Finite Group Databasebeta

← C59×#23901 #28436010

Subgroups — C59×#23901 #28436010

Order 15104.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3484C211yesmaximal, characteristic
2C22×#123×C59C211yesmaximal, characteristic, complemented
3C2×C59×#458C212yesmaximal, complemented
4C59×#2849C212yesmaximal, complemented
5C59×#2022C214yesmaximal, complemented
6C59×#1898C212yesmaximal, complemented
7C59×#1890C212yesmaximal, complemented
8C59×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C59C2211yescharacteristic
10C23×D4×C59C2211yescharacteristic
11C24×C4×C59C2211yescharacteristic, complemented
12C59×#546—24no—
13D42×C59—24no—
14C59×#535—21no—
15C59×#534—21no—
16C59×#534C2212yes—
17C59×#525—21no—
18C2×#123×C59—21no—
19C2×#123×C59—21no—
20C2×#123×C59—28no—
21C2×#123×C59C2212yes—
22C2×#123×C59C2212yescomplemented
23C2×#123×C59C2214yescomplemented
24C2×#120×C59—21no—
25C2×#120×C59C2212yes—
26C59×#518C2211yescharacteristic
27C22×C4⋊C4×C59—21no—
28C22×C22⋊C4×C59—21no—
29C22×C22⋊C4×C59C2211yescharacteristic, complemented
30C59×#458—28no—
31C2×#99×C59C2212yes—
32C2×#99×C59C2214yescomplemented
33C2×#99×C59C2212yescomplemented
34C59×#386—24no—
35C59×#385—24no—
36C59×#381C2214yescomplemented
37C59×#379C2212yescomplemented
38C59×#342C2214yescomplemented
39C25×C59D411yescharacteristic
40C25×C59C2311yescharacteristic
41H(5,𝔽2)×C59—24no—
42H(5,𝔽2)×C59—42no—
43C2×#41×C59—21no—
44C2×#41×C59—22no—
45C22×D4×C59—21no—
46C22×D4×C59—22no—
47C22×D4×C59—24no—
48C22×D4×C59—22no—
49C22×D4×C59—24no—
50C22×D4×C59—42no—
51C22×D4×C59D411yescharacteristic
52C22×D4×C59D411yescharacteristic
53C22×D4×C59D414yes—
54C22×D4×C59C2312yes—
55C22×D4×C59C2311yescharacteristic
56C22×D4×C59C2311yescharacteristic
57C23×C4×C59—21no—
58C23×C4×C59—21no—
59C23×C4×C59—21no—
60C23×C4×C59—24no—
61C23×C4×C59—24no—
62C23×C4×C59—42no—
63C23×C4×C59D411yescharacteristic
64C23×C4×C59C2312yes—
65#127×C59—22no—
66#124×C59—22no—
67#123×C59—28no—
68#123×C59—28no—
69#123×C59—22no—
70#123×C59—24no—
71#123×C59—22no—
72#123×C59—22no—
73#123×C59—24no—
74#123×C59—24no—
75#123×C59—28no—
76#123×C59—44no—
77#123×C59D418yescomplemented
78C22⋊Q8×C59—22no—
79#121×C59—22no—
80#121×C59—24no—
81#121×C59—24no—
82#121×C59—24no—
83#120×C59—24no—
84#120×C59—22no—
85#120×C59—22no—
86#120×C59—22no—
87#120×C59—22no—
88#120×C59—24no—
89#120×C59—22no—
90C4×D4×C59—24no—
91C4×D4×C59—24no—
92#117×C59—21no—
93C2×C4⋊C4×C59—28no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59—24no—
97C2×C22⋊C4×C59—28no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59—22no—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—24no—
104C2×C22⋊C4×C59—24no—
105C2×C22⋊C4×C59—24no—
106C2×C22⋊C4×C59D414yes—
107C2×C22⋊C4×C59C2314yes—
108C2×C22⋊C4×C59C2312yes—
109C2×C22⋊C4×C59C2312yes—
110#99×C59—24no—
111#99×C59—28no—
112#99×C59—42no—
113#95×C59—28no—
114C24×C59—22no—
115C24×C59—22no—
116C24×C59—21no—
117C24×C59—22no—
118C24×C59—22no—
119C24×C59—44no—
120C24×C59—41no—
121C24×C59—44no—
122C24×C59C2×D411yescharacteristic
123C24×C59C2×D414yes—
124C24×C59C2×D411yescharacteristic
125C24×C59C2×D412yes—
126C24×C59C2×D411yescharacteristic
127C24×C59C2×D411yescharacteristic
128C24×C59C2411yescharacteristic
129#41×C59—48no—
130#41×C59—42no—
131#41×C59—42no—
132C2×Q8×C59—21no—
133C2×D4×C59—22no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—21no—
137C2×D4×C59—24no—
138C2×D4×C59—22no—
139C2×D4×C59—22no—
140C2×D4×C59—22no—
141C2×D4×C59—24no—
142C2×D4×C59—22no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—22no—
146C2×D4×C59—22no—
147C2×D4×C59—24no—
148C2×D4×C59—24no—
149C2×D4×C59—24no—
150C2×D4×C59—22no—
151C2×D4×C59—41no—
152C2×D4×C59—41no—
153C2×D4×C59—416no—
154C2×D4×C59—48no—
155C2×D4×C59—42no—
156C2×D4×C59—42no—
157C2×D4×C59—48no—
158C2×D4×C59—84no—
159C2×D4×C59C2×D414yes—
160C22×C4×C59—24no—
161C22×C4×C59—24no—
162C22×C4×C59—21no—
163C22×C4×C59—22no—
164C22×C4×C59—24no—
165C22×C4×C59—22no—
166C22×C4×C59—24no—
167C22×C4×C59—22no—
168C22×C4×C59—22no—
169C22×C4×C59—22no—
170C22×C4×C59—24no—
171C22×C4×C59—22no—
172C22×C4×C59—22no—
173C22×C4×C59—24no—
174C22×C4×C59—41no—
175C22×C4×C59—416no—
176C22×C4×C59—44no—
177C22×C4×C59—42no—
178C22×C4×C59—42no—
179C22×C4×C59—42no—
180C22×C4×C59—42no—
181C22×C4×C59C2×D412yes—
182C22×C4×C59C2×D411yescharacteristic
183C22×C4×C59C2×D411yescharacteristic
184C22×C4×C59C2×D414yes—
185C22×C4×C59C2×D414yescomplemented
186C4⋊C4×C59—28no—
187C4⋊C4×C59—22no—
188C4⋊C4×C59—24no—
189C4⋊C4×C59—28no—
190C4⋊C4×C59—48no—
191C22⋊C4×C59—24no—
192C22⋊C4×C59—28no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—24no—
195C22⋊C4×C59—24no—
196C22⋊C4×C59—28no—
197C22⋊C4×C59—24no—
198C22⋊C4×C59—24no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—22no—
201C22⋊C4×C59—22no—
202C22⋊C4×C59—24no—
203C22⋊C4×C59—24no—
204C22⋊C4×C59—48no—
205C22⋊C4×C59—48no—
206C22⋊C4×C59C2×D414yescomplemented
207C42×C59—22no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—24no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—28no—
215C23×C59—28no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—22no—
220C23×C59—21no—
221C23×C59—24no—
222C23×C59—44no—
223C23×C59—42no—
224C23×C59—42no—
225C23×C59—44no—
226C23×C59—48no—
227C23×C59—48no—
228C23×C59—42no—
229C23×C59—42no—
230C23×C59—48no—
231C23×C59—42no—
232C23×C59—44no—
233C23×C59—44no—
234C23×C59—82no—
235C23×C59—84no—
236C23×C59—82no—
237C23×C59#12012yes—
238C23×C59#12012yes—
239C23×C59#12012yes—
240C23×C59C22×D411yescharacteristic
241C23×C59C22×D412yes—
242C23×C59C22×D412yes—
243C23×C59H(5,𝔽2)12yes—
244Q8×C59—42no—
245D4×C59—28no—
246D4×C59—44no—
247D4×C59—42no—
248D4×C59—44no—
249D4×C59—44no—
250D4×C59—42no—
251D4×C59—42no—
252D4×C59—44no—
253D4×C59—44no—
254D4×C59—48no—
255D4×C59—84no—
256D4×C59—84no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—24no—
260C2×C4×C59—24no—
261C2×C4×C59—21no—
262C2×C4×C59—24no—
263C2×C4×C59—22no—
264C2×C4×C59—21no—
265C2×C4×C59—21no—
266C2×C4×C59—28no—
267C2×C4×C59—24no—
268C2×C4×C59—44no—
269C2×C4×C59—44no—
270C2×C4×C59—48no—
271C2×C4×C59—44no—
272C2×C4×C59—48no—
273C2×C4×C59—48no—
274C2×C4×C59—42no—
275C2×C4×C59—42no—
276C2×C4×C59—42no—
277C2×C4×C59—44no—
278C2×C4×C59—44no—
279C2×C4×C59—48no—
280C2×C4×C59—44no—
281C2×C4×C59—82no—
282C2×C4×C59#12012yes—
283#23901C5911yesmaximal, characteristic, direct product factor, Sylow
284C22×C59—24no—
285C22×C59—24no—
286C22×C59—22no—
287C22×C59—24no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—24no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—416no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—42no—
300C22×C59—42no—
301C22×C59—42no—
302C22×C59—41no—
303C22×C59—44no—
304C22×C59—42no—
305C22×C59—81no—
306C22×C59—88no—
307C22×C59—84no—
308C22×C59—81no—
309C22×C59—162no—
310C22×C59#45814yescomplemented
311C22×C59C2×#12011yescharacteristic
312C22×C59C2×#12011yescharacteristic
313C22×C59#53411yescharacteristic
314C22×C59D4214yes—
315C4×C59—22no—
316C4×C59—44no—
317C4×C59—44no—
318C4×C59—41no—
319C4×C59—42no—
320C4×C59—44no—
321C4×C59—84no—
322#3484C2×C5911yescharacteristic
323C22×#123C2×C5911yescharacteristic, complemented
324C2×#458C2×C5912yescomplemented
325#2849C2×C5912yescomplemented
326#2022C2×C5914yescomplemented
327#1898C2×C5912yescomplemented
328#1890C2×C5912yescomplemented
329#1781C2×C5911yescharacteristic, complemented
330C2×C59—21no—
331C2×C59—24no—
332C2×C59—21no—
333C2×C59—24no—
334C2×C59—42no—
335C2×C59—42no—
336C2×C59—81no—
337C2×C59—82no—
338C2×C59#240311yescharacteristic
339C2×C59C2×#45812yes—
340C2×H(5,𝔽2)C22×C5911yescharacteristic
341C23×D4C22×C5911yescharacteristic
342C24×C4C22×C5911yescharacteristic, complemented
343#546—24no—
344D42—24no—
345#535—21no—
346#534—21no—
347#534C22×C5912yes—
348#525—21no—
349C2×#123—21no—
350C2×#123—21no—
351C2×#123—28no—
352C2×#123C22×C5912yes—
353C2×#123C22×C5912yescomplemented
354C2×#123C22×C5914yescomplemented
355C2×#120—21no—
356C2×#120C22×C5912yes—
357#518C22×C5911yescharacteristic
358C22×C4⋊C4—21no—
359C22×C22⋊C4—21no—
360C22×C22⋊C4C22×C5911yescharacteristic, complemented
361#458—28no—
362C2×#99C22×C5912yes—
363C2×#99C22×C5914yescomplemented
364C2×#99C22×C5912yescomplemented
365#386—24no—
366#385—24no—
367#381C22×C5914yescomplemented
368#379C22×C5912yescomplemented
369#342C22×C5914yescomplemented
370C59#2390111yescharacteristic, direct product factor, Sylow
371C25D4×C5911yescharacteristic
372C25C23×C5911yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C5911yescharacteristic
384C22×D4D4×C5911yescharacteristic
385C22×D4D4×C5914yes—
386C22×D4C23×C5912yes—
387C22×D4C23×C5911yescharacteristic
388C22×D4C23×C5911yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C5911yescharacteristic
396C23×C4C23×C5912yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C5918yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C5914yes—
439C2×C22⋊C4C23×C5914yes—
440C2×C22⋊C4C23×C5912yes—
441C2×C22⋊C4C23×C5912yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C5911yescharacteristic
455C24C2×D4×C5914yes—
456C24C2×D4×C5911yescharacteristic
457C24C2×D4×C5912yes—
458C24C2×D4×C5911yescharacteristic
459C24C2×D4×C5911yescharacteristic
460C24C24×C5911yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C5914yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C5912yes—
514C22×C4C2×D4×C5911yescharacteristic
515C22×C4C2×D4×C5911yescharacteristic
516C22×C4C2×D4×C5914yes—
517C22×C4C2×D4×C5914yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C5914yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C5912yes—
570C23#120×C5912yes—
571C23#120×C5912yes—
572C23C22×D4×C5911yescharacteristic
573C23C22×D4×C5912yes—
574C23C22×D4×C5912yes—
575C23H(5,𝔽2)×C5912yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C5912yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C59×#45814yescomplemented
642C22C2×#120×C5911yescharacteristic
643C22C2×#120×C5911yescharacteristic
644C22C59×#53411yescharacteristic
645C22D42×C5914yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C59×#240311yescharacteristic
662C2C2×C59×#45812yes—
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC24C24×C59
prime radicalC59×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C25×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C59C2semidirect
3. C2×C59×#458C2semidirect
4. C59×#2849C2semidirect
5. C59×#2022C2semidirect
6. C59×#1898C2semidirect
7. C59×#1890C2semidirect
8. C59×#1781C2semidirect
11. C24×C4×C59C22semidirect
22. C2×#123×C59C22semidirect
23. C2×#123×C59C22semidirect
29. C22×C22⋊C4×C59C22semidirect
32. C2×#99×C59C22semidirect
33. C2×#99×C59C22semidirect
36. C59×#381C22semidirect
37. C59×#379C22semidirect
38. C59×#342C22semidirect
77. #123×C59D4semidirect
185. C22×C4×C59C2×D4semidirect
206. C22⋊C4×C59C2×D4semidirect
283. #23901C59direct (N × H)
310. C22×C59#458semidirect
323. C22×#123C2×C59semidirect
324. C2×#458C2×C59semidirect
325. #2849C2×C59semidirect
326. #2022C2×C59semidirect
327. #1898C2×C59semidirect
328. #1890C2×C59semidirect
329. #1781C2×C59semidirect
342. C24×C4C22×C59semidirect
353. C2×#123C22×C59semidirect
354. C2×#123C22×C59semidirect
360. C22×C22⋊C4C22×C59semidirect
363. C2×#99C22×C59semidirect
364. C2×#99C22×C59semidirect
367. #381C22×C59semidirect
368. #379C22×C59semidirect
369. #342C22×C59semidirect
370. C59#23901direct (N × H)
409. #123D4×C59semidirect
517. C22×C4C2×D4×C59semidirect
538. C22⋊C4C2×D4×C59semidirect
641. C22C59×#458semidirect

Maximal subgroups

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

subgroupindex
C59×#34842
C22×#123×C592
C2×C59×#4582
C59×#28492
C59×#20222
C59×#18982
C59×#18902
C59×#17812
#2390159