Finite Group Databasebeta

← C59×#18298 #28430407

Subgroups — C59×#18298 #28430407

Order 15104.

1024 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3460C211yesmaximal, characteristic, complemented
2C59×#3188C212yesmaximal, complemented
3C59×#2635C211yesmaximal, characteristic, complemented
4C59×#1618C212yesmaximal, complemented
5C59×#1617C211yesmaximal, characteristic, complemented
6C59×#1616C211yesmaximal, characteristic, complemented
7C59×#1600C212yesmaximal, complemented
8C59×#1599C211yesmaximal, characteristic, complemented
9C59×#1598C211yesmaximal, characteristic, complemented
10C59×#1516C212yesmaximal, complemented
11C59×#1459C211yesmaximal, 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×C59C2211yescharacteristic
24C4×#41×C59—21no—
25C2×C4×D4×C59C2211yescharacteristic
26C59×#484C2212yes—
27C59×#482—22no—
28C59×#481C2212yes—
29C59×#466—22no—
30C59×#465—22no—
31C59×#450C2212yes—
32C59×#449—22no—
33C59×#447C2212yes—
34C2×C22⋊C8×C59C2212yes—
35C59×#400C2211yescharacteristic
36C59×#399C2211yescharacteristic
37C59×#394—21no—
38C59×#394C2211yescharacteristic
39C59×#392—21no—
40C59×#392C2211yescharacteristic
41C59×#388—21no—
42C59×#386—21no—
43C59×#381C2211yescharacteristic
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×C59D411yescharacteristic
59C23×C4×C59C2311yescharacteristic
60C2×SD16×C59—22no—
61C2×D8×C59—22no—
62C22×C8×C59—22no—
63C4⋊Q8×C59—22no—
64#127×C59—22no—
65#126×C59—22no—
66#126×C59—22no—
67#125×C59—22no—
68#124×C59—22no—
69#124×C59—22no—
70#124×C59—22no—
71#123×C59—22no—
72#123×C59—22no—
73#123×C59—22no—
74#123×C59—22no—
75C22⋊Q8×C59—21no—
76C22⋊Q8×C59—21no—
77C22⋊Q8×C59—22no—
78C22⋊Q8×C59D412yescomplemented
79C22⋊Q8×C59C2312yes—
80#121×C59—21no—
81#121×C59—22no—
82#121×C59—22no—
83#121×C59—22no—
84#121×C59—22no—
85#121×C59—22no—
86#121×C59—21no—
87#121×C59—22no—
88#121×C59—22no—
89#121×C59D412yescomplemented
90#121×C59C2312yes—
91#120×C59—22no—
92#120×C59—22no—
93C4×Q8×C59—21no—
94C4×Q8×C59—21no—
95C4×D4×C59—21no—
96C4×D4×C59—21no—
97C4×D4×C59—22no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—22no—
103#117×C59—22no—
104#117×C59—21no—
105#117×C59—21no—
106#117×C59D412yescomplemented
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59C2311yescharacteristic
118C2×C42×C59—22no—
119C2×C42×C59D411yescharacteristic
120#107×C59—22no—
121#106×C59—22no—
122#103×C59—22no—
123#102×C59—22no—
124#102×C59—22no—
125#102×C59—22no—
126C22⋊C8×C59—22no—
127C22⋊C8×C59C2312yes—
128C22⋊C8×C59C2312yes—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134C24×C59—21no—
135C24×C59C2×D411yescharacteristic
136C24×C59C2×D411yescharacteristic
137#41×C59—42no—
138#41×C59—42no—
139#41×C59—82no—
140#41×C59—82no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C2×D412yes—
144C2×D4×C59—21no—
145C2×D4×C59—22no—
146C2×D4×C59—22no—
147C2×D4×C59—22no—
148C2×D4×C59—22no—
149C2×D4×C59—21no—
150C2×D4×C59—22no—
151C2×D4×C59—22no—
152C2×D4×C59—22no—
153C2×D4×C59—22no—
154C2×D4×C59—41no—
155C2×D4×C59—42no—
156C2×D4×C59—42no—
157C2×D4×C59—81no—
158C2×D4×C59—81no—
159C2×D4×C59—81no—
160C2×D4×C59—81no—
161C2×D4×C59—81no—
162C2×D4×C59C2×D412yes—
163C22×C4×C59—22no—
164C22×C4×C59—22no—
165C22×C4×C59—22no—
166C22×C4×C59—21no—
167C22×C4×C59—21no—
168C22×C4×C59—21no—
169C22×C4×C59—21no—
170C22×C4×C59—21no—
171C22×C4×C59—81no—
172C22×C4×C59—81no—
173C22×C4×C59C2×D411yescharacteristic
174C22×C4×C59C2×D411yescharacteristic
175C22×C4×C59C2×D411yescharacteristic
176C22×C4×C59C2×D411yescharacteristic
177C22×C4×C59C2×D412yes—
178C22×C4×C59C2×D412yes—
179C22×C4×C59C2411yescharacteristic
180SD16×C59—82no—
181D8×C59—82no—
182C2×C8×C59—22no—
183C2×C8×C59—22no—
184C2×C8×C59—82no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—22no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—22no—
193C4⋊C4×C59—22no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—81no—
196C4⋊C4×C59C2×D412yes—
197C4⋊C4×C59C2×D412yes—
198C22⋊C4×C59—22no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—22no—
201C22⋊C4×C59—22no—
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—41no—
210C22⋊C4×C59—41no—
211C22⋊C4×C59—42no—
212C22⋊C4×C59—81no—
213C22⋊C4×C59—81no—
214C22⋊C4×C59—81no—
215C22⋊C4×C59—81no—
216C42×C59—22no—
217C42×C59—22no—
218C42×C59—21no—
219C42×C59—22no—
220C42×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—22no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
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#12012yes—
239C23×C59C22×D411yescharacteristic
240Q8×C59—42no—
241Q8×C59—42no—
242Q8×C59—81no—
243D4×C59—44no—
244D4×C59—42no—
245D4×C59—42no—
246D4×C59—42no—
247D4×C59—42no—
248D4×C59—42no—
249D4×C59—42no—
250D4×C59—82no—
251D4×C59—82no—
252D4×C59—82no—
253D4×C59—81no—
254D4×C59—82no—
255D4×C59—84no—
256D4×C59—84no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—21no—
260C2×C4×C59—22no—
261C2×C4×C59—22no—
262C2×C4×C59—22no—
263C2×C4×C59—21no—
264C2×C4×C59—42no—
265C2×C4×C59—41no—
266C2×C4×C59—42no—
267C2×C4×C59—41no—
268C2×C4×C59—42no—
269C2×C4×C59—44no—
270C2×C4×C59—44no—
271C2×C4×C59—41no—
272C2×C4×C59—41no—
273C2×C4×C59—41no—
274C2×C4×C59—41no—
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—81no—
283C2×C4×C59—82no—
284C2×C4×C59#12011yescharacteristic
285C2×C4×C59#12011yescharacteristic
286C2×C4×C59#13512yes—
287C2×C4×C59C22×D412yes—
288C8×C59—42no—
289C8×C59—42no—
290#18298C5911yesmaximal, characteristic, direct product factor, Sylow
291C22×C59—21no—
292C22×C59—21no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—41no—
299C22×C59—42no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—42no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59—81no—
308C22×C59—81no—
309C22×C59—81no—
310C22×C59—81no—
311C22×C59—81no—
312C22×C59—81no—
313C22×C59C2×#12011yescharacteristic
314C22×C59C2×#13511yescharacteristic
315C22×C59#57511yescharacteristic
316C4×C59—22no—
317C4×C59—22no—
318C4×C59—21no—
319C4×C59—21no—
320C4×C59—42no—
321C4×C59—41no—
322C4×C59—42no—
323C4×C59—41no—
324C4×C59—42no—
325C4×C59—81no—
326C4×C59—81no—
327C4×C59—81no—
328#3460C2×C5911yescharacteristic, complemented
329#3188C2×C5912yescomplemented
330#2635C2×C5911yescharacteristic, complemented
331#1618C2×C5912yescomplemented
332#1617C2×C5911yescharacteristic, complemented
333#1616C2×C5911yescharacteristic, complemented
334#1600C2×C5912yescomplemented
335#1599C2×C5911yescharacteristic, complemented
336#1598C2×C5911yescharacteristic, complemented
337#1516C2×C5912yescomplemented
338#1459C2×C5911yescharacteristic, complemented
339C2×C59—21no—
340C2×C59—21no—
341C2×C59—42no—
342C2×C59—82no—
343C2×C59—81no—
344C2×C59—161no—
345C2×C59#300211yescharacteristic
346C2×C59#302111yescharacteristic
347C2×C59#302611yescharacteristic
348#544C22×C5911yescharacteristic
349#542—22no—
350#540—22no—
351#538—21no—
352#538C22×C5912yes—
353#537C22×C5911yescharacteristic
354#535—21no—
355#532—22no—
356#529—22no—
357#525—21no—
358#525C22×C5912yes—
359C2×#121C22×C5911yescharacteristic
360C4×#41—21no—
361C2×C4×D4C22×C5911yescharacteristic
362#484C22×C5912yes—
363#482—22no—
364#481C22×C5912yes—
365#466—22no—
366#465—22no—
367#450C22×C5912yes—
368#449—22no—
369#447C22×C5912yes—
370C2×C22⋊C8C22×C5912yes—
371#400C22×C5911yescharacteristic
372#399C22×C5911yescharacteristic
373#394—21no—
374#394C22×C5911yescharacteristic
375#392—21no—
376#392C22×C5911yescharacteristic
377#388—21no—
378#386—21no—
379#381C22×C5911yescharacteristic
380#328C22×C5912yes—
381#328C22×C5912yes—
382#327C22×C5912yes—
383#327C22×C5912yes—
384#324C22×C5911yescharacteristic
385#324C22×C5911yescharacteristic
386#323C22×C5912yes—
387C59#1829811yescharacteristic, direct product factor, Sylow
388C2×#41—21no—
389C2×#41—21no—
390C2×#41D4×C5912yescomplemented
391C22×D4—21no—
392C22×D4—21no—
393C22×D4D4×C5911yescharacteristic
394C22×D4C23×C5911yescharacteristic
395C23×C4D4×C5911yescharacteristic
396C23×C4C23×C5911yescharacteristic
397C2×SD16—22no—
398C2×D8—22no—
399C22×C8—22no—
400C4⋊Q8—22no—
401#127—22no—
402#126—22no—
403#126—22no—
404#125—22no—
405#124—22no—
406#124—22no—
407#124—22no—
408#123—22no—
409#123—22no—
410#123—22no—
411#123—22no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—22no—
415C22⋊Q8D4×C5912yescomplemented
416C22⋊Q8C23×C5912yes—
417#121—21no—
418#121—22no—
419#121—22no—
420#121—22no—
421#121—22no—
422#121—22no—
423#121—21no—
424#121—22no—
425#121—22no—
426#121D4×C5912yescomplemented
427#121C23×C5912yes—
428#120—22no—
429#120—22no—
430C4×Q8—21no—
431C4×Q8—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—22no—
439C4×D4—22no—
440#117—22no—
441#117—21no—
442#117—21no—
443#117D4×C5912yescomplemented
444C2×C4⋊C4—21no—
445C2×C4⋊C4C23×C5911yescharacteristic
446C2×C4⋊C4C23×C5911yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4D4×C5911yescharacteristic
454C2×C22⋊C4C23×C5911yescharacteristic
455C2×C42—22no—
456C2×C42D4×C5911yescharacteristic
457#107—22no—
458#106—22no—
459#103—22no—
460#102—22no—
461#102—22no—
462#102—22no—
463C22⋊C8—22no—
464C22⋊C8C23×C5912yes—
465C22⋊C8C23×C5912yes—
466#95—21no—
467#95—21no—
468#95—21no—
469#95C23×C5911yescharacteristic
470#95C23×C5911yescharacteristic
471C24—21no—
472C24C2×D4×C5911yescharacteristic
473C24C2×D4×C5911yescharacteristic
474#41—42no—
475#41—42no—
476#41—82no—
477#41—82no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×D4×C5912yes—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—41no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—81no—
495C2×D4—81no—
496C2×D4—81no—
497C2×D4—81no—
498C2×D4—81no—
499C2×D4C2×D4×C5912yes—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4C2×D4×C5911yescharacteristic
511C22×C4C2×D4×C5911yescharacteristic
512C22×C4C2×D4×C5911yescharacteristic
513C22×C4C2×D4×C5911yescharacteristic
514C22×C4C2×D4×C5912yes—
515C22×C4C2×D4×C5912yes—
516C22×C4C24×C5911yescharacteristic
517SD16—82no—
518D8—82no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—82no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—81no—
533C4⋊C4C2×D4×C5912yes—
534C4⋊C4C2×D4×C5912yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—81no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—22no—
557C42—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23#120×C5912yes—
576C23C22×D4×C5911yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—81no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—81no—
591D4—82no—
592D4—84no—
593D4—84no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4#120×C5911yescharacteristic
622C2×C4#120×C5911yescharacteristic
623C2×C4#135×C5912yes—
624C2×C4C22×D4×C5912yes—
625C8—42no—
626C8—42no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×#120×C5911yescharacteristic
650C22C2×#135×C5911yescharacteristic
651C22C59×#57511yescharacteristic
652C4—22no—
653C4—22no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—81no—
668C2—82no—
669C2—161no—
670C2C59×#300211yescharacteristic
671C2C59×#302111yescharacteristic
672C2C59×#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×#3188C2semidirect
3. C59×#2635C2semidirect
4. C59×#1618C2semidirect
5. C59×#1617C2semidirect
6. C59×#1616C2semidirect
7. C59×#1600C2semidirect
8. C59×#1599C2semidirect
9. C59×#1598C2semidirect
10. C59×#1516C2semidirect
11. C59×#1459C2semidirect
53. C2×#41×C59D4semidirect
78. C22⋊Q8×C59D4semidirect
89. #121×C59D4semidirect
106. #117×C59D4semidirect
290. #18298C59direct (N × H)
328. #3460C2×C59semidirect
329. #3188C2×C59semidirect
330. #2635C2×C59semidirect
331. #1618C2×C59semidirect
332. #1617C2×C59semidirect
333. #1616C2×C59semidirect
334. #1600C2×C59semidirect
335. #1599C2×C59semidirect
336. #1598C2×C59semidirect
337. #1516C2×C59semidirect
338. #1459C2×C59semidirect
387. C59#18298direct (N × H)
390. C2×#41D4×C59semidirect
415. C22⋊Q8D4×C59semidirect
426. #121D4×C59semidirect
443. #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×#31882
C59×#26352
C59×#16182
C59×#16172
C59×#16162
C59×#16002
C59×#15992
C59×#15982
C59×#15162
C59×#14592
#1829859