Finite Group Databasebeta

← C59×#18565 #28430674

Subgroups — C59×#18565 #28430674

Order 15104.

986 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×#3522C211yesmaximal, characteristic
2C59×#3238C211yesmaximal, characteristic, complemented
3C59×#3152C211yesmaximal, characteristic, complemented
4C2×C59×#477C211yesmaximal, characteristic, complemented
5C59×#1633C212yesmaximal, complemented
6C59×#1632C212yesmaximal, complemented
7C59×#1631C212yesmaximal, complemented
8C59×#1630C212yesmaximal, complemented
9C59×#1551C211yesmaximal, characteristic, complemented
10C59×#1491C211yesmaximal, characteristic, complemented
11C59×#1483C211yesmaximal, characteristic, complemented
12C2×#137×C59—22no—
13C59×#559—21no—
14C59×#559C2211yescharacteristic
15Q82×C59C2211yescharacteristic
16C59×#554—21no—
17C59×#554C2211yescharacteristic
18C59×#550C2211yescharacteristic
19C59×#548—21no—
20C59×#547—22no—
21C59×#547—21no—
22C59×#542—22no—
23C59×#542—21no—
24C59×#532—21no—
25C59×#532C2211yescharacteristic
26C2×C4⋊Q8×C59C2211yescharacteristic
27C59×#529—21no—
28C59×#529C2211yescharacteristic
29C59×#520—21no—
30C2×C4×Q8×C59C2211yescharacteristic
31C59×#501—21no—
32C59×#497—22no—
33C59×#494C2212yes—
34C59×#492C2212yes—
35C59×#489—21no—
36C59×#483—21no—
37C59×#480—21no—
38C59×#477—21no—
39C59×#477—21no—
40C59×#477C2212yes—
41C59×#477C2212yes—
42C59×#477C2212yes—
43C59×#476C2212yes—
44C59×#432C2211yescharacteristic
45C2×#107×C59—21no—
46C2×C4⋊C8×C59C2211yescharacteristic
47C2×#103×C59—21no—
48C4×M16×C59C2211yescharacteristic
49C59×#333C2212yes—
50C59×#331C2212yes—
51C59×Q8⋊C8C2212yes—
52C59×Q8⋊C8C2212yes—
53C59×Q8⋊C8C2212yes—
54C59×D4⋊C8C2212yes—
55C2×#41×C59—22no—
56C2×#41×C59—21no—
57C2×#41×C59D412yes—
58C22×Q8×C59—21no—
59C22×Q8×C59D411yescharacteristic
60#137×C59—48no—
61C2×Q16×C59—24no—
62C2×SD16×C59—24no—
63C2×M16×C59—21no—
64C22×C8×C59—21no—
65C4⋊Q8×C59—22no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—21no—
69C4⋊Q8×C59—21no—
70C4⋊Q8×C59—21no—
71C4⋊Q8×C59C2312yes—
72C4⋊Q8×C59C2311yescharacteristic
73#127×C59—21no—
74#127×C59C2311yescharacteristic
75#126×C59—22no—
76#126×C59—21no—
77#125×C59—21no—
78#125×C59D411yescharacteristic
79#124×C59—22no—
80#124×C59—22no—
81#124×C59—21no—
82#124×C59—21no—
83#124×C59D411yescharacteristic
84#124×C59D412yes—
85#123×C59—22no—
86#123×C59—22no—
87#123×C59—22no—
88#123×C59—22no—
89#123×C59—21no—
90C22⋊Q8×C59—22no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—22no—
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59D411yescharacteristic
96#121×C59—22no—
97#121×C59—22no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—21no—
101#121×C59D411yescharacteristic
102C4×Q8×C59—21no—
103C4×Q8×C59—21no—
104C4×Q8×C59—21no—
105C4×Q8×C59—21no—
106C4×Q8×C59D411yescharacteristic
107C4×Q8×C59D411yescharacteristic
108C4×Q8×C59C2311yescharacteristic
109C4×Q8×C59C2311yescharacteristic
110C4×Q8×C59C2311yescharacteristic
111C4×D4×C59—22no—
112C4×D4×C59—22no—
113C4×D4×C59—22no—
114C4×D4×C59—21no—
115C4×D4×C59—21no—
116C4×D4×C59—21no—
117C4×D4×C59—21no—
118C4×D4×C59—21no—
119C4×D4×C59—21no—
120C4×D4×C59—21no—
121C4×D4×C59C2311yescharacteristic
122#117×C59—22no—
123#117×C59—21no—
124C2×C4⋊C4×C59—22no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59—21no—
129C2×C4⋊C4×C59D411yescharacteristic
130C2×C42×C59—21no—
131C2×C42×C59C2311yescharacteristic
132#107×C59—22no—
133#107×C59—22no—
134#107×C59—22no—
135#106×C59—22no—
136C4⋊C8×C59—21no—
137C4⋊C8×C59C2312yes—
138C4⋊C8×C59C2312yes—
139#103×C59—22no—
140#103×C59—22no—
141#103×C59—22no—
142#102×C59—22no—
143C22⋊C8×C59—21no—
144#97×C59—21no—
145#97×C59—21no—
146C4×C8×C59C2312yes—
147#41×C59—44no—
148#41×C59—42no—
149#41×C59—42no—
150#41×C59—84no—
151C2×Q8×C59—22no—
152C2×Q8×C59—21no—
153C2×Q8×C59—22no—
154C2×Q8×C59—22no—
155C2×Q8×C59—41no—
156C2×Q8×C59—44no—
157C2×Q8×C59—41no—
158C2×Q8×C59C2×D411yescharacteristic
159C2×Q8×C59C2×D411yescharacteristic
160C2×Q8×C59C2×D411yescharacteristic
161C2×Q8×C59C2×D412yes—
162C2×D4×C59—22no—
163C2×D4×C59—22no—
164C2×D4×C59—21no—
165C2×D4×C59—22no—
166C2×D4×C59—21no—
167C2×D4×C59—21no—
168C2×D4×C59C2×D411yescharacteristic
169C2×D4×C59C2×D412yes—
170C22×C4×C59—22no—
171C22×C4×C59—21no—
172C22×C4×C59—21no—
173C22×C4×C59—22no—
174C22×C4×C59—22no—
175C22×C4×C59—21no—
176C22×C4×C59—22no—
177C22×C4×C59—41no—
178C22×C4×C59C2×D411yescharacteristic
179C22×C4×C59C2×D411yescharacteristic
180C22×C4×C59C2×D411yescharacteristic
181Q16×C59—48no—
182SD16×C59—48no—
183M16×C59—24no—
184C2×C8×C59—22no—
185C2×C8×C59—22no—
186C2×C8×C59—22no—
187C2×C8×C59—42no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—22no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—21no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—21no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—21no—
200C4⋊C4×C59—21no—
201C4⋊C4×C59—21no—
202C4⋊C4×C59—21no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59—22no—
205C4⋊C4×C59—41no—
206C4⋊C4×C59—41no—
207C4⋊C4×C59C2×D411yescharacteristic
208C4⋊C4×C59C2×D411yescharacteristic
209C4⋊C4×C59C2×D411yescharacteristic
210C4⋊C4×C59C2×D411yescharacteristic
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—22no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—22no—
218C22⋊C4×C59—21no—
219C22⋊C4×C59—21no—
220C22⋊C4×C59—22no—
221C22⋊C4×C59—21no—
222C22⋊C4×C59—21no—
223C42×C59—22no—
224C42×C59—21no—
225C42×C59—21no—
226C42×C59—21no—
227C42×C59—21no—
228C42×C59—21no—
229C42×C59C2×D411yescharacteristic
230C42×C59C2×D411yescharacteristic
231C42×C59C2×D411yescharacteristic
232C42×C59C2411yescharacteristic
233C23×C59—21no—
234C23×C59—22no—
235C23×C59#12011yescharacteristic
236Q8×C59—21no—
237Q8×C59—21no—
238Q8×C59—21no—
239Q8×C59—22no—
240Q8×C59—42no—
241Q8×C59—42no—
242Q8×C59—44no—
243Q8×C59—82no—
244Q8×C59—82no—
245Q8×C59#13712yescomplemented
246D4×C59—22no—
247D4×C59—42no—
248D4×C59—44no—
249D4×C59—44no—
250D4×C59—42no—
251D4×C59—41no—
252D4×C59—41no—
253D4×C59—42no—
254D4×C59—82no—
255C2×C4×C59—21no—
256C2×C4×C59—22no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—22no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—21no—
263C2×C4×C59—21no—
264C2×C4×C59—22no—
265C2×C4×C59—21no—
266C2×C4×C59—21no—
267C2×C4×C59—42no—
268C2×C4×C59—44no—
269C2×C4×C59—42no—
270C2×C4×C59—41no—
271C2×C4×C59—41no—
272C2×C4×C59—42no—
273C2×C4×C59—82no—
274C2×C4×C59—82no—
275C2×C4×C59—82no—
276C2×C4×C59—82no—
277C2×C4×C59#12011yescharacteristic
278C2×C4×C59#12011yescharacteristic
279C2×C4×C59#12011yescharacteristic
280C2×C4×C59C22×D411yescharacteristic
281C2×C4×C59C22×D411yescharacteristic
282C2×C4×C59C22×D411yescharacteristic
283C8×C59—24no—
284C8×C59—42no—
285C8×C59—42no—
286#18565C5911yesmaximal, characteristic, direct product factor, Sylow
287C22×C59—21no—
288C22×C59—21no—
289C22×C59—42no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—41no—
295C22×C59#45312yes—
296C22×C59C2×#12011yescharacteristic
297C4×C59—21no—
298C4×C59—22no—
299C4×C59—21no—
300C4×C59—21no—
301C4×C59—21no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—42no—
308C4×C59—41no—
309C4×C59—82no—
310C4×C59C2×#13711yescharacteristic
311C4×C59#57511yescharacteristic
312C4×C59#57812yes—
313#3522C2×C5911yescharacteristic
314#3238C2×C5911yescharacteristic, complemented
315#3152C2×C5911yescharacteristic, complemented
316C2×#477C2×C5911yescharacteristic, complemented
317#1633C2×C5912yescomplemented
318#1632C2×C5912yescomplemented
319#1631C2×C5912yescomplemented
320#1630C2×C5912yescomplemented
321#1551C2×C5911yescharacteristic, complemented
322#1491C2×C5911yescharacteristic, complemented
323#1483C2×C5911yescharacteristic, complemented
324C2×C59—22no—
325C2×C59—81no—
326C2×C59—82no—
327C2×C59#300711yescharacteristic
328C2×C59#301111yescharacteristic
329C2×C59C2×#45311yescharacteristic
330C2×#137—22no—
331#559—21no—
332#559C22×C5911yescharacteristic
333Q82C22×C5911yescharacteristic
334#554—21no—
335#554C22×C5911yescharacteristic
336#550C22×C5911yescharacteristic
337#548—21no—
338#547—22no—
339#547—21no—
340#542—22no—
341#542—21no—
342#532—21no—
343#532C22×C5911yescharacteristic
344C2×C4⋊Q8C22×C5911yescharacteristic
345#529—21no—
346#529C22×C5911yescharacteristic
347#520—21no—
348C2×C4×Q8C22×C5911yescharacteristic
349#501—21no—
350#497—22no—
351#494C22×C5912yes—
352#492C22×C5912yes—
353#489—21no—
354#483—21no—
355#480—21no—
356#477—21no—
357#477—21no—
358#477C22×C5912yes—
359#477C22×C5912yes—
360#477C22×C5912yes—
361#476C22×C5912yes—
362#432C22×C5911yescharacteristic
363C2×#107—21no—
364C2×C4⋊C8C22×C5911yescharacteristic
365C2×#103—21no—
366C4×M16C22×C5911yescharacteristic
367#333C22×C5912yes—
368#331C22×C5912yes—
369Q8⋊C8C22×C5912yes—
370Q8⋊C8C22×C5912yes—
371Q8⋊C8C22×C5912yes—
372D4⋊C8C22×C5912yes—
373C59#1856511yescharacteristic, direct product factor, Sylow
374C2×#41—22no—
375C2×#41—21no—
376C2×#41D4×C5912yes—
377C22×Q8—21no—
378C22×Q8D4×C5911yescharacteristic
379#137—48no—
380C2×Q16—24no—
381C2×SD16—24no—
382C2×M16—21no—
383C22×C8—21no—
384C4⋊Q8—22no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—21no—
390C4⋊Q8C23×C5912yes—
391C4⋊Q8C23×C5911yescharacteristic
392#127—21no—
393#127C23×C5911yescharacteristic
394#126—22no—
395#126—21no—
396#125—21no—
397#125D4×C5911yescharacteristic
398#124—22no—
399#124—22no—
400#124—21no—
401#124—21no—
402#124D4×C5911yescharacteristic
403#124D4×C5912yes—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—22no—
408#123—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—21no—
414C22⋊Q8D4×C5911yescharacteristic
415#121—22no—
416#121—22no—
417#121—21no—
418#121—21no—
419#121—21no—
420#121D4×C5911yescharacteristic
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C5911yescharacteristic
426C4×Q8D4×C5911yescharacteristic
427C4×Q8C23×C5911yescharacteristic
428C4×Q8C23×C5911yescharacteristic
429C4×Q8C23×C5911yescharacteristic
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4—21no—
438C4×D4—21no—
439C4×D4—21no—
440C4×D4C23×C5911yescharacteristic
441#117—22no—
442#117—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4D4×C5911yescharacteristic
449C2×C42—21no—
450C2×C42C23×C5911yescharacteristic
451#107—22no—
452#107—22no—
453#107—22no—
454#106—22no—
455C4⋊C8—21no—
456C4⋊C8C23×C5912yes—
457C4⋊C8C23×C5912yes—
458#103—22no—
459#103—22no—
460#103—22no—
461#102—22no—
462C22⋊C8—21no—
463#97—21no—
464#97—21no—
465C4×C8C23×C5912yes—
466#41—44no—
467#41—42no—
468#41—42no—
469#41—84no—
470C2×Q8—22no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—41no—
475C2×Q8—44no—
476C2×Q8—41no—
477C2×Q8C2×D4×C5911yescharacteristic
478C2×Q8C2×D4×C5911yescharacteristic
479C2×Q8C2×D4×C5911yescharacteristic
480C2×Q8C2×D4×C5912yes—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C2×D4×C5911yescharacteristic
488C2×D4C2×D4×C5912yes—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—41no—
497C22×C4C2×D4×C5911yescharacteristic
498C22×C4C2×D4×C5911yescharacteristic
499C22×C4C2×D4×C5911yescharacteristic
500Q16—48no—
501SD16—48no—
502M16—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4C2×D4×C5911yescharacteristic
527C4⋊C4C2×D4×C5911yescharacteristic
528C4⋊C4C2×D4×C5911yescharacteristic
529C4⋊C4C2×D4×C5911yescharacteristic
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42C2×D4×C5911yescharacteristic
549C42C2×D4×C5911yescharacteristic
550C42C2×D4×C5911yescharacteristic
551C42C24×C5911yescharacteristic
552C23—21no—
553C23—22no—
554C23#120×C5911yescharacteristic
555Q8—21no—
556Q8—21no—
557Q8—21no—
558Q8—22no—
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—82no—
563Q8—82no—
564Q8#137×C5912yescomplemented
565D4—22no—
566D4—42no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—41no—
571D4—41no—
572D4—42no—
573D4—82no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4#120×C5911yescharacteristic
597C2×C4#120×C5911yescharacteristic
598C2×C4#120×C5911yescharacteristic
599C2×C4C22×D4×C5911yescharacteristic
600C2×C4C22×D4×C5911yescharacteristic
601C2×C4C22×D4×C5911yescharacteristic
602C8—24no—
603C8—42no—
604C8—42no—
605C22—21no—
606C22—21no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22C59×#45312yes—
614C22C2×#120×C5911yescharacteristic
615C4—21no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—82no—
628C4C2×#137×C5911yescharacteristic
629C4C59×#57511yescharacteristic
630C4C59×#57812yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2C59×#300711yescharacteristic
635C2C59×#301111yescharacteristic
636C2C2×C59×#45311yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3522C2
LCSG ▷ C42 ▷ 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×#3238C2semidirect
3. C59×#3152C2semidirect
4. C2×C59×#477C2semidirect
5. C59×#1633C2semidirect
6. C59×#1632C2semidirect
7. C59×#1631C2semidirect
8. C59×#1630C2semidirect
9. C59×#1551C2semidirect
10. C59×#1491C2semidirect
11. C59×#1483C2semidirect
245. Q8×C59#137semidirect
286. #18565C59direct (N × H)
314. #3238C2×C59semidirect
315. #3152C2×C59semidirect
316. C2×#477C2×C59semidirect
317. #1633C2×C59semidirect
318. #1632C2×C59semidirect
319. #1631C2×C59semidirect
320. #1630C2×C59semidirect
321. #1551C2×C59semidirect
322. #1491C2×C59semidirect
323. #1483C2×C59semidirect
373. C59#18565direct (N × H)
564. Q8#137×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#35222
C59×#32382
C59×#31522
C2×C59×#4772
C59×#16332
C59×#16322
C59×#16312
C59×#16302
C59×#15512
C59×#14912
C59×#14832
#1856559