Finite Group Databasebeta

← C59×#18386 #28430495

Subgroups — C59×#18386 #28430495

Order 15104.

984 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3491C211yesmaximal, characteristic, complemented
2C59×#3211C212yesmaximal, complemented
3C59×#2610C211yesmaximal, characteristic, complemented
4C59×#1603C212yesmaximal, complemented
5C59×#1602C211yesmaximal, characteristic, complemented
6C59×#1601C211yesmaximal, characteristic, complemented
7C59×#1597C212yesmaximal, complemented
8C59×#1596C211yesmaximal, characteristic, complemented
9C59×#1595C211yesmaximal, 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×#124×C59C2211yescharacteristic, complemented
23C2×#123×C59—21no—
24C59×#518—21no—
25C2×#117×C59C2211yescharacteristic
26C59×#495—22no—
27C59×#488—22no—
28C59×#464—22no—
29C59×#463—22no—
30C59×#451C2212yescomplemented
31C59×#450C2212yescomplemented
32C59×#448C2212yes—
33C59×#447C2212yes—
34C59×#431C2212yescomplemented
35C59×#395C2211yescharacteristic, complemented
36C59×#394C2211yescharacteristic, complemented
37C59×#393C2211yescharacteristic
38C59×#392C2211yescharacteristic
39C59×#390—21no—
40C59×#388—21no—
41C59×#388—21no—
42C59×#386—21no—
43C59×#376C2211yescharacteristic, complemented
44C59×#328C2212yescomplemented
45C59×#328C2212yescomplemented
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2212yescomplemented
49C59×#323C2211yescharacteristic
50C59×#323C2211yescharacteristic
51C22×Q8×C59C2311yescharacteristic
52C22×D4×C59—21no—
53C22×D4×C59—21no—
54C22×D4×C59C2311yescharacteristic
55C23×C4×C59D411yescharacteristic
56C2×Q16×C59—22no—
57C2×SD16×C59—22no—
58C2×SD16×C59—22no—
59C2×D8×C59—22no—
60#126×C59—22no—
61#126×C59—22no—
62#126×C59—22no—
63#126×C59—22no—
64#125×C59—21no—
65#125×C59—22no—
66#125×C59—22no—
67#125×C59D412yescomplemented
68#124×C59—21no—
69#124×C59—22no—
70#124×C59—22no—
71#124×C59D412yescomplemented
72#123×C59—22no—
73#123×C59—22no—
74#123×C59—22no—
75#123×C59—22no—
76#123×C59—22no—
77#123×C59—22no—
78#123×C59—22no—
79C22⋊Q8×C59—22no—
80C22⋊Q8×C59—22no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—21no—
83C22⋊Q8×C59—22no—
84C22⋊Q8×C59C2312yes—
85#121×C59—21no—
86#121×C59C2312yes—
87#120×C59—22no—
88C4×Q8×C59—21no—
89C4×Q8×C59D412yescomplemented
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92C4×D4×C59—21no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C4×D4×C59D412yescomplemented
96#117×C59—22no—
97#117×C59—22no—
98#117×C59—22no—
99#117×C59—22no—
100#117×C59—22no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C22⋊C4×C59—22no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
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×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59C2311yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59D411yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C4⋊C8×C59—22no—
122#103×C59—22no—
123#103×C59—22no—
124#102×C59—22no—
125#102×C59—22no—
126C22⋊C8×C59C2312yes—
127C22⋊C8×C59C2312yes—
128C4×C8×C59—22no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134C24×C59—21no—
135C24×C59—21no—
136C24×C59C2×D411yescharacteristic
137C2×Q8×C59—21no—
138C2×Q8×C59—22no—
139C2×Q8×C59—81no—
140C2×Q8×C59C2×D412yes—
141C2×D4×C59—22no—
142C2×D4×C59—22no—
143C2×D4×C59—21no—
144C2×D4×C59—22no—
145C2×D4×C59—42no—
146C2×D4×C59—42no—
147C2×D4×C59—42no—
148C2×D4×C59—42no—
149C2×D4×C59—81no—
150C2×D4×C59—81no—
151C2×D4×C59C2×D412yes—
152C22×C4×C59—21no—
153C22×C4×C59—22no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—22no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—22no—
161C22×C4×C59—21no—
162C22×C4×C59—21no—
163C22×C4×C59—41no—
164C22×C4×C59C2×D411yescharacteristic
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2411yescharacteristic
170Q16×C59—82no—
171SD16×C59—82no—
172SD16×C59—82no—
173D8×C59—82no—
174C2×C8×C59—22no—
175C2×C8×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
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—22no—
203C22⋊C4×C59—42no—
204C22⋊C4×C59—42no—
205C22⋊C4×C59—41no—
206C22⋊C4×C59—41no—
207C22⋊C4×C59—42no—
208C22⋊C4×C59—81no—
209C22⋊C4×C59—81no—
210C42×C59—22no—
211C42×C59—21no—
212C42×C59—21no—
213C42×C59—22no—
214C42×C59—42no—
215C42×C59C2×D412yes—
216C42×C59C2×D412yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—22no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—42no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59—81no—
231C23×C59—81no—
232C23×C59#13512yes—
233C23×C59C22×D411yescharacteristic
234Q8×C59—42no—
235Q8×C59—42no—
236Q8×C59—42no—
237Q8×C59—81no—
238D4×C59—42no—
239D4×C59—42no—
240D4×C59—42no—
241D4×C59—44no—
242D4×C59—42no—
243D4×C59—42no—
244D4×C59—81no—
245D4×C59—84no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—22no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—22no—
252C2×C4×C59—22no—
253C2×C4×C59—22no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—42no—
260C2×C4×C59—42no—
261C2×C4×C59—41no—
262C2×C4×C59—42no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59—41no—
269C2×C4×C59—41no—
270C2×C4×C59—41no—
271C2×C4×C59—41no—
272C2×C4×C59—82no—
273C2×C4×C59—81no—
274C2×C4×C59—82no—
275C2×C4×C59—81no—
276C2×C4×C59#12011yescharacteristic
277C2×C4×C59#12011yescharacteristic
278C2×C4×C59#12012yes—
279C2×C4×C59C22×D412yes—
280C8×C59—42no—
281C8×C59—42no—
282#18386C5911yesmaximal, characteristic, direct product factor, Sylow
283C22×C59—21no—
284C22×C59—21no—
285C22×C59—22no—
286C22×C59—42no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—42no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59—162no—
301C22×C59C2×#12011yescharacteristic
302C22×C59C2×#13511yescharacteristic
303C22×C59#57511yescharacteristic
304C4×C59—21no—
305C4×C59—21no—
306C4×C59—22no—
307C4×C59—22no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—42no—
313C4×C59—41no—
314C4×C59—42no—
315C4×C59—81no—
316C4×C59—81no—
317#3491C2×C5911yescharacteristic, complemented
318#3211C2×C5912yescomplemented
319#2610C2×C5911yescharacteristic, complemented
320#1603C2×C5912yescomplemented
321#1602C2×C5911yescharacteristic, complemented
322#1601C2×C5911yescharacteristic, complemented
323#1597C2×C5912yescomplemented
324#1596C2×C5911yescharacteristic, complemented
325#1595C2×C5911yescharacteristic
326#1500C2×C5912yescomplemented
327#1458C2×C5911yescharacteristic, complemented
328C2×C59—21no—
329C2×C59—21no—
330C2×C59—42no—
331C2×C59—82no—
332C2×C59—161no—
333C2×C59#300211yescharacteristic
334C2×C59#301811yescharacteristic
335C2×C59#302711yescharacteristic
336#552—22no—
337#552—22no—
338#551—22no—
339#551—22no—
340#543C22×C5911yescharacteristic
341#539—21no—
342#539C22×C5912yes—
343#534C22×C5911yescharacteristic
344#529—21no—
345#529C22×C5912yes—
346C2×#124C22×C5911yescharacteristic, complemented
347C2×#123—21no—
348#518—21no—
349C2×#117C22×C5911yescharacteristic
350#495—22no—
351#488—22no—
352#464—22no—
353#463—22no—
354#451C22×C5912yescomplemented
355#450C22×C5912yescomplemented
356#448C22×C5912yes—
357#447C22×C5912yes—
358#431C22×C5912yescomplemented
359#395C22×C5911yescharacteristic, complemented
360#394C22×C5911yescharacteristic, complemented
361#393C22×C5911yescharacteristic
362#392C22×C5911yescharacteristic
363#390—21no—
364#388—21no—
365#388—21no—
366#386—21no—
367#376C22×C5911yescharacteristic, complemented
368#328C22×C5912yescomplemented
369#328C22×C5912yescomplemented
370#327C22×C5912yes—
371#327C22×C5912yes—
372#324C22×C5912yescomplemented
373#323C22×C5911yescharacteristic
374#323C22×C5911yescharacteristic
375C59#1838611yescharacteristic, direct product factor, Sylow
376C22×Q8C23×C5911yescharacteristic
377C22×D4—21no—
378C22×D4—21no—
379C22×D4C23×C5911yescharacteristic
380C23×C4D4×C5911yescharacteristic
381C2×Q16—22no—
382C2×SD16—22no—
383C2×SD16—22no—
384C2×D8—22no—
385#126—22no—
386#126—22no—
387#126—22no—
388#126—22no—
389#125—21no—
390#125—22no—
391#125—22no—
392#125D4×C5912yescomplemented
393#124—21no—
394#124—22no—
395#124—22no—
396#124D4×C5912yescomplemented
397#123—22no—
398#123—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—22no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—21no—
408C22⋊Q8—22no—
409C22⋊Q8C23×C5912yes—
410#121—21no—
411#121C23×C5912yes—
412#120—22no—
413C4×Q8—21no—
414C4×Q8D4×C5912yescomplemented
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4D4×C5912yescomplemented
421#117—22no—
422#117—22no—
423#117—22no—
424#117—22no—
425#117—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4D4×C5911yescharacteristic
430C2×C4⋊C4C23×C5911yescharacteristic
431C2×C22⋊C4—22no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4D4×C5911yescharacteristic
442C2×C22⋊C4C23×C5911yescharacteristic
443C2×C42—21no—
444C2×C42D4×C5911yescharacteristic
445C2×C42C23×C5911yescharacteristic
446C4⋊C8—22no—
447#103—22no—
448#103—22no—
449#102—22no—
450#102—22no—
451C22⋊C8C23×C5912yes—
452C22⋊C8C23×C5912yes—
453C4×C8—22no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95C23×C5911yescharacteristic
458#95C23×C5911yescharacteristic
459C24—21no—
460C24—21no—
461C24C2×D4×C5911yescharacteristic
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—81no—
465C2×Q8C2×D4×C5912yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4C2×D4×C5912yes—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4C2×D4×C5911yescharacteristic
490C22×C4C2×D4×C5911yescharacteristic
491C22×C4C2×D4×C5911yescharacteristic
492C22×C4C2×D4×C5911yescharacteristic
493C22×C4C2×D4×C5911yescharacteristic
494C22×C4C24×C5911yescharacteristic
495Q16—82no—
496SD16—82no—
497SD16—82no—
498D8—82no—
499C2×C8—22no—
500C2×C8—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—81no—
514C4⋊C4C2×D4×C5912yes—
515C4⋊C4C2×D4×C5912yes—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—42no—
540C42C2×D4×C5912yes—
541C42C2×D4×C5912yes—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—81no—
556C23—81no—
557C23#135×C5912yes—
558C23C22×D4×C5911yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—81no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—42no—
568D4—42no—
569D4—81no—
570D4—84no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—82no—
598C2×C4—81no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4#120×C5911yescharacteristic
602C2×C4#120×C5911yescharacteristic
603C2×C4#120×C5912yes—
604C2×C4C22×D4×C5912yes—
605C8—42no—
606C8—42no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—42no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—162no—
625C22C2×#120×C5911yescharacteristic
626C22C2×#135×C5911yescharacteristic
627C22C59×#57511yescharacteristic
628C4—21no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—81no—
640C4—81no—
641C2—21no—
642C2—21no—
643C2—42no—
644C2—82no—
645C2—161no—
646C2C59×#300211yescharacteristic
647C2C59×#301811yescharacteristic
648C2C59×#302711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#1595C2
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×#3491C2semidirect
2. C59×#3211C2semidirect
3. C59×#2610C2semidirect
4. C59×#1603C2semidirect
5. C59×#1602C2semidirect
6. C59×#1601C2semidirect
7. C59×#1597C2semidirect
8. C59×#1596C2semidirect
10. C59×#1500C2semidirect
11. C59×#1458C2semidirect
22. C2×#124×C59C22semidirect
30. C59×#451C22semidirect
31. C59×#450C22semidirect
34. C59×#431C22semidirect
35. C59×#395C22semidirect
36. C59×#394C22semidirect
43. C59×#376C22semidirect
44. C59×#328C22semidirect
45. C59×#328C22semidirect
48. C59×#324C22semidirect
67. #125×C59D4semidirect
71. #124×C59D4semidirect
89. C4×Q8×C59D4semidirect
95. C4×D4×C59D4semidirect
282. #18386C59direct (N × H)
317. #3491C2×C59semidirect
318. #3211C2×C59semidirect
319. #2610C2×C59semidirect
320. #1603C2×C59semidirect
321. #1602C2×C59semidirect
322. #1601C2×C59semidirect
323. #1597C2×C59semidirect
324. #1596C2×C59semidirect
326. #1500C2×C59semidirect
327. #1458C2×C59semidirect
346. C2×#124C22×C59semidirect
354. #451C22×C59semidirect
355. #450C22×C59semidirect
358. #431C22×C59semidirect
359. #395C22×C59semidirect
360. #394C22×C59semidirect
367. #376C22×C59semidirect
368. #328C22×C59semidirect
369. #328C22×C59semidirect
372. #324C22×C59semidirect
375. C59#18386direct (N × H)
392. #125D4×C59semidirect
396. #124D4×C59semidirect
414. C4×Q8D4×C59semidirect
420. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34912
C59×#32112
C59×#26102
C59×#16032
C59×#16022
C59×#16012
C59×#15972
C59×#15962
C59×#15952
C59×#15002
C59×#14582
#1838659