Finite Group Databasebeta

← C59×#25517 #28437626

Subgroups — C59×#25517 #28437626

Order 15104.

1174 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3492C211yesmaximal, characteristic, complemented
2C59×#3055C212yesmaximal, complemented
3C59×#2501C211yesmaximal, characteristic, complemented
4C59×#2060C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic, complemented
6C59×#2027C211yesmaximal, characteristic, complemented
7C59×#2006C211yesmaximal, characteristic, complemented
8C59×#1883C212yesmaximal, complemented
9C59×#1871C212yesmaximal, complemented
10C59×#1859C212yesmaximal, complemented
11C2×C59×#343C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C59C2212yes—
13C22×M16×C59C2212yes—
14C59×#554—22no—
15C59×#553—22no—
16C59×#552—22no—
17C59×#547—22no—
18C59×#544C2211yescharacteristic
19C59×#539—22no—
20C59×#537C2211yescharacteristic
21C59×#525—22no—
22C59×#525C2212yes—
23C2×#123×C59—22no—
24C2×#121×C59C2211yescharacteristic
25C59×#520C2212yes—
26C22×C4⋊C4×C59C2211yescharacteristic
27C59×#469—22no—
28C59×#468—22no—
29C59×#466—22no—
30C59×#465—22no—
31C59×#428C2212yes—
32C2×#104×C59C2212yes—
33C2×#104×C59C2212yes—
34C59×#418C2212yes—
35C59×#417C2212yes—
36C59×#399—22no—
37C59×#390C2211yescharacteristic
38C59×#390C2211yescharacteristic
39C59×#384C2211yescharacteristic
40C59×#384C2211yescharacteristic
41C4×C22⋊C4×C59C2211yescharacteristic
42C59×#343C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2212yes—
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59—22no—
48C2×#41×C59C2312yes—
49C22×D4×C59—21no—
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D412yes—
52C23×C4×C59C2311yescharacteristic
53AGL(1,ℤ/8)×C59—24no—
54AGL(1,ℤ/8)×C59—42no—
55C2×SD16×C59—22no—
56C2×D8×C59—22no—
57C2×M16×C59—22no—
58C2×M16×C59—24no—
59C2×M16×C59—42no—
60C2×M16×C59C2312yes—
61C2×M16×C59C2312yes—
62C22×C8×C59—22no—
63C4⋊Q8×C59—22no—
64#126×C59—24no—
65#125×C59—24no—
66#125×C59—22no—
67#123×C59—24no—
68#123×C59—24no—
69#123×C59—22no—
70#123×C59—48no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—24no—
73C22⋊Q8×C59—22no—
74C22⋊Q8×C59—22no—
75C22⋊Q8×C59D412yes—
76C22⋊Q8×C59D412yes—
77#121×C59—22no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—22no—
81#121×C59—22no—
82#121×C59D412yes—
83#121×C59D412yes—
84#120×C59—22no—
85C4×Q8×C59—22no—
86C4×D4×C59—24no—
87C4×D4×C59—22no—
88C4×D4×C59—24no—
89C4×D4×C59—22no—
90C4×D4×C59—22no—
91#117×C59—22no—
92#117×C59—22no—
93#117×C59C2312yes—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59D412yes—
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—22no—
105C2×C22⋊C4×C59—22no—
106C2×C22⋊C4×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109#107×C59—22no—
110#106×C59—22no—
111#104×C59—24no—
112#104×C59—42no—
113#103×C59—22no—
114#102×C59—22no—
115#102×C59—22no—
116#102×C59—22no—
117#95×C59—21no—
118C24×C59—21no—
119C24×C59C2×D411yescharacteristic
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123#41×C59—22no—
124#41×C59—84no—
125C2×Q8×C59—22no—
126C2×Q8×C59C2×D412yes—
127C2×D4×C59—22no—
128C2×D4×C59—22no—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—22no—
132C2×D4×C59—22no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—44no—
136C2×D4×C59—41no—
137C2×D4×C59—41no—
138C2×D4×C59C2×D412yes—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—24no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—21no—
149C22×C4×C59—44no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D412yes—
155C22×C4×C59C2411yescharacteristic
156SD16×C59—44no—
157D8×C59—44no—
158M16×C59—22no—
159M16×C59—22no—
160M16×C59—22no—
161M16×C59—44no—
162M16×C59#4114yescomplemented
163C2×C8×C59—22no—
164C2×C8×C59—22no—
165C2×C8×C59—44no—
166C4⋊C4×C59—24no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—24no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—44no—
177C4⋊C4×C59—44no—
178C4⋊C4×C59C2×D412yes—
179C4⋊C4×C59C2×D412yes—
180C22⋊C4×C59—24no—
181C22⋊C4×C59—24no—
182C22⋊C4×C59—24no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—22no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—44no—
189C22⋊C4×C59—44no—
190C22⋊C4×C59—42no—
191C42×C59—24no—
192C42×C59—22no—
193C42×C59—22no—
194C42×C59—21no—
195C42×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—41no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59#12011yescharacteristic
207C23×C59#12011yescharacteristic
208C23×C59C22×D411yescharacteristic
209Q8×C59—22no—
210Q8×C59—22no—
211Q8×C59—82no—
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—44no—
215D4×C59—42no—
216D4×C59—42no—
217D4×C59—44no—
218D4×C59—42no—
219D4×C59—42no—
220D4×C59—82no—
221D4×C59—82no—
222D4×C59—82no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—44no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—41no—
247C2×C4×C59—82no—
248C2×C4×C59—84no—
249C2×C4×C59#12012yes—
250C2×C4×C59C22×D412yes—
251C2×C4×C59C2×#4112yes—
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C8×C59—24no—
255C8×C59—42no—
256#25517C5911yesmaximal, characteristic, direct product factor, Sylow
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—21no—
262C22×C59—21no—
263C22×C59—21no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—82no—
270C22×C59—82no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59#45312yes—
275C22×C59C2×#12011yescharacteristic
276C22×C59#53211yescharacteristic
277C22×C59#53711yescharacteristic
278C4×C59—22no—
279C4×C59—42no—
280C4×C59—42no—
281C4×C59—42no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—81no—
287C4×C59#54612yes—
288C4×C59#54712yes—
289#3492C2×C5911yescharacteristic, complemented
290#3055C2×C5912yescomplemented
291#2501C2×C5911yescharacteristic, complemented
292#2060C2×C5911yescharacteristic, complemented
293#2039C2×C5911yescharacteristic, complemented
294#2027C2×C5911yescharacteristic, complemented
295#2006C2×C5911yescharacteristic, complemented
296#1883C2×C5912yescomplemented
297#1871C2×C5912yescomplemented
298#1859C2×C5912yescomplemented
299C2×#343C2×C5911yescharacteristic, complemented
300C2×C59—21no—
301C2×C59—21no—
302C2×C59—22no—
303C2×C59—41no—
304C2×C59—82no—
305C2×C59—161no—
306C2×C59#243011yescharacteristic
307C2×C59C2×#45311yescharacteristic
308C2×C59#302011yescharacteristic
309C2×AGL(1,ℤ/8)C22×C5912yes—
310C22×M16C22×C5912yes—
311#554—22no—
312#553—22no—
313#552—22no—
314#547—22no—
315#544C22×C5911yescharacteristic
316#539—22no—
317#537C22×C5911yescharacteristic
318#525—22no—
319#525C22×C5912yes—
320C2×#123—22no—
321C2×#121C22×C5911yescharacteristic
322#520C22×C5912yes—
323C22×C4⋊C4C22×C5911yescharacteristic
324#469—22no—
325#468—22no—
326#466—22no—
327#465—22no—
328#428C22×C5912yes—
329C2×#104C22×C5912yes—
330C2×#104C22×C5912yes—
331#418C22×C5912yes—
332#417C22×C5912yes—
333#399—22no—
334#390C22×C5911yescharacteristic
335#390C22×C5911yescharacteristic
336#384C22×C5911yescharacteristic
337#384C22×C5911yescharacteristic
338C4×C22⋊C4C22×C5911yescharacteristic
339#343C22×C5911yescharacteristic
340#343C22×C5912yes—
341#343C22×C5911yescharacteristic
342#339C22×C5912yes—
343#339C22×C5912yes—
344C59#2551711yescharacteristic, direct product factor, Sylow
345C2×#41—22no—
346C2×#41C23×C5912yes—
347C22×D4—21no—
348C22×D4C23×C5911yescharacteristic
349C23×C4D4×C5912yes—
350C23×C4C23×C5911yescharacteristic
351AGL(1,ℤ/8)—24no—
352AGL(1,ℤ/8)—42no—
353C2×SD16—22no—
354C2×D8—22no—
355C2×M16—22no—
356C2×M16—24no—
357C2×M16—42no—
358C2×M16C23×C5912yes—
359C2×M16C23×C5912yes—
360C22×C8—22no—
361C4⋊Q8—22no—
362#126—24no—
363#125—24no—
364#125—22no—
365#123—24no—
366#123—24no—
367#123—22no—
368#123—48no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8D4×C5912yes—
374C22⋊Q8D4×C5912yes—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121D4×C5912yes—
381#121D4×C5912yes—
382#120—22no—
383C4×Q8—22no—
384C4×D4—24no—
385C4×D4—22no—
386C4×D4—24no—
387C4×D4—22no—
388C4×D4—22no—
389#117—22no—
390#117—22no—
391#117C23×C5912yes—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4D4×C5912yes—
398C2×C4⋊C4C23×C5911yescharacteristic
399C2×C4⋊C4C23×C5911yescharacteristic
400C2×C4⋊C4C23×C5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C42C23×C5911yescharacteristic
406C2×C42C23×C5911yescharacteristic
407#107—22no—
408#106—22no—
409#104—24no—
410#104—42no—
411#103—22no—
412#102—22no—
413#102—22no—
414#102—22no—
415#95—21no—
416C24—21no—
417C24C2×D4×C5911yescharacteristic
418#41—22no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—84no—
423C2×Q8—22no—
424C2×Q8C2×D4×C5912yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—44no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4C2×D4×C5912yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—44no—
448C22×C4C2×D4×C5911yescharacteristic
449C22×C4C2×D4×C5912yes—
450C22×C4C2×D4×C5912yes—
451C22×C4C2×D4×C5912yes—
452C22×C4C2×D4×C5912yes—
453C22×C4C24×C5911yescharacteristic
454SD16—44no—
455D8—44no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—44no—
460M16#41×C5914yescomplemented
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4C2×D4×C5912yes—
477C4⋊C4C2×D4×C5912yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—21no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23#120×C5911yescharacteristic
505C23#120×C5911yescharacteristic
506C23C22×D4×C5911yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—82no—
510D4—22no—
511D4—22no—
512D4—44no—
513D4—44no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—82no—
519D4—82no—
520D4—82no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4#120×C5912yes—
548C2×C4C22×D4×C5912yes—
549C2×C4C2×#41×C5912yes—
550C2×C4H(5,𝔽2)×C5911yescharacteristic
551C2×C42−1+4×C5911yescharacteristic
552C8—24no—
553C8—42no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C59×#45312yes—
572C22C2×#120×C5911yescharacteristic
573C22C59×#53211yescharacteristic
574C22C59×#53711yescharacteristic
575C4—22no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4C59×#54612yes—
585C4C59×#54712yes—
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—82no—
591C2—161no—
592C2C59×#243011yescharacteristic
593C2C2×C59×#45311yescharacteristic
594C2C59×#302011yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
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
1. C59×#3492C2semidirect
2. C59×#3055C2semidirect
3. C59×#2501C2semidirect
4. C59×#2060C2semidirect
5. C59×#2039C2semidirect
6. C59×#2027C2semidirect
7. C59×#2006C2semidirect
8. C59×#1883C2semidirect
9. C59×#1871C2semidirect
10. C59×#1859C2semidirect
11. C2×C59×#343C2semidirect
162. M16×C59#41semidirect
256. #25517C59direct (N × H)
289. #3492C2×C59semidirect
290. #3055C2×C59semidirect
291. #2501C2×C59semidirect
292. #2060C2×C59semidirect
293. #2039C2×C59semidirect
294. #2027C2×C59semidirect
295. #2006C2×C59semidirect
296. #1883C2×C59semidirect
297. #1871C2×C59semidirect
298. #1859C2×C59semidirect
299. C2×#343C2×C59semidirect
344. C59#25517direct (N × H)
460. M16#41×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34922
C59×#30552
C59×#25012
C59×#20602
C59×#20392
C59×#20272
C59×#20062
C59×#18832
C59×#18712
C59×#18592
C2×C59×#3432
#2551759