Finite Group Databasebeta

← C59×#18695 #28430804

Subgroups — C59×#18695 #28430804

Order 15104.

866 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×#3552C211yesmaximal, characteristic
2C59×#3238C211yesmaximal, characteristic, complemented
3C59×#3238C211yesmaximal, characteristic, complemented
4C2×C59×#487C211yesmaximal, characteristic, complemented
5C59×#1638C212yesmaximal, complemented
6C59×#1637C212yesmaximal, complemented
7C59×#1636C212yesmaximal, complemented
8C59×#1635C212yesmaximal, complemented
9C59×#1556C211yesmaximal, characteristic, complemented
10C59×#1494C211yesmaximal, characteristic, complemented
11C59×#1494C211yesmaximal, characteristic, complemented
12C59×#563—22no—
13C59×#557C2211yescharacteristic
14C59×#556C2211yescharacteristic
15C59×#554—21no—
16C59×#553—21no—
17D4×Q8×C59—21no—
18D4×Q8×C59C2211yescharacteristic
19C59×#548—21no—
20C59×#547C2211yescharacteristic
21C59×#544—21no—
22C59×#542—21no—
23C59×#541—21no—
24C59×#539—22no—
25C59×#536—21no—
26C2×C4⋊Q8×C59C2211yescharacteristic
27C2×C4⋊Q8×C59C2211yescharacteristic
28C59×#529C2211yescharacteristic
29C59×#529C2211yescharacteristic
30C59×#501—21no—
31C59×#501—21no—
32C59×#489—21no—
33C59×#489—21no—
34C59×#487—22no—
35C59×#487C2212yes—
36C59×#487C2212yes—
37C59×#483—21no—
38C59×#483—21no—
39C59×#480—21no—
40C59×#480—21no—
41C59×#477C2212yes—
42C59×#477C2212yes—
43C59×#476C2212yes—
44C59×#476C2212yes—
45C59×#432C2211yescharacteristic
46C59×#432C2211yescharacteristic
47C2×#103×C59—22no—
48C2×C4×C8×C59C2211yescharacteristic
49C59×#332C2212yes—
50C59×#332C2212yes—
51C59×Q8⋊C8C2212yes—
52C59×Q8⋊C8C2212yes—
53C59×D4⋊C8C2212yes—
54C59×D4⋊C8C2212yes—
55C2×#41×C59—21no—
56C22×Q8×C59—21no—
57C22×Q8×C59—21no—
58C22×Q8×C59—21no—
59C22×C8×C59—21no—
60C4⋊Q8×C59—22no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59—21no—
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59—21no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59D412yes—
69C4⋊Q8×C59C2312yes—
70C4⋊Q8×C59C2311yescharacteristic
71C4⋊Q8×C59C2311yescharacteristic
72#126×C59—22no—
73#126×C59—22no—
74#126×C59—22no—
75#126×C59—21no—
76#126×C59—21no—
77#125×C59—22no—
78#125×C59—22no—
79#125×C59—22no—
80#125×C59—21no—
81#125×C59D411yescharacteristic
82#125×C59D411yescharacteristic
83#124×C59—22no—
84#124×C59—21no—
85#124×C59D411yescharacteristic
86#124×C59D411yescharacteristic
87#123×C59—22no—
88#123×C59—22no—
89#123×C59—21no—
90#123×C59—21no—
91#123×C59—22no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—22no—
94C22⋊Q8×C59—22no—
95C22⋊Q8×C59—22no—
96C22⋊Q8×C59—22no—
97C22⋊Q8×C59—21no—
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—21no—
101C22⋊Q8×C59D411yescharacteristic
102C22⋊Q8×C59D411yescharacteristic
103#121×C59—22no—
104#121×C59—21no—
105#121×C59D411yescharacteristic
106#121×C59D411yescharacteristic
107C4×Q8×C59—21no—
108C4×Q8×C59—21no—
109C4×Q8×C59—21no—
110C4×Q8×C59C2311yescharacteristic
111C4×Q8×C59C2311yescharacteristic
112C4×D4×C59—22no—
113C4×D4×C59—22no—
114C4×D4×C59—21no—
115C4×D4×C59—21no—
116C4×D4×C59—21no—
117C4×D4×C59C2311yescharacteristic
118C4×D4×C59C2311yescharacteristic
119#117×C59—21no—
120#117×C59—21no—
121#117×C59—21no—
122C2×C4⋊C4×C59—21no—
123C2×C4⋊C4×C59—21no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59D412yes—
127C2×C42×C59C2311yescharacteristic
128#107×C59—22no—
129#107×C59—22no—
130#106×C59—22no—
131#106×C59—22no—
132C4⋊C8×C59C2312yes—
133C4⋊C8×C59C2312yes—
134#103×C59—22no—
135#103×C59—22no—
136#103×C59—24no—
137#103×C59—24no—
138#102×C59—22no—
139#102×C59—22no—
140C22⋊C8×C59—21no—
141C22⋊C8×C59—21no—
142#97×C59—21no—
143#97×C59—21no—
144C4×C8×C59—21no—
145C4×C8×C59C2312yes—
146#41×C59—82no—
147C2×Q8×C59—22no—
148C2×Q8×C59—22no—
149C2×Q8×C59—21no—
150C2×Q8×C59—21no—
151C2×Q8×C59—22no—
152C2×Q8×C59—44no—
153C2×Q8×C59—41no—
154C2×Q8×C59—82no—
155C2×Q8×C59—82no—
156C2×Q8×C59C2×D411yescharacteristic
157C2×Q8×C59C2×D411yescharacteristic
158C2×D4×C59—21no—
159C2×D4×C59—21no—
160C2×D4×C59—22no—
161C2×D4×C59C2×D411yescharacteristic
162C2×D4×C59C2×D411yescharacteristic
163C22×C4×C59—22no—
164C22×C4×C59—22no—
165C22×C4×C59—22no—
166C22×C4×C59—21no—
167C22×C4×C59—21no—
168C22×C4×C59—41no—
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D411yescharacteristic
171C22×C4×C59C2×D411yescharacteristic
172C2×C8×C59—22no—
173C2×C8×C59—22no—
174C2×C8×C59—22no—
175C2×C8×C59—42no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59—21no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—41no—
197C4⋊C4×C59—41no—
198C4⋊C4×C59C2×D412yes—
199C4⋊C4×C59C2×D412yes—
200C4⋊C4×C59C2×D411yescharacteristic
201C4⋊C4×C59C2×D411yescharacteristic
202C4⋊C4×C59C2×D411yescharacteristic
203C4⋊C4×C59C2×D411yescharacteristic
204C22⋊C4×C59—22no—
205C22⋊C4×C59—22no—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—21no—
209C22⋊C4×C59—21no—
210C22⋊C4×C59—21no—
211C22⋊C4×C59—21no—
212C22⋊C4×C59—21no—
213C22⋊C4×C59—21no—
214C22⋊C4×C59—21no—
215C22⋊C4×C59—21no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—21no—
218C42×C59—22no—
219C42×C59—22no—
220C42×C59—21no—
221C42×C59—21no—
222C42×C59C2×D411yescharacteristic
223C42×C59C2×D411yescharacteristic
224C42×C59C2×D411yescharacteristic
225C42×C59C2411yescharacteristic
226C23×C59—21no—
227C23×C59—21no—
228C23×C59#12011yescharacteristic
229Q8×C59—22no—
230Q8×C59—22no—
231Q8×C59—44no—
232Q8×C59—42no—
233Q8×C59—42no—
234Q8×C59—82no—
235Q8×C59—82no—
236Q8×C59—82no—
237D4×C59—22no—
238D4×C59—22no—
239D4×C59—42no—
240D4×C59—42no—
241D4×C59—82no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—21no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—82no—
257C2×C4×C59—82no—
258C2×C4×C59—82no—
259C2×C4×C59—82no—
260C2×C4×C59#12011yescharacteristic
261C2×C4×C59#12011yescharacteristic
262C2×C4×C59#12011yescharacteristic
263C2×C4×C59C22×D411yescharacteristic
264C2×C4×C59C22×D411yescharacteristic
265C2×C4×C59C22×D411yescharacteristic
266C8×C59—42no—
267C8×C59—42no—
268C8×C59—42no—
269#18695C5911yesmaximal, characteristic, direct product factor, Sylow
270C22×C59—21no—
271C22×C59—21no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59#45412yes—
279C22×C59C2×#12011yescharacteristic
280C4×C59—21no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—21no—
284C4×C59—41no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—41no—
292C4×C59—82no—
293C4×C59#57712yes—
294C4×C59#57812yes—
295#3552C2×C5911yescharacteristic
296#3238C2×C5911yescharacteristic, complemented
297#3238C2×C5911yescharacteristic, complemented
298C2×#487C2×C5911yescharacteristic, complemented
299#1638C2×C5912yescomplemented
300#1637C2×C5912yescomplemented
301#1636C2×C5912yescomplemented
302#1635C2×C5912yescomplemented
303#1556C2×C5911yescharacteristic, complemented
304#1494C2×C5911yescharacteristic, complemented
305#1494C2×C5911yescharacteristic, complemented
306C2×C59—22no—
307C2×C59—81no—
308C2×C59—81no—
309C2×C59#300911yescharacteristic
310C2×C59#301111yescharacteristic
311C2×C59C2×#45411yescharacteristic
312#563—22no—
313#557C22×C5911yescharacteristic
314#556C22×C5911yescharacteristic
315#554—21no—
316#553—21no—
317D4×Q8—21no—
318D4×Q8C22×C5911yescharacteristic
319#548—21no—
320#547C22×C5911yescharacteristic
321#544—21no—
322#542—21no—
323#541—21no—
324#539—22no—
325#536—21no—
326C2×C4⋊Q8C22×C5911yescharacteristic
327C2×C4⋊Q8C22×C5911yescharacteristic
328#529C22×C5911yescharacteristic
329#529C22×C5911yescharacteristic
330#501—21no—
331#501—21no—
332#489—21no—
333#489—21no—
334#487—22no—
335#487C22×C5912yes—
336#487C22×C5912yes—
337#483—21no—
338#483—21no—
339#480—21no—
340#480—21no—
341#477C22×C5912yes—
342#477C22×C5912yes—
343#476C22×C5912yes—
344#476C22×C5912yes—
345#432C22×C5911yescharacteristic
346#432C22×C5911yescharacteristic
347C2×#103—22no—
348C2×C4×C8C22×C5911yescharacteristic
349#332C22×C5912yes—
350#332C22×C5912yes—
351Q8⋊C8C22×C5912yes—
352Q8⋊C8C22×C5912yes—
353D4⋊C8C22×C5912yes—
354D4⋊C8C22×C5912yes—
355C59#1869511yescharacteristic, direct product factor, Sylow
356C2×#41—21no—
357C22×Q8—21no—
358C22×Q8—21no—
359C22×Q8—21no—
360C22×C8—21no—
361C4⋊Q8—22no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8—21no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—21no—
369C4⋊Q8D4×C5912yes—
370C4⋊Q8C23×C5912yes—
371C4⋊Q8C23×C5911yescharacteristic
372C4⋊Q8C23×C5911yescharacteristic
373#126—22no—
374#126—22no—
375#126—22no—
376#126—21no—
377#126—21no—
378#125—22no—
379#125—22no—
380#125—22no—
381#125—21no—
382#125D4×C5911yescharacteristic
383#125D4×C5911yescharacteristic
384#124—22no—
385#124—21no—
386#124D4×C5911yescharacteristic
387#124D4×C5911yescharacteristic
388#123—22no—
389#123—22no—
390#123—21no—
391#123—21no—
392#123—22no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8D4×C5911yescharacteristic
403C22⋊Q8D4×C5911yescharacteristic
404#121—22no—
405#121—21no—
406#121D4×C5911yescharacteristic
407#121D4×C5911yescharacteristic
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8C23×C5911yescharacteristic
412C4×Q8C23×C5911yescharacteristic
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4C23×C5911yescharacteristic
419C4×D4C23×C5911yescharacteristic
420#117—21no—
421#117—21no—
422#117—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4D4×C5912yes—
428C2×C42C23×C5911yescharacteristic
429#107—22no—
430#107—22no—
431#106—22no—
432#106—22no—
433C4⋊C8C23×C5912yes—
434C4⋊C8C23×C5912yes—
435#103—22no—
436#103—22no—
437#103—24no—
438#103—24no—
439#102—22no—
440#102—22no—
441C22⋊C8—21no—
442C22⋊C8—21no—
443#97—21no—
444#97—21no—
445C4×C8—21no—
446C4×C8C23×C5912yes—
447#41—82no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—44no—
454C2×Q8—41no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8C2×D4×C5911yescharacteristic
458C2×Q8C2×D4×C5911yescharacteristic
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4C2×D4×C5911yescharacteristic
463C2×D4C2×D4×C5911yescharacteristic
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4C2×D4×C5911yescharacteristic
471C22×C4C2×D4×C5911yescharacteristic
472C22×C4C2×D4×C5911yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—42no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4C2×D4×C5912yes—
500C4⋊C4C2×D4×C5912yes—
501C4⋊C4C2×D4×C5911yescharacteristic
502C4⋊C4C2×D4×C5911yescharacteristic
503C4⋊C4C2×D4×C5911yescharacteristic
504C4⋊C4C2×D4×C5911yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42C2×D4×C5911yescharacteristic
524C42C2×D4×C5911yescharacteristic
525C42C2×D4×C5911yescharacteristic
526C42C24×C5911yescharacteristic
527C23—21no—
528C23—21no—
529C23#120×C5911yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—44no—
533Q8—42no—
534Q8—42no—
535Q8—82no—
536Q8—82no—
537Q8—82no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4#120×C5911yescharacteristic
562C2×C4#120×C5911yescharacteristic
563C2×C4#120×C5911yescharacteristic
564C2×C4C22×D4×C5911yescharacteristic
565C2×C4C22×D4×C5911yescharacteristic
566C2×C4C22×D4×C5911yescharacteristic
567C8—42no—
568C8—42no—
569C8—42no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C59×#45412yes—
579C22C2×#120×C5911yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4C59×#57712yes—
594C4C59×#57812yes—
595C2—22no—
596C2—81no—
597C2—81no—
598C2C59×#300911yescharacteristic
599C2C59×#301111yescharacteristic
600C2C2×C59×#45411yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC42C24×C59
prime radicalC59×#3552C2
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×#3238C2semidirect
4. C2×C59×#487C2semidirect
5. C59×#1638C2semidirect
6. C59×#1637C2semidirect
7. C59×#1636C2semidirect
8. C59×#1635C2semidirect
9. C59×#1556C2semidirect
10. C59×#1494C2semidirect
11. C59×#1494C2semidirect
269. #18695C59direct (N × H)
296. #3238C2×C59semidirect
297. #3238C2×C59semidirect
298. C2×#487C2×C59semidirect
299. #1638C2×C59semidirect
300. #1637C2×C59semidirect
301. #1636C2×C59semidirect
302. #1635C2×C59semidirect
303. #1556C2×C59semidirect
304. #1494C2×C59semidirect
305. #1494C2×C59semidirect
355. C59#18695direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35522
C59×#32382
C2×C59×#4872
C59×#16382
C59×#16372
C59×#16362
C59×#16352
C59×#15562
C59×#14942
#1869559