Finite Group Databasebeta

← C59×#24058 #28436167

Subgroups — C59×#24058 #28436167

Order 15104.

1556 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3484C211yesmaximal, characteristic
2C59×#2622C212yesmaximal, complemented
3C59×#2308C211yesmaximal, characteristic, complemented
4C59×#2044C214yesmaximal, complemented
5C59×#1898C212yesmaximal, complemented
6C59×#1867C212yesmaximal, complemented
7C59×#1790C212yesmaximal, complemented
8C59×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C59C2211yescharacteristic
10C23×D4×C59C2211yescharacteristic
11C59×#546—24no—
12D42×C59—24no—
13C59×#535—21no—
14C59×#534—21no—
15C59×#534C2212yes—
16C59×#525—21no—
17C2×#120×C59—21no—
18C2×#120×C59C2212yes—
19C59×#518C2211yescharacteristic
20C22×C22⋊C4×C59C2211yescharacteristic, complemented
21C22×C22⋊C4×C59C2211yescharacteristic, complemented
22C2×#99×C59C2212yes—
23C2×#99×C59C2212yescomplemented
24C2×#99×C59C2214yescomplemented
25C59×#397—24no—
26C59×#394—24no—
27C59×#386—21no—
28C59×#386—21no—
29C59×#386C2212yes—
30C59×#386C2212yescomplemented
31C59×#386C2214yescomplemented
32C59×#385C2214yescomplemented
33C59×#381—21no—
34C59×#379—21no—
35C59×#379C2212yescomplemented
36C59×#342C2214yescomplemented
37C25×C59D411yescharacteristic
38C25×C59C2311yescharacteristic
39H(5,𝔽2)×C59—24no—
40H(5,𝔽2)×C59—42no—
41C2×#41×C59—21no—
42C2×#41×C59—22no—
43C22×D4×C59—21no—
44C22×D4×C59—22no—
45C22×D4×C59—24no—
46C22×D4×C59—22no—
47C22×D4×C59—24no—
48C22×D4×C59—42no—
49C22×D4×C59D411yescharacteristic
50C22×D4×C59D411yescharacteristic
51C22×D4×C59D414yes—
52C22×D4×C59C2312yes—
53C22×D4×C59C2311yescharacteristic
54C22×D4×C59C2311yescharacteristic
55C23×C4×C59—21no—
56C23×C4×C59D411yescharacteristic
57C23×C4×C59C2312yes—
58#127×C59—22no—
59#124×C59—22no—
60#123×C59—22no—
61#123×C59—22no—
62#123×C59—22no—
63C22⋊Q8×C59—22no—
64#121×C59—22no—
65#121×C59—24no—
66#121×C59—24no—
67#121×C59—24no—
68#120×C59—24no—
69#120×C59—22no—
70#120×C59—22no—
71#120×C59—22no—
72#120×C59—22no—
73#120×C59—24no—
74#120×C59—22no—
75C4×D4×C59—24no—
76C4×D4×C59—24no—
77#117×C59—21no—
78C2×C4⋊C4×C59—24no—
79C2×C22⋊C4×C59—22no—
80C2×C22⋊C4×C59—21no—
81C2×C22⋊C4×C59—21no—
82C2×C22⋊C4×C59—21no—
83C2×C22⋊C4×C59—21no—
84C2×C22⋊C4×C59—24no—
85C2×C22⋊C4×C59—22no—
86C2×C22⋊C4×C59—22no—
87C2×C22⋊C4×C59—24no—
88C2×C22⋊C4×C59—24no—
89C2×C22⋊C4×C59—42no—
90C2×C22⋊C4×C59D414yes—
91C2×C22⋊C4×C59C2312yes—
92C2×C22⋊C4×C59C2314yes—
93C2×C22⋊C4×C59C2312yes—
94#99×C59—24no—
95#99×C59—42no—
96#99×C59—44no—
97#95×C59—22no—
98#95×C59—22no—
99#95×C59—24no—
100#95×C59—24no—
101C24×C59—22no—
102C24×C59—22no—
103C24×C59—21no—
104C24×C59—22no—
105C24×C59—22no—
106C24×C59—41no—
107C24×C59—82no—
108C24×C59—82no—
109C24×C59C2×D411yescharacteristic
110C24×C59C2×D412yes—
111C24×C59C2×D414yes—
112C24×C59C2×D411yescharacteristic
113C24×C59C2×D411yescharacteristic
114C24×C59C2×D411yescharacteristic
115C24×C59C2411yescharacteristic
116#41×C59—48no—
117#41×C59—42no—
118#41×C59—42no—
119C2×Q8×C59—21no—
120C2×D4×C59—22no—
121C2×D4×C59—22no—
122C2×D4×C59—22no—
123C2×D4×C59—21no—
124C2×D4×C59—24no—
125C2×D4×C59—22no—
126C2×D4×C59—22no—
127C2×D4×C59—22no—
128C2×D4×C59—22no—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—22no—
132C2×D4×C59—24no—
133C2×D4×C59—22no—
134C2×D4×C59—24no—
135C2×D4×C59—24no—
136C2×D4×C59—24no—
137C2×D4×C59—22no—
138C2×D4×C59—41no—
139C2×D4×C59—41no—
140C2×D4×C59—416no—
141C2×D4×C59—48no—
142C2×D4×C59—42no—
143C2×D4×C59—42no—
144C2×D4×C59—84no—
145C2×D4×C59—84no—
146C2×D4×C59#4114yes—
147C22×C4×C59—21no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—22no—
151C22×C4×C59—22no—
152C22×C4×C59—22no—
153C22×C4×C59—24no—
154C22×C4×C59—22no—
155C22×C4×C59—24no—
156C22×C4×C59—41no—
157C22×C4×C59—42no—
158C22×C4×C59—82no—
159C22×C4×C59—82no—
160C22×C4×C59C2×D412yes—
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2×D414yes—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—84no—
167C22⋊C4×C59—22no—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—84no—
181C22⋊C4×C59—84no—
182C22⋊C4×C59—82no—
183C22⋊C4×C59—82no—
184C42×C59—22no—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—28no—
192C23×C59—21no—
193C23×C59—22no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—24no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—44no—
201C23×C59—44no—
202C23×C59—44no—
203C23×C59—42no—
204C23×C59—42no—
205C23×C59—44no—
206C23×C59—42no—
207C23×C59—44no—
208C23×C59—82no—
209C23×C59—84no—
210C23×C59—84no—
211C23×C59—84no—
212C23×C59—82no—
213C23×C59—84no—
214C23×C59#12012yes—
215C23×C59#12012yes—
216C23×C59#12312yes—
217C23×C59C22×D411yescharacteristic
218C23×C59C22×D412yes—
219C23×C59C2×#4112yes—
220C23×C59H(5,𝔽2)12yes—
221Q8×C59—42no—
222D4×C59—28no—
223D4×C59—44no—
224D4×C59—42no—
225D4×C59—44no—
226D4×C59—44no—
227D4×C59—48no—
228D4×C59—42no—
229D4×C59—42no—
230D4×C59—44no—
231D4×C59—44no—
232D4×C59—84no—
233D4×C59—84no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—24no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—24no—
242C2×C4×C59—44no—
243C2×C4×C59—44no—
244C2×C4×C59—42no—
245C2×C4×C59—44no—
246C2×C4×C59—44no—
247C2×C4×C59—44no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—44no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—44no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—44no—
257C2×C4×C59—82no—
258C2×C4×C59—84no—
259C2×C4×C59#12312yes—
260#24058C5911yesmaximal, characteristic, direct product factor, Sylow
261C22×C59—24no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—24no—
268C22×C59—22no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—41no—
280C22×C59—44no—
281C22×C59—42no—
282C22×C59—81no—
283C22×C59—88no—
284C22×C59—84no—
285C22×C59—81no—
286C22×C59—88no—
287C22×C59—162no—
288C22×C59C2×#12011yescharacteristic
289C22×C59C2×#12311yescharacteristic
290C22×C59#53411yescharacteristic
291C22×C59#54614yes—
292C4×C59—22no—
293C4×C59—41no—
294C4×C59—42no—
295C4×C59—44no—
296C4×C59—84no—
297C4×C59—82no—
298C4×C59—82no—
299#3484C2×C5911yescharacteristic
300#2622C2×C5912yescomplemented
301#2308C2×C5911yescharacteristic, complemented
302#2044C2×C5914yescomplemented
303#1898C2×C5912yescomplemented
304#1867C2×C5912yescomplemented
305#1790C2×C5912yescomplemented
306#1781C2×C5911yescharacteristic, complemented
307C2×C59—21no—
308C2×C59—24no—
309C2×C59—21no—
310C2×C59—42no—
311C2×C59—42no—
312C2×C59—42no—
313C2×C59—81no—
314C2×C59—82no—
315C2×C59#240511yescharacteristic
316C2×C59#302612yes—
317C2×H(5,𝔽2)C22×C5911yescharacteristic
318C23×D4C22×C5911yescharacteristic
319#546—24no—
320D42—24no—
321#535—21no—
322#534—21no—
323#534C22×C5912yes—
324#525—21no—
325C2×#120—21no—
326C2×#120C22×C5912yes—
327#518C22×C5911yescharacteristic
328C22×C22⋊C4C22×C5911yescharacteristic, complemented
329C22×C22⋊C4C22×C5911yescharacteristic, complemented
330C2×#99C22×C5912yes—
331C2×#99C22×C5914yescomplemented
332C2×#99C22×C5912yescomplemented
333#397—24no—
334#394—24no—
335#386—21no—
336#386—21no—
337#386C22×C5912yes—
338#386C22×C5912yescomplemented
339#386C22×C5914yescomplemented
340#385C22×C5914yescomplemented
341#381—21no—
342#379—21no—
343#379C22×C5912yescomplemented
344#342C22×C5914yescomplemented
345C59#2405811yescharacteristic, direct product factor, Sylow
346C25D4×C5911yescharacteristic
347C25C23×C5911yescharacteristic
348H(5,𝔽2)—24no—
349H(5,𝔽2)—42no—
350C2×#41—21no—
351C2×#41—22no—
352C22×D4—21no—
353C22×D4—22no—
354C22×D4—24no—
355C22×D4—22no—
356C22×D4—24no—
357C22×D4—42no—
358C22×D4D4×C5911yescharacteristic
359C22×D4D4×C5911yescharacteristic
360C22×D4D4×C5914yes—
361C22×D4C23×C5912yes—
362C22×D4C23×C5911yescharacteristic
363C22×D4C23×C5911yescharacteristic
364C23×C4—21no—
365C23×C4D4×C5911yescharacteristic
366C23×C4C23×C5912yes—
367#127—22no—
368#124—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372C22⋊Q8—22no—
373#121—22no—
374#121—24no—
375#121—24no—
376#121—24no—
377#120—24no—
378#120—22no—
379#120—22no—
380#120—22no—
381#120—22no—
382#120—24no—
383#120—22no—
384C4×D4—24no—
385C4×D4—24no—
386#117—21no—
387C2×C4⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4D4×C5914yes—
400C2×C22⋊C4C23×C5912yes—
401C2×C22⋊C4C23×C5914yes—
402C2×C22⋊C4C23×C5912yes—
403#99—24no—
404#99—42no—
405#99—44no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—41no—
416C24—82no—
417C24—82no—
418C24C2×D4×C5911yescharacteristic
419C24C2×D4×C5912yes—
420C24C2×D4×C5914yes—
421C24C2×D4×C5911yescharacteristic
422C24C2×D4×C5911yescharacteristic
423C24C2×D4×C5911yescharacteristic
424C24C24×C5911yescharacteristic
425#41—48no—
426#41—42no—
427#41—42no—
428C2×Q8—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—416no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4#41×C5914yes—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C2×D4×C5912yes—
470C22×C4C2×D4×C5911yescharacteristic
471C22×C4C2×D4×C5911yescharacteristic
472C22×C4C2×D4×C5914yes—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—84no—
476C22⋊C4—22no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—22no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—28no—
501C23—21no—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—82no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—82no—
522C23—84no—
523C23#120×C5912yes—
524C23#120×C5912yes—
525C23#123×C5912yes—
526C23C22×D4×C5911yescharacteristic
527C23C22×D4×C5912yes—
528C23C2×#41×C5912yes—
529C23H(5,𝔽2)×C5912yes—
530Q8—42no—
531D4—28no—
532D4—44no—
533D4—42no—
534D4—44no—
535D4—44no—
536D4—48no—
537D4—42no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—84no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—82no—
567C2×C4—84no—
568C2×C4#123×C5912yes—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—81no—
591C22—88no—
592C22—81no—
593C22—84no—
594C22—88no—
595C22—162no—
596C22C2×#120×C5911yescharacteristic
597C22C2×#123×C5911yescharacteristic
598C22C59×#53411yescharacteristic
599C22C59×#54614yes—
600C4—22no—
601C4—44no—
602C4—41no—
603C4—42no—
604C4—82no—
605C4—84no—
606C4—82no—
607C2—21no—
608C2—24no—
609C2—21no—
610C2—42no—
611C2—42no—
612C2—42no—
613C2—81no—
614C2—82no—
615C2C59×#240511yescharacteristic
616C2C59×#302612yes—
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. C59×#2622C2semidirect
3. C59×#2308C2semidirect
4. C59×#2044C2semidirect
5. C59×#1898C2semidirect
6. C59×#1867C2semidirect
7. C59×#1790C2semidirect
8. C59×#1781C2semidirect
20. C22×C22⋊C4×C59C22semidirect
21. C22×C22⋊C4×C59C22semidirect
23. C2×#99×C59C22semidirect
24. C2×#99×C59C22semidirect
30. C59×#386C22semidirect
31. C59×#386C22semidirect
32. C59×#385C22semidirect
35. C59×#379C22semidirect
36. C59×#342C22semidirect
260. #24058C59direct (N × H)
300. #2622C2×C59semidirect
301. #2308C2×C59semidirect
302. #2044C2×C59semidirect
303. #1898C2×C59semidirect
304. #1867C2×C59semidirect
305. #1790C2×C59semidirect
306. #1781C2×C59semidirect
328. C22×C22⋊C4C22×C59semidirect
329. C22×C22⋊C4C22×C59semidirect
331. C2×#99C22×C59semidirect
332. C2×#99C22×C59semidirect
338. #386C22×C59semidirect
339. #386C22×C59semidirect
340. #385C22×C59semidirect
343. #379C22×C59semidirect
344. #342C22×C59semidirect
345. C59#24058direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34842
C59×#26222
C59×#23082
C59×#20442
C59×#18982
C59×#18672
C59×#17902
C59×#17812
#2405859