Finite Group Databasebeta

← C59×#18588 #28430697

Subgroups — C59×#18588 #28430697

Order 15104.

898 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3523C211yesmaximal, characteristic
2C59×#3239C211yesmaximal, characteristic
3C59×#3165C211yesmaximal, characteristic, complemented
4C2×C59×#462C211yesmaximal, characteristic, complemented
5C59×#1653C212yesmaximal, complemented
6C59×#1652C212yesmaximal, complemented
7C59×#1649C212yesmaximal, complemented
8C59×#1648C212yesmaximal, complemented
9C59×#1552C211yesmaximal, characteristic, complemented
10C59×#1491C211yesmaximal, characteristic, complemented
11C59×#1483C211yesmaximal, characteristic
12C22×Q16×C59—21no—
13C59×#557—21no—
14C59×#557C2211yescharacteristic
15C59×#557C2211yescharacteristic
16C59×#556—21no—
17C59×#556C2211yescharacteristic
18C59×#552—22no—
19C59×#552—21no—
20D4×Q8×C59—21no—
21D4×Q8×C59C2211yescharacteristic
22C59×#541—22no—
23C59×#541—21no—
24C59×#533—21no—
25C59×#533C2211yescharacteristic
26C2×C4⋊Q8×C59C2211yescharacteristic
27C59×#529—21no—
28C59×#529C2211yescharacteristic
29C59×#519—21no—
30C2×C4×Q8×C59C2211yescharacteristic
31C59×#501—22no—
32C59×#501—21no—
33C59×#500C2212yes—
34C59×#498C2212yes—
35C59×#489—21no—
36C59×#484—21no—
37C59×#481—21no—
38C59×#475C2212yes—
39C59×#474C2212yes—
40C59×#462—21no—
41C59×#462—21no—
42C59×#462C2212yes—
43C59×#462C2212yes—
44C59×#432C2211yescharacteristic
45C59×#428—22no—
46C2×C4⋊C8×C59C2211yescharacteristic
47C2×#103×C59—21no—
48C4×M16×C59C2211yescharacteristic
49C59×#332C2212yes—
50C59×#332C2212yes—
51C59×Q8⋊C8C2212yes—
52C59×Q8⋊C8C2212yes—
53C59×Q8⋊C8C2212yes—
54C59×D4⋊C8C2212yes—
55C22×Q8×C59—21no—
56C22×Q8×C59—21no—
57C22×Q8×C59D411yescharacteristic
58C2×Q16×C59—22no—
59C2×Q16×C59—22no—
60C2×Q16×C59—44no—
61C2×M16×C59—21no—
62C22×C8×C59—21no—
63C4⋊Q8×C59—22no—
64C4⋊Q8×C59—21no—
65C4⋊Q8×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—22no—
76#126×C59—22no—
77#126×C59—21no—
78#125×C59—22no—
79#125×C59—22no—
80#125×C59—22no—
81#125×C59—21no—
82#125×C59—22no—
83#125×C59—22no—
84#125×C59—21no—
85#125×C59D411yescharacteristic
86#125×C59D412yes—
87#124×C59—21no—
88#124×C59D411yescharacteristic
89#123×C59—22no—
90#123×C59—21no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59—21no—
96C22⋊Q8×C59—21no—
97C22⋊Q8×C59—22no—
98C22⋊Q8×C59D411yescharacteristic
99#121×C59D411yescharacteristic
100C4×Q8×C59—22no—
101C4×Q8×C59—22no—
102C4×Q8×C59—21no—
103C4×Q8×C59—21no—
104C4×Q8×C59—21no—
105C4×Q8×C59—21no—
106C4×Q8×C59—21no—
107C4×Q8×C59—21no—
108C4×Q8×C59—21no—
109C4×Q8×C59—21no—
110C4×Q8×C59—21no—
111C4×Q8×C59D411yescharacteristic
112C4×Q8×C59D411yescharacteristic
113C4×Q8×C59C2311yescharacteristic
114C4×Q8×C59C2311yescharacteristic
115C4×Q8×C59C2311yescharacteristic
116C4×D4×C59—22no—
117C4×D4×C59—21no—
118C4×D4×C59—21no—
119C4×D4×C59C2311yescharacteristic
120#117×C59—22no—
121#117×C59—22no—
122#117×C59—22no—
123#117×C59—21no—
124#117×C59—21no—
125#117×C59D412yes—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59—21no—
129C2×C4⋊C4×C59D411yescharacteristic
130C2×C42×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—
147C2×Q8×C59—22no—
148C2×Q8×C59—21no—
149C2×Q8×C59—21no—
150C2×Q8×C59—21no—
151C2×Q8×C59—22no—
152C2×Q8×C59—22no—
153C2×Q8×C59—41no—
154C2×Q8×C59—42no—
155C2×Q8×C59—42no—
156C2×Q8×C59—42no—
157C2×Q8×C59—42no—
158C2×Q8×C59—41no—
159C2×Q8×C59C2×D411yescharacteristic
160C2×Q8×C59C2×D411yescharacteristic
161C2×Q8×C59C2×D411yescharacteristic
162C2×D4×C59—21no—
163C2×D4×C59C2×D411yescharacteristic
164C22×C4×C59—22no—
165C22×C4×C59—21no—
166C22×C4×C59—21no—
167C22×C4×C59—21no—
168C22×C4×C59—41no—
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D411yescharacteristic
171C22×C4×C59C2×D411yescharacteristic
172Q16×C59—44no—
173Q16×C59—44no—
174M16×C59—24no—
175C2×C8×C59—22no—
176C2×C8×C59—22no—
177C2×C8×C59—22no—
178C2×C8×C59—42no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—22no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—21no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—21no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—21no—
200C4⋊C4×C59—22no—
201C4⋊C4×C59—22no—
202C4⋊C4×C59—41no—
203C4⋊C4×C59—41no—
204C4⋊C4×C59C2×D411yescharacteristic
205C4⋊C4×C59C2×D411yescharacteristic
206C4⋊C4×C59C2×D412yes—
207C4⋊C4×C59C2×D411yescharacteristic
208C4⋊C4×C59C2×D412yes—
209C4⋊C4×C59C2×D411yescharacteristic
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—21no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—21no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—21no—
218C22⋊C4×C59—21no—
219C42×C59—22no—
220C42×C59—22no—
221C42×C59—22no—
222C42×C59—21no—
223C42×C59—21no—
224C42×C59—22no—
225C42×C59—21no—
226C42×C59—21no—
227C42×C59—21no—
228C42×C59C2×D411yescharacteristic
229C42×C59C2×D411yescharacteristic
230C42×C59C2×D411yescharacteristic
231C42×C59C2411yescharacteristic
232C23×C59—21no—
233C23×C59#12011yescharacteristic
234Q8×C59—21no—
235Q8×C59—21no—
236Q8×C59—21no—
237Q8×C59—22no—
238Q8×C59—42no—
239Q8×C59—44no—
240Q8×C59—41no—
241Q8×C59—41no—
242Q8×C59—42no—
243Q8×C59—42no—
244Q8×C59—42no—
245Q8×C59—82no—
246Q8×C59AGL(1,ℤ/8)12yes—
247D4×C59—22no—
248D4×C59—42no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—22no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—22no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—21no—
260C2×C4×C59—21no—
261C2×C4×C59—22no—
262C2×C4×C59—42no—
263C2×C4×C59—44no—
264C2×C4×C59—42no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—42no—
268C2×C4×C59—82no—
269C2×C4×C59#12011yescharacteristic
270C2×C4×C59#12011yescharacteristic
271C2×C4×C59#12011yescharacteristic
272C2×C4×C59C22×D411yescharacteristic
273C2×C4×C59C22×D411yescharacteristic
274C2×C4×C59C22×D411yescharacteristic
275C8×C59—24no—
276C8×C59—42no—
277C8×C59—42no—
278#18588C5911yesmaximal, characteristic, direct product factor, Sylow
279C22×C59—21no—
280C22×C59—21no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59#45612yes—
285C22×C59C2×#12011yescharacteristic
286C4×C59—21no—
287C4×C59—22no—
288C4×C59—21no—
289C4×C59—21no—
290C4×C59—21no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—42no—
296C4×C59—41no—
297C4×C59—41no—
298C4×C59—41no—
299C4×C59—82no—
300C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C59#57511yescharacteristic
302C4×C59#57812yes—
303#3523C2×C5911yescharacteristic
304#3239C2×C5911yescharacteristic
305#3165C2×C5911yescharacteristic, complemented
306C2×#462C2×C5911yescharacteristic, complemented
307#1653C2×C5912yescomplemented
308#1652C2×C5912yescomplemented
309#1649C2×C5912yescomplemented
310#1648C2×C5912yescomplemented
311#1552C2×C5911yescharacteristic, complemented
312#1491C2×C5911yescharacteristic, complemented
313#1483C2×C5911yescharacteristic
314C2×C59—22no—
315C2×C59—81no—
316C2×C59#300611yescharacteristic
317C2×C59#301111yescharacteristic
318C2×C59C2×#45611yescharacteristic
319C22×Q16—21no—
320#557—21no—
321#557C22×C5911yescharacteristic
322#557C22×C5911yescharacteristic
323#556—21no—
324#556C22×C5911yescharacteristic
325#552—22no—
326#552—21no—
327D4×Q8—21no—
328D4×Q8C22×C5911yescharacteristic
329#541—22no—
330#541—21no—
331#533—21no—
332#533C22×C5911yescharacteristic
333C2×C4⋊Q8C22×C5911yescharacteristic
334#529—21no—
335#529C22×C5911yescharacteristic
336#519—21no—
337C2×C4×Q8C22×C5911yescharacteristic
338#501—22no—
339#501—21no—
340#500C22×C5912yes—
341#498C22×C5912yes—
342#489—21no—
343#484—21no—
344#481—21no—
345#475C22×C5912yes—
346#474C22×C5912yes—
347#462—21no—
348#462—21no—
349#462C22×C5912yes—
350#462C22×C5912yes—
351#432C22×C5911yescharacteristic
352#428—22no—
353C2×C4⋊C8C22×C5911yescharacteristic
354C2×#103—21no—
355C4×M16C22×C5911yescharacteristic
356#332C22×C5912yes—
357#332C22×C5912yes—
358Q8⋊C8C22×C5912yes—
359Q8⋊C8C22×C5912yes—
360Q8⋊C8C22×C5912yes—
361D4⋊C8C22×C5912yes—
362C59#1858811yescharacteristic, direct product factor, Sylow
363C22×Q8—21no—
364C22×Q8—21no—
365C22×Q8D4×C5911yescharacteristic
366C2×Q16—22no—
367C2×Q16—22no—
368C2×Q16—44no—
369C2×M16—21no—
370C22×C8—21no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8C23×C5912yes—
379C4⋊Q8C23×C5911yescharacteristic
380C4⋊Q8C23×C5911yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#126—21no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—22no—
391#125—22no—
392#125—21no—
393#125D4×C5911yescharacteristic
394#125D4×C5912yes—
395#124—21no—
396#124D4×C5911yescharacteristic
397#123—22no—
398#123—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8D4×C5911yescharacteristic
407#121D4×C5911yescharacteristic
408C4×Q8—22no—
409C4×Q8—22no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8D4×C5911yescharacteristic
420C4×Q8D4×C5911yescharacteristic
421C4×Q8C23×C5911yescharacteristic
422C4×Q8C23×C5911yescharacteristic
423C4×Q8C23×C5911yescharacteristic
424C4×D4—22no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4C23×C5911yescharacteristic
428#117—22no—
429#117—22no—
430#117—22no—
431#117—21no—
432#117—21no—
433#117D4×C5912yes—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4D4×C5911yescharacteristic
438C2×C42—21no—
439C2×C42C23×C5911yescharacteristic
440#107—24no—
441#107—22no—
442#106—24no—
443#106—22no—
444C4⋊C8—21no—
445C4⋊C8C23×C5912yes—
446C4⋊C8C23×C5912yes—
447#103—22no—
448#103—22no—
449#103—22no—
450#102—22no—
451C22⋊C8—21no—
452#97—21no—
453#97—21no—
454C4×C8C23×C5912yes—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—41no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—41no—
467C2×Q8C2×D4×C5911yescharacteristic
468C2×Q8C2×D4×C5911yescharacteristic
469C2×Q8C2×D4×C5911yescharacteristic
470C2×D4—21no—
471C2×D4C2×D4×C5911yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—41no—
477C22×C4C2×D4×C5911yescharacteristic
478C22×C4C2×D4×C5911yescharacteristic
479C22×C4C2×D4×C5911yescharacteristic
480Q16—44no—
481Q16—44no—
482M16—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—42no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4C2×D4×C5911yescharacteristic
513C4⋊C4C2×D4×C5911yescharacteristic
514C4⋊C4C2×D4×C5912yes—
515C4⋊C4C2×D4×C5911yescharacteristic
516C4⋊C4C2×D4×C5912yes—
517C4⋊C4C2×D4×C5911yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42C2×D4×C5911yescharacteristic
537C42C2×D4×C5911yescharacteristic
538C42C2×D4×C5911yescharacteristic
539C42C24×C5911yescharacteristic
540C23—21no—
541C23#120×C5911yescharacteristic
542Q8—21no—
543Q8—21no—
544Q8—21no—
545Q8—22no—
546Q8—42no—
547Q8—44no—
548Q8—41no—
549Q8—41no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—82no—
554Q8AGL(1,ℤ/8)×C5912yes—
555D4—22no—
556D4—42no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—82no—
577C2×C4#120×C5911yescharacteristic
578C2×C4#120×C5911yescharacteristic
579C2×C4#120×C5911yescharacteristic
580C2×C4C22×D4×C5911yescharacteristic
581C2×C4C22×D4×C5911yescharacteristic
582C2×C4C22×D4×C5911yescharacteristic
583C8—24no—
584C8—42no—
585C8—42no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C59×#45612yes—
592C22C2×#120×C5911yescharacteristic
593C4—21no—
594C4—22no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—82no—
607C4C2×AGL(1,ℤ/8)×C5911yescharacteristic
608C4C59×#57511yescharacteristic
609C4C59×#57812yes—
610C2—22no—
611C2—81no—
612C2C59×#300611yescharacteristic
613C2C59×#301111yescharacteristic
614C2C2×C59×#45611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radical#121×C59D4
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
3. C59×#3165C2semidirect
4. C2×C59×#462C2semidirect
5. C59×#1653C2semidirect
6. C59×#1652C2semidirect
7. C59×#1649C2semidirect
8. C59×#1648C2semidirect
9. C59×#1552C2semidirect
10. C59×#1491C2semidirect
278. #18588C59direct (N × H)
305. #3165C2×C59semidirect
306. C2×#462C2×C59semidirect
307. #1653C2×C59semidirect
308. #1652C2×C59semidirect
309. #1649C2×C59semidirect
310. #1648C2×C59semidirect
311. #1552C2×C59semidirect
312. #1491C2×C59semidirect
362. C59#18588direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35232
C59×#32392
C59×#31652
C2×C59×#4622
C59×#16532
C59×#16522
C59×#16492
C59×#16482
C59×#15522
C59×#14912
C59×#14832
#1858859