Finite Group Databasebeta

← C59×#18216 #28430325

Subgroups — C59×#18216 #28430325

Order 15104.

988 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3493C211yesmaximal, characteristic, complemented
2C59×#3105C212yesmaximal, complemented
3C59×#2598C211yesmaximal, 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×#1521C212yesmaximal, complemented
11C59×#1458C211yesmaximal, characteristic, complemented
12C59×#556—22no—
13C59×#555—22no—
14C59×#553—22no—
15C59×#552—22no—
16C59×#544C2211yescharacteristic
17C59×#538—21no—
18C59×#538C2212yes—
19C59×#537C2211yescharacteristic
20C59×#529—21no—
21C59×#529C2212yes—
22C59×#525—21no—
23C59×#520—21no—
24C2×C4×D4×C59C2211yescharacteristic
25C2×#117×C59C2211yescharacteristic
26C59×#484C2212yes—
27C59×#481C2212yes—
28C59×#478—22no—
29C59×#463—22no—
30C59×#450C2212yes—
31C59×#447C2212yes—
32C4×SD16×C59—22no—
33C4×D8×C59—22no—
34C59×#431C2212yes—
35C59×#400C2211yescharacteristic
36C59×#399C2211yescharacteristic
37C59×#394C2211yescharacteristic
38C59×#392C2211yescharacteristic
39C59×#388—21no—
40C59×#386—21no—
41C59×#386—21no—
42C59×#385—21no—
43C59×#383C2211yescharacteristic
44C59×#328C2212yes—
45C59×#328C2212yes—
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2211yescharacteristic
49C59×#324C2211yescharacteristic
50C59×#323C2212yes—
51C2×#41×C59—21no—
52C22×D4×C59—21no—
53C22×D4×C59C2311yescharacteristic
54C23×C4×C59—21no—
55C23×C4×C59D411yescharacteristic
56C2×SD16×C59—22no—
57C2×D8×C59—22no—
58C4⋊Q8×C59—22no—
59#126×C59—22no—
60#126×C59—22no—
61#126×C59—22no—
62#126×C59—22no—
63#125×C59—21no—
64#125×C59—22no—
65#125×C59D412yescomplemented
66#124×C59—21no—
67#124×C59—22no—
68#124×C59—22no—
69#124×C59D412yescomplemented
70#123×C59—22no—
71#123×C59—22no—
72#123×C59—22no—
73C22⋊Q8×C59—21no—
74C22⋊Q8×C59—21no—
75C22⋊Q8×C59—21no—
76C22⋊Q8×C59—22no—
77C22⋊Q8×C59C2312yes—
78#121×C59—21no—
79#121×C59—21no—
80#121×C59—22no—
81#121×C59—22no—
82#121×C59—22no—
83#121×C59—21no—
84#121×C59C2312yes—
85#120×C59—22no—
86C4×Q8×C59—22no—
87C4×Q8×C59—21no—
88C4×Q8×C59D412yescomplemented
89C4×D4×C59—22no—
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92C4×D4×C59—21no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C4×D4×C59—22no—
96C4×D4×C59—22no—
97C4×D4×C59D412yescomplemented
98#117×C59—22no—
99#117×C59—22no—
100#117×C59—21no—
101#117×C59—22no—
102#117×C59—22no—
103#117×C59—22no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59C2311yescharacteristic
116C2×C42×C59D411yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#107×C59—22no—
119#106×C59—22no—
120C4⋊C8×C59—22no—
121#103×C59—22no—
122#102×C59—22no—
123#102×C59—22no—
124#102×C59—22no—
125C22⋊C8×C59C2312yes—
126C22⋊C8×C59C2312yes—
127C4×C8×C59—22no—
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
136#41×C59—42no—
137#41×C59—42no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59C2×D412yes—
141C2×D4×C59—21no—
142C2×D4×C59—22no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—21no—
146C2×D4×C59—22no—
147C2×D4×C59—42no—
148C2×D4×C59—42no—
149C2×D4×C59—42no—
150C2×D4×C59—81no—
151C2×D4×C59C2×D412yes—
152C22×C4×C59—22no—
153C22×C4×C59—22no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—22no—
157C22×C4×C59—22no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59—41no—
162C22×C4×C59—42no—
163C22×C4×C59—81no—
164C22×C4×C59—81no—
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2411yescharacteristic
171SD16×C59—44no—
172D8×C59—44no—
173C2×C8×C59—22no—
174C2×C8×C59—22no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—22no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—22no—
188C4⋊C4×C59—81no—
189C4⋊C4×C59C2×D412yes—
190C4⋊C4×C59C2×D412yes—
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—22no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—22no—
201C22⋊C4×C59—22no—
202C22⋊C4×C59—42no—
203C22⋊C4×C59—41no—
204C22⋊C4×C59—41no—
205C22⋊C4×C59—42no—
206C22⋊C4×C59—81no—
207C22⋊C4×C59—81no—
208C42×C59—22no—
209C42×C59—22no—
210C42×C59—21no—
211C42×C59—21no—
212C42×C59—22no—
213C42×C59C2×D412yes—
214C42×C59C2×D412yes—
215C23×C59—22no—
216C23×C59—21no—
217C23×C59—22no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59C22×D411yescharacteristic
229Q8×C59—42no—
230Q8×C59—42no—
231Q8×C59—81no—
232D4×C59—24no—
233D4×C59—42no—
234D4×C59—42no—
235D4×C59—42no—
236D4×C59—42no—
237D4×C59—42no—
238D4×C59—82no—
239D4×C59—81no—
240D4×C59—82no—
241D4×C59—84no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—22no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—44no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—82no—
268C2×C4×C59—84no—
269C2×C4×C59—82no—
270C2×C4×C59#12011yescharacteristic
271C2×C4×C59#12011yescharacteristic
272C2×C4×C59#12012yes—
273C2×C4×C59#13512yes—
274C2×C4×C59C22×D412yes—
275C8×C59—42no—
276C8×C59—42no—
277#18216C5911yesmaximal, characteristic, direct product factor, Sylow
278C22×C59—21no—
279C22×C59—21no—
280C22×C59—42no—
281C22×C59—42no—
282C22×C59—42no—
283C22×C59—41no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—41no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—41no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59C2×#12011yescharacteristic
296C22×C59C2×#13511yescharacteristic
297C22×C59#57511yescharacteristic
298C4×C59—22no—
299C4×C59—21no—
300C4×C59—21no—
301C4×C59—22no—
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—41no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—81no—
309C4×C59—81no—
310C4×C59—81no—
311#3493C2×C5911yescharacteristic, complemented
312#3105C2×C5912yescomplemented
313#2598C2×C5911yescharacteristic, complemented
314#1618C2×C5912yescomplemented
315#1617C2×C5911yescharacteristic, complemented
316#1616C2×C5911yescharacteristic, complemented
317#1600C2×C5912yescomplemented
318#1599C2×C5911yescharacteristic, complemented
319#1598C2×C5911yescharacteristic, complemented
320#1521C2×C5912yescomplemented
321#1458C2×C5911yescharacteristic, complemented
322C2×C59—21no—
323C2×C59—21no—
324C2×C59—82no—
325C2×C59—81no—
326C2×C59—82no—
327C2×C59#300211yescharacteristic
328C2×C59#302111yescharacteristic
329C2×C59#302511yescharacteristic
330#556—22no—
331#555—22no—
332#553—22no—
333#552—22no—
334#544C22×C5911yescharacteristic
335#538—21no—
336#538C22×C5912yes—
337#537C22×C5911yescharacteristic
338#529—21no—
339#529C22×C5912yes—
340#525—21no—
341#520—21no—
342C2×C4×D4C22×C5911yescharacteristic
343C2×#117C22×C5911yescharacteristic
344#484C22×C5912yes—
345#481C22×C5912yes—
346#478—22no—
347#463—22no—
348#450C22×C5912yes—
349#447C22×C5912yes—
350C4×SD16—22no—
351C4×D8—22no—
352#431C22×C5912yes—
353#400C22×C5911yescharacteristic
354#399C22×C5911yescharacteristic
355#394C22×C5911yescharacteristic
356#392C22×C5911yescharacteristic
357#388—21no—
358#386—21no—
359#386—21no—
360#385—21no—
361#383C22×C5911yescharacteristic
362#328C22×C5912yes—
363#328C22×C5912yes—
364#327C22×C5912yes—
365#327C22×C5912yes—
366#324C22×C5911yescharacteristic
367#324C22×C5911yescharacteristic
368#323C22×C5912yes—
369C59#1821611yescharacteristic, direct product factor, Sylow
370C2×#41—21no—
371C22×D4—21no—
372C22×D4C23×C5911yescharacteristic
373C23×C4—21no—
374C23×C4D4×C5911yescharacteristic
375C2×SD16—22no—
376C2×D8—22no—
377C4⋊Q8—22no—
378#126—22no—
379#126—22no—
380#126—22no—
381#126—22no—
382#125—21no—
383#125—22no—
384#125D4×C5912yescomplemented
385#124—21no—
386#124—22no—
387#124—22no—
388#124D4×C5912yescomplemented
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—22no—
396C22⋊Q8C23×C5912yes—
397#121—21no—
398#121—21no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121—21no—
403#121C23×C5912yes—
404#120—22no—
405C4×Q8—22no—
406C4×Q8—21no—
407C4×Q8D4×C5912yescomplemented
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4D4×C5912yescomplemented
417#117—22no—
418#117—22no—
419#117—21no—
420#117—22no—
421#117—22no—
422#117—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4D4×C5911yescharacteristic
426C2×C4⋊C4C23×C5911yescharacteristic
427C2×C4⋊C4C23×C5911yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4D4×C5911yescharacteristic
434C2×C22⋊C4C23×C5911yescharacteristic
435C2×C42D4×C5911yescharacteristic
436C2×C42C23×C5911yescharacteristic
437#107—22no—
438#106—22no—
439C4⋊C8—22no—
440#103—22no—
441#102—22no—
442#102—22no—
443#102—22no—
444C22⋊C8C23×C5912yes—
445C22⋊C8C23×C5912yes—
446C4×C8—22no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95C23×C5911yescharacteristic
452#95C23×C5911yescharacteristic
453C24—21no—
454C24C2×D4×C5911yescharacteristic
455#41—42no—
456#41—42no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8C2×D4×C5912yes—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—81no—
470C2×D4C2×D4×C5912yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—81no—
484C22×C4C2×D4×C5911yescharacteristic
485C22×C4C2×D4×C5911yescharacteristic
486C22×C4C2×D4×C5911yescharacteristic
487C22×C4C2×D4×C5911yescharacteristic
488C22×C4C2×D4×C5911yescharacteristic
489C22×C4C24×C5911yescharacteristic
490SD16—44no—
491D8—44no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—81no—
508C4⋊C4C2×D4×C5912yes—
509C4⋊C4C2×D4×C5912yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—22no—
532C42C2×D4×C5912yes—
533C42C2×D4×C5912yes—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×D4×C5911yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—81no—
551D4—24no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—82no—
558D4—81no—
559D4—82no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4—82no—
589C2×C4#120×C5911yescharacteristic
590C2×C4#120×C5911yescharacteristic
591C2×C4#120×C5912yes—
592C2×C4#135×C5912yes—
593C2×C4C22×D4×C5912yes—
594C8—42no—
595C8—42no—
596C22—21no—
597C22—21no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22C2×#120×C5911yescharacteristic
614C22C2×#135×C5911yescharacteristic
615C22C59×#57511yescharacteristic
616C4—22no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—22no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—81no—
629C2—21no—
630C2—21no—
631C2—81no—
632C2—82no—
633C2—82no—
634C2C59×#300211yescharacteristic
635C2C59×#302111yescharacteristic
636C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
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
1. C59×#3493C2semidirect
2. C59×#3105C2semidirect
3. C59×#2598C2semidirect
4. C59×#1618C2semidirect
5. C59×#1617C2semidirect
6. C59×#1616C2semidirect
7. C59×#1600C2semidirect
8. C59×#1599C2semidirect
9. C59×#1598C2semidirect
10. C59×#1521C2semidirect
11. C59×#1458C2semidirect
65. #125×C59D4semidirect
69. #124×C59D4semidirect
88. C4×Q8×C59D4semidirect
97. C4×D4×C59D4semidirect
277. #18216C59direct (N × H)
311. #3493C2×C59semidirect
312. #3105C2×C59semidirect
313. #2598C2×C59semidirect
314. #1618C2×C59semidirect
315. #1617C2×C59semidirect
316. #1616C2×C59semidirect
317. #1600C2×C59semidirect
318. #1599C2×C59semidirect
319. #1598C2×C59semidirect
320. #1521C2×C59semidirect
321. #1458C2×C59semidirect
369. C59#18216direct (N × H)
384. #125D4×C59semidirect
388. #124D4×C59semidirect
407. C4×Q8D4×C59semidirect
416. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34932
C59×#31052
C59×#25982
C59×#16182
C59×#16172
C59×#16162
C59×#16002
C59×#15992
C59×#15982
C59×#15212
C59×#14582
#1821659