Finite Group Databasebeta

← C59×#25522 #28437631

Subgroups — C59×#25522 #28437631

Order 15104.

1084 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3493C211yesmaximal, characteristic, complemented
2C59×#3055C212yesmaximal, complemented
3C59×#2500C211yesmaximal, characteristic, complemented
4C59×#2060C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic, complemented
6C59×#2027C211yesmaximal, characteristic, complemented
7C59×#2006C211yesmaximal, characteristic, complemented
8C59×#1883C212yesmaximal, complemented
9C59×#1871C212yesmaximal, complemented
10C59×#1860C212yesmaximal, complemented
11C59×#1742C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C59C2212yes—
13C22×M16×C59C2212yes—
14C59×#556—22no—
15C59×#555—22no—
16C59×#553—22no—
17C59×#552—22no—
18C59×#544C2211yescharacteristic
19C59×#538—22no—
20C59×#537C2211yescharacteristic
21C59×#529—22no—
22C59×#525C2212yes—
23C2×#121×C59C2211yescharacteristic
24C59×#520C2212yes—
25C2×#117×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×#394—22no—
36C59×#390C2211yescharacteristic
37C59×#390C2211yescharacteristic
38C59×#388—22no—
39C59×#384C2211yescharacteristic
40C59×#382C2211yescharacteristic
41C4×C22⋊C4×C59C2211yescharacteristic
42C59×#343C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2212yes—
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59C2312yes—
48C22×D4×C59—21no—
49C22×D4×C59C2311yescharacteristic
50C23×C4×C59C2311yescharacteristic
51AGL(1,ℤ/8)×C59—24no—
52AGL(1,ℤ/8)×C59—42no—
53C2×SD16×C59—22no—
54C2×D8×C59—22no—
55C2×M16×C59—22no—
56C2×M16×C59—24no—
57C2×M16×C59—42no—
58C2×M16×C59C2312yes—
59C2×M16×C59C2312yes—
60C22×C8×C59—22no—
61C4⋊Q8×C59—22no—
62#126×C59—24no—
63#126×C59—24no—
64#125×C59—22no—
65#125×C59—22no—
66#124×C59—22no—
67#124×C59—24no—
68#123×C59—24no—
69#123×C59—22no—
70C22⋊Q8×C59—22no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59D412yes—
73C22⋊Q8×C59D412yes—
74#121×C59—24no—
75#121×C59—22no—
76#121×C59—22no—
77#121×C59—22no—
78#121×C59—22no—
79#121×C59D412yes—
80#121×C59D412yes—
81#120×C59—22no—
82C4×Q8×C59—22no—
83C4×Q8×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—24no—
87C4×D4×C59—22no—
88C4×D4×C59—22no—
89#117×C59—24no—
90#117×C59—22no—
91#117×C59—22no—
92#117×C59—22no—
93#117×C59C2312yes—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—22no—
96C2×C4⋊C4×C59C2311yescharacteristic
97C2×C4⋊C4×C59C2311yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59D412yes—
104C2×C42×C59D412yes—
105C2×C42×C59C2311yescharacteristic
106C2×C42×C59C2311yescharacteristic
107#107×C59—22no—
108#106×C59—22no—
109#104×C59—24no—
110#104×C59—42no—
111#103×C59—22no—
112#102×C59—22no—
113#102×C59—22no—
114#102×C59—22no—
115#95×C59—21no—
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—
123C2×Q8×C59—22no—
124C2×Q8×C59C2×D412yes—
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—21no—
132C2×D4×C59—21no—
133C2×D4×C59—42no—
134C2×D4×C59—82no—
135C2×D4×C59C2×D412yes—
136C22×C4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—24no—
141C22×C4×C59—22no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D412yes—
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2411yescharacteristic
150SD16×C59—44no—
151D8×C59—44no—
152M16×C59—22no—
153M16×C59—22no—
154M16×C59—22no—
155M16×C59—44no—
156M16×C59#4114yes—
157C2×C8×C59—22no—
158C2×C8×C59—22no—
159C2×C8×C59—44no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—82no—
170C4⋊C4×C59C2×D412yes—
171C4⋊C4×C59C2×D412yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—44no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—41no—
183C22⋊C4×C59—41no—
184C22⋊C4×C59—82no—
185C42×C59—24no—
186C42×C59—22no—
187C42×C59—22no—
188C42×C59—22no—
189C42×C59—22no—
190C42×C59—21no—
191C42×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—22no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59#12012yes—
203C23×C59C22×D411yescharacteristic
204Q8×C59—22no—
205Q8×C59—22no—
206Q8×C59—82no—
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—42no—
210D4×C59—42no—
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—82no—
214D4×C59—82no—
215D4×C59—82no—
216D4×C59—84no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
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—41no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—84no—
241C2×C4×C59#12011yescharacteristic
242C2×C4×C59#12011yescharacteristic
243C2×C4×C59C22×D412yes—
244C2×C4×C59C2×#4112yes—
245C2×C4×C59H(5,𝔽2)11yescharacteristic
246C2×C4×C592−1+411yescharacteristic
247C8×C59—24no—
248C8×C59—42no—
249#25522C5911yesmaximal, characteristic, direct product factor, Sylow
250C22×C59—21no—
251C22×C59—21no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—82no—
261C22×C59—82no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59C2×#12011yescharacteristic
266C22×C59#53211yescharacteristic
267C22×C59#53711yescharacteristic
268C4×C59—21no—
269C4×C59—21no—
270C4×C59—42no—
271C4×C59—42no—
272C4×C59—42no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59—42no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59#54612yes—
279C4×C59#54712yes—
280#3493C2×C5911yescharacteristic, complemented
281#3055C2×C5912yescomplemented
282#2500C2×C5911yescharacteristic, complemented
283#2060C2×C5911yescharacteristic, complemented
284#2039C2×C5911yescharacteristic, complemented
285#2027C2×C5911yescharacteristic, complemented
286#2006C2×C5911yescharacteristic, complemented
287#1883C2×C5912yescomplemented
288#1871C2×C5912yescomplemented
289#1860C2×C5912yescomplemented
290#1742C2×C5911yescharacteristic, complemented
291C2×C59—21no—
292C2×C59—21no—
293C2×C59—42no—
294C2×C59—82no—
295C2×C59—161no—
296C2×C59#243011yescharacteristic
297C2×C59#301811yescharacteristic
298C2×C59#302111yescharacteristic
299C2×AGL(1,ℤ/8)C22×C5912yes—
300C22×M16C22×C5912yes—
301#556—22no—
302#555—22no—
303#553—22no—
304#552—22no—
305#544C22×C5911yescharacteristic
306#538—22no—
307#537C22×C5911yescharacteristic
308#529—22no—
309#525C22×C5912yes—
310C2×#121C22×C5911yescharacteristic
311#520C22×C5912yes—
312C2×#117C22×C5911yescharacteristic
313#469—22no—
314#468—22no—
315#466—22no—
316#465—22no—
317#428C22×C5912yes—
318C2×#104C22×C5912yes—
319C2×#104C22×C5912yes—
320#418C22×C5912yes—
321#417C22×C5912yes—
322#394—22no—
323#390C22×C5911yescharacteristic
324#390C22×C5911yescharacteristic
325#388—22no—
326#384C22×C5911yescharacteristic
327#382C22×C5911yescharacteristic
328C4×C22⋊C4C22×C5911yescharacteristic
329#343C22×C5911yescharacteristic
330#343C22×C5912yes—
331#343C22×C5911yescharacteristic
332#339C22×C5912yes—
333#339C22×C5912yes—
334C59#2552211yescharacteristic, direct product factor, Sylow
335C2×#41C23×C5912yes—
336C22×D4—21no—
337C22×D4C23×C5911yescharacteristic
338C23×C4C23×C5911yescharacteristic
339AGL(1,ℤ/8)—24no—
340AGL(1,ℤ/8)—42no—
341C2×SD16—22no—
342C2×D8—22no—
343C2×M16—22no—
344C2×M16—24no—
345C2×M16—42no—
346C2×M16C23×C5912yes—
347C2×M16C23×C5912yes—
348C22×C8—22no—
349C4⋊Q8—22no—
350#126—24no—
351#126—24no—
352#125—22no—
353#125—22no—
354#124—22no—
355#124—24no—
356#123—24no—
357#123—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8D4×C5912yes—
361C22⋊Q8D4×C5912yes—
362#121—24no—
363#121—22no—
364#121—22no—
365#121—22no—
366#121—22no—
367#121D4×C5912yes—
368#121D4×C5912yes—
369#120—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—24no—
375C4×D4—22no—
376C4×D4—22no—
377#117—24no—
378#117—22no—
379#117—22no—
380#117—22no—
381#117C23×C5912yes—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4C23×C5911yescharacteristic
385C2×C4⋊C4C23×C5911yescharacteristic
386C2×C4⋊C4C23×C5911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4D4×C5912yes—
392C2×C42D4×C5912yes—
393C2×C42C23×C5911yescharacteristic
394C2×C42C23×C5911yescharacteristic
395#107—22no—
396#106—22no—
397#104—24no—
398#104—42no—
399#103—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403#95—21no—
404#95—22no—
405C24—21no—
406C24C2×D4×C5911yescharacteristic
407#41—22no—
408#41—22no—
409#41—22no—
410#41—22no—
411C2×Q8—22no—
412C2×Q8C2×D4×C5912yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4C2×D4×C5912yes—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4C2×D4×C5911yescharacteristic
433C22×C4C2×D4×C5912yes—
434C22×C4C2×D4×C5912yes—
435C22×C4C2×D4×C5912yes—
436C22×C4C2×D4×C5912yes—
437C22×C4C24×C5911yescharacteristic
438SD16—44no—
439D8—44no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—44no—
444M16#41×C5914yes—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—44no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—82no—
458C4⋊C4C2×D4×C5912yes—
459C4⋊C4C2×D4×C5912yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23#120×C5912yes—
491C23C22×D4×C5911yescharacteristic
492Q8—22no—
493Q8—22no—
494Q8—82no—
495D4—22no—
496D4—22no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4#120×C5911yescharacteristic
530C2×C4#120×C5911yescharacteristic
531C2×C4C22×D4×C5912yes—
532C2×C4C2×#41×C5912yes—
533C2×C4H(5,𝔽2)×C5911yescharacteristic
534C2×C42−1+4×C5911yescharacteristic
535C8—24no—
536C8—42no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22C2×#120×C5911yescharacteristic
553C22C59×#53211yescharacteristic
554C22C59×#53711yescharacteristic
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4C59×#54612yes—
566C4C59×#54712yes—
567C2—21no—
568C2—21no—
569C2—42no—
570C2—82no—
571C2—161no—
572C2C59×#243011yescharacteristic
573C2C59×#301811yescharacteristic
574C2C59×#302111yescharacteristic
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×#3493C2semidirect
2. C59×#3055C2semidirect
3. C59×#2500C2semidirect
4. C59×#2060C2semidirect
5. C59×#2039C2semidirect
6. C59×#2027C2semidirect
7. C59×#2006C2semidirect
8. C59×#1883C2semidirect
9. C59×#1871C2semidirect
10. C59×#1860C2semidirect
11. C59×#1742C2semidirect
249. #25522C59direct (N × H)
280. #3493C2×C59semidirect
281. #3055C2×C59semidirect
282. #2500C2×C59semidirect
283. #2060C2×C59semidirect
284. #2039C2×C59semidirect
285. #2027C2×C59semidirect
286. #2006C2×C59semidirect
287. #1883C2×C59semidirect
288. #1871C2×C59semidirect
289. #1860C2×C59semidirect
290. #1742C2×C59semidirect
334. C59#25522direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34932
C59×#30552
C59×#25002
C59×#20602
C59×#20392
C59×#20272
C59×#20062
C59×#18832
C59×#18712
C59×#18602
C59×#17422
#2552259