Finite Group Databasebeta

← C59×#19037 #28431146

Subgroups — C59×#19037 #28431146

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3563C211yesmaximal, characteristic
2C59×#3237C211yesmaximal, characteristic, complemented
3C59×#3172C211yesmaximal, characteristic
4C59×#3166C211yesmaximal, characteristic, complemented
5C59×#1664C212yesmaximal, complemented
6C59×#1663C212yesmaximal, complemented
7C59×#1663C212yesmaximal, complemented
8C59×#1660C212yesmaximal, complemented
9C59×#1529C211yesmaximal, characteristic
10C59×#1528C211yesmaximal, characteristic, complemented
11C59×#1526C211yesmaximal, characteristic, complemented
12C59×#564C2212yes—
13C59×#563C2212yes—
14C59×#557—21no—
15C59×#554—21no—
16C59×#552—21no—
17C59×#552—22no—
18C59×#552—21no—
19C59×#544—21no—
20C59×#544—21no—
21C59×#541—21no—
22C59×#541—22no—
23C59×#539—21no—
24C59×#533C2211yescharacteristic
25C59×#533C2211yescharacteristic
26C59×#529C2211yescharacteristic
27C2×#125×C59C2211yescharacteristic
28C59×#501C2212yes—
29C59×#501C2212yes—
30C59×#501C2211yescharacteristic
31C59×#501C2212yes—
32C59×#499—22no—
33C59×#497C2211yescharacteristic
34C59×#491C2211yescharacteristic
35C59×#491C2212yes—
36C59×#489C2211yescharacteristic
37C59×#485—22no—
38C59×#477—21no—
39C59×#475—21no—
40C59×#470—21no—
41C59×#468—21no—
42C4×Q16×C59—21no—
43C4×SD16×C59—21no—
44C59×#432C2211yescharacteristic
45C59×#428—21no—
46C59×#428—21no—
47C4×M16×C59C2211yescharacteristic
48C4×M16×C59C2211yescharacteristic
49C59×#330C2212yes—
50C59×#330C2212yes—
51C59×#330C2212yes—
52C59×#330C2212yes—
53C59×#330C2212yes—
54C59×#329C2212yes—
55C2×Q16×C59—21no—
56C2×SD16×C59—21no—
57C2×M16×C59—21no—
58C2×M16×C59—21no—
59C4⋊Q8×C59—22no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—22no—
62C4⋊Q8×C59D411yescharacteristic
63C4⋊Q8×C59D411yescharacteristic
64C4⋊Q8×C59C2311yescharacteristic
65C4⋊Q8×C59C2311yescharacteristic
66#126×C59—22no—
67#126×C59—22no—
68#126×C59—22no—
69#126×C59—22no—
70#126×C59—21no—
71#125×C59—22no—
72#125×C59—22no—
73#125×C59—21no—
74#125×C59—22no—
75#125×C59—22no—
76#125×C59—22no—
77#125×C59—22no—
78#125×C59—22no—
79#125×C59—22no—
80#125×C59—21no—
81#125×C59—21no—
82#125×C59—21no—
83#125×C59D412yes—
84#125×C59C2311yescharacteristic
85#125×C59C2312yes—
86#125×C59C2312yes—
87#124×C59C2311yescharacteristic
88#123×C59—22no—
89#123×C59—22no—
90#123×C59—21no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—21no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—22no—
95C22⋊Q8×C59—21no—
96C22⋊Q8×C59—21no—
97C22⋊Q8×C59—22no—
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—21no—
101C22⋊Q8×C59D411yescharacteristic
102#121×C59D411yescharacteristic
103C4×Q8×C59—21no—
104C4×Q8×C59—22no—
105C4×Q8×C59—21no—
106C4×Q8×C59—21no—
107C4×Q8×C59—21no—
108C4×Q8×C59—21no—
109C4×Q8×C59—21no—
110C4×Q8×C59D411yescharacteristic
111C4×D4×C59—22no—
112C4×D4×C59—21no—
113C4×D4×C59D411yescharacteristic
114#117×C59—22no—
115#117×C59—21no—
116#117×C59—22no—
117#117×C59—21no—
118#117×C59—21no—
119#117×C59D412yes—
120C2×C4⋊C4×C59—21no—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59—21no—
123C2×C4⋊C4×C59D411yescharacteristic
124C2×C4⋊C4×C59D411yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#107×C59—22no—
127#107×C59—22no—
128#107×C59—21no—
129#107×C59—22no—
130#106×C59—22no—
131#106×C59—22no—
132#106×C59—21no—
133#106×C59—22no—
134C4⋊C8×C59—21no—
135#103×C59—21no—
136#103×C59—21no—
137#103×C59—21no—
138#103×C59—22no—
139#103×C59—22no—
140#102×C59—21no—
141C22⋊C8×C59—21no—
142#97×C59C2311yescharacteristic
143#97×C59C2311yescharacteristic
144#97×C59C2312yes—
145#97×C59C2312yes—
146C4×C8×C59—21no—
147C4×C8×C59—21no—
148C2×Q8×C59—22no—
149C2×Q8×C59—21no—
150C2×Q8×C59—21no—
151C2×Q8×C59—21no—
152C2×Q8×C59—21no—
153C2×Q8×C59C2×D411yescharacteristic
154C2×D4×C59—21no—
155C2×D4×C59C2×D411yescharacteristic
156C22×C4×C59—21no—
157C22×C4×C59—22no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163Q16×C59—42no—
164SD16×C59—42no—
165M16×C59—24no—
166M16×C59—42no—
167C2×C8×C59—22no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—22no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—22no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—21no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—41no—
198C4⋊C4×C59—41no—
199C4⋊C4×C59C2×D412yes—
200C4⋊C4×C59C2×D411yescharacteristic
201C4⋊C4×C59C2×D412yes—
202C4⋊C4×C59C2×D411yescharacteristic
203C4⋊C4×C59C2×D412yes—
204C4⋊C4×C59C2×D412yes—
205C22⋊C4×C59—22no—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—21no—
209C22⋊C4×C59—22no—
210C22⋊C4×C59—21no—
211C22⋊C4×C59—21no—
212C22⋊C4×C59—21no—
213C22⋊C4×C59—21no—
214C22⋊C4×C59—41no—
215C42×C59—22no—
216C42×C59—22no—
217C42×C59—22no—
218C42×C59—22no—
219C42×C59—21no—
220C42×C59—41no—
221C42×C59C2×D411yescharacteristic
222C42×C59C2×D411yescharacteristic
223C42×C59C2×D411yescharacteristic
224C42×C59C2411yescharacteristic
225C23×C59—21no—
226C23×C59#12011yescharacteristic
227Q8×C59—41no—
228Q8×C59—41no—
229Q8×C59—41no—
230Q8×C59—42no—
231Q8×C59—42no—
232Q8×C59—81no—
233Q8×C59—82no—
234D4×C59—41no—
235D4×C59—42no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—81no—
251C2×C4×C59—82no—
252C2×C4×C59#12011yescharacteristic
253C2×C4×C59#12011yescharacteristic
254C2×C4×C59#12011yescharacteristic
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C22×D411yescharacteristic
257C2×C4×C59C22×D411yescharacteristic
258C8×C59—24no—
259C8×C59—22no—
260C8×C59—42no—
261C8×C59—41no—
262#19037C5911yesmaximal, characteristic, direct product factor, Sylow
263C22×C59—21no—
264C22×C59—21no—
265C22×C59—21no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59C2×#12011yescharacteristic
270C4×C59—21no—
271C4×C59—21no—
272C4×C59—21no—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—41no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—41no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#45612yes—
285#3563C2×C5911yescharacteristic
286#3237C2×C5911yescharacteristic, complemented
287#3172C2×C5911yescharacteristic
288#3166C2×C5911yescharacteristic, complemented
289#1664C2×C5912yescomplemented
290#1663C2×C5912yescomplemented
291#1663C2×C5912yescomplemented
292#1660C2×C5912yescomplemented
293#1529C2×C5911yescharacteristic
294#1528C2×C5911yescharacteristic, complemented
295#1526C2×C5911yescharacteristic, complemented
296C2×C59—41no—
297C2×C59—81no—
298C2×C59C2×#45611yescharacteristic
299C2×C59#301811yescharacteristic
300C2×C59#302011yescharacteristic
301#564C22×C5912yes—
302#563C22×C5912yes—
303#557—21no—
304#554—21no—
305#552—21no—
306#552—22no—
307#552—21no—
308#544—21no—
309#544—21no—
310#541—21no—
311#541—22no—
312#539—21no—
313#533C22×C5911yescharacteristic
314#533C22×C5911yescharacteristic
315#529C22×C5911yescharacteristic
316C2×#125C22×C5911yescharacteristic
317#501C22×C5912yes—
318#501C22×C5912yes—
319#501C22×C5911yescharacteristic
320#501C22×C5912yes—
321#499—22no—
322#497C22×C5911yescharacteristic
323#491C22×C5911yescharacteristic
324#491C22×C5912yes—
325#489C22×C5911yescharacteristic
326#485—22no—
327#477—21no—
328#475—21no—
329#470—21no—
330#468—21no—
331C4×Q16—21no—
332C4×SD16—21no—
333#432C22×C5911yescharacteristic
334#428—21no—
335#428—21no—
336C4×M16C22×C5911yescharacteristic
337C4×M16C22×C5911yescharacteristic
338#330C22×C5912yes—
339#330C22×C5912yes—
340#330C22×C5912yes—
341#330C22×C5912yes—
342#330C22×C5912yes—
343#329C22×C5912yes—
344C59#1903711yescharacteristic, direct product factor, Sylow
345C2×Q16—21no—
346C2×SD16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C4⋊Q8—22no—
350C4⋊Q8—21no—
351C4⋊Q8—22no—
352C4⋊Q8D4×C5911yescharacteristic
353C4⋊Q8D4×C5911yescharacteristic
354C4⋊Q8C23×C5911yescharacteristic
355C4⋊Q8C23×C5911yescharacteristic
356#126—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#126—21no—
361#125—22no—
362#125—22no—
363#125—21no—
364#125—22no—
365#125—22no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—21no—
371#125—21no—
372#125—21no—
373#125D4×C5912yes—
374#125C23×C5912yes—
375#125C23×C5911yescharacteristic
376#125C23×C5912yes—
377#124C23×C5911yescharacteristic
378#123—22no—
379#123—22no—
380#123—21no—
381C22⋊Q8—22no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8D4×C5911yescharacteristic
392#121D4×C5911yescharacteristic
393C4×Q8—21no—
394C4×Q8—22no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8D4×C5911yescharacteristic
401C4×D4—22no—
402C4×D4—21no—
403C4×D4D4×C5911yescharacteristic
404#117—22no—
405#117—21no—
406#117—22no—
407#117—21no—
408#117—21no—
409#117D4×C5912yes—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4D4×C5911yescharacteristic
414C2×C4⋊C4D4×C5911yescharacteristic
415C2×C42C23×C5911yescharacteristic
416#107—22no—
417#107—22no—
418#107—21no—
419#107—22no—
420#106—22no—
421#106—22no—
422#106—21no—
423#106—22no—
424C4⋊C8—21no—
425#103—21no—
426#103—21no—
427#103—21no—
428#103—22no—
429#103—22no—
430#102—21no—
431C22⋊C8—21no—
432#97C23×C5912yes—
433#97C23×C5912yes—
434#97C23×C5911yescharacteristic
435#97C23×C5911yescharacteristic
436C4×C8—21no—
437C4×C8—21no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×D4×C5911yescharacteristic
444C2×D4—21no—
445C2×D4C2×D4×C5911yescharacteristic
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4C2×D4×C5911yescharacteristic
451C22×C4C2×D4×C5911yescharacteristic
452C22×C4C2×D4×C5911yescharacteristic
453Q16—42no—
454SD16—42no—
455M16—24no—
456M16—42no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C5912yes—
490C4⋊C4C2×D4×C5911yescharacteristic
491C4⋊C4C2×D4×C5912yes—
492C4⋊C4C2×D4×C5911yescharacteristic
493C4⋊C4C2×D4×C5912yes—
494C4⋊C4C2×D4×C5912yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—41no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—41no—
511C42C2×D4×C5911yescharacteristic
512C42C2×D4×C5911yescharacteristic
513C42C2×D4×C5911yescharacteristic
514C42C24×C5911yescharacteristic
515C23—21no—
516C23#120×C5911yescharacteristic
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—42no—
521Q8—42no—
522Q8—81no—
523Q8—82no—
524D4—41no—
525D4—42no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4#120×C5911yescharacteristic
543C2×C4#120×C5911yescharacteristic
544C2×C4#120×C5911yescharacteristic
545C2×C4C22×D4×C5911yescharacteristic
546C2×C4C22×D4×C5911yescharacteristic
547C2×C4C22×D4×C5911yescharacteristic
548C8—24no—
549C8—22no—
550C8—42no—
551C8—41no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22C2×#120×C5911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4C59×#45612yes—
574C2—41no—
575C2—81no—
576C2C2×C59×#45611yescharacteristic
577C2C59×#301811yescharacteristic
578C2C59×#302011yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radical#121×C59D4
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×#3237C2semidirect
4. C59×#3166C2semidirect
5. C59×#1664C2semidirect
6. C59×#1663C2semidirect
7. C59×#1663C2semidirect
8. C59×#1660C2semidirect
10. C59×#1528C2semidirect
11. C59×#1526C2semidirect
262. #19037C59direct (N × H)
286. #3237C2×C59semidirect
288. #3166C2×C59semidirect
289. #1664C2×C59semidirect
290. #1663C2×C59semidirect
291. #1663C2×C59semidirect
292. #1660C2×C59semidirect
294. #1528C2×C59semidirect
295. #1526C2×C59semidirect
344. C59#19037direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35632
C59×#32372
C59×#31722
C59×#31662
C59×#16642
C59×#16632
C59×#16602
C59×#15292
C59×#15282
C59×#15262
#1903759