Finite Group Databasebeta

← C59×#33569 #28445678

Subgroups — C59×#33569 #28445678

Order 15104.

1082 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3544C211yesmaximal, characteristic, complemented
2C59×#3020C212yesmaximal, complemented
3C2×C59×#456C211yesmaximal, characteristic, complemented
4C59×#2205C212yesmaximal, complemented
5C59×#2204C212yesmaximal, complemented
6C59×#2203C214yesmaximal, complemented
7C59×#2133C212yesmaximal, complemented
8C2×C59×#356C211yesmaximal, characteristic, complemented
9C2×2−1+4×C59C2211yescharacteristic
10C59×#575—22no—
11C2×#137×C59—21no—
12C59×#563—21no—
13C59×#563C2212yescomplemented
14Q82×C59C2211yescharacteristic, complemented
15C59×#555C2211yescharacteristic, complemented
16C59×#554—21no—
17C59×#552—22no—
18C59×#547—22no—
19C59×#544—22no—
20C59×#542—22no—
21C59×#536—21no—
22C59×#536C2212yes—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C59×#529—21no—
25C59×#456—21no—
26C59×#456—21no—
27C59×#456—22no—
28C59×#456C2212yes—
29C59×#456C2212yes—
30C59×#456C2214yes—
31C59×#454—22no—
32C59×#454C2214yes—
33C59×#421—22no—
34C2×#104×C59—21no—
35C2×#101×C59C2211yescharacteristic
36C2×#101×C59C2212yes—
37C59×#356C2212yescomplemented
38C59×#356C2212yescomplemented
39C59×#356C2214yescomplemented
40C59×#355C2214yescomplemented
412−1+4×C59—22no—
422−1+4×C59—21no—
432−1+4×C59—42no—
442−1+4×C59C2312yes—
45C2×#41×C59—21no—
46C2×#41×C59—21no—
47C2×#41×C59—22no—
48C2×#41×C59—41no—
49C2×#41×C59D412yes—
50C22×Q8×C59—41no—
51C22×Q8×C59D411yescharacteristic
52C22×Q8×C59C2311yescharacteristic
53#137×C59—22no—
54#137×C59—22no—
55#137×C59—24no—
56#137×C59—42no—
57#137×C59—42no—
58AGL(1,ℤ/8)×C59—24no—
59AGL(1,ℤ/8)×C59—42no—
60#135×C59—44no—
61C2×Q16×C59—41no—
62C2×SD16×C59—41no—
63C2×M16×C59—21no—
64C2×M16×C59—22no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—22no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—22no—
69C4⋊Q8×C59D411yescharacteristic
70C4⋊Q8×C59D411yescharacteristic
71C4⋊Q8×C59D412yes—
72C4⋊Q8×C59C2311yescharacteristic
73C4⋊Q8×C59C2311yescharacteristic
74C4⋊Q8×C59C2312yes—
75#126×C59—22no—
76#126×C59—22no—
77#126×C59—22no—
78#125×C59—21no—
79#125×C59—22no—
80#125×C59—22no—
81#125×C59—22no—
82#124×C59—21no—
83#124×C59—21no—
84#124×C59—21no—
85#124×C59D412yes—
86#124×C59C2312yes—
87#123×C59—22no—
88#123×C59—22no—
89#123×C59—22no—
90#123×C59—22no—
91#123×C59—22no—
92C22⋊Q8×C59—21no—
93C22⋊Q8×C59—22no—
94C22⋊Q8×C59—22no—
95C22⋊Q8×C59—22no—
96C22⋊Q8×C59—22no—
97C22⋊Q8×C59—22no—
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59—22no—
100#121×C59—21no—
101#121×C59—22no—
102C4×Q8×C59—21no—
103C4×Q8×C59—22no—
104C4×D4×C59—22no—
105C4×D4×C59—22no—
106C4×D4×C59—21no—
107#117×C59—22no—
108#117×C59—21no—
109#117×C59—22no—
110#117×C59D412yes—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—22no—
113C2×C42×C59D411yescharacteristic
114#104×C59—22no—
115#104×C59—22no—
116#104×C59—24no—
117#104×C59—24no—
118#101×C59—22no—
119#101×C59—21no—
120#101×C59C2312yes—
121#101×C59C2314yes—
122#41×C59—24no—
123#41×C59—22no—
124#41×C59—24no—
125#41×C59—22no—
126#41×C59—42no—
127#41×C59—41no—
128#41×C59—42no—
129#41×C59—42no—
130#41×C59—42no—
131#41×C59—44no—
132#41×C59—42no—
133#41×C59—41no—
134#41×C59—42no—
135#41×C59—81no—
136#41×C59—81no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—22no—
142C2×Q8×C59—41no—
143C2×Q8×C59—41no—
144C2×Q8×C59—41no—
145C2×Q8×C59—41no—
146C2×Q8×C59—42no—
147C2×Q8×C59—44no—
148C2×Q8×C59—42no—
149C2×Q8×C59C2×D411yescharacteristic
150C2×Q8×C59C2×D412yes—
151C2×Q8×C59C2×D411yescharacteristic
152C2×Q8×C59C2×D411yescharacteristic
153C2×Q8×C59C2×D412yes—
154C2×Q8×C59C2411yescharacteristic
155C2×D4×C59—21no—
156C2×D4×C59—22no—
157C2×D4×C59—21no—
158C2×D4×C59—22no—
159C2×D4×C59—21no—
160C2×D4×C59—41no—
161C2×D4×C59C2×D412yes—
162C22×C4×C59—22no—
163C22×C4×C59—21no—
164C22×C4×C59—21no—
165C22×C4×C59—22no—
166C22×C4×C59—22no—
167C22×C4×C59—41no—
168C22×C4×C59—42no—
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D412yes—
171Q16×C59—42no—
172Q16×C59—42no—
173Q16×C59—44no—
174SD16×C59—42no—
175SD16×C59—42no—
176SD16×C59—44no—
177SD16×C59—44no—
178D8×C59—44no—
179M16×C59—24no—
180M16×C59—22no—
181M16×C59—42no—
182M16×C59—41no—
183C2×C8×C59—41no—
184C2×C8×C59—42no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—22no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—22no—
193C4⋊C4×C59—22no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—22no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—22no—
199C4⋊C4×C59—22no—
200C4⋊C4×C59—22no—
201C22⋊C4×C59—22no—
202C22⋊C4×C59—22no—
203C22⋊C4×C59—22no—
204C22⋊C4×C59—22no—
205C22⋊C4×C59—22no—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—22no—
209C22⋊C4×C59—22no—
210C42×C59—21no—
211C42×C59—22no—
212C42×C59—21no—
213C42×C59—21no—
214C42×C59C2×D411yescharacteristic
215C42×C59C2×D411yescharacteristic
216C42×C59C2×D412yes—
217C42×C59C2×D412yes—
218C23×C59—22no—
219C23×C59—41no—
220C23×C59#12011yescharacteristic
221Q8×C59—24no—
222Q8×C59—22no—
223Q8×C59—42no—
224Q8×C59—41no—
225Q8×C59—41no—
226Q8×C59—41no—
227Q8×C59—41no—
228Q8×C59—42no—
229Q8×C59—42no—
230Q8×C59—42no—
231Q8×C59—41no—
232Q8×C59—81no—
233Q8×C59—81no—
234D4×C59—24no—
235D4×C59—22no—
236D4×C59—42no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—41no—
240D4×C59—42no—
241D4×C59—42no—
242D4×C59—42no—
243D4×C59—42no—
244D4×C59—81no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—21no—
252C2×C4×C59—22no—
253C2×C4×C59—21no—
254C2×C4×C59—22no—
255C2×C4×C59—21no—
256C2×C4×C59—42no—
257C2×C4×C59—44no—
258C2×C4×C59—44no—
259C2×C4×C59—42no—
260C2×C4×C59—44no—
261C2×C4×C59—42no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—82no—
266C2×C4×C59—82no—
267C2×C4×C59#12011yescharacteristic
268C2×C4×C59#12012yes—
269C2×C4×C59C22×D411yescharacteristic
270C2×C4×C59C22×D412yes—
271C8×C59—44no—
272C8×C59—42no—
273#33569C5911yesmaximal, characteristic, direct product factor, Sylow
274C22×C59—41no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—82no—
280C22×C59B(4,𝔽2)11yescharacteristic
281C22×C59B(4,𝔽2)11yescharacteristic
282C22×C59C2×#12011yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—21no—
286C4×C59—22no—
287C4×C59—21no—
288C4×C59—42no—
289C4×C59—41no—
290C4×C59—42no—
291C4×C59—41no—
292C4×C59—41no—
293C4×C59—42no—
294C4×C59—42no—
295#3544C2×C5911yescharacteristic, complemented
296#3020C2×C5912yescomplemented
297C2×#456C2×C5911yescharacteristic, complemented
298#2205C2×C5912yescomplemented
299#2204C2×C5912yescomplemented
300#2203C2×C5914yescomplemented
301#2133C2×C5912yescomplemented
302C2×#356C2×C5911yescharacteristic, complemented
303C2×C59—21no—
304C2×C59—21no—
305C2×C59—21no—
306C2×C59—82no—
307C2×C59—82no—
308C2×C59C2×B(4,𝔽2)11yescharacteristic
309C2×2−1+4C22×C5911yescharacteristic
310#575—22no—
311C2×#137—21no—
312#563—21no—
313#563C22×C5912yescomplemented
314Q82C22×C5911yescharacteristic, complemented
315#555C22×C5911yescharacteristic, complemented
316#554—21no—
317#552—22no—
318#547—22no—
319#544—22no—
320#542—22no—
321#536—21no—
322#536C22×C5912yes—
323C2×C4⋊Q8C22×C5911yescharacteristic
324#529—21no—
325#456—21no—
326#456—21no—
327#456—22no—
328#456C22×C5912yes—
329#456C22×C5914yes—
330#456C22×C5912yes—
331#454—22no—
332#454C22×C5914yes—
333#421—22no—
334C2×#104—21no—
335C2×#101C22×C5912yes—
336C2×#101C22×C5911yescharacteristic
337#356C22×C5912yescomplemented
338#356C22×C5912yescomplemented
339#356C22×C5914yescomplemented
340#355C22×C5914yescomplemented
341C59#3356911yescharacteristic, direct product factor, Sylow
3422−1+4—22no—
3432−1+4—21no—
3442−1+4—42no—
3452−1+4C23×C5912yes—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—22no—
349C2×#41—41no—
350C2×#41D4×C5912yes—
351C22×Q8—41no—
352C22×Q8D4×C5911yescharacteristic
353C22×Q8C23×C5911yescharacteristic
354#137—22no—
355#137—22no—
356#137—24no—
357#137—42no—
358#137—42no—
359AGL(1,ℤ/8)—24no—
360AGL(1,ℤ/8)—42no—
361#135—44no—
362C2×Q16—41no—
363C2×SD16—41no—
364C2×M16—21no—
365C2×M16—22no—
366C4⋊Q8—21no—
367C4⋊Q8—22no—
368C4⋊Q8—21no—
369C4⋊Q8—22no—
370C4⋊Q8D4×C5911yescharacteristic
371C4⋊Q8D4×C5911yescharacteristic
372C4⋊Q8D4×C5912yes—
373C4⋊Q8C23×C5911yescharacteristic
374C4⋊Q8C23×C5911yescharacteristic
375C4⋊Q8C23×C5912yes—
376#126—22no—
377#126—22no—
378#126—22no—
379#125—21no—
380#125—22no—
381#125—22no—
382#125—22no—
383#124—21no—
384#124—21no—
385#124—21no—
386#124D4×C5912yes—
387#124C23×C5912yes—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393C22⋊Q8—21no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401#121—21no—
402#121—22no—
403C4×Q8—21no—
404C4×Q8—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—21no—
408#117—22no—
409#117—21no—
410#117—22no—
411#117D4×C5912yes—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—22no—
414C2×C42D4×C5911yescharacteristic
415#104—22no—
416#104—22no—
417#104—24no—
418#104—24no—
419#101—22no—
420#101—21no—
421#101C23×C5912yes—
422#101C23×C5914yes—
423#41—24no—
424#41—22no—
425#41—24no—
426#41—22no—
427#41—42no—
428#41—41no—
429#41—42no—
430#41—42no—
431#41—42no—
432#41—44no—
433#41—42no—
434#41—41no—
435#41—42no—
436#41—81no—
437#41—81no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8C2×D4×C5911yescharacteristic
451C2×Q8C2×D4×C5912yes—
452C2×Q8C2×D4×C5911yescharacteristic
453C2×Q8C2×D4×C5911yescharacteristic
454C2×Q8C2×D4×C5912yes—
455C2×Q8C24×C5911yescharacteristic
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—41no—
462C2×D4C2×D4×C5912yes—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4C2×D4×C5911yescharacteristic
471C22×C4C2×D4×C5912yes—
472Q16—42no—
473Q16—42no—
474Q16—44no—
475SD16—42no—
476SD16—42no—
477SD16—44no—
478SD16—44no—
479D8—44no—
480M16—24no—
481M16—22no—
482M16—42no—
483M16—41no—
484C2×C8—41no—
485C2×C8—42no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C42—21no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42C2×D4×C5911yescharacteristic
516C42C2×D4×C5911yescharacteristic
517C42C2×D4×C5912yes—
518C42C2×D4×C5912yes—
519C23—22no—
520C23—41no—
521C23#120×C5911yescharacteristic
522Q8—24no—
523Q8—22no—
524Q8—42no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—41no—
533Q8—81no—
534Q8—81no—
535D4—24no—
536D4—22no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—41no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—81no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4#120×C5911yescharacteristic
569C2×C4#120×C5912yes—
570C2×C4C22×D4×C5911yescharacteristic
571C2×C4C22×D4×C5912yes—
572C8—44no—
573C8—42no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22C59×B(4,𝔽2)11yescharacteristic
581C22C59×B(4,𝔽2)11yescharacteristic
582C22C2×#120×C5911yescharacteristic
583C4—22no—
584C4—22no—
585C4—21no—
586C4—22no—
587C4—21no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—42no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—82no—
599C2—82no—
600C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C59C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C59
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C23×C59 ◁ C22×Q8×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3544C2semidirect
2. C59×#3020C2semidirect
3. C2×C59×#456C2semidirect
4. C59×#2205C2semidirect
5. C59×#2204C2semidirect
6. C59×#2203C2semidirect
7. C59×#2133C2semidirect
8. C2×C59×#356C2semidirect
13. C59×#563C22semidirect
14. Q82×C59C22semidirect
15. C59×#555C22semidirect
37. C59×#356C22semidirect
38. C59×#356C22semidirect
39. C59×#356C22semidirect
40. C59×#355C22semidirect
273. #33569C59direct (N × H)
295. #3544C2×C59semidirect
296. #3020C2×C59semidirect
297. C2×#456C2×C59semidirect
298. #2205C2×C59semidirect
299. #2204C2×C59semidirect
300. #2203C2×C59semidirect
301. #2133C2×C59semidirect
302. C2×#356C2×C59semidirect
313. #563C22×C59semidirect
314. Q82C22×C59semidirect
315. #555C22×C59semidirect
337. #356C22×C59semidirect
338. #356C22×C59semidirect
339. #356C22×C59semidirect
340. #355C22×C59semidirect
341. C59#33569direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35442
C59×#30202
C2×C59×#4562
C59×#22052
C59×#22042
C59×#22032
C59×#21332
C2×C59×#3562
#3356959