×SD16×C59—22no—
62C2×SD16×C59—22no—
63C2×SD16×C59—44no—
64C2×M16×C59—21no—
65C22×C8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59—21no—
70C4⋊Q8×C59C2312yes—
71C4⋊Q8×C59C2311yescharacteristic
72C4⋊Q8×C59C2311yescharacteristic
73#126×C59—22no—
74#126×C59—22no—
75#126×C59—21no—
76#125×C59—22no—
77#125×C59D411yescharacteristic
78#125×C59D412yes—
79#124×C59—22no—
80#124×C59—21no—
81#124×C59D411yescharacteristic
82#123×C59—22no—
83#123×C59—22no—
84#123×C59—21no—
85#123×C59—22no—
86#123×C59—22no—
87C22⋊Q8×C59—22no—
88C22⋊Q8×C59—22no—
89C22⋊Q8×C59—22no—
90C22⋊Q8×C59—21no—
91C22⋊Q8×C59—21no—
92C22⋊Q8×C59—21no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59—22no—
96C22⋊Q8×C59—21no—
97C22⋊Q8×C59D411yescharacteristic
98#121×C59—22no—
99#121×C59—21no—
100#121×C59—21no—
101#121×C59D411yescharacteristic
102#120×C59—22no—
103C4×Q8×C59—21no—
104C4×Q8×C59—21no—
105C4×Q8×C59—21no—
106C4×Q8×C59C2311yescharacteristic
107C4×D4×C59—22no—
108C4×D4×C59—22no—
109C4×D4×C59—22no—
110C4×D4×C59—21no—
111C4×D4×C59—21no—
112C4×D4×C59—21no—
113C4×D4×C59—21no—
114C4×D4×C59D411yescharacteristic
115C4×D4×C59D411yescharacteristic
116C4×D4×C59C2311yescharacteristic
117C4×D4×C59C2311yescharacteristic
118C4×D4×C59C2311yescharacteristic
119#117×C59—22no—
120#117×C59—21no—
121#117×C59—21no—
122#117×C59D412yes—
123C2×C4⋊C4×C59—22no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59D411yescharacteristic
129C2×C22⋊C4×C59—22no—
130C2×C22⋊C4×C59—21no—
131C2×C42×C59C2311yescharacteristic
132#107×C59—24no—
133#107×C59—22no—
134#106×C59—24no—
135#106×C59—22no—
136C4⋊C8×C59—21no—
137C4⋊C8×C59C2312yes—
138C4⋊C8×C59C2312yes—
139#103×C59—22no—
140#103×C59—22no—
141#103×C59—22no—
142#102×C59—22no—
143C22⋊C8×C59—21no—
144#97×C59—21no—
145#97×C59—21no—
146C4×C8×C59C2312yes—
147C24×C59—21no—
148#41×C59—42no—
149#41×C59—42no—
150#41×C59—82no—
151C2×Q8×C59—22no—
152C2×Q8×C59—21no—
153C2×Q8×C59—21no—
154C2×Q8×C59—22no—
155C2×Q8×C59—42no—
156C2×Q8×C59—42no—
157C2×Q8×C59—41no—
158C2×Q8×C59—82no—
159C2×Q8×C59C2×D411yescharacteristic
160C2×D4×C59—22no—
161C2×D4×C59—22no—
162C2×D4×C59—21no—
163C2×D4×C59—22no—
164C2×D4×C59—21no—
165C2×D4×C59—41no—
166C2×D4×C59C2×D411yescharacteristic
167C2×D4×C59C2×D411yescharacteristic
168C2×D4×C59C2×D411yescharacteristic
169C22×C4×C59—22no—
170C22×C4×C59—22no—
171C22×C4×C59—22no—
172C22×C4×C59—21no—
173C22×C4×C59—21no—
174C22×C4×C59—21no—
175C22×C4×C59—21no—
176C22×C4×C59—21no—
177C22×C4×C59—21no—
178C22×C4×C59—41no—
179C22×C4×C59—82no—
180C22×C4×C59C2×D411yescharacteristic
181C22×C4×C59C2×D411yescharacteristic
182C22×C4×C59C2×D411yescharacteristic
183SD16×C59—44no—
184SD16×C59—44no—
185M16×C59—24no—
186C2×C8×C59—22no—
187C2×C8×C59—22no—
188C2×C8×C59—22no—
189C2×C8×C59—42no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—22no—
193C4⋊C4×C59—22no—
194C4⋊C4×C59—21no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—21no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—22no—
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×D412yes—
211C4⋊C4×C59C2×D411yescharacteristic
212C4⋊C4×C59C2×D412yes—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—22no—
216C22⋊C4×C59—22no—
217C22⋊C4×C59—22no—
218C22⋊C4×C59—21no—
219C22⋊C4×C59—22no—
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—
227C22⋊C4×C59—21no—
228C42×C59—22no—
229C42×C59—22no—
230C42×C59—21no—
231C42×C59C2×D411yescharacteristic
232C42×C59C2×D411yescharacteristic
233C42×C59C2×D411yescharacteristic
234C42×C59C2411yescharacteristic
235C23×C59—21no—
236C23×C59—21no—
237C23×C59—21no—
238C23×C59—42no—
239C23×C59—42no—
240C23×C59—81no—
241C23×C59#12011yescharacteristic
242Q8×C59—22no—
243Q8×C59—41no—
244Q8×C59—41no—
245Q8×C59—42no—
246Q8×C59—42no—
247Q8×C59—42no—
248Q8×C59—82no—
249Q8×C59—82no—
250D4×C59—21no—
251D4×C59—21no—
252D4×C59—21no—
253D4×C59—22no—
254D4×C59—42no—
255D4×C59—44no—
256D4×C59—42no—
257D4×C59—82no—
258D4×C59AGL(1,ℤ/8)12yes—
259C2×C4×C59—22no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—22no—
263C2×C4×C59—21no—
264C2×C4×C59—22no—
265C2×C4×C59—21no—
266C2×C4×C59—21no—
267C2×C4×C59—22no—
268C2×C4×C59—21no—
269C2×C4×C59—21no—
270C2×C4×C59—44no—
271C2×C4×C59—41no—
272C2×C4×C59—41no—
273C2×C4×C59—42no—
274C2×C4×C59—42no—
275C2×C4×C59—42no—
276C2×C4×C59—81no—
277C2×C4×C59—81no—
278C2×C4×C59—81no—
279C2×C4×C59—82no—
280C2×C4×C59—82no—
281C2×C4×C59—82no—
282C2×C4×C59#12011yescharacteristic
283C2×C4×C59#12011yescharacteristic
284C2×C4×C59#12011yescharacteristic
285C2×C4×C59C22×D411yescharacteristic
286C2×C4×C59C22×D411yescharacteristic
287C2×C4×C59C22×D411yescharacteristic
288C8×C59—24no—
289C8×C59—42no—
290C8×C59—42no—
291#18586C5911yesmaximal, characteristic, direct product factor, Sylow
292C22×C59—21no—
293C22×C59—21no—
294C22×C59—22no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—84no—
304C22×C59#45612yes—
305C22×C59C2×#12011yescharacteristic
306C4×C59—21no—
307C4×C59—21no—
308C4×C59—21no—
309C4×C59—21no—
310C4×C59—41no—
311C4×C59—42no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—42no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—82no—
318C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
319C4×C59#57511yescharacteristic
320C4×C59#57712yes—
321#3520C2×C5911yescharacteristic
322#3239C2×C5911yescharacteristic, complemented
323#3165C2×C5911yescharacteristic, complemented
324C2×#461C2×C5911yescharacteristic, complemented
325#1654C2×C5912yescomplemented
326#1651C2×C5912yescomplemented
327#1650C2×C5912yescomplemented
328#1647C2×C5912yescomplemented
329#1552C2×C5911yescharacteristic, complemented
330#1490C2×C5911yescharacteristic, complemented
331#1483C2×C5911yescharacteristic, complemented
332C2×C59—22no—
333C2×C59—42no—
334C2×C59—81no—
335C2×C59—81no—
336C2×C59#300611yescharacteristic
337C2×C59#300911yescharacteristic
338C2×C59C2×#45611yescharacteristic
339C22×SD16—21no—
340#556C22×C5911yescharacteristic
341#554—21no—
342#552—21no—
343#551—22no—
344D4×Q8C22×C5911yescharacteristic
345#548—21no—
346#547C22×C5911yescharacteristic
347#547C22×C5911yescharacteristic
348#546—21no—
349#542—21no—
350#539—22no—
351#533C22×C5911yescharacteristic
352C2×C4⋊Q8C22×C5911yescharacteristic
353#529C22×C5911yescharacteristic
354#525—21no—
355C2×C22⋊Q8—21no—
356#520—21no—
357C2×C4×D4C22×C5911yescharacteristic
358#501—22no—
359#501—21no—
360#500C22×C5912yescomplemented
361#498C22×C5912yescomplemented
362#489—21no—
363#484—21no—
364#481—21no—
365#475C22×C5912yescomplemented
366#474C22×C5912yescomplemented
367#461—21no—
368#461—21no—
369#461C22×C5912yes—
370#461C22×C5912yes—
371#432C22×C5911yescharacteristic
372#428—22no—
373C2×C4⋊C8C22×C5911yescharacteristic
374C2×#103—21no—
375C4×M16C22×C5911yescharacteristic
376#332C22×C5912yescomplemented
377#332C22×C5912yescomplemented
378Q8⋊C8C22×C5912yescomplemented
379D4⋊C8C22×C5912yes—
380D4⋊C8C22×C5912yes—
381D4⋊C8C22×C5912yescomplemented
382C59#1858611yescharacteristic, direct product factor, Sylow
383C2×#41—21no—
384C2×#41—21no—
385C22×Q8—21no—
386C22×Q8—21no—
387C22×D4D4×C5911yescharacteristic
388C23×C4—21no—
389C2×SD16—22no—
390C2×SD16—22no—
391C2×SD16—44no—
392C2×M16—21no—
393C22×C8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398C4⋊Q8C23×C5912yes—
399C4⋊Q8C23×C5911yescharacteristic
400C4⋊Q8C23×C5911yescharacteristic
401#126—22no—
402#126—22no—
403#126—21no—
404#125—22no—
405#125D4×C5911yescharacteristic
406#125D4×C5912yes—
407#124—22no—
408#124—21no—
409#124D4×C5911yescharacteristic
410#123—22no—
411#123—22no—
412#123—21no—
413#123—22no—
414#123—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—22no—
418C22⋊Q8—21no—
419C22⋊Q8—21no—
420C22⋊Q8—21no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—21no—
425C22⋊Q8D4×C5911yescharacteristic
426#121—22no—
427#121—21no—
428#121—21no—
429#121D4×C5911yescharacteristic
430#120—22no—
431C4×Q8—21no—
432C4×Q8—21no—
433C4×Q8—21no—
434C4×Q8C23×C5911yescharacteristic
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4D4×C5911yescharacteristic
443C4×D4D4×C5911yescharacteristic
444C4×D4C23×C5911yescharacteristic
445C4×D4C23×C5911yescharacteristic
446C4×D4C23×C5911yescharacteristic
447#117—22no—
448#117—21no—
449#117—21no—
450#117D4×C5912yes—
451C2×C4⋊C4—22no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4D4×C5911yescharacteristic
457C2×C22⋊C4—22no—
458C2×C22⋊C4—21no—
459C2×C42C23×C5911yescharacteristic
460#107—24no—
461#107—22no—
462#106—24no—
463#106—22no—
464C4⋊C8—21no—
465C4⋊C8C23×C5912yes—
466C4⋊C8C23×C5912yes—
467#103—22no—
468#103—22no—
469#103—22no—
470#102—22no—
471C22⋊C8—21no—
472#97—21no—
473#97—21no—
474C4×C8C23×C5912yes—
475C24—21no—
476#41—42no—
477#41—42no—
478#41—82no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—41no—
486C2×Q8—82no—
487C2×Q8C2×D4×C5911yescharacteristic
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—41no—
494C2×D4C2×D4×C5911yescharacteristic
495C2×D4C2×D4×C5911yescharacteristic
496C2×D4C2×D4×C5911yescharacteristic
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—41no—
507C22×C4—82no—
508C22×C4C2×D4×C5911yescharacteristic
509C22×C4C2×D4×C5911yescharacteristic
510C22×C4C2×D4×C5911yescharacteristic
511SD16—44no—
512SD16—44no—
513M16—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—42no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4C2×D4×C5911yescharacteristic
536C4⋊C4C2×D4×C5911yescharacteristic
537C4⋊C4C2×D4×C5911yescharacteristic
538C4⋊C4C2×D4×C5912yes—
539C4⋊C4C2×D4×C5911yescharacteristic
540C4⋊C4C2×D4×C5912yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—21no—
556C42—22no—
557C42—22no—
558C42—21no—
559C42C2×D4×C5911yescharacteristic
560C42C2×D4×C5911yescharacteristic
561C42C2×D4×C5911yescharacteristic
562C42C24×C5911yescharacteristic
563C23—21no—
564C23—21no—
565C23—21no—
566C23—42no—
567C23—42no—
568C23—81no—
569C23#120×C5911yescharacteristic
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—82no—
577Q8—82no—
578D4—21no—
579D4—21no—
580D4—21no—
581D4—22no—
582D4—42no—
583D4—44no—
584D4—42no—
585D4—82no—
586D4AGL(1,ℤ/8)×C5912yes—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4#120×C5911yescharacteristic
611C2×C4#120×C5911yescharacteristic
612C2×C4#120×C5911yescharacteristic
613C2×C4C22×D4×C5911yescharacteristic
614C2×C4C22×D4×C5911yescharacteristic
615C2×C4C22×D4×C5911yescharacteristic
616C8—24no—
617C8—42no—
618C8—42no—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—84no—
631C22C59×#45612yes—
632C22C2×#120×C5911yescharacteristic
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4C2×AGL(1,ℤ/8)×C5911yescharacteristic
646C4C59×#57511yescharacteristic
647C4C59×#57712yes—
648C2—22no—
649C2—42no—
650C2—81no—
651C2—81no—
652C2C59×#300611yescharacteristic
653C2C59×#300911yescharacteristic
654C2C2×C59×#45611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3520C2
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×#3165C2semidirect
4. C2×C59×#461C2semidirect
5. C59×#1654C2semidirect
6. C59×#1651C2semidirect
7. C59×#1650C2semidirect
8. C59×#1647C2semidirect
9. C59×#1552C2semidirect
10. C59×#1490C2semidirect
11. C59×#1483C2semidirect
33. C59×#500C22semidirect
34. C59×#498C22semidirect
38. C59×#475C22semidirect
39. C59×#474C22semidirect
49. C59×#332C22semidirect
50. C59×#332C22semidirect
51. C59×Q8⋊C8C22semidirect
54. C59×D4⋊C8C22semidirect
291. #18586C59direct (N × H)
322. #3239C2×C59semidirect
323. #3165C2×C59semidirect
324. C2×#461C2×C59semidirect
325. #1654C2×C59semidirect
326. #1651C2×C59semidirect
327. #1650C2×C59semidirect
328. #1647C2×C59semidirect
329. #1552C2×C59semidirect
330. #1490C2×C59semidirect
331. #1483C2×C59semidirect
360. #500C22×C59semidirect
361. #498C22×C59semidirect
365. #475C22×C59semidirect
366. #474C22×C59semidirect
376. #332C22×C59semidirect
377. #332C22×C59semidirect
378. Q8⋊C8C22×C59semidirect
381. D4⋊C8C22×C59semidirect
382. C59#18586direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35202
C59×#32392
C59×#31652
C2×C59×#4612
C59×#16542
C59×#16512
C59×#16502
C59×#16472
C59×#15522
C59×#14902
C59×#14832
#1858659