Finite Group Databasebeta

← C59×#18199 #28430308

Subgroups — C59×#18199 #28430308

Order 15104.

1200 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3505C211yesmaximal, characteristic
2C2×C59×#451C211yesmaximal, characteristic, complemented
3C2×C59×#448C211yesmaximal, characteristic, complemented
4C59×#2409C211yesmaximal, characteristic, complemented
5C59×#1606C214yesmaximal, complemented
6C59×#1603C214yesmaximal, complemented
7C59×#1520C211yesmaximal, characteristic, complemented
8C59×#1519C211yesmaximal, characteristic, complemented
9C59×#1459C211yesmaximal, characteristic, complemented
10C23×Q8×C59C2211yescharacteristic
11C22×Q16×C59—21no—
12C22×SD16×C59—21no—
13C59×#553—22no—
14C59×#553—21no—
15C59×#552—21no—
16C59×#551—22no—
17C59×#548—21no—
18C59×#546—21no—
19C59×#539C2212yes—
20C59×#538C2212yes—
21C59×#534—21no—
22C59×#529—21no—
23C59×#525—21no—
24C2×C22⋊Q8×C59—21no—
25C2×C22⋊Q8×C59C2211yescharacteristic
26C2×#121×C59C2211yescharacteristic
27C59×#518—21no—
28C2×C4×D4×C59C2211yescharacteristic
29C59×#451—22no—
30C59×#451C2214yescomplemented
31C59×#448—22no—
32C59×#448C2214yes—
33C2×#103×C59—21no—
34C2×#103×C59—21no—
35C2×C22⋊C8×C59C2211yescharacteristic
36C2×C22⋊C8×C59C2211yescharacteristic
37C59×#395C2212yescomplemented
38C59×#393C2212yescomplemented
39C59×#391—21no—
40C59×#386—21no—
41C59×#385—21no—
42C59×#380C2211yescharacteristic
43C59×#328C2214yescomplemented
44C59×#327C2214yes—
45C59×#324C2212yescomplemented
46C59×#323C2212yescomplemented
47C2×#41×C59—21no—
48C22×Q8×C59—21no—
49C22×Q8×C59—21no—
50C22×Q8×C59—21no—
51C22×Q8×C59—21no—
52C22×Q8×C59—21no—
53C22×Q8×C59—44no—
54C22×Q8×C59D411yescharacteristic
55C22×Q8×C59C2312yes—
56C22×D4×C59—21no—
57C22×D4×C59D411yescharacteristic
58C22×D4×C59D411yescharacteristic
59C23×C4×C59—21no—
60C23×C4×C59D411yescharacteristic
61C23×C4×C59C2311yescharacteristic
62C2×Q16×C59—24no—
63C2×Q16×C59—44no—
64C2×SD16×C59—24no—
65C2×SD16×C59—44no—
66C22×C8×C59—21no—
67C22×C8×C59—21no—
68#126×C59—22no—
69#126×C59—22no—
70#126×C59—22no—
71#125×C59—21no—
72#125×C59—22no—
73#124×C59—21no—
74#124×C59—22no—
75#123×C59—22no—
76#123×C59—22no—
77#123×C59—22no—
78#123×C59—22no—
79#123×C59—22no—
80C22⋊Q8×C59—21no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—21no—
83C22⋊Q8×C59—21no—
84C22⋊Q8×C59—44no—
85C22⋊Q8×C59D412yes—
86C22⋊Q8×C59C2312yes—
87#121×C59—21no—
88#121×C59—22no—
89#121×C59—22no—
90#121×C59—22no—
91#121×C59—21no—
92#121×C59—21no—
93#121×C59D412yes—
94#121×C59C2312yes—
95#120×C59—22no—
96#120×C59—22no—
97C4×Q8×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—21no—
102C4×D4×C59—21no—
103C4×D4×C59—21no—
104C4×D4×C59—21no—
105C4×D4×C59—21no—
106C4×D4×C59—22no—
107C4×D4×C59—22no—
108#117×C59—22no—
109#117×C59—21no—
110#117×C59—22no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—22no—
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59D411yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—22no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—22no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59C2311yescharacteristic
124C2×C42×C59D411yescharacteristic
125#103×C59—24no—
126#103×C59—24no—
127C22⋊C8×C59—21no—
128C22⋊C8×C59—21no—
129C22⋊C8×C59C2312yes—
130C22⋊C8×C59C2312yes—
131#95×C59—22no—
132#95×C59—22no—
133#95×C59C2312yes—
134C24×C59—21no—
135C24×C59C2×D411yescharacteristic
136C24×C59C2×D411yescharacteristic
137#41×C59—82no—
138C2×Q8×C59—24no—
139C2×Q8×C59—21no—
140C2×Q8×C59—24no—
141C2×Q8×C59—24no—
142C2×Q8×C59—22no—
143C2×Q8×C59—22no—
144C2×Q8×C59—44no—
145C2×Q8×C59—42no—
146C2×Q8×C59—42no—
147C2×Q8×C59—42no—
148C2×Q8×C59—42no—
149C2×Q8×C59—44no—
150C2×Q8×C59—44no—
151C2×Q8×C59—41no—
152C2×Q8×C59—41no—
153C2×Q8×C59C2×D412yes—
154C2×D4×C59—21no—
155C2×D4×C59—21no—
156C2×D4×C59—22no—
157C2×D4×C59—22no—
158C2×D4×C59—22no—
159C2×D4×C59—22no—
160C2×D4×C59—22no—
161C2×D4×C59—22no—
162C2×D4×C59—21no—
163C2×D4×C59—42no—
164C2×D4×C59—42no—
165C2×D4×C59—42no—
166C2×D4×C59C2×D412yes—
167C22×C4×C59—21no—
168C22×C4×C59—22no—
169C22×C4×C59—22no—
170C22×C4×C59—22no—
171C22×C4×C59—21no—
172C22×C4×C59—21no—
173C22×C4×C59—21no—
174C22×C4×C59—21no—
175C22×C4×C59—22no—
176C22×C4×C59—42no—
177C22×C4×C59—44no—
178C22×C4×C59—42no—
179C22×C4×C59—41no—
180C22×C4×C59—41no—
181C22×C4×C59C2×D411yescharacteristic
182C22×C4×C59C2×D411yescharacteristic
183C22×C4×C59C2×D411yescharacteristic
184C22×C4×C59C2×D411yescharacteristic
185C22×C4×C59C2×D411yescharacteristic
186C22×C4×C59C2×D411yescharacteristic
187C22×C4×C59C2×D411yescharacteristic
188C22×C4×C59C2×D411yescharacteristic
189C22×C4×C59C2411yescharacteristic
190Q16×C59—48no—
191SD16×C59—48no—
192C2×C8×C59—22no—
193C2×C8×C59—22no—
194C2×C8×C59—42no—
195C2×C8×C59—42no—
196C4⋊C4×C59—22no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—22no—
199C4⋊C4×C59—22no—
200C4⋊C4×C59—22no—
201C4⋊C4×C59—21no—
202C4⋊C4×C59—21no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59—21no—
205C4⋊C4×C59—21no—
206C4⋊C4×C59—42no—
207C4⋊C4×C59—44no—
208C4⋊C4×C59C2×D412yes—
209C4⋊C4×C59C2×D412yes—
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—22no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—22no—
218C22⋊C4×C59—21no—
219C22⋊C4×C59—22no—
220C22⋊C4×C59—22no—
221C22⋊C4×C59—22no—
222C22⋊C4×C59—21no—
223C22⋊C4×C59—21no—
224C22⋊C4×C59—44no—
225C22⋊C4×C59—42no—
226C42×C59—22no—
227C42×C59—22no—
228C42×C59—21no—
229C42×C59—21no—
230C23×C59—21no—
231C23×C59—22no—
232C23×C59—21no—
233C23×C59—21no—
234C23×C59—41no—
235C23×C59—41no—
236C23×C59—41no—
237C23×C59—41no—
238C23×C59—42no—
239C23×C59—81no—
240C23×C59—81no—
241C23×C59#12011yescharacteristic
242C23×C59#12011yescharacteristic
243C23×C59#13512yes—
244C23×C59#13712yes—
245C23×C59C22×D411yescharacteristic
246Q8×C59—24no—
247Q8×C59—24no—
248Q8×C59—42no—
249Q8×C59—42no—
250Q8×C59—42no—
251Q8×C59—44no—
252Q8×C59—44no—
253Q8×C59—42no—
254D4×C59—42no—
255D4×C59—42no—
256D4×C59—42no—
257D4×C59—42no—
258D4×C59—42no—
259D4×C59—44no—
260D4×C59—42no—
261D4×C59—82no—
262D4×C59—82no—
263C2×C4×C59—22no—
264C2×C4×C59—22no—
265C2×C4×C59—22no—
266C2×C4×C59—24no—
267C2×C4×C59—21no—
268C2×C4×C59—21no—
269C2×C4×C59—22no—
270C2×C4×C59—21no—
271C2×C4×C59—22no—
272C2×C4×C59—22no—
273C2×C4×C59—21no—
274C2×C4×C59—21no—
275C2×C4×C59—41no—
276C2×C4×C59—41no—
277C2×C4×C59—41no—
278C2×C4×C59—42no—
279C2×C4×C59—44no—
280C2×C4×C59—44no—
281C2×C4×C59—42no—
282C2×C4×C59—42no—
283C2×C4×C59—42no—
284C2×C4×C59—41no—
285C2×C4×C59—41no—
286C2×C4×C59—42no—
287C2×C4×C59—42no—
288C2×C4×C59—42no—
289C2×C4×C59—81no—
290C2×C4×C59—81no—
291C2×C4×C59—81no—
292C2×C4×C59—82no—
293C2×C4×C59—82no—
294C2×C4×C59#12011yescharacteristic
295C2×C4×C59#12011yescharacteristic
296C2×C4×C59C22×D411yescharacteristic
297C2×C4×C59C22×D411yescharacteristic
298C8×C59—42no—
299C8×C59—42no—
300#18199C5911yesmaximal, characteristic, direct product factor, Sylow
301C22×C59—21no—
302C22×C59—21no—
303C22×C59—21no—
304C22×C59—21no—
305C22×C59—21no—
306C22×C59—21no—
307C22×C59—42no—
308C22×C59—42no—
309C22×C59—42no—
310C22×C59—41no—
311C22×C59—42no—
312C22×C59—42no—
313C22×C59—41no—
314C22×C59—41no—
315C22×C59—42no—
316C22×C59—81no—
317C22×C59—162no—
318C22×C59#45212yes—
319C22×C59#45212yes—
320C22×C59C2×#12011yescharacteristic
321C22×C59C2×#13511yescharacteristic
322C22×C59C2×#13711yescharacteristic
323C4×C59—21no—
324C4×C59—21no—
325C4×C59—21no—
326C4×C59—41no—
327C4×C59—42no—
328C4×C59—41no—
329C4×C59—42no—
330C4×C59—44no—
331C4×C59—41no—
332C4×C59—42no—
333C4×C59—82no—
334C4×C59B(4,𝔽2)12yes—
335#3505C2×C5911yescharacteristic
336C2×#451C2×C5911yescharacteristic, complemented
337C2×#448C2×C5911yescharacteristic, complemented
338#2409C2×C5911yescharacteristic, complemented
339#1606C2×C5914yescomplemented
340#1603C2×C5914yescomplemented
341#1520C2×C5911yescharacteristic, complemented
342#1519C2×C5911yescharacteristic, complemented
343#1459C2×C5911yescharacteristic, complemented
344C2×C59—21no—
345C2×C59—21no—
346C2×C59—22no—
347C2×C59—22no—
348C2×C59—81no—
349C2×C59—82no—
350C2×C59C2×#45211yescharacteristic
351C2×C59#302211yescharacteristic
352C2×C59C2×B(4,𝔽2)11yescharacteristic
353C23×Q8C22×C5911yescharacteristic
354C22×Q16—21no—
355C22×SD16—21no—
356#553—22no—
357#553—21no—
358#552—21no—
359#551—22no—
360#548—21no—
361#546—21no—
362#539C22×C5912yes—
363#538C22×C5912yes—
364#534—21no—
365#529—21no—
366#525—21no—
367C2×C22⋊Q8—21no—
368C2×C22⋊Q8C22×C5911yescharacteristic
369C2×#121C22×C5911yescharacteristic
370#518—21no—
371C2×C4×D4C22×C5911yescharacteristic
372#451—22no—
373#451C22×C5914yescomplemented
374#448—22no—
375#448C22×C5914yes—
376C2×#103—21no—
377C2×#103—21no—
378C2×C22⋊C8C22×C5911yescharacteristic
379C2×C22⋊C8C22×C5911yescharacteristic
380#395C22×C5912yescomplemented
381#393C22×C5912yescomplemented
382#391—21no—
383#386—21no—
384#385—21no—
385#380C22×C5911yescharacteristic
386#328C22×C5914yescomplemented
387#327C22×C5914yes—
388#324C22×C5912yescomplemented
389#323C22×C5912yescomplemented
390C59#1819911yescharacteristic, direct product factor, Sylow
391C2×#41—21no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—44no—
398C22×Q8D4×C5911yescharacteristic
399C22×Q8C23×C5912yes—
400C22×D4—21no—
401C22×D4D4×C5911yescharacteristic
402C22×D4D4×C5911yescharacteristic
403C23×C4—21no—
404C23×C4D4×C5911yescharacteristic
405C23×C4C23×C5911yescharacteristic
406C2×Q16—24no—
407C2×Q16—44no—
408C2×SD16—24no—
409C2×SD16—44no—
410C22×C8—21no—
411C22×C8—21no—
412#126—22no—
413#126—22no—
414#126—22no—
415#125—21no—
416#125—22no—
417#124—21no—
418#124—22no—
419#123—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424C22⋊Q8—21no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—44no—
429C22⋊Q8D4×C5912yes—
430C22⋊Q8C23×C5912yes—
431#121—21no—
432#121—22no—
433#121—22no—
434#121—22no—
435#121—21no—
436#121—21no—
437#121D4×C5912yes—
438#121C23×C5912yes—
439#120—22no—
440#120—22no—
441C4×Q8—21no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—22no—
451C4×D4—22no—
452#117—22no—
453#117—21no—
454#117—22no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4D4×C5911yescharacteristic
459C2×C4⋊C4D4×C5911yescharacteristic
460C2×C4⋊C4C23×C5911yescharacteristic
461C2×C22⋊C4—22no—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4—22no—
465C2×C22⋊C4—21no—
466C2×C22⋊C4D4×C5911yescharacteristic
467C2×C22⋊C4C23×C5911yescharacteristic
468C2×C42D4×C5911yescharacteristic
469#103—24no—
470#103—24no—
471C22⋊C8—21no—
472C22⋊C8—21no—
473C22⋊C8C23×C5912yes—
474C22⋊C8C23×C5912yes—
475#95—22no—
476#95—22no—
477#95C23×C5912yes—
478C24—21no—
479C24C2×D4×C5911yescharacteristic
480C24C2×D4×C5911yescharacteristic
481#41—82no—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—24no—
485C2×Q8—24no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—44no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—41no—
496C2×Q8—41no—
497C2×Q8C2×D4×C5912yes—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—21no—
507C2×D4—42no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×D4×C5912yes—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—41no—
524C22×C4—41no—
525C22×C4C2×D4×C5911yescharacteristic
526C22×C4C2×D4×C5911yescharacteristic
527C22×C4C2×D4×C5911yescharacteristic
528C22×C4C2×D4×C5911yescharacteristic
529C22×C4C2×D4×C5911yescharacteristic
530C22×C4C2×D4×C5911yescharacteristic
531C22×C4C2×D4×C5911yescharacteristic
532C22×C4C2×D4×C5911yescharacteristic
533C22×C4C24×C5911yescharacteristic
534Q16—48no—
535SD16—48no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—42no—
539C2×C8—42no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4C2×D4×C5912yes—
553C4⋊C4C2×D4×C5912yes—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—44no—
569C22⋊C4—42no—
570C42—22no—
571C42—22no—
572C42—21no—
573C42—21no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—21no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—81no—
584C23—81no—
585C23#120×C5911yescharacteristic
586C23#120×C5911yescharacteristic
587C23#135×C5912yes—
588C23#137×C5912yes—
589C23C22×D4×C5911yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—44no—
596Q8—44no—
597Q8—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—44no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—82no—
606D4—82no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4#120×C5911yescharacteristic
639C2×C4#120×C5911yescharacteristic
640C2×C4C22×D4×C5911yescharacteristic
641C2×C4C22×D4×C5911yescharacteristic
642C8—42no—
643C8—42no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—81no—
660C22—162no—
661C22C59×#45212yes—
662C22C59×#45212yes—
663C22C2×#120×C5911yescharacteristic
664C22C2×#135×C5911yescharacteristic
665C22C2×#137×C5911yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—41no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—41no—
675C4—42no—
676C4—82no—
677C4C59×B(4,𝔽2)12yes—
678C2—21no—
679C2—21no—
680C2—22no—
681C2—22no—
682C2—81no—
683C2—82no—
684C2C2×C59×#45211yescharacteristic
685C2C59×#302211yescharacteristic
686C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3505C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#451C2semidirect
3. C2×C59×#448C2semidirect
4. C59×#2409C2semidirect
5. C59×#1606C2semidirect
6. C59×#1603C2semidirect
7. C59×#1520C2semidirect
8. C59×#1519C2semidirect
9. C59×#1459C2semidirect
30. C59×#451C22semidirect
37. C59×#395C22semidirect
38. C59×#393C22semidirect
43. C59×#328C22semidirect
45. C59×#324C22semidirect
46. C59×#323C22semidirect
300. #18199C59direct (N × H)
336. C2×#451C2×C59semidirect
337. C2×#448C2×C59semidirect
338. #2409C2×C59semidirect
339. #1606C2×C59semidirect
340. #1603C2×C59semidirect
341. #1520C2×C59semidirect
342. #1519C2×C59semidirect
343. #1459C2×C59semidirect
373. #451C22×C59semidirect
380. #395C22×C59semidirect
381. #393C22×C59semidirect
386. #328C22×C59semidirect
388. #324C22×C59semidirect
389. #323C22×C59semidirect
390. C59#18199direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35052
C2×C59×#4512
C2×C59×#4482
C59×#24092
C59×#16062
C59×#16032
C59×#15202
C59×#15192
C59×#14592
#1819959