Finite Group Databasebeta

← C59×#25521 #28437630

Subgroups — C59×#25521 #28437630

Order 15104.

1118 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3492C211yesmaximal, characteristic, complemented
2C59×#3055C212yesmaximal, complemented
3C59×#2504C211yesmaximal, characteristic, complemented
4C59×#2061C211yesmaximal, characteristic, complemented
5C59×#2040C211yesmaximal, characteristic, complemented
6C59×#2027C211yesmaximal, characteristic, complemented
7C59×#2006C211yesmaximal, characteristic, complemented
8C59×#1883C212yesmaximal, complemented
9C59×#1871C212yesmaximal, complemented
10C59×#1860C212yesmaximal, complemented
11C2×C59×#343C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C59C2212yes—
13C22×M16×C59C2212yes—
14C59×#554—22no—
15C59×#553—22no—
16C59×#552—22no—
17C59×#547—22no—
18C59×#544C2211yescharacteristic
19C59×#539—22no—
20C59×#537C2211yescharacteristic
21C59×#525—22no—
22C59×#525C2212yes—
23C2×#121×C59C2211yescharacteristic
24C59×#520C2212yes—
25C22×C4⋊C4×C59C2211yescharacteristic
26C59×#469—22no—
27C59×#468—22no—
28C59×#466—22no—
29C59×#465—22no—
30C59×#428C2212yes—
31C2×#104×C59C2212yes—
32C2×#104×C59C2212yes—
33C59×#418C2212yes—
34C59×#417C2212yes—
35C59×#397—22no—
36C59×#390C2211yescharacteristic
37C59×#390C2211yescharacteristic
38C59×#386—22no—
39C59×#382C2211yescharacteristic
40C59×#382C2211yescharacteristic
41C4×C22⋊C4×C59C2211yescharacteristic
42C59×#343C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2212yes—
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59—22no—
48C2×#41×C59C2312yes—
49C22×D4×C59—21no—
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D412yes—
52C23×C4×C59C2311yescharacteristic
53AGL(1,ℤ/8)×C59—24no—
54AGL(1,ℤ/8)×C59—42no—
55C2×SD16×C59—22no—
56C2×D8×C59—22no—
57C2×M16×C59—22no—
58C2×M16×C59—24no—
59C2×M16×C59—42no—
60C2×M16×C59C2312yes—
61C2×M16×C59C2312yes—
62C22×C8×C59—22no—
63C4⋊Q8×C59—22no—
64#126×C59—24no—
65#125×C59—24no—
66#125×C59—22no—
67#123×C59—24no—
68#123×C59—24no—
69#123×C59—22no—
70C22⋊Q8×C59—22no—
71C22⋊Q8×C59—24no—
72C22⋊Q8×C59—22no—
73C22⋊Q8×C59—22no—
74C22⋊Q8×C59D412yes—
75C22⋊Q8×C59D412yes—
76#121×C59—22no—
77#121×C59—22no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—22no—
81#121×C59D412yes—
82#121×C59D412yes—
83#120×C59—22no—
84C4×Q8×C59—22no—
85C4×D4×C59—24no—
86C4×D4×C59—22no—
87C4×D4×C59—24no—
88C4×D4×C59—22no—
89C4×D4×C59—22no—
90#117×C59—22no—
91#117×C59—22no—
92#117×C59C2312yes—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59D412yes—
97C2×C4⋊C4×C59C2311yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59—22no—
104C2×C42×C59C2311yescharacteristic
105C2×C42×C59C2311yescharacteristic
106#107×C59—22no—
107#106×C59—22no—
108#104×C59—24no—
109#104×C59—42no—
110#103×C59—22no—
111#102×C59—22no—
112#102×C59—22no—
113#102×C59—22no—
114#95×C59—21no—
115#95×C59—22no—
116#95×C59—22no—
117C24×C59—21no—
118C24×C59C2×D411yescharacteristic
119#41×C59—22no—
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123#41×C59—84no—
124C2×Q8×C59—22no—
125C2×Q8×C59C2×D412yes—
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—21no—
133C2×D4×C59—21no—
134C2×D4×C59—44no—
135C2×D4×C59—41no—
136C2×D4×C59—41no—
137C2×D4×C59C2×D412yes—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—22no—
142C22×C4×C59—24no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—82no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D412yes—
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2411yescharacteristic
155SD16×C59—44no—
156D8×C59—44no—
157M16×C59—22no—
158M16×C59—22no—
159M16×C59—22no—
160M16×C59—44no—
161M16×C59#4114yescomplemented
162C2×C8×C59—22no—
163C2×C8×C59—22no—
164C2×C8×C59—44no—
165C4⋊C4×C59—24no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59C2×D412yes—
176C4⋊C4×C59C2×D412yes—
177C22⋊C4×C59—24no—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—42no—
185C22⋊C4×C59—42no—
186C22⋊C4×C59—82no—
187C22⋊C4×C59—82no—
188C42×C59—24no—
189C42×C59—22no—
190C42×C59—22no—
191C42×C59—21no—
192C42×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—22no—
196C23×C59—22no—
197C23×C59—41no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59#12011yescharacteristic
204C23×C59#12011yescharacteristic
205C23×C59C22×D411yescharacteristic
206Q8×C59—22no—
207Q8×C59—22no—
208Q8×C59—82no—
209D4×C59—22no—
210D4×C59—22no—
211D4×C59—44no—
212D4×C59—42no—
213D4×C59—42no—
214D4×C59—44no—
215D4×C59—42no—
216D4×C59—42no—
217D4×C59—82no—
218D4×C59—82no—
219D4×C59—82no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—44no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—42no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—82no—
245C2×C4×C59—84no—
246C2×C4×C59#12012yes—
247C2×C4×C59C22×D412yes—
248C2×C4×C59C2×#4112yes—
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C8×C59—24no—
252C8×C59—42no—
253#25521C5911yesmaximal, characteristic, direct product factor, Sylow
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—82no—
267C22×C59—82no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59#45412yes—
272C22×C59C2×#12011yescharacteristic
273C22×C59#53211yescharacteristic
274C22×C59#53711yescharacteristic
275C4×C59—22no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—42no—
280C4×C59—42no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#54612yes—
285C4×C59#54712yes—
286#3492C2×C5911yescharacteristic, complemented
287#3055C2×C5912yescomplemented
288#2504C2×C5911yescharacteristic, complemented
289#2061C2×C5911yescharacteristic, complemented
290#2040C2×C5911yescharacteristic, complemented
291#2027C2×C5911yescharacteristic, complemented
292#2006C2×C5911yescharacteristic, complemented
293#1883C2×C5912yescomplemented
294#1871C2×C5912yescomplemented
295#1860C2×C5912yescomplemented
296C2×#343C2×C5911yescharacteristic, complemented
297C2×C59—21no—
298C2×C59—21no—
299C2×C59—22no—
300C2×C59—41no—
301C2×C59—82no—
302C2×C59—161no—
303C2×C59#243011yescharacteristic
304C2×C59C2×#45411yescharacteristic
305C2×C59#301911yescharacteristic
306C2×AGL(1,ℤ/8)C22×C5912yes—
307C22×M16C22×C5912yes—
308#554—22no—
309#553—22no—
310#552—22no—
311#547—22no—
312#544C22×C5911yescharacteristic
313#539—22no—
314#537C22×C5911yescharacteristic
315#525—22no—
316#525C22×C5912yes—
317C2×#121C22×C5911yescharacteristic
318#520C22×C5912yes—
319C22×C4⋊C4C22×C5911yescharacteristic
320#469—22no—
321#468—22no—
322#466—22no—
323#465—22no—
324#428C22×C5912yes—
325C2×#104C22×C5912yes—
326C2×#104C22×C5912yes—
327#418C22×C5912yes—
328#417C22×C5912yes—
329#397—22no—
330#390C22×C5911yescharacteristic
331#390C22×C5911yescharacteristic
332#386—22no—
333#382C22×C5911yescharacteristic
334#382C22×C5911yescharacteristic
335C4×C22⋊C4C22×C5911yescharacteristic
336#343C22×C5911yescharacteristic
337#343C22×C5912yes—
338#343C22×C5911yescharacteristic
339#339C22×C5912yes—
340#339C22×C5912yes—
341C59#2552111yescharacteristic, direct product factor, Sylow
342C2×#41—22no—
343C2×#41C23×C5912yes—
344C22×D4—21no—
345C22×D4C23×C5911yescharacteristic
346C23×C4D4×C5912yes—
347C23×C4C23×C5911yescharacteristic
348AGL(1,ℤ/8)—24no—
349AGL(1,ℤ/8)—42no—
350C2×SD16—22no—
351C2×D8—22no—
352C2×M16—22no—
353C2×M16—24no—
354C2×M16—42no—
355C2×M16C23×C5912yes—
356C2×M16C23×C5912yes—
357C22×C8—22no—
358C4⋊Q8—22no—
359#126—24no—
360#125—24no—
361#125—22no—
362#123—24no—
363#123—24no—
364#123—22no—
365C22⋊Q8—22no—
366C22⋊Q8—24no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8D4×C5912yes—
370C22⋊Q8D4×C5912yes—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121D4×C5912yes—
377#121D4×C5912yes—
378#120—22no—
379C4×Q8—22no—
380C4×D4—24no—
381C4×D4—22no—
382C4×D4—24no—
383C4×D4—22no—
384C4×D4—22no—
385#117—22no—
386#117—22no—
387#117C23×C5912yes—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4D4×C5912yes—
392C2×C4⋊C4C23×C5911yescharacteristic
393C2×C4⋊C4C23×C5911yescharacteristic
394C2×C4⋊C4C23×C5911yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C42C23×C5911yescharacteristic
400C2×C42C23×C5911yescharacteristic
401#107—22no—
402#106—22no—
403#104—24no—
404#104—42no—
405#103—22no—
406#102—22no—
407#102—22no—
408#102—22no—
409#95—21no—
410#95—22no—
411#95—22no—
412C24—21no—
413C24C2×D4×C5911yescharacteristic
414#41—22no—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—84no—
419C2×Q8—22no—
420C2×Q8C2×D4×C5912yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—44no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4C2×D4×C5912yes—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—82no—
444C22×C4C2×D4×C5911yescharacteristic
445C22×C4C2×D4×C5912yes—
446C22×C4C2×D4×C5912yes—
447C22×C4C2×D4×C5912yes—
448C22×C4C2×D4×C5912yes—
449C22×C4C24×C5911yescharacteristic
450SD16—44no—
451D8—44no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—44no—
456M16#41×C5914yescomplemented
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—44no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4C2×D4×C5912yes—
471C4⋊C4C2×D4×C5912yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—21no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23#120×C5911yescharacteristic
499C23#120×C5911yescharacteristic
500C23C22×D4×C5911yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—82no—
504D4—22no—
505D4—22no—
506D4—44no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—82no—
513D4—82no—
514D4—82no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4#120×C5912yes—
542C2×C4C22×D4×C5912yes—
543C2×C4C2×#41×C5912yes—
544C2×C4H(5,𝔽2)×C5911yescharacteristic
545C2×C42−1+4×C5911yescharacteristic
546C8—24no—
547C8—42no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—82no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C59×#45412yes—
566C22C2×#120×C5911yescharacteristic
567C22C59×#53211yescharacteristic
568C22C59×#53711yescharacteristic
569C4—22no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4C59×#54612yes—
579C4C59×#54712yes—
580C2—21no—
581C2—21no—
582C2—22no—
583C2—41no—
584C2—82no—
585C2—161no—
586C2C59×#243011yescharacteristic
587C2C2×C59×#45411yescharacteristic
588C2C59×#301911yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3492C2semidirect
2. C59×#3055C2semidirect
3. C59×#2504C2semidirect
4. C59×#2061C2semidirect
5. C59×#2040C2semidirect
6. C59×#2027C2semidirect
7. C59×#2006C2semidirect
8. C59×#1883C2semidirect
9. C59×#1871C2semidirect
10. C59×#1860C2semidirect
11. C2×C59×#343C2semidirect
161. M16×C59#41semidirect
253. #25521C59direct (N × H)
286. #3492C2×C59semidirect
287. #3055C2×C59semidirect
288. #2504C2×C59semidirect
289. #2061C2×C59semidirect
290. #2040C2×C59semidirect
291. #2027C2×C59semidirect
292. #2006C2×C59semidirect
293. #1883C2×C59semidirect
294. #1871C2×C59semidirect
295. #1860C2×C59semidirect
296. C2×#343C2×C59semidirect
341. C59#25521direct (N × H)
456. M16#41×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34922
C59×#30552
C59×#25042
C59×#20612
C59×#20402
C59×#20272
C59×#20062
C59×#18832
C59×#18712
C59×#18602
C2×C59×#3432
#2552159