Finite Group Databasebeta

← C59×#19015 #28431124

Subgroups — C59×#19015 #28431124

Order 15104.

1012 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3558C211yesmaximal, characteristic
2C59×#3238C211yesmaximal, characteristic, complemented
3C59×#3152C211yesmaximal, characteristic, complemented
4C2×C59×#489C211yesmaximal, characteristic, complemented
5C59×#1661C214yesmaximal, complemented
6C59×#1659C214yesmaximal, complemented
7C59×#1530C211yesmaximal, characteristic, complemented
8C59×#1527C211yesmaximal, characteristic, complemented
9C59×#1524C211yesmaximal, characteristic, complemented
10C2×#137×C59—22no—
11C59×#562C2212yes—
12C59×H(3,𝔽4)C2212yes—
13C59×#559—21no—
14C59×#548—21no—
15C59×#546—22no—
16C59×#546—21no—
17D42×C59—21no—
18C59×#542—21no—
19C59×#538—22no—
20C59×#538—21no—
21C59×#535—21no—
22C59×#534—21no—
23C59×#532C2211yescharacteristic
24C2×C4⋊Q8×C59C2211yescharacteristic, complemented
25C59×#529C2211yescharacteristic
26C2×#124×C59C2211yescharacteristic
27C59×#501C2212yescomplemented
28C59×#497C2214yescomplemented
29C59×#494—21no—
30C59×#492—21no—
31C59×#489—22no—
32C59×#489C2214yescomplemented
33C59×#489C2212yescomplemented
34C59×#483—21no—
35C59×#480—21no—
36C59×#477—21no—
37C59×#476—21no—
38C59×#432C2211yescharacteristic, complemented
39C2×#103×C59—21no—
40C2×#102×C59—21no—
41C4×M16×C59C2211yescharacteristic, complemented
42C2×#97×C59C2211yescharacteristic, complemented
43C59×#330C2212yescomplemented
44C59×#329C2212yescomplemented
45C59×#329C2214yescomplemented
46C59×#329C2214yescomplemented
47C2×#41×C59—21no—
48C2×#41×C59D412yes—
49C22×Q8×C59—21no—
50C22×Q8×C59D411yescharacteristic
51C22×D4×C59—22no—
52C22×D4×C59—21no—
53C22×D4×C59D411yescharacteristic
54#137×C59—48no—
55C2×Q16×C59—24no—
56C2×SD16×C59—24no—
57C2×M16×C59—21no—
58C22×C8×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59D411yescharacteristic
63C4⋊Q8×C59C2312yes—
64#127×C59—21no—
65#127×C59D411yescharacteristic
66#126×C59—22no—
67#126×C59—21no—
68#125×C59C2311yescharacteristic
69#124×C59—21no—
70#124×C59—22no—
71#124×C59—22no—
72#124×C59—22no—
73#124×C59—21no—
74#124×C59—21no—
75#124×C59D412yes—
76#124×C59C2312yes—
77#124×C59C2311yescharacteristic
78#124×C59C2312yes—
79#123×C59—22no—
80#123×C59—22no—
81#123×C59—21no—
82#123×C59—22no—
83#123×C59—22no—
84C22⋊Q8×C59—21no—
85C22⋊Q8×C59—22no—
86C22⋊Q8×C59—22no—
87C22⋊Q8×C59D411yescharacteristic
88#121×C59—22no—
89#121×C59—22no—
90#121×C59—22no—
91#121×C59—21no—
92#121×C59—22no—
93#121×C59—22no—
94#121×C59—21no—
95#121×C59—21no—
96#121×C59D411yescharacteristic
97#120×C59—22no—
98#120×C59—22no—
99#120×C59—22no—
100#120×C59—22no—
101#120×C59—22no—
102C4×Q8×C59—21no—
103C4×Q8×C59D411yescharacteristic
104C4×D4×C59—22no—
105C4×D4×C59—22no—
106C4×D4×C59—21no—
107C4×D4×C59—21no—
108C4×D4×C59—21no—
109C4×D4×C59—21no—
110C4×D4×C59D411yescharacteristic
111#117×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C22⋊C4×C59—22no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C42×C59C2311yescharacteristic
118#107×C59—22no—
119#106×C59—22no—
120C4⋊C8×C59—21no—
121#103×C59—24no—
122#103×C59—22no—
123#102×C59—24no—
124#102×C59—22no—
125C22⋊C8×C59—21no—
126#97×C59—21no—
127#97×C59C2312yes—
128#97×C59C2312yes—
129#97×C59C2312yes—
130C4×C8×C59—21no—
131C24×C59—22no—
132C24×C59—21no—
133#41×C59—44no—
134#41×C59—82no—
135C2×Q8×C59—21no—
136C2×Q8×C59—22no—
137C2×Q8×C59—42no—
138C2×Q8×C59—44no—
139C2×Q8×C59—41no—
140C2×Q8×C59C2×D411yescharacteristic
141C2×Q8×C59C2×D412yes—
142C2×Q8×C59C2×D412yes—
143C2×D4×C59—22no—
144C2×D4×C59—21no—
145C2×D4×C59—22no—
146C2×D4×C59—22no—
147C2×D4×C59—22no—
148C2×D4×C59—22no—
149C2×D4×C59—22no—
150C2×D4×C59—21no—
151C2×D4×C59—21no—
152C2×D4×C59—21no—
153C2×D4×C59—42no—
154C2×D4×C59—82no—
155C2×D4×C59—84no—
156C2×D4×C59C2×D411yescharacteristic
157C2×D4×C59C2×D412yes—
158C2×D4×C59C2×D412yes—
159C22×C4×C59—22no—
160C22×C4×C59—22no—
161C22×C4×C59—21no—
162C22×C4×C59—21no—
163C22×C4×C59—22no—
164C22×C4×C59—41no—
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168Q16×C59—48no—
169SD16×C59—48no—
170M16×C59—24no—
171C2×C8×C59—22no—
172C2×C8×C59—22no—
173C2×C8×C59—22no—
174C2×C8×C59—42no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—41no—
186C4⋊C4×C59—41no—
187C4⋊C4×C59C2×D411yescharacteristic
188C4⋊C4×C59C2×D411yescharacteristic
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—21no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—21no—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—21no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—22no—
201C22⋊C4×C59—21no—
202C22⋊C4×C59—21no—
203C22⋊C4×C59—22no—
204C22⋊C4×C59—22no—
205C42×C59—22no—
206C42×C59—21no—
207C42×C59C2×D411yescharacteristic
208C42×C59C2×D411yescharacteristic
209C42×C59C2×D411yescharacteristic
210C42×C59C2411yescharacteristic
211C23×C59—22no—
212C23×C59—21no—
213C23×C59—22no—
214C23×C59—42no—
215C23×C59—81no—
216C23×C59—81no—
217C23×C59—82no—
218C23×C59—82no—
219C23×C59—82no—
220C23×C59#12011yescharacteristic
221Q8×C59—42no—
222Q8×C59—42no—
223Q8×C59—41no—
224Q8×C59—44no—
225Q8×C59—42no—
226Q8×C59—82no—
227D4×C59—42no—
228D4×C59—42no—
229D4×C59—41no—
230D4×C59—42no—
231D4×C59—44no—
232D4×C59—42no—
233D4×C59—42no—
234D4×C59—81no—
235D4×C59—82no—
236D4×C59—82no—
237D4×C59—82no—
238D4×C59—82no—
239D4×C59—82no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—41no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—82no—
251C2×C4×C59—81no—
252C2×C4×C59—82no—
253C2×C4×C59—82no—
254C2×C4×C59—82no—
255C2×C4×C59#12011yescharacteristic
256C2×C4×C59#12011yescharacteristic
257C2×C4×C59#12011yescharacteristic
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C22×D411yescharacteristic
260C2×C4×C59C22×D411yescharacteristic
261C8×C59—24no—
262C8×C59—42no—
263C8×C59—42no—
264#19015C5911yesmaximal, characteristic, direct product factor, Sylow
265C22×C59—21no—
266C22×C59—21no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—41no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—162no—
277C22×C59—164no—
278C22×C59#45312yes—
279C22×C59C2×#12011yescharacteristic
280C4×C59—21no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—42no—
289C4×C59—41no—
290C4×C59—82no—
291C4×C59#45512yes—
292#3558C2×C5911yescharacteristic
293#3238C2×C5911yescharacteristic, complemented
294#3152C2×C5911yescharacteristic, complemented
295C2×#489C2×C5911yescharacteristic, complemented
296#1661C2×C5914yescomplemented
297#1659C2×C5914yescomplemented
298#1530C2×C5911yescharacteristic, complemented
299#1527C2×C5911yescharacteristic, complemented
300#1524C2×C5911yescharacteristic, complemented
301C2×C59—22no—
302C2×C59—82no—
303C2×C59—82no—
304C2×C59—81no—
305C2×C59C2×#45311yescharacteristic
306C2×C59C2×#45511yescharacteristic
307C2×C59#302211yescharacteristic
308C2×#137—22no—
309#562C22×C5912yes—
310H(3,𝔽4)C22×C5912yes—
311#559—21no—
312#548—21no—
313#546—22no—
314#546—21no—
315D42—21no—
316#542—21no—
317#538—22no—
318#538—21no—
319#535—21no—
320#534—21no—
321#532C22×C5911yescharacteristic
322C2×C4⋊Q8C22×C5911yescharacteristic, complemented
323#529C22×C5911yescharacteristic
324C2×#124C22×C5911yescharacteristic
325#501C22×C5912yescomplemented
326#497C22×C5914yescomplemented
327#494—21no—
328#492—21no—
329#489—22no—
330#489C22×C5914yescomplemented
331#489C22×C5912yescomplemented
332#483—21no—
333#480—21no—
334#477—21no—
335#476—21no—
336#432C22×C5911yescharacteristic, complemented
337C2×#103—21no—
338C2×#102—21no—
339C4×M16C22×C5911yescharacteristic, complemented
340C2×#97C22×C5911yescharacteristic, complemented
341#330C22×C5912yescomplemented
342#329C22×C5912yescomplemented
343#329C22×C5914yescomplemented
344#329C22×C5914yescomplemented
345C59#1901511yescharacteristic, direct product factor, Sylow
346C2×#41—21no—
347C2×#41D4×C5912yes—
348C22×Q8—21no—
349C22×Q8D4×C5911yescharacteristic
350C22×D4—22no—
351C22×D4—21no—
352C22×D4D4×C5911yescharacteristic
353#137—48no—
354C2×Q16—24no—
355C2×SD16—24no—
356C2×M16—21no—
357C22×C8—21no—
358C4⋊Q8—21no—
359C4⋊Q8—21no—
360C4⋊Q8—21no—
361C4⋊Q8D4×C5911yescharacteristic
362C4⋊Q8C23×C5912yes—
363#127—21no—
364#127D4×C5911yescharacteristic
365#126—22no—
366#126—21no—
367#125C23×C5911yescharacteristic
368#124—21no—
369#124—22no—
370#124—22no—
371#124—22no—
372#124—21no—
373#124—21no—
374#124D4×C5912yes—
375#124C23×C5912yes—
376#124C23×C5911yescharacteristic
377#124C23×C5912yes—
378#123—22no—
379#123—22no—
380#123—21no—
381#123—22no—
382#123—22no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C5911yescharacteristic
387#121—22no—
388#121—22no—
389#121—22no—
390#121—21no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121D4×C5911yescharacteristic
396#120—22no—
397#120—22no—
398#120—22no—
399#120—22no—
400#120—22no—
401C4×Q8—21no—
402C4×Q8D4×C5911yescharacteristic
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4D4×C5911yescharacteristic
410#117—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42C23×C5911yescharacteristic
417#107—22no—
418#106—22no—
419C4⋊C8—21no—
420#103—24no—
421#103—22no—
422#102—24no—
423#102—22no—
424C22⋊C8—21no—
425#97—21no—
426#97C23×C5912yes—
427#97C23×C5912yes—
428#97C23×C5912yes—
429C4×C8—21no—
430C24—22no—
431C24—21no—
432#41—44no—
433#41—82no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—41no—
439C2×Q8C2×D4×C5911yescharacteristic
440C2×Q8C2×D4×C5912yes—
441C2×Q8C2×D4×C5912yes—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—84no—
455C2×D4C2×D4×C5911yescharacteristic
456C2×D4C2×D4×C5912yes—
457C2×D4C2×D4×C5912yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—41no—
464C22×C4C2×D4×C5911yescharacteristic
465C22×C4C2×D4×C5911yescharacteristic
466C22×C4C2×D4×C5911yescharacteristic
467Q16—48no—
468SD16—48no—
469M16—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4C2×D4×C5911yescharacteristic
487C4⋊C4C2×D4×C5911yescharacteristic
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—21no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C42—22no—
505C42—21no—
506C42C2×D4×C5911yescharacteristic
507C42C2×D4×C5911yescharacteristic
508C42C2×D4×C5911yescharacteristic
509C42C24×C5911yescharacteristic
510C23—22no—
511C23—21no—
512C23—22no—
513C23—42no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23#120×C5911yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—41no—
523Q8—44no—
524Q8—42no—
525Q8—82no—
526D4—42no—
527D4—42no—
528D4—41no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—81no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#120×C5911yescharacteristic
555C2×C4#120×C5911yescharacteristic
556C2×C4#120×C5911yescharacteristic
557C2×C4C22×D4×C5911yescharacteristic
558C2×C4C22×D4×C5911yescharacteristic
559C2×C4C22×D4×C5911yescharacteristic
560C8—24no—
561C8—42no—
562C8—42no—
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22C59×#45312yes—
577C22C2×#120×C5911yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4C59×#45512yes—
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2C2×C59×#45311yescharacteristic
595C2C2×C59×#45511yescharacteristic
596C2C59×#302211yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3558C2
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×#489C2semidirect
5. C59×#1661C2semidirect
6. C59×#1659C2semidirect
7. C59×#1530C2semidirect
8. C59×#1527C2semidirect
9. C59×#1524C2semidirect
24. C2×C4⋊Q8×C59C22semidirect
27. C59×#501C22semidirect
28. C59×#497C22semidirect
32. C59×#489C22semidirect
33. C59×#489C22semidirect
38. C59×#432C22semidirect
41. C4×M16×C59C22semidirect
42. C2×#97×C59C22semidirect
43. C59×#330C22semidirect
44. C59×#329C22semidirect
45. C59×#329C22semidirect
46. C59×#329C22semidirect
264. #19015C59direct (N × H)
293. #3238C2×C59semidirect
294. #3152C2×C59semidirect
295. C2×#489C2×C59semidirect
296. #1661C2×C59semidirect
297. #1659C2×C59semidirect
298. #1530C2×C59semidirect
299. #1527C2×C59semidirect
300. #1524C2×C59semidirect
322. C2×C4⋊Q8C22×C59semidirect
325. #501C22×C59semidirect
326. #497C22×C59semidirect
330. #489C22×C59semidirect
331. #489C22×C59semidirect
336. #432C22×C59semidirect
339. C4×M16C22×C59semidirect
340. C2×#97C22×C59semidirect
341. #330C22×C59semidirect
342. #329C22×C59semidirect
343. #329C22×C59semidirect
344. #329C22×C59semidirect
345. C59#19015direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35582
C59×#32382
C59×#31522
C2×C59×#4892
C59×#16612
C59×#16592
C59×#15302
C59×#15272
C59×#15242
#1901559