Finite Group Databasebeta

← C59×#18579 #28430688

Subgroups — C59×#18579 #28430688

Order 15104.

932 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3520C211yesmaximal, characteristic
2C59×#3238C211yesmaximal, characteristic, complemented
3C59×#3139C211yesmaximal, characteristic, complemented
4C2×C59×#476C211yesmaximal, characteristic, complemented
5C59×#1638C212yesmaximal, complemented
6C59×#1637C212yesmaximal, complemented
7C59×#1635C212yesmaximal, complemented
8C59×#1635C212yesmaximal, complemented
9C59×#1552C211yesmaximal, characteristic, complemented
10C59×#1490C211yesmaximal, characteristic, complemented
11C59×#1483C211yesmaximal, characteristic, complemented
12C59×#556C2211yescharacteristic
13C59×#554—21no—
14C59×#552—21no—
15C59×#551—22no—
16D4×Q8×C59C2211yescharacteristic
17C59×#548—21no—
18C59×#547C2211yescharacteristic
19C59×#547C2211yescharacteristic
20C59×#546—21no—
21C59×#542—21no—
22C59×#539—22no—
23C59×#533C2211yescharacteristic
24C2×C4⋊Q8×C59C2211yescharacteristic
25C59×#529C2211yescharacteristic
26C59×#525—21no—
27C2×C22⋊Q8×C59—21no—
28C59×#520—21no—
29C2×C4×D4×C59C2211yescharacteristic
30C59×#501—21no—
31C59×#491—22no—
32C59×#489—21no—
33C59×#487C2212yescomplemented
34C59×#487C2212yescomplemented
35C59×#483—21no—
36C59×#480—21no—
37C59×#477C2212yescomplemented
38C59×#476—21no—
39C59×#476—21no—
40C59×#476C2212yes—
41C59×#476C2212yes—
42C59×#476C2212yescomplemented
43C59×#432C2211yescharacteristic
44C2×#106×C59—21no—
45C2×C4⋊C8×C59C2211yescharacteristic
46C59×#419—22no—
47C2×#102×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×M16×C59—21no—
62C22×C8×C59—21no—
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59—21no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59C2312yes—
68C4⋊Q8×C59C2311yescharacteristic
69C4⋊Q8×C59C2311yescharacteristic
70#126×C59—22no—
71#126×C59—22no—
72#126×C59—21no—
73#125×C59—22no—
74#125×C59D412yes—
75#125×C59D411yescharacteristic
76#124×C59—22no—
77#124×C59—21no—
78#124×C59D411yescharacteristic
79#123×C59—22no—
80#123×C59—22no—
81#123×C59—21no—
82#123×C59—22no—
83#123×C59—22no—
84C22⋊Q8×C59—22no—
85C22⋊Q8×C59—22no—
86C22⋊Q8×C59—22no—
87C22⋊Q8×C59—21no—
88C22⋊Q8×C59—21no—
89C22⋊Q8×C59—21no—
90C22⋊Q8×C59—21no—
91C22⋊Q8×C59—21no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59D411yescharacteristic
95#121×C59—22no—
96#121×C59—21no—
97#121×C59—21no—
98#121×C59D411yescharacteristic
99#120×C59—22no—
100C4×Q8×C59—21no—
101C4×Q8×C59—21no—
102C4×Q8×C59—21no—
103C4×Q8×C59C2311yescharacteristic
104C4×D4×C59—22no—
105C4×D4×C59—22no—
106C4×D4×C59—22no—
107C4×D4×C59—21no—
108C4×D4×C59—21no—
109C4×D4×C59—21no—
110C4×D4×C59—21no—
111C4×D4×C59D411yescharacteristic
112C4×D4×C59D411yescharacteristic
113C4×D4×C59C2311yescharacteristic
114C4×D4×C59C2311yescharacteristic
115C4×D4×C59C2311yescharacteristic
116#117×C59—22no—
117#117×C59—21no—
118#117×C59—21no—
119#117×C59D412yes—
120C2×C4⋊C4×C59—22no—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59—21no—
123C2×C4⋊C4×C59—21no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59—22no—
127C2×C22⋊C4×C59—21no—
128C2×C42×C59C2311yescharacteristic
129#107×C59—22no—
130#106×C59—22no—
131#106×C59—22no—
132#106×C59—22no—
133C4⋊C8×C59—21no—
134C4⋊C8×C59C2312yes—
135C4⋊C8×C59C2312yes—
136#103×C59—22no—
137#103×C59—24no—
138#103×C59—24no—
139#102×C59—22no—
140#102×C59—22no—
141#102×C59—22no—
142C22⋊C8×C59—21no—
143#97×C59—21no—
144#97×C59—21no—
145C4×C8×C59C2312yes—
146C24×C59—21no—
147#41×C59—42no—
148#41×C59—42no—
149#41×C59—82no—
150C2×Q8×C59—22no—
151C2×Q8×C59—21no—
152C2×Q8×C59—21no—
153C2×Q8×C59—22no—
154C2×Q8×C59—44no—
155C2×Q8×C59—41no—
156C2×Q8×C59—82no—
157C2×Q8×C59C2×D411yescharacteristic
158C2×D4×C59—22no—
159C2×D4×C59—22no—
160C2×D4×C59—21no—
161C2×D4×C59—22no—
162C2×D4×C59—21no—
163C2×D4×C59—41no—
164C2×D4×C59C2×D411yescharacteristic
165C2×D4×C59C2×D411yescharacteristic
166C2×D4×C59C2×D411yescharacteristic
167C22×C4×C59—22no—
168C22×C4×C59—22no—
169C22×C4×C59—22no—
170C22×C4×C59—21no—
171C22×C4×C59—21no—
172C22×C4×C59—21no—
173C22×C4×C59—21no—
174C22×C4×C59—21no—
175C22×C4×C59—21no—
176C22×C4×C59—41no—
177C22×C4×C59—82no—
178C22×C4×C59C2×D411yescharacteristic
179C22×C4×C59C2×D411yescharacteristic
180C22×C4×C59C2×D411yescharacteristic
181M16×C59—42no—
182C2×C8×C59—22no—
183C2×C8×C59—22no—
184C2×C8×C59—22no—
185C2×C8×C59—42no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—22no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—22no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
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—22no—
201C4⋊C4×C59—41no—
202C4⋊C4×C59—41no—
203C4⋊C4×C59C2×D412yes—
204C4⋊C4×C59C2×D412yes—
205C4⋊C4×C59C2×D411yescharacteristic
206C4⋊C4×C59C2×D411yescharacteristic
207C4⋊C4×C59C2×D411yescharacteristic
208C4⋊C4×C59C2×D411yescharacteristic
209C22⋊C4×C59—22no—
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—21no—
215C22⋊C4×C59—22no—
216C22⋊C4×C59—21no—
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—
224C42×C59—22no—
225C42×C59—22no—
226C42×C59—21no—
227C42×C59C2×D411yescharacteristic
228C42×C59C2×D411yescharacteristic
229C42×C59C2×D411yescharacteristic
230C42×C59C2411yescharacteristic
231C23×C59—21no—
232C23×C59—21no—
233C23×C59—21no—
234C23×C59—42no—
235C23×C59—42no—
236C23×C59—81no—
237C23×C59#12011yescharacteristic
238Q8×C59—22no—
239Q8×C59—44no—
240Q8×C59—41no—
241Q8×C59—41no—
242Q8×C59—42no—
243Q8×C59—82no—
244Q8×C59—82no—
245D4×C59—21no—
246D4×C59—21no—
247D4×C59—21no—
248D4×C59—22no—
249D4×C59—42no—
250D4×C59—44no—
251D4×C59—42no—
252D4×C59—82no—
253D4×C59#13712yes—
254C2×C4×C59—22no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—22no—
258C2×C4×C59—21no—
259C2×C4×C59—22no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—22no—
263C2×C4×C59—21no—
264C2×C4×C59—21no—
265C2×C4×C59—44no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59—42no—
269C2×C4×C59—42no—
270C2×C4×C59—42no—
271C2×C4×C59—81no—
272C2×C4×C59—81no—
273C2×C4×C59—81no—
274C2×C4×C59—82no—
275C2×C4×C59—82no—
276C2×C4×C59—82no—
277C2×C4×C59#12011yescharacteristic
278C2×C4×C59#12011yescharacteristic
279C2×C4×C59#12011yescharacteristic
280C2×C4×C59C22×D411yescharacteristic
281C2×C4×C59C22×D411yescharacteristic
282C2×C4×C59C22×D411yescharacteristic
283C8×C59—42no—
284C8×C59—42no—
285C8×C59—42no—
286#18579C5911yesmaximal, characteristic, direct product factor, Sylow
287C22×C59—21no—
288C22×C59—21no—
289C22×C59—22no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—84no—
299C22×C59#45412yes—
300C22×C59C2×#12011yescharacteristic
301C4×C59—21no—
302C4×C59—21no—
303C4×C59—21no—
304C4×C59—21no—
305C4×C59—41no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59—82no—
313C4×C59C2×#13711yescharacteristic
314C4×C59#57511yescharacteristic
315C4×C59#57712yes—
316#3520C2×C5911yescharacteristic
317#3238C2×C5911yescharacteristic, complemented
318#3139C2×C5911yescharacteristic, complemented
319C2×#476C2×C5911yescharacteristic, complemented
320#1638C2×C5912yescomplemented
321#1637C2×C5912yescomplemented
322#1635C2×C5912yescomplemented
323#1635C2×C5912yescomplemented
324#1552C2×C5911yescharacteristic, complemented
325#1490C2×C5911yescharacteristic, complemented
326#1483C2×C5911yescharacteristic, complemented
327C2×C59—22no—
328C2×C59—42no—
329C2×C59—81no—
330C2×C59—81no—
331C2×C59#300711yescharacteristic
332C2×C59#300911yescharacteristic
333C2×C59C2×#45411yescharacteristic
334#556C22×C5911yescharacteristic
335#554—21no—
336#552—21no—
337#551—22no—
338D4×Q8C22×C5911yescharacteristic
339#548—21no—
340#547C22×C5911yescharacteristic
341#547C22×C5911yescharacteristic
342#546—21no—
343#542—21no—
344#539—22no—
345#533C22×C5911yescharacteristic
346C2×C4⋊Q8C22×C5911yescharacteristic
347#529C22×C5911yescharacteristic
348#525—21no—
349C2×C22⋊Q8—21no—
350#520—21no—
351C2×C4×D4C22×C5911yescharacteristic
352#501—21no—
353#491—22no—
354#489—21no—
355#487C22×C5912yescomplemented
356#487C22×C5912yescomplemented
357#483—21no—
358#480—21no—
359#477C22×C5912yescomplemented
360#476—21no—
361#476—21no—
362#476C22×C5912yes—
363#476C22×C5912yes—
364#476C22×C5912yescomplemented
365#432C22×C5911yescharacteristic
366C2×#106—21no—
367C2×C4⋊C8C22×C5911yescharacteristic
368#419—22no—
369C2×#102—21no—
370C4×M16C22×C5911yescharacteristic
371#332C22×C5912yescomplemented
372#332C22×C5912yescomplemented
373Q8⋊C8C22×C5912yescomplemented
374D4⋊C8C22×C5912yes—
375D4⋊C8C22×C5912yes—
376D4⋊C8C22×C5912yescomplemented
377C59#1857911yescharacteristic, direct product factor, Sylow
378C2×#41—21no—
379C2×#41—21no—
380C22×Q8—21no—
381C22×Q8—21no—
382C22×D4D4×C5911yescharacteristic
383C23×C4—21no—
384C2×M16—21no—
385C22×C8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C23×C5912yes—
391C4⋊Q8C23×C5911yescharacteristic
392C4⋊Q8C23×C5911yescharacteristic
393#126—22no—
394#126—22no—
395#126—21no—
396#125—22no—
397#125D4×C5911yescharacteristic
398#125D4×C5912yes—
399#124—22no—
400#124—21no—
401#124D4×C5911yescharacteristic
402#123—22no—
403#123—22no—
404#123—21no—
405#123—22no—
406#123—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8D4×C5911yescharacteristic
418#121—22no—
419#121—21no—
420#121—21no—
421#121D4×C5911yescharacteristic
422#120—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8C23×C5911yescharacteristic
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4D4×C5911yescharacteristic
435C4×D4D4×C5911yescharacteristic
436C4×D4C23×C5911yescharacteristic
437C4×D4C23×C5911yescharacteristic
438C4×D4C23×C5911yescharacteristic
439#117—22no—
440#117—21no—
441#117—21no—
442#117D4×C5912yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4D4×C5911yescharacteristic
449C2×C22⋊C4—22no—
450C2×C22⋊C4—21no—
451C2×C42C23×C5911yescharacteristic
452#107—22no—
453#106—22no—
454#106—22no—
455#106—22no—
456C4⋊C8—21no—
457C4⋊C8C23×C5912yes—
458C4⋊C8C23×C5912yes—
459#103—22no—
460#103—24no—
461#103—24no—
462#102—22no—
463#102—22no—
464#102—22no—
465C22⋊C8—21no—
466#97—21no—
467#97—21no—
468C4×C8C23×C5912yes—
469C24—21no—
470#41—42no—
471#41—42no—
472#41—82no—
473C2×Q8—22no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—22no—
477C2×Q8—44no—
478C2×Q8—41no—
479C2×Q8—82no—
480C2×Q8C2×D4×C5911yescharacteristic
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—41no—
487C2×D4C2×D4×C5911yescharacteristic
488C2×D4C2×D4×C5911yescharacteristic
489C2×D4C2×D4×C5911yescharacteristic
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—82no—
501C22×C4C2×D4×C5911yescharacteristic
502C22×C4C2×D4×C5911yescharacteristic
503C22×C4C2×D4×C5911yescharacteristic
504M16—42no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—42no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4C2×D4×C5912yes—
527C4⋊C4C2×D4×C5912yes—
528C4⋊C4C2×D4×C5911yescharacteristic
529C4⋊C4C2×D4×C5911yescharacteristic
530C4⋊C4C2×D4×C5911yescharacteristic
531C4⋊C4C2×D4×C5911yescharacteristic
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42C2×D4×C5911yescharacteristic
551C42C2×D4×C5911yescharacteristic
552C42C2×D4×C5911yescharacteristic
553C42C24×C5911yescharacteristic
554C23—21no—
555C23—21no—
556C23—21no—
557C23—42no—
558C23—42no—
559C23—81no—
560C23#120×C5911yescharacteristic
561Q8—22no—
562Q8—44no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—82no—
567Q8—82no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—22no—
572D4—42no—
573D4—44no—
574D4—42no—
575D4—82no—
576D4#137×C5912yes—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#120×C5911yescharacteristic
601C2×C4#120×C5911yescharacteristic
602C2×C4#120×C5911yescharacteristic
603C2×C4C22×D4×C5911yescharacteristic
604C2×C4C22×D4×C5911yescharacteristic
605C2×C4C22×D4×C5911yescharacteristic
606C8—42no—
607C8—42no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—84no—
621C22C59×#45412yes—
622C22C2×#120×C5911yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—41no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—82no—
635C4C2×#137×C5911yescharacteristic
636C4C59×#57511yescharacteristic
637C4C59×#57712yes—
638C2—22no—
639C2—42no—
640C2—81no—
641C2—81no—
642C2C59×#300711yescharacteristic
643C2C59×#300911yescharacteristic
644C2C2×C59×#45411yescharacteristic
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×#3238C2semidirect
3. C59×#3139C2semidirect
4. C2×C59×#476C2semidirect
5. C59×#1638C2semidirect
6. C59×#1637C2semidirect
7. C59×#1635C2semidirect
8. C59×#1635C2semidirect
9. C59×#1552C2semidirect
10. C59×#1490C2semidirect
11. C59×#1483C2semidirect
33. C59×#487C22semidirect
34. C59×#487C22semidirect
37. C59×#477C22semidirect
42. C59×#476C22semidirect
49. C59×#332C22semidirect
50. C59×#332C22semidirect
51. C59×Q8⋊C8C22semidirect
54. C59×D4⋊C8C22semidirect
286. #18579C59direct (N × H)
317. #3238C2×C59semidirect
318. #3139C2×C59semidirect
319. C2×#476C2×C59semidirect
320. #1638C2×C59semidirect
321. #1637C2×C59semidirect
322. #1635C2×C59semidirect
323. #1635C2×C59semidirect
324. #1552C2×C59semidirect
325. #1490C2×C59semidirect
326. #1483C2×C59semidirect
355. #487C22×C59semidirect
356. #487C22×C59semidirect
359. #477C22×C59semidirect
364. #476C22×C59semidirect
371. #332C22×C59semidirect
372. #332C22×C59semidirect
373. Q8⋊C8C22×C59semidirect
376. D4⋊C8C22×C59semidirect
377. C59#18579direct (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×#32382
C59×#31392
C2×C59×#4762
C59×#16382
C59×#16372
C59×#16352
C59×#15522
C59×#14902
C59×#14832
#1857959