Finite Group Databasebeta

← C59×#18393 #28430502

Subgroups — C59×#18393 #28430502

Order 15104.

910 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3491C211yesmaximal, characteristic
2C59×#3232C212yesmaximal, complemented
3C59×#2611C211yesmaximal, characteristic, complemented
4C59×#1615C212yesmaximal, complemented
5C59×#1614C211yesmaximal, characteristic, complemented
6C59×#1613C211yesmaximal, characteristic
7C59×#1609C212yesmaximal, complemented
8C59×#1608C211yesmaximal, characteristic, complemented
9C59×#1607C211yesmaximal, characteristic
10C59×#1500C212yesmaximal, complemented
11C59×#1458C211yesmaximal, characteristic, complemented
12C59×#552—22no—
13C59×#552—22no—
14C59×#551—22no—
15C59×#551—22no—
16C59×#543C2211yescharacteristic
17C59×#539—21no—
18C59×#539C2212yes—
19C59×#534C2211yescharacteristic
20C59×#529—21no—
21C59×#529C2212yes—
22C2×#125×C59C2211yescharacteristic, complemented
23C2×#123×C59—21no—
24C59×#518—21no—
25C2×#117×C59C2211yescharacteristic
26C59×#499—22no—
27C59×#488—22no—
28C59×#484C2212yescomplemented
29C59×#483C2212yescomplemented
30C59×#481C2212yes—
31C59×#480C2212yes—
32C59×#479—22no—
33C59×#478—22no—
34C59×#431C2212yescomplemented
35C59×#400C2211yescharacteristic, complemented
36C59×#399C2211yescharacteristic
37C59×#398C2211yescharacteristic, complemented
38C59×#397C2211yescharacteristic
39C59×#389—21no—
40C59×#388—21no—
41C59×#387—21no—
42C59×#385—21no—
43C59×#376C2211yescharacteristic, complemented
44C59×#328C2212yescomplemented
45C59×#328C2212yescomplemented
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2212yescomplemented
49C59×#323C2211yescharacteristic
50C59×#323C2211yescharacteristic
51C22×D4×C59—21no—
52C23×C4×C59D411yescharacteristic
53#126×C59—22no—
54#126×C59—22no—
55#126×C59—22no—
56#126×C59—22no—
57#125×C59—21no—
58#125×C59—22no—
59#125×C59—22no—
60#125×C59—22no—
61#125×C59—22no—
62#125×C59D412yes—
63#124×C59—21no—
64#124×C59D412yes—
65#123×C59—22no—
66#123×C59—22no—
67#123×C59—22no—
68#123×C59—22no—
69#123×C59—22no—
70#123×C59—22no—
71#123×C59—22no—
72C22⋊Q8×C59—22no—
73C22⋊Q8×C59—22no—
74C22⋊Q8×C59—22no—
75C22⋊Q8×C59—21no—
76C22⋊Q8×C59—22no—
77C22⋊Q8×C59C2312yes—
78#121×C59—21no—
79#121×C59C2312yes—
80#120×C59—22no—
81C4×Q8×C59—21no—
82C4×Q8×C59D412yes—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—21no—
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88C4×D4×C59D412yes—
89#117×C59—22no—
90#117×C59—22no—
91#117×C59—22no—
92#117×C59—22no—
93#117×C59—22no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59D411yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C22⋊C4×C59—22no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59C2311yescharacteristic
112C2×C42×C59—21no—
113C2×C42×C59D411yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#107×C59—22no—
116#107×C59—22no—
117#106×C59—22no—
118#106×C59—22no—
119C4⋊C8×C59—22no—
120#103×C59—22no—
121#103×C59—22no—
122#102×C59—22no—
123#102×C59—22no—
124C22⋊C8×C59C2312yes—
125C22⋊C8×C59C2312yes—
126C4×C8×C59—22no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134C24×C59—21no—
135C24×C59C2×D411yescharacteristic
136C2×Q8×C59—21no—
137C2×Q8×C59C2×D412yes—
138C2×D4×C59—22no—
139C2×D4×C59—22no—
140C2×D4×C59—21no—
141C2×D4×C59—42no—
142C2×D4×C59—42no—
143C2×D4×C59C2×D412yes—
144C22×C4×C59—21no—
145C22×C4×C59—22no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—22no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—22no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—41no—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2411yescharacteristic
163C2×C8×C59—22no—
164C2×C8×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59—42no—
181C4⋊C4×C59—41no—
182C4⋊C4×C59—41no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59C2×D412yes—
185C4⋊C4×C59C2×D412yes—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—22no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—42no—
199C22⋊C4×C59—42no—
200C42×C59—22no—
201C42×C59—21no—
202C42×C59—21no—
203C42×C59—22no—
204C42×C59—42no—
205C42×C59C2×D412yes—
206C42×C59C2×D412yes—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—22no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—42no—
213C23×C59—81no—
214C23×C59—81no—
215C23×C59#13512yes—
216C23×C59C22×D411yescharacteristic
217Q8×C59—42no—
218Q8×C59—81no—
219D4×C59—42no—
220D4×C59—44no—
221D4×C59—42no—
222D4×C59—42no—
223D4×C59—81no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
240C2×C4×C59—42no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—82no—
254C2×C4×C59—81no—
255C2×C4×C59—82no—
256C2×C4×C59—81no—
257C2×C4×C59#12011yescharacteristic
258C2×C4×C59#12011yescharacteristic
259C2×C4×C59#12012yes—
260C2×C4×C59C22×D412yes—
261C8×C59—42no—
262C8×C59—42no—
263#18393C5911yesmaximal, characteristic, direct product factor, Sylow
264C22×C59—21no—
265C22×C59—21no—
266C22×C59—22no—
267C22×C59—42no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—81no—
274C22×C59—162no—
275C22×C59C2×#12011yescharacteristic
276C22×C59C2×#13511yescharacteristic
277C22×C59#57511yescharacteristic
278C4×C59—21no—
279C4×C59—21no—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—42no—
283C4×C59—41no—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—42no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59—81no—
292#3491C2×C5911yescharacteristic
293#3232C2×C5912yescomplemented
294#2611C2×C5911yescharacteristic, complemented
295#1615C2×C5912yescomplemented
296#1614C2×C5911yescharacteristic, complemented
297#1613C2×C5911yescharacteristic
298#1609C2×C5912yescomplemented
299#1608C2×C5911yescharacteristic, complemented
300#1607C2×C5911yescharacteristic
301#1500C2×C5912yescomplemented
302#1458C2×C5911yescharacteristic, complemented
303C2×C59—21no—
304C2×C59—21no—
305C2×C59—42no—
306C2×C59—82no—
307C2×C59#300211yescharacteristic
308C2×C59#301811yescharacteristic
309C2×C59#302811yescharacteristic
310#552—22no—
311#552—22no—
312#551—22no—
313#551—22no—
314#543C22×C5911yescharacteristic
315#539—21no—
316#539C22×C5912yes—
317#534C22×C5911yescharacteristic
318#529—21no—
319#529C22×C5912yes—
320C2×#125C22×C5911yescharacteristic, complemented
321C2×#123—21no—
322#518—21no—
323C2×#117C22×C5911yescharacteristic
324#499—22no—
325#488—22no—
326#484C22×C5912yescomplemented
327#483C22×C5912yescomplemented
328#481C22×C5912yes—
329#480C22×C5912yes—
330#479—22no—
331#478—22no—
332#431C22×C5912yescomplemented
333#400C22×C5911yescharacteristic, complemented
334#399C22×C5911yescharacteristic
335#398C22×C5911yescharacteristic, complemented
336#397C22×C5911yescharacteristic
337#389—21no—
338#388—21no—
339#387—21no—
340#385—21no—
341#376C22×C5911yescharacteristic, complemented
342#328C22×C5912yescomplemented
343#328C22×C5912yescomplemented
344#327C22×C5912yes—
345#327C22×C5912yes—
346#324C22×C5912yescomplemented
347#323C22×C5911yescharacteristic
348#323C22×C5911yescharacteristic
349C59#1839311yescharacteristic, direct product factor, Sylow
350C22×D4—21no—
351C23×C4D4×C5911yescharacteristic
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#125—21no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—22no—
361#125D4×C5912yes—
362#124—21no—
363#124D4×C5912yes—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8C23×C5912yes—
377#121—21no—
378#121C23×C5912yes—
379#120—22no—
380C4×Q8—21no—
381C4×Q8D4×C5912yes—
382C4×D4—22no—
383C4×D4—22no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4D4×C5912yes—
388#117—22no—
389#117—22no—
390#117—22no—
391#117—22no—
392#117—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4D4×C5911yescharacteristic
400C2×C4⋊C4C23×C5911yescharacteristic
401C2×C4⋊C4C23×C5911yescharacteristic
402C2×C4⋊C4C23×C5911yescharacteristic
403C2×C22⋊C4—22no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4D4×C5911yescharacteristic
410C2×C22⋊C4C23×C5911yescharacteristic
411C2×C42—21no—
412C2×C42D4×C5911yescharacteristic
413C2×C42C23×C5911yescharacteristic
414#107—22no—
415#107—22no—
416#106—22no—
417#106—22no—
418C4⋊C8—22no—
419#103—22no—
420#103—22no—
421#102—22no—
422#102—22no—
423C22⋊C8C23×C5912yes—
424C22⋊C8C23×C5912yes—
425C4×C8—22no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95C23×C5911yescharacteristic
432#95C23×C5911yescharacteristic
433C24—21no—
434C24C2×D4×C5911yescharacteristic
435C2×Q8—21no—
436C2×Q8C2×D4×C5912yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4C2×D4×C5912yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4C2×D4×C5911yescharacteristic
457C22×C4C2×D4×C5911yescharacteristic
458C22×C4C2×D4×C5911yescharacteristic
459C22×C4C2×D4×C5911yescharacteristic
460C22×C4C2×D4×C5911yescharacteristic
461C22×C4C24×C5911yescharacteristic
462C2×C8—22no—
463C2×C8—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4C2×D4×C5912yes—
484C4⋊C4C2×D4×C5912yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—22no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—42no—
504C42C2×D4×C5912yes—
505C42C2×D4×C5912yes—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—81no—
513C23—81no—
514C23#135×C5912yes—
515C23C22×D4×C5911yescharacteristic
516Q8—42no—
517Q8—81no—
518D4—42no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—81no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4#120×C5911yescharacteristic
557C2×C4#120×C5911yescharacteristic
558C2×C4#120×C5912yes—
559C2×C4C22×D4×C5912yes—
560C8—42no—
561C8—42no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—162no—
573C22C2×#120×C5911yescharacteristic
574C22C2×#135×C5911yescharacteristic
575C22C59×#57511yescharacteristic
576C4—21no—
577C4—21no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—81no—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—82no—
594C2C59×#300211yescharacteristic
595C2C59×#301811yescharacteristic
596C2C59×#302811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#534C22
LCSG ▷ C22×C4 ▷ 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×#3232C2semidirect
3. C59×#2611C2semidirect
4. C59×#1615C2semidirect
5. C59×#1614C2semidirect
7. C59×#1609C2semidirect
8. C59×#1608C2semidirect
10. C59×#1500C2semidirect
11. C59×#1458C2semidirect
22. C2×#125×C59C22semidirect
28. C59×#484C22semidirect
29. C59×#483C22semidirect
34. C59×#431C22semidirect
35. C59×#400C22semidirect
37. C59×#398C22semidirect
43. C59×#376C22semidirect
44. C59×#328C22semidirect
45. C59×#328C22semidirect
48. C59×#324C22semidirect
263. #18393C59direct (N × H)
293. #3232C2×C59semidirect
294. #2611C2×C59semidirect
295. #1615C2×C59semidirect
296. #1614C2×C59semidirect
298. #1609C2×C59semidirect
299. #1608C2×C59semidirect
301. #1500C2×C59semidirect
302. #1458C2×C59semidirect
320. C2×#125C22×C59semidirect
326. #484C22×C59semidirect
327. #483C22×C59semidirect
332. #431C22×C59semidirect
333. #400C22×C59semidirect
335. #398C22×C59semidirect
341. #376C22×C59semidirect
342. #328C22×C59semidirect
343. #328C22×C59semidirect
346. #324C22×C59semidirect
349. C59#18393direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34912
C59×#32322
C59×#26112
C59×#16152
C59×#16142
C59×#16132
C59×#16092
C59×#16082
C59×#16072
C59×#15002
C59×#14582
#1839359