Finite Group Databasebeta

← C59×#9113 #28421222

Subgroups — C59×#9113 #28421222

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C59C211yesmaximal, characteristic, complemented
2C59×#2847C211yesmaximal, characteristic
3C2×C59×#327C212yesmaximal, complemented
4C59×C22⋊#95C211yesmaximal, characteristic, complemented
5C59×#1294C212yesmaximal, complemented
6C24×C4×C59C2211yescharacteristic
7C23×C8×C59—21no—
8C2×#121×C59C411yescharacteristic, complemented
9C2×#121×C59C411yescharacteristic, complemented
10C2×#121×C59C2211yescharacteristic
11C2×#120×C59—21no—
12C22×C22⋊C4×C59—21no—
13C2×C22⋊C8×C59—21no—
14C2×C22⋊C8×C59—24no—
15C2×C22⋊C8×C59C2212yes—
16C59×#386—21no—
17C59×#386—21no—
18C59×#386—21no—
19C59×#380C411yescharacteristic, complemented
20C59×#380C411yescharacteristic, complemented
21C59×#380C2211yescharacteristic
22C2×#95×C59—21no—
23C2×#95×C59C2212yes—
24C59×#327—24no—
25C25×C59D411yescharacteristic
26C22×D4×C59—21no—
27C22×D4×C59—21no—
28C22×D4×C59—21no—
29C22×D4×C59C2×C411yescharacteristic
30C23×C4×C59—22no—
31C23×C4×C59—22no—
32C23×C4×C59—22no—
33C23×C4×C59—42no—
34C23×C4×C59—42no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59C2311yescharacteristic
41C22×C8×C59—24no—
42C22×C8×C59—22no—
43C22×C8×C59—42no—
44C22×C8×C59—42no—
45#121×C59—22no—
46#121×C59—22no—
47#121×C59—42no—
48#121×C59—42no—
49#120×C59—42no—
50#120×C59—42no—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C4⋊C4×C59C2×C411yescharacteristic
54C2×C22⋊C4×C59—24no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—42no—
62C2×C22⋊C4×C59—42no—
63C22⋊C8×C59—24no—
64C22⋊C8×C59—22no—
65C22⋊C8×C59D414yescomplemented
66#95×C59—21no—
67#95×C59—21no—
68#95×C59—21no—
69#95×C59—22no—
70#95×C59—22no—
71#95×C59—42no—
72C24×C59—22no—
73C24×C59—22no—
74C24×C59—22no—
75C24×C59—22no—
76C24×C59—21no—
77C24×C59—41no—
78C24×C59—41no—
79C24×C59C22⋊C411yescharacteristic
80C24×C59C22⋊C411yescharacteristic
81C24×C59D812yes—
82C24×C59SD1612yes—
83C24×C59C2×D411yescharacteristic
84C2×D4×C59—22no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—42no—
92C2×D4×C59—42no—
93C2×D4×C59—42no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C2×D4×C59—42no—
97C22×C4×C59—22no—
98C22×C4×C59—22no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—21no—
108C22×C4×C59—21no—
109C22×C4×C59—24no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—41no—
114C22×C4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59—41no—
117C22×C4×C59—41no—
118C22×C4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59—48no—
121C22×C4×C59—48no—
122C22×C4×C59—42no—
123C22×C4×C59—42no—
124C22×C4×C59—42no—
125C22×C4×C59—42no—
126C22×C4×C59—42no—
127C22×C4×C59—42no—
128C22×C4×C59C22⋊C411yescharacteristic
129C22×C4×C59C22⋊C411yescharacteristic
130C22×C4×C59C22×C411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D412yes—
133C22×C4×C59#4112yes—
134C2×C8×C59—24no—
135C2×C8×C59—48no—
136C2×C8×C59—42no—
137C2×C8×C59—42no—
138C4⋊C4×C59—22no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C22⋊C4×C59—42no—
142C22⋊C4×C59—48no—
143C22⋊C4×C59—42no—
144C22⋊C4×C59—42no—
145C22⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—42no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—21no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—24no—
159C23×C59—24no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—24no—
167C23×C59—24no—
168C23×C59—42no—
169C23×C59—42no—
170C23×C59—44no—
171C23×C59—44no—
172C23×C59—44no—
173C23×C59—44no—
174C23×C59—41no—
175C23×C59—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—41no—
181C23×C59#9912yes—
182C23×C59#10212yes—
183C23×C59#10212yes—
184C23×C59#10412yes—
185C23×C59C2×C22⋊C411yescharacteristic
186C23×C59#12112yes—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×SD1611yescharacteristic
189C23×C59#13512yes—
190D4×C59—82no—
191D4×C59—84no—
192D4×C59—84no—
193D4×C59—82no—
194D4×C59—82no—
195C2×C4×C59—24no—
196C2×C4×C59—24no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59—44no—
209C2×C4×C59—44no—
210C2×C4×C59—44no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—44no—
214C2×C4×C59—44no—
215C2×C4×C59—44no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—82no—
226C2×C4×C59—82no—
227C2×C4×C59—82no—
228C2×C4×C59—82no—
229C2×C4×C59#9912yes—
230C2×C4×C59C4×D412yes—
231C2×C4×C59#12011yescharacteristic
232C2×C4×C59#12311yescharacteristic
233C8×C59—44no—
234#9113C5911yesmaximal, characteristic, direct product factor, Sylow
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—22no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—21no—
247C22×C59—22no—
248C22×C59—21no—
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—48no—
253C22×C59—48no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59#32714yescomplemented
266C22×C59#38611yescharacteristic
267C22×C59C2×#9911yescharacteristic
268C22×C59C2×#9911yescharacteristic
269C22×C59C2×#10211yescharacteristic
270C22×C59#41611yescharacteristic
271C22×C59C2×#10411yescharacteristic
272C22×C59#42111yescharacteristic
273C22×C59#46512yes—
274C22×C59#46612yes—
275C4×C59—21no—
276C4×C59—22no—
277C4×C59—21no—
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—84no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59—81no—
284C22×C22⋊C8C2×C5911yescharacteristic, complemented
285#2847C2×C5911yescharacteristic
286C2×#327C2×C5912yescomplemented
287C22⋊#95C2×C5911yescharacteristic, complemented
288#1294C2×C5912yescomplemented
289C2×C59—22no—
290C2×C59—24no—
291C2×C59—22no—
292C2×C59—42no—
293C2×C59—161no—
294C2×C59C2×#32711yescharacteristic
295C2×C59#150111yescharacteristic
296C2×C59#151611yescharacteristic
297C2×C59#151911yescharacteristic
298C2×C59#187111yescharacteristic
299C2×C59#189311yescharacteristic
300C2×C59#189811yescharacteristic
301C24×C4C22×C5911yescharacteristic
302C23×C8—21no—
303C2×#121C4×C5911yescharacteristic, complemented
304C2×#121C4×C5911yescharacteristic, complemented
305C2×#121C22×C5911yescharacteristic
306C2×#120—21no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C22×C5912yes—
311#386—21no—
312#386—21no—
313#386—21no—
314#380C4×C5911yescharacteristic, complemented
315#380C4×C5911yescharacteristic, complemented
316#380C22×C5911yescharacteristic
317C2×#95—21no—
318C2×#95C22×C5912yes—
319#327—24no—
320C59#911311yescharacteristic, direct product factor, Sylow
321C25D4×C5911yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×C4×C5911yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C2×C4×C5911yescharacteristic
332C23×C4C2×C4×C5911yescharacteristic
333C23×C4D4×C5911yescharacteristic
334C23×C4D4×C5911yescharacteristic
335C23×C4D4×C5911yescharacteristic
336C23×C4C23×C5911yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×C4×C5911yescharacteristic
348C2×C4⋊C4C2×C4×C5911yescharacteristic
349C2×C4⋊C4C2×C4×C5911yescharacteristic
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8D4×C5914yescomplemented
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—42no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C5911yescharacteristic
376C24C22⋊C4×C5911yescharacteristic
377C24D8×C5912yes—
378C24SD16×C5912yes—
379C24C2×D4×C5911yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C5911yescharacteristic
425C22×C4C22⋊C4×C5911yescharacteristic
426C22×C4C22×C4×C5911yescharacteristic
427C22×C4C2×D4×C5911yescharacteristic
428C22×C4C2×D4×C5912yes—
429C22×C4#41×C5912yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#99×C5912yes—
478C23#102×C5912yes—
479C23#102×C5912yes—
480C23#104×C5912yes—
481C23C2×C22⋊C4×C5911yescharacteristic
482C23#121×C5912yes—
483C23C2×D8×C5911yescharacteristic
484C23C2×SD16×C5911yescharacteristic
485C23#135×C5912yes—
486D4—82no—
487D4—84no—
488D4—82no—
489D4—82no—
490D4—84no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#99×C5912yes—
526C2×C4C4×D4×C5912yes—
527C2×C4#120×C5911yescharacteristic
528C2×C4#123×C5911yescharacteristic
529C8—44no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C59×#32714yescomplemented
561C22C59×#38611yescharacteristic
562C22C2×#99×C5911yescharacteristic
563C22C2×#99×C5911yescharacteristic
564C22C2×#102×C5911yescharacteristic
565C22C59×#41611yescharacteristic
566C22C2×#104×C5911yescharacteristic
567C22C59×#42111yescharacteristic
568C22C59×#46512yes—
569C22C59×#46612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—161no—
584C2C2×C59×#32711yescharacteristic
585C2C59×#150111yescharacteristic
586C2C59×#151611yescharacteristic
587C2C59×#151911yescharacteristic
588C2C59×#187111yescharacteristic
589C2C59×#189311yescharacteristic
590C2C59×#189811yescharacteristic
subgroupG/H
center, socleC23×C59C2×C22⋊C4
commutatorC22×C4C22×C4×C59
Frattini, cosocleC23×C4C23×C59
prime radicalC59×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C59 ◁ C24×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C59C2semidirect
3. C2×C59×#327C2semidirect
4. C59×C22⋊#95C2semidirect
5. C59×#1294C2semidirect
8. C2×#121×C59C4semidirect
9. C2×#121×C59C4semidirect
19. C59×#380C4semidirect
20. C59×#380C4semidirect
65. C22⋊C8×C59D4semidirect
234. #9113C59direct (N × H)
265. C22×C59#327semidirect
284. C22×C22⋊C8C2×C59semidirect
286. C2×#327C2×C59semidirect
287. C22⋊#95C2×C59semidirect
288. #1294C2×C59semidirect
303. C2×#121C4×C59semidirect
304. C2×#121C4×C59semidirect
314. #380C4×C59semidirect
315. #380C4×C59semidirect
320. C59#9113direct (N × H)
361. C22⋊C8D4×C59semidirect
560. C22C59×#327semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C592
C59×#28472
C2×C59×#3272
C59×C22⋊#952
C59×#12942
#911359