Finite Group Databasebeta

← C59×#18330 #28430439

Subgroups — C59×#18330 #28430439

Order 15104.

1068 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3460C211yesmaximal, characteristic, complemented
2C59×#3194C212yesmaximal, complemented
3C59×#2543C211yesmaximal, characteristic, complemented
4C59×#1612C212yesmaximal, complemented
5C59×#1611C211yesmaximal, characteristic, complemented
6C59×#1610C211yesmaximal, characteristic, complemented
7C59×#1600C212yesmaximal, complemented
8C59×#1599C211yesmaximal, characteristic, complemented
9C59×#1598C211yesmaximal, characteristic, complemented
10C59×#1504C212yesmaximal, complemented
11C59×#1465C211yesmaximal, characteristic, complemented
12C59×#544C2211yescharacteristic
13C59×#542—22no—
14C59×#540—22no—
15C59×#538—21no—
16C59×#538C2212yes—
17C59×#537C2211yescharacteristic
18C59×#535—21no—
19C59×#532—22no—
20C59×#529—22no—
21C59×#525—21no—
22C59×#525C2212yes—
23C2×#121×C59—21no—
24C2×#121×C59C2211yescharacteristic
25C4×#41×C59—21no—
26C2×C4×D4×C59C2211yescharacteristic
27C59×#483C2212yes—
28C59×#482—22no—
29C59×#480C2212yes—
30C59×#469—22no—
31C59×#468—22no—
32C59×#450C2212yes—
33C59×#449—22no—
34C59×#447C2212yes—
35C59×#407C2212yes—
36C59×#398C2211yescharacteristic
37C59×#397C2211yescharacteristic
38C59×#394C2211yescharacteristic
39C59×#392C2211yescharacteristic
40C59×#388—21no—
41C59×#386—21no—
42C59×#386—21no—
43C2×#95×C59C2211yescharacteristic
44C59×#328C2212yes—
45C59×#328C2212yes—
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2211yescharacteristic
49C59×#324C2211yescharacteristic
50C59×#323C2212yes—
51C2×#41×C59—21no—
52C2×#41×C59—21no—
53C2×#41×C59D412yescomplemented
54C22×D4×C59—21no—
55C22×D4×C59—21no—
56C22×D4×C59D411yescharacteristic
57C22×D4×C59C2311yescharacteristic
58C23×C4×C59—21no—
59C23×C4×C59D411yescharacteristic
60C23×C4×C59C2311yescharacteristic
61C2×SD16×C59—22no—
62C2×D8×C59—22no—
63C2×M16×C59—22no—
64C4⋊Q8×C59—22no—
65#127×C59—22no—
66#126×C59—22no—
67#126×C59—22no—
68#125×C59—22no—
69#124×C59—22no—
70#124×C59—22no—
71#124×C59—22no—
72#123×C59—22no—
73#123×C59—22no—
74#123×C59—22no—
75#123×C59—22no—
76C22⋊Q8×C59—21no—
77C22⋊Q8×C59—21no—
78C22⋊Q8×C59—22no—
79C22⋊Q8×C59D412yescomplemented
80C22⋊Q8×C59C2312yes—
81#121×C59—21no—
82#121×C59—22no—
83#121×C59—22no—
84#121×C59—22no—
85#121×C59—22no—
86#121×C59—22no—
87#121×C59—21no—
88#121×C59—22no—
89#121×C59—22no—
90#121×C59—44no—
91#121×C59D412yescomplemented
92#121×C59C2312yes—
93#120×C59—22no—
94#120×C59—22no—
95C4×Q8×C59—21no—
96C4×Q8×C59—21no—
97C4×D4×C59—21no—
98C4×D4×C59—21no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—22no—
105#117×C59—22no—
106#117×C59—21no—
107#117×C59—21no—
108#117×C59D412yescomplemented
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C22⋊C4×C59C2311yescharacteristic
119C2×C42×C59—22no—
120C2×C42×C59D411yescharacteristic
121#107×C59—22no—
122#106×C59—22no—
123#103×C59—22no—
124#102×C59—22no—
125#102×C59—22no—
126#102×C59—22no—
127C22⋊C8×C59—22no—
128C22⋊C8×C59C2312yes—
129C22⋊C8×C59C2312yes—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59—21no—
133#95×C59C2311yescharacteristic
134#95×C59C2311yescharacteristic
135C24×C59—21no—
136C24×C59C2×D411yescharacteristic
137C24×C59C2×D411yescharacteristic
138#41×C59—42no—
139#41×C59—42no—
140#41×C59—82no—
141#41×C59—82no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59C2×D412yes—
145C2×D4×C59—21no—
146C2×D4×C59—22no—
147C2×D4×C59—22no—
148C2×D4×C59—22no—
149C2×D4×C59—22no—
150C2×D4×C59—21no—
151C2×D4×C59—22no—
152C2×D4×C59—22no—
153C2×D4×C59—22no—
154C2×D4×C59—22no—
155C2×D4×C59—41no—
156C2×D4×C59—42no—
157C2×D4×C59—42no—
158C2×D4×C59—42no—
159C2×D4×C59—42no—
160C2×D4×C59—42no—
161C2×D4×C59—81no—
162C2×D4×C59—81no—
163C2×D4×C59C2×D412yes—
164C22×C4×C59—22no—
165C22×C4×C59—21no—
166C22×C4×C59—22no—
167C22×C4×C59—22no—
168C22×C4×C59—21no—
169C22×C4×C59—21no—
170C22×C4×C59—21no—
171C22×C4×C59—21no—
172C22×C4×C59—42no—
173C22×C4×C59—42no—
174C22×C4×C59—81no—
175C22×C4×C59—81no—
176C22×C4×C59—81no—
177C22×C4×C59C2×D411yescharacteristic
178C22×C4×C59C2×D411yescharacteristic
179C22×C4×C59C2×D411yescharacteristic
180C22×C4×C59C2×D411yescharacteristic
181C22×C4×C59C2×D412yes—
182C22×C4×C59C2×D412yes—
183C22×C4×C59C2411yescharacteristic
184SD16×C59—82no—
185D8×C59—82no—
186M16×C59—82no—
187C2×C8×C59—22no—
188C2×C8×C59—22no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—22no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—22no—
199C4⋊C4×C59—42no—
200C4⋊C4×C59C2×D412yes—
201C4⋊C4×C59C2×D412yes—
202C22⋊C4×C59—22no—
203C22⋊C4×C59—22no—
204C22⋊C4×C59—22no—
205C22⋊C4×C59—22no—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—22no—
209C22⋊C4×C59—22no—
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—42no—
214C22⋊C4×C59—42no—
215C22⋊C4×C59—41no—
216C22⋊C4×C59—41no—
217C22⋊C4×C59—42no—
218C22⋊C4×C59—81no—
219C42×C59—22no—
220C42×C59—22no—
221C42×C59—21no—
222C42×C59—22no—
223C42×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—22no—
228C23×C59—21no—
229C23×C59—21no—
230C23×C59—41no—
231C23×C59—41no—
232C23×C59—41no—
233C23×C59—41no—
234C23×C59—41no—
235C23×C59—41no—
236C23×C59—41no—
237C23×C59—41no—
238C23×C59—41no—
239C23×C59—41no—
240C23×C59—41no—
241C23×C59#12012yes—
242C23×C59C22×D411yescharacteristic
243Q8×C59—42no—
244Q8×C59—42no—
245Q8×C59—81no—
246D4×C59—44no—
247D4×C59—42no—
248D4×C59—42no—
249D4×C59—42no—
250D4×C59—42no—
251D4×C59—42no—
252D4×C59—42no—
253D4×C59—82no—
254D4×C59—82no—
255D4×C59—82no—
256D4×C59—81no—
257D4×C59—82no—
258D4×C59—84no—
259D4×C59—84no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—22no—
263C2×C4×C59—21no—
264C2×C4×C59—22no—
265C2×C4×C59—22no—
266C2×C4×C59—22no—
267C2×C4×C59—21no—
268C2×C4×C59—42no—
269C2×C4×C59—41no—
270C2×C4×C59—42no—
271C2×C4×C59—41no—
272C2×C4×C59—42no—
273C2×C4×C59—44no—
274C2×C4×C59—44no—
275C2×C4×C59—41no—
276C2×C4×C59—41no—
277C2×C4×C59—41no—
278C2×C4×C59—41no—
279C2×C4×C59—41no—
280C2×C4×C59—41no—
281C2×C4×C59—41no—
282C2×C4×C59—41no—
283C2×C4×C59—41no—
284C2×C4×C59—41no—
285C2×C4×C59—81no—
286C2×C4×C59—82no—
287C2×C4×C59—84no—
288C2×C4×C59#12011yescharacteristic
289C2×C4×C59#12011yescharacteristic
290C2×C4×C59AGL(1,ℤ/8)12yes—
291C2×C4×C59C22×D412yes—
292C8×C59—42no—
293C8×C59—42no—
294#18330C5911yesmaximal, characteristic, direct product factor, Sylow
295C22×C59—21no—
296C22×C59—21no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—42no—
301C22×C59—42no—
302C22×C59—42no—
303C22×C59—42no—
304C22×C59—41no—
305C22×C59—42no—
306C22×C59—41no—
307C22×C59—41no—
308C22×C59—42no—
309C22×C59—81no—
310C22×C59—81no—
311C22×C59—81no—
312C22×C59—81no—
313C22×C59—81no—
314C22×C59—81no—
315C22×C59—81no—
316C22×C59—81no—
317C22×C59C2×#12011yescharacteristic
318C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C59#57511yescharacteristic
320C4×C59—22no—
321C4×C59—22no—
322C4×C59—21no—
323C4×C59—21no—
324C4×C59—42no—
325C4×C59—42no—
326C4×C59—41no—
327C4×C59—42no—
328C4×C59—41no—
329C4×C59—42no—
330C4×C59—81no—
331C4×C59—81no—
332#3460C2×C5911yescharacteristic, complemented
333#3194C2×C5912yescomplemented
334#2543C2×C5911yescharacteristic, complemented
335#1612C2×C5912yescomplemented
336#1611C2×C5911yescharacteristic, complemented
337#1610C2×C5911yescharacteristic, complemented
338#1600C2×C5912yescomplemented
339#1599C2×C5911yescharacteristic, complemented
340#1598C2×C5911yescharacteristic, complemented
341#1504C2×C5912yescomplemented
342#1465C2×C5911yescharacteristic, complemented
343C2×C59—21no—
344C2×C59—21no—
345C2×C59—42no—
346C2×C59—82no—
347C2×C59—81no—
348C2×C59—161no—
349C2×C59#300611yescharacteristic
350C2×C59#301811yescharacteristic
351C2×C59#302611yescharacteristic
352#544C22×C5911yescharacteristic
353#542—22no—
354#540—22no—
355#538—21no—
356#538C22×C5912yes—
357#537C22×C5911yescharacteristic
358#535—21no—
359#532—22no—
360#529—22no—
361#525—21no—
362#525C22×C5912yes—
363C2×#121—21no—
364C2×#121C22×C5911yescharacteristic
365C4×#41—21no—
366C2×C4×D4C22×C5911yescharacteristic
367#483C22×C5912yes—
368#482—22no—
369#480C22×C5912yes—
370#469—22no—
371#468—22no—
372#450C22×C5912yes—
373#449—22no—
374#447C22×C5912yes—
375#407C22×C5912yes—
376#398C22×C5911yescharacteristic
377#397C22×C5911yescharacteristic
378#394C22×C5911yescharacteristic
379#392C22×C5911yescharacteristic
380#388—21no—
381#386—21no—
382#386—21no—
383C2×#95C22×C5911yescharacteristic
384#328C22×C5912yes—
385#328C22×C5912yes—
386#327C22×C5912yes—
387#327C22×C5912yes—
388#324C22×C5911yescharacteristic
389#324C22×C5911yescharacteristic
390#323C22×C5912yes—
391C59#1833011yescharacteristic, direct product factor, Sylow
392C2×#41—21no—
393C2×#41—21no—
394C2×#41D4×C5912yescomplemented
395C22×D4—21no—
396C22×D4—21no—
397C22×D4D4×C5911yescharacteristic
398C22×D4C23×C5911yescharacteristic
399C23×C4—21no—
400C23×C4D4×C5911yescharacteristic
401C23×C4C23×C5911yescharacteristic
402C2×SD16—22no—
403C2×D8—22no—
404C2×M16—22no—
405C4⋊Q8—22no—
406#127—22no—
407#126—22no—
408#126—22no—
409#125—22no—
410#124—22no—
411#124—22no—
412#124—22no—
413#123—22no—
414#123—22no—
415#123—22no—
416#123—22no—
417C22⋊Q8—21no—
418C22⋊Q8—21no—
419C22⋊Q8—22no—
420C22⋊Q8D4×C5912yescomplemented
421C22⋊Q8C23×C5912yes—
422#121—21no—
423#121—22no—
424#121—22no—
425#121—22no—
426#121—22no—
427#121—22no—
428#121—21no—
429#121—22no—
430#121—22no—
431#121—44no—
432#121D4×C5912yescomplemented
433#121C23×C5912yes—
434#120—22no—
435#120—22no—
436C4×Q8—21no—
437C4×Q8—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4—22no—
441C4×D4—22no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—22no—
446#117—22no—
447#117—21no—
448#117—21no—
449#117D4×C5912yescomplemented
450C2×C4⋊C4—21no—
451C2×C4⋊C4C23×C5911yescharacteristic
452C2×C4⋊C4C23×C5911yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4D4×C5911yescharacteristic
459C2×C22⋊C4C23×C5911yescharacteristic
460C2×C42—22no—
461C2×C42D4×C5911yescharacteristic
462#107—22no—
463#106—22no—
464#103—22no—
465#102—22no—
466#102—22no—
467#102—22no—
468C22⋊C8—22no—
469C22⋊C8C23×C5912yes—
470C22⋊C8C23×C5912yes—
471#95—21no—
472#95—21no—
473#95—21no—
474#95C23×C5911yescharacteristic
475#95C23×C5911yescharacteristic
476C24—21no—
477C24C2×D4×C5911yescharacteristic
478C24C2×D4×C5911yescharacteristic
479#41—42no—
480#41—42no—
481#41—82no—
482#41—82no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8C2×D4×C5912yes—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—21no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—22no—
496C2×D4—41no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—81no—
503C2×D4—81no—
504C2×D4C2×D4×C5912yes—
505C22×C4—22no—
506C22×C4—21no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—81no—
516C22×C4—81no—
517C22×C4—81no—
518C22×C4C2×D4×C5911yescharacteristic
519C22×C4C2×D4×C5911yescharacteristic
520C22×C4C2×D4×C5911yescharacteristic
521C22×C4C2×D4×C5911yescharacteristic
522C22×C4C2×D4×C5912yes—
523C22×C4C2×D4×C5912yes—
524C22×C4C24×C5911yescharacteristic
525SD16—82no—
526D8—82no—
527M16—82no—
528C2×C8—22no—
529C2×C8—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4C2×D4×C5912yes—
542C4⋊C4C2×D4×C5912yes—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—22no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—42no—
559C22⋊C4—81no—
560C42—22no—
561C42—22no—
562C42—21no—
563C42—22no—
564C42—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—21no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23#120×C5912yes—
583C23C22×D4×C5911yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—81no—
587D4—44no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—81no—
598D4—82no—
599D4—84no—
600D4—84no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—81no—
627C2×C4—82no—
628C2×C4—84no—
629C2×C4#120×C5911yescharacteristic
630C2×C4#120×C5911yescharacteristic
631C2×C4AGL(1,ℤ/8)×C5912yes—
632C2×C4C22×D4×C5912yes—
633C8—42no—
634C8—42no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C2×#120×C5911yescharacteristic
658C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
659C22C59×#57511yescharacteristic
660C4—22no—
661C4—22no—
662C4—21no—
663C4—21no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C2—21no—
673C2—21no—
674C2—42no—
675C2—81no—
676C2—82no—
677C2—161no—
678C2C59×#300611yescharacteristic
679C2C59×#301811yescharacteristic
680C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3460C2semidirect
2. C59×#3194C2semidirect
3. C59×#2543C2semidirect
4. C59×#1612C2semidirect
5. C59×#1611C2semidirect
6. C59×#1610C2semidirect
7. C59×#1600C2semidirect
8. C59×#1599C2semidirect
9. C59×#1598C2semidirect
10. C59×#1504C2semidirect
11. C59×#1465C2semidirect
53. C2×#41×C59D4semidirect
79. C22⋊Q8×C59D4semidirect
91. #121×C59D4semidirect
108. #117×C59D4semidirect
294. #18330C59direct (N × H)
332. #3460C2×C59semidirect
333. #3194C2×C59semidirect
334. #2543C2×C59semidirect
335. #1612C2×C59semidirect
336. #1611C2×C59semidirect
337. #1610C2×C59semidirect
338. #1600C2×C59semidirect
339. #1599C2×C59semidirect
340. #1598C2×C59semidirect
341. #1504C2×C59semidirect
342. #1465C2×C59semidirect
391. C59#18330direct (N × H)
394. C2×#41D4×C59semidirect
420. C22⋊Q8D4×C59semidirect
432. #121D4×C59semidirect
449. #117D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34602
C59×#31942
C59×#25432
C59×#16122
C59×#16112
C59×#16102
C59×#16002
C59×#15992
C59×#15982
C59×#15042
C59×#14652
#1833059