Finite Group Databasebeta

← C59×#24082 #28436191

Subgroups — C59×#24082 #28436191

Order 15104.

1130 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3494C211yesmaximal, characteristic
2C59×#3092C211yesmaximal, characteristic, complemented
3C2×C59×#428C211yesmaximal, characteristic, complemented
4C59×#2501C211yesmaximal, characteristic, complemented
5C59×#2060C212yesmaximal, complemented
6C59×#2037C212yesmaximal, complemented
7C59×#1900C211yesmaximal, characteristic, complemented
8C59×#1871C211yesmaximal, characteristic, complemented
9C59×#1859C212yesmaximal, complemented
10C59×#1856C212yesmaximal, complemented
11C59×#1826C211yesmaximal, characteristic, complemented
12C22×M16×C59C2211yescharacteristic
13C59×#555—21no—
14C59×#553—21no—
15C59×#553—21no—
16C59×#551—22no—
17C59×#547—21no—
18C59×#546—22no—
19C59×#544C2211yescharacteristic
20C59×#542—21no—
21C59×#540—21no—
22C59×#535C2211yescharacteristic
23C59×#532—21no—
24C59×#529—21no—
25C59×#525C2211yescharacteristic
26C2×#123×C59—22no—
27C2×#121×C59C2211yescharacteristic
28C59×#520C2211yescharacteristic
29C59×#518C2211yescharacteristic
30C2×#117×C59C2211yescharacteristic
31C22×C4⋊C4×C59C2211yescharacteristic
32C59×#484—21no—
33C59×#483—21no—
34C59×#481—21no—
35C59×#480—21no—
36C59×#428—21no—
37C59×#428—21no—
38C59×#428C2212yes—
39C59×#428C2212yes—
40C59×#428C2212yes—
41C2×#107×C59—21no—
42C2×#106×C59—21no—
43C2×#104×C59C2212yes—
44C59×#417C2212yes—
45C2×#100×C59C2212yes—
46C59×#407C2211yescharacteristic
47C59×#399—22no—
48C59×#390C2212yes—
49C59×#384C2212yes—
50C4×C22⋊C4×C59C2211yescharacteristic
51C59×#343C2212yes—
52C59×#339C2212yes—
53C59×#339C2212yes—
54C59×#339C2212yes—
55C2×#41×C59—22no—
56C2×#41×C59C2311yescharacteristic
57C22×D4×C59D412yes—
58C22×D4×C59C2311yescharacteristic
59C23×C4×C59—21no—
60C23×C4×C59C2311yescharacteristic
61C2×M16×C59—21no—
62C2×M16×C59—24no—
63C2×M16×C59C2312yes—
64C2×M16×C59C2312yes—
65C22×C8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68#127×C59—21no—
69#126×C59—22no—
70#126×C59—22no—
71#125×C59—21no—
72#125×C59—21no—
73#124×C59—22no—
74#124×C59—22no—
75#124×C59—21no—
76#123×C59—22no—
77#123×C59—22no—
78#123×C59—22no—
79#123×C59—22no—
80#123×C59—21no—
81#123×C59—48no—
82C22⋊Q8×C59—22no—
83C22⋊Q8×C59—22no—
84C22⋊Q8×C59D411yescharacteristic
85C22⋊Q8×C59D411yescharacteristic
86#121×C59—22no—
87#121×C59—22no—
88#121×C59—22no—
89#121×C59—21no—
90#121×C59—21no—
91#121×C59—21no—
92#121×C59—21no—
93#121×C59—21no—
94#121×C59—21no—
95#121×C59D411yescharacteristic
96#121×C59D411yescharacteristic
97#120×C59—21no—
98#120×C59—22no—
99C4×Q8×C59—21no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—21no—
103C4×D4×C59—22no—
104C4×D4×C59—21no—
105C4×D4×C59—21no—
106C4×D4×C59—21no—
107C4×D4×C59—21no—
108C4×D4×C59—21no—
109C4×D4×C59—21no—
110#117×C59—22no—
111#117×C59—21no—
112#117×C59—21no—
113#117×C59—21no—
114#117×C59D411yescharacteristic
115#117×C59D411yescharacteristic
116#117×C59C2311yescharacteristic
117#117×C59C2311yescharacteristic
118#117×C59C2311yescharacteristic
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—21no—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59—21no—
123C2×C4⋊C4×C59—21no—
124C2×C4⋊C4×C59—24no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59D411yescharacteristic
127C2×C4⋊C4×C59D411yescharacteristic
128C2×C4⋊C4×C59C2312yes—
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C22⋊C4×C59—21no—
131C2×C22⋊C4×C59—21no—
132C2×C22⋊C4×C59—22no—
133C2×C22⋊C4×C59—24no—
134C2×C22⋊C4×C59D412yes—
135C2×C42×C59—21no—
136C2×C42×C59C2312yes—
137#107×C59—22no—
138#107×C59—22no—
139#107×C59—22no—
140#106×C59—22no—
141#106×C59—22no—
142#106×C59—22no—
143#104×C59—24no—
144#103×C59—22no—
145#102×C59—22no—
146#100×C59—24no—
147C22⋊C8×C59—21no—
148C22⋊C8×C59—21no—
149#95×C59—21no—
150C24×C59C2×D411yescharacteristic
151C24×C59C2×D412yes—
152#41×C59—22no—
153#41×C59—22no—
154#41×C59—84no—
155C2×Q8×C59—22no—
156C2×Q8×C59C2×D411yescharacteristic
157C2×D4×C59—21no—
158C2×D4×C59—22no—
159C2×D4×C59—21no—
160C2×D4×C59—22no—
161C2×D4×C59—22no—
162C2×D4×C59—21no—
163C2×D4×C59—22no—
164C2×D4×C59—21no—
165C2×D4×C59—21no—
166C2×D4×C59—22no—
167C2×D4×C59—22no—
168C2×D4×C59—44no—
169C2×D4×C59C2×D411yescharacteristic
170C22×C4×C59—22no—
171C22×C4×C59—22no—
172C22×C4×C59—21no—
173C22×C4×C59—22no—
174C22×C4×C59—22no—
175C22×C4×C59—22no—
176C22×C4×C59—22no—
177C22×C4×C59—21no—
178C22×C4×C59—21no—
179C22×C4×C59—22no—
180C22×C4×C59—44no—
181C22×C4×C59—41no—
182C22×C4×C59—41no—
183C22×C4×C59C2×D411yescharacteristic
184C22×C4×C59C2×D411yescharacteristic
185C22×C4×C59C2×D411yescharacteristic
186C22×C4×C59C2×D411yescharacteristic
187C22×C4×C59C2×D411yescharacteristic
188C22×C4×C59C2×D412yes—
189C22×C4×C59C2411yescharacteristic
190M16×C59—24no—
191M16×C59—22no—
192M16×C59—22no—
193M16×C59—22no—
194M16×C59#4114yescomplemented
195C2×C8×C59—24no—
196C2×C8×C59—22no—
197C2×C8×C59—22no—
198C4⋊C4×C59—22no—
199C4⋊C4×C59—22no—
200C4⋊C4×C59—22no—
201C4⋊C4×C59—22no—
202C4⋊C4×C59—21no—
203C4⋊C4×C59—21no—
204C4⋊C4×C59—21no—
205C4⋊C4×C59—21no—
206C4⋊C4×C59—21no—
207C4⋊C4×C59—21no—
208C4⋊C4×C59—22no—
209C4⋊C4×C59—22no—
210C4⋊C4×C59—48no—
211C4⋊C4×C59—41no—
212C4⋊C4×C59—41no—
213C4⋊C4×C59C2×D411yescharacteristic
214C4⋊C4×C59C2×D411yescharacteristic
215C4⋊C4×C59C2×D411yescharacteristic
216C4⋊C4×C59C2×D411yescharacteristic
217C4⋊C4×C59C2×D411yescharacteristic
218C4⋊C4×C59C2×D411yescharacteristic
219C22⋊C4×C59—22no—
220C22⋊C4×C59—22no—
221C22⋊C4×C59—24no—
222C22⋊C4×C59—22no—
223C22⋊C4×C59—22no—
224C22⋊C4×C59—22no—
225C22⋊C4×C59—21no—
226C22⋊C4×C59—21no—
227C22⋊C4×C59—21no—
228C22⋊C4×C59—21no—
229C22⋊C4×C59—21no—
230C22⋊C4×C59—21no—
231C22⋊C4×C59—21no—
232C22⋊C4×C59—22no—
233C22⋊C4×C59—22no—
234C22⋊C4×C59—21no—
235C22⋊C4×C59—21no—
236C22⋊C4×C59—48no—
237C42×C59—22no—
238C42×C59—21no—
239C42×C59—21no—
240C42×C59—21no—
241C42×C59—21no—
242C42×C59—21no—
243C42×C59—22no—
244C42×C59—22no—
245C23×C59—22no—
246C23×C59—22no—
247C23×C59—22no—
248C23×C59—21no—
249C23×C59—24no—
250C23×C59—41no—
251C23×C59—42no—
252C23×C59—42no—
253C23×C59#12011yescharacteristic
254C23×C59#12011yescharacteristic
255C23×C59C22×D411yescharacteristic
256Q8×C59—22no—
257Q8×C59—82no—
258D4×C59—22no—
259D4×C59—42no—
260D4×C59—44no—
261D4×C59—42no—
262D4×C59—41no—
263D4×C59—41no—
264D4×C59—42no—
265D4×C59—81no—
266D4×C59—82no—
267D4×C59—82no—
268D4×C59—82no—
269C2×C4×C59—22no—
270C2×C4×C59—21no—
271C2×C4×C59—21no—
272C2×C4×C59—21no—
273C2×C4×C59—21no—
274C2×C4×C59—22no—
275C2×C4×C59—22no—
276C2×C4×C59—24no—
277C2×C4×C59—21no—
278C2×C4×C59—21no—
279C2×C4×C59—21no—
280C2×C4×C59—21no—
281C2×C4×C59—21no—
282C2×C4×C59—21no—
283C2×C4×C59—21no—
284C2×C4×C59—22no—
285C2×C4×C59—42no—
286C2×C4×C59—42no—
287C2×C4×C59—42no—
288C2×C4×C59—41no—
289C2×C4×C59—41no—
290C2×C4×C59—42no—
291C2×C4×C59—44no—
292C2×C4×C59—42no—
293C2×C4×C59—81no—
294C2×C4×C59—82no—
295C2×C4×C59—82no—
296C2×C4×C59—82no—
297C2×C4×C59—82no—
298C2×C4×C59—82no—
299C2×C4×C59#12011yescharacteristic
300C2×C4×C59#12011yescharacteristic
301C2×C4×C59C22×D411yescharacteristic
302C2×C4×C59C22×D411yescharacteristic
303C2×C4×C59C2×#4112yes—
304C2×C4×C592−1+412yes—
305C8×C59—24no—
306C8×C59—42no—
307#24082C5911yesmaximal, characteristic, direct product factor, Sylow
308C22×C59—21no—
309C22×C59—21no—
310C22×C59—21no—
311C22×C59—21no—
312C22×C59—21no—
313C22×C59—21no—
314C22×C59—21no—
315C22×C59—44no—
316C22×C59—41no—
317C22×C59—42no—
318C22×C59—41no—
319C22×C59—42no—
320C22×C59—42no—
321C22×C59—41no—
322C22×C59—84no—
323C22×C59#45312yes—
324C22×C59C2×#12011yescharacteristic
325C22×C59#54712yes—
326C4×C59—21no—
327C4×C59—21no—
328C4×C59—44no—
329C4×C59—41no—
330C4×C59—41no—
331C4×C59—41no—
332C4×C59—42no—
333C4×C59—41no—
334C4×C59—41no—
335C4×C59—41no—
336C4×C59—41no—
337C4×C59—82no—
338C4×C59#52511yescharacteristic
339C4×C59#53611yescharacteristic
340C4×C59#54712yes—
341#3494C2×C5911yescharacteristic
342#3092C2×C5911yescharacteristic, complemented
343C2×#428C2×C5911yescharacteristic, complemented
344#2501C2×C5911yescharacteristic, complemented
345#2060C2×C5912yescomplemented
346#2037C2×C5912yescomplemented
347#1900C2×C5911yescharacteristic, complemented
348#1871C2×C5911yescharacteristic, complemented
349#1859C2×C5912yescomplemented
350#1856C2×C5912yescomplemented
351#1826C2×C5911yescharacteristic, complemented
352C2×C59—22no—
353C2×C59—22no—
354C2×C59—41no—
355C2×C59—81no—
356C2×C59—82no—
357C2×C59#240611yescharacteristic
358C2×C59C2×#45311yescharacteristic
359C2×C59#302211yescharacteristic
360C22×M16C22×C5911yescharacteristic
361#555—21no—
362#553—21no—
363#553—21no—
364#551—22no—
365#547—21no—
366#546—22no—
367#544C22×C5911yescharacteristic
368#542—21no—
369#540—21no—
370#535C22×C5911yescharacteristic
371#532—21no—
372#529—21no—
373#525C22×C5911yescharacteristic
374C2×#123—22no—
375C2×#121C22×C5911yescharacteristic
376#520C22×C5911yescharacteristic
377#518C22×C5911yescharacteristic
378C2×#117C22×C5911yescharacteristic
379C22×C4⋊C4C22×C5911yescharacteristic
380#484—21no—
381#483—21no—
382#481—21no—
383#480—21no—
384#428—21no—
385#428—21no—
386#428C22×C5912yes—
387#428C22×C5912yes—
388#428C22×C5912yes—
389C2×#107—21no—
390C2×#106—21no—
391C2×#104C22×C5912yes—
392#417C22×C5912yes—
393C2×#100C22×C5912yes—
394#407C22×C5911yescharacteristic
395#399—22no—
396#390C22×C5912yes—
397#384C22×C5912yes—
398C4×C22⋊C4C22×C5911yescharacteristic
399#343C22×C5912yes—
400#339C22×C5912yes—
401#339C22×C5912yes—
402#339C22×C5912yes—
403C59#2408211yescharacteristic, direct product factor, Sylow
404C2×#41—22no—
405C2×#41C23×C5911yescharacteristic
406C22×D4D4×C5912yes—
407C22×D4C23×C5911yescharacteristic
408C23×C4—21no—
409C23×C4C23×C5911yescharacteristic
410C2×M16—21no—
411C2×M16—24no—
412C2×M16C23×C5912yes—
413C2×M16C23×C5912yes—
414C22×C8—21no—
415C4⋊Q8—21no—
416C4⋊Q8—21no—
417#127—21no—
418#126—22no—
419#126—22no—
420#125—21no—
421#125—21no—
422#124—22no—
423#124—22no—
424#124—21no—
425#123—22no—
426#123—22no—
427#123—22no—
428#123—22no—
429#123—21no—
430#123—48no—
431C22⋊Q8—22no—
432C22⋊Q8—22no—
433C22⋊Q8D4×C5911yescharacteristic
434C22⋊Q8D4×C5911yescharacteristic
435#121—22no—
436#121—22no—
437#121—22no—
438#121—21no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—21no—
443#121—21no—
444#121D4×C5911yescharacteristic
445#121D4×C5911yescharacteristic
446#120—21no—
447#120—22no—
448C4×Q8—21no—
449C4×D4—22no—
450C4×D4—22no—
451C4×D4—21no—
452C4×D4—22no—
453C4×D4—21no—
454C4×D4—21no—
455C4×D4—21no—
456C4×D4—21no—
457C4×D4—21no—
458C4×D4—21no—
459#117—22no—
460#117—21no—
461#117—21no—
462#117—21no—
463#117D4×C5911yescharacteristic
464#117D4×C5911yescharacteristic
465#117C23×C5911yescharacteristic
466#117C23×C5911yescharacteristic
467#117C23×C5911yescharacteristic
468C2×C4⋊C4—21no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—21no—
473C2×C4⋊C4—21no—
474C2×C4⋊C4—24no—
475C2×C4⋊C4D4×C5911yescharacteristic
476C2×C4⋊C4D4×C5911yescharacteristic
477C2×C4⋊C4C23×C5912yes—
478C2×C4⋊C4C23×C5911yescharacteristic
479C2×C22⋊C4—21no—
480C2×C22⋊C4—22no—
481C2×C22⋊C4—21no—
482C2×C22⋊C4—24no—
483C2×C22⋊C4D4×C5912yes—
484C2×C42—21no—
485C2×C42C23×C5912yes—
486#107—22no—
487#107—22no—
488#107—22no—
489#106—22no—
490#106—22no—
491#106—22no—
492#104—24no—
493#103—22no—
494#102—22no—
495#100—24no—
496C22⋊C8—21no—
497C22⋊C8—21no—
498#95—21no—
499C24C2×D4×C5911yescharacteristic
500C24C2×D4×C5912yes—
501#41—22no—
502#41—22no—
503#41—84no—
504C2×Q8—22no—
505C2×Q8C2×D4×C5911yescharacteristic
506C2×D4—21no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—21no—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—22no—
516C2×D4—22no—
517C2×D4—44no—
518C2×D4C2×D4×C5911yescharacteristic
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—44no—
530C22×C4—41no—
531C22×C4—41no—
532C22×C4C2×D4×C5911yescharacteristic
533C22×C4C2×D4×C5911yescharacteristic
534C22×C4C2×D4×C5911yescharacteristic
535C22×C4C2×D4×C5911yescharacteristic
536C22×C4C2×D4×C5911yescharacteristic
537C22×C4C2×D4×C5912yes—
538C22×C4C24×C5911yescharacteristic
539M16—24no—
540M16—22no—
541M16—22no—
542M16—22no—
543M16#41×C5914yescomplemented
544C2×C8—24no—
545C2×C8—22no—
546C2×C8—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—48no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4C2×D4×C5911yescharacteristic
563C4⋊C4C2×D4×C5911yescharacteristic
564C4⋊C4C2×D4×C5911yescharacteristic
565C4⋊C4C2×D4×C5911yescharacteristic
566C4⋊C4C2×D4×C5911yescharacteristic
567C4⋊C4C2×D4×C5911yescharacteristic
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—24no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—21no—
577C22⋊C4—21no—
578C22⋊C4—21no—
579C22⋊C4—21no—
580C22⋊C4—21no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—21no—
584C22⋊C4—21no—
585C22⋊C4—48no—
586C42—22no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—22no—
593C42—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—21no—
598C23—24no—
599C23—41no—
600C23—42no—
601C23—42no—
602C23#120×C5911yescharacteristic
603C23#120×C5911yescharacteristic
604C23C22×D4×C5911yescharacteristic
605Q8—22no—
606Q8—82no—
607D4—22no—
608D4—44no—
609D4—42no—
610D4—42no—
611D4—41no—
612D4—41no—
613D4—42no—
614D4—81no—
615D4—82no—
616D4—82no—
617D4—82no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4#120×C5911yescharacteristic
649C2×C4#120×C5911yescharacteristic
650C2×C4C22×D4×C5911yescharacteristic
651C2×C4C22×D4×C5911yescharacteristic
652C2×C4C2×#41×C5912yes—
653C2×C42−1+4×C5912yes—
654C8—24no—
655C8—42no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—41no—
668C22—41no—
669C22—42no—
670C22—84no—
671C22C59×#45312yes—
672C22C2×#120×C5911yescharacteristic
673C22C59×#54712yes—
674C4—21no—
675C4—21no—
676C4—44no—
677C4—41no—
678C4—41no—
679C4—42no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—82no—
686C4C59×#52511yescharacteristic
687C4C59×#53611yescharacteristic
688C4C59×#54712yes—
689C2—22no—
690C2—22no—
691C2—41no—
692C2—82no—
693C2—81no—
694C2C59×#240611yescharacteristic
695C2C2×C59×#45311yescharacteristic
696C2C59×#302211yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3494C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3092C2semidirect
3. C2×C59×#428C2semidirect
4. C59×#2501C2semidirect
5. C59×#2060C2semidirect
6. C59×#2037C2semidirect
7. C59×#1900C2semidirect
8. C59×#1871C2semidirect
9. C59×#1859C2semidirect
10. C59×#1856C2semidirect
11. C59×#1826C2semidirect
194. M16×C59#41semidirect
307. #24082C59direct (N × H)
342. #3092C2×C59semidirect
343. C2×#428C2×C59semidirect
344. #2501C2×C59semidirect
345. #2060C2×C59semidirect
346. #2037C2×C59semidirect
347. #1900C2×C59semidirect
348. #1871C2×C59semidirect
349. #1859C2×C59semidirect
350. #1856C2×C59semidirect
351. #1826C2×C59semidirect
403. C59#24082direct (N × H)
543. M16#41×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34942
C59×#30922
C2×C59×#4282
C59×#25012
C59×#20602
C59×#20372
C59×#19002
C59×#18712
C59×#18592
C59×#18562
C59×#18262
#2408259