Finite Group Databasebeta

← C59×#25582 #28437691

Subgroups — C59×#25582 #28437691

Order 15104.

1190 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#553C211yesmaximal, characteristic
2C2×C59×#491C211yesmaximal, characteristic, complemented
3C2×C59×#461C211yesmaximal, characteristic, complemented
4C59×#2542C211yesmaximal, characteristic, complemented
5C59×#2000C212yesmaximal, complemented
6C59×#2000C212yesmaximal, complemented
7C59×#1989C211yesmaximal, characteristic, complemented
8C59×#1955C211yesmaximal, characteristic, complemented
9C59×#1873C212yesmaximal, complemented
10C59×#1872C212yesmaximal, complemented
11C59×#1762C211yesmaximal, characteristic, complemented
12C22×SD16×C59C2212yes—
13C59×#553—24no—
14C59×#553—24no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C2×#126×C59—21no—
17C2×#125×C59C2211yescharacteristic
18C2×#123×C59—21no—
19C2×C22⋊Q8×C59C2212yes—
20C2×C22⋊Q8×C59C2212yes—
21C2×#121×C59—21no—
22C2×#121×C59C2211yescharacteristic
23C2×#121×C59C2211yescharacteristic
24C2×C4×D4×C59—21no—
25C2×C4×D4×C59C2211yescharacteristic
26C2×C4×D4×C59C2211yescharacteristic
27C2×#117×C59C2211yescharacteristic
28C22×C4⋊C4×C59C2211yescharacteristic
29C59×#491—24no—
30C59×#461—24no—
31C2×C4⋊C8×C59C2211yescharacteristic
32C2×#103×C59C2212yes—
33C2×#103×C59C2212yes—
34C2×#103×C59C2212yes—
35C2×#102×C59C2212yes—
36C2×C22⋊C8×C59C2212yes—
37C2×C22⋊C8×C59C2212yes—
38C2×#97×C59C2211yescharacteristic
39C59×#399—21no—
40C59×#399—21no—
41C59×#390—21no—
42C59×#384—21no—
43C4×C4⋊C4×C59C2211yescharacteristic
44C59×#340C2212yes—
45C59×#340C2212yes—
46C59×#340C2212yes—
47C22×Q8×C59C2312yes—
48C22×D4×C59—21no—
49C22×D4×C59—21no—
50C22×D4×C59D411yescharacteristic
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59—21no—
53C23×C4×C59—21no—
54C23×C4×C59C2311yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C2×SD16×C59—24no—
57C2×SD16×C59—42no—
58C2×SD16×C59—42no—
59C22×C8×C59C2312yes—
60C22×C8×C59C2312yes—
61C4⋊Q8×C59—22no—
62C4⋊Q8×C59—22no—
63#126×C59—28no—
64#125×C59—22no—
65#125×C59D414yes—
66#123×C59—28no—
67C22⋊Q8×C59—24no—
68C22⋊Q8×C59—24no—
69C22⋊Q8×C59—42no—
70C22⋊Q8×C59—42no—
71#121×C59—28no—
72#121×C59—22no—
73#121×C59—22no—
74#121×C59—22no—
75#121×C59—22no—
76C4×D4×C59—28no—
77C4×D4×C59—22no—
78C4×D4×C59—22no—
79C4×D4×C59—22no—
80C4×D4×C59D414yes—
81#117×C59—22no—
82#117×C59—22no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59D411yescharacteristic
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C4⋊C4×C59C2312yes—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59D411yescharacteristic
104C2×C22⋊C4×C59D411yescharacteristic
105C2×C42×C59—21no—
106C2×C42×C59—21no—
107C2×C42×C59C2311yescharacteristic
108C4⋊C8×C59—22no—
109#103×C59—24no—
110#103×C59—24no—
111#103×C59—24no—
112#102×C59—24no—
113C22⋊C8×C59—42no—
114C22⋊C8×C59—42no—
115#97×C59—22no—
116#95×C59—21no—
117C24×C59—21no—
118C24×C59C2×D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—22no—
126C2×D4×C59—24no—
127C2×D4×C59—22no—
128C2×D4×C59—24no—
129C2×D4×C59—22no—
130C2×D4×C59—21no—
131C2×D4×C59—22no—
132C2×D4×C59—22no—
133C2×D4×C59—41no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59—82no—
137C2×D4×C59C2×D412yes—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—24no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—22no—
151C22×C4×C59—24no—
152C22×C4×C59—22no—
153C22×C4×C59—82no—
154C22×C4×C59—82no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2411yescharacteristic
160SD16×C59—48no—
161C2×C8×C59—22no—
162C2×C8×C59—22no—
163C2×C8×C59—22no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—44no—
176C4⋊C4×C59—44no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59C2×D412yes—
181C4⋊C4×C59C2×D412yes—
182C4⋊C4×C59#4114yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—24no—
185C22⋊C4×C59—24no—
186C22⋊C4×C59—24no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—44no—
190C22⋊C4×C59—44no—
191C42×C59—24no—
192C42×C59—42no—
193C42×C59C2×D412yes—
194C42×C59C2×D412yes—
195C23×C59—21no—
196C23×C59—22no—
197C23×C59—21no—
198C23×C59—22no—
199C23×C59—21no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—22no—
205C23×C59—42no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—42no—
209C23×C59#13712yes—
210C23×C59C22×D411yescharacteristic
211Q8×C59—42no—
212Q8×C59—42no—
213Q8×C59—42no—
214Q8×C59—42no—
215D4×C59—22no—
216D4×C59—22no—
217D4×C59—44no—
218D4×C59—48no—
219D4×C59—42no—
220D4×C59—84no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
229C2×C4×C59—24no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—41no—
233C2×C4×C59—42no—
234C2×C4×C59—41no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—84no—
248C2×C4×C59—84no—
249C2×C4×C59#12012yes—
250C2×C4×C59#12012yes—
251C2×C4×C59C22×D411yescharacteristic
252C2×C4×C59C22×D411yescharacteristic
253C2×C4×C59C2×#4112yes—
254C2×C4×C59H(5,𝔽2)12yes—
255C8×C59—42no—
256C8×C59—42no—
257#25582C5911yesmaximal, characteristic, direct product factor, Sylow
258C22×C59—22no—
259C22×C59—24no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—42no—
263C22×C59—41no—
264C22×C59—42no—
265C22×C59—41no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—41no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—82no—
272C22×C59—82no—
273C22×C59C2×#12011yescharacteristic
274C22×C59#54612yes—
275C22×C59C2×#13711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57712yes—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—42no—
285C4×C59—44no—
286C4×C59—82no—
287C4×C59#53211yescharacteristic
288C4×C59#53511yescharacteristic
289C4×C59#55012yes—
290C2×#553C2×C5911yescharacteristic
291C2×#491C2×C5911yescharacteristic, complemented
292C2×#461C2×C5911yescharacteristic, complemented
293#2542C2×C5911yescharacteristic, complemented
294#2000C2×C5912yescomplemented
295#2000C2×C5912yescomplemented
296#1989C2×C5911yescharacteristic, complemented
297#1955C2×C5911yescharacteristic, complemented
298#1873C2×C5912yescomplemented
299#1872C2×C5912yescomplemented
300#1762C2×C5911yescharacteristic, complemented
301C2×C59—42no—
302C2×C59—81no—
303C2×C59—82no—
304C2×C59#243111yescharacteristic
305C2×C59#300711yescharacteristic
306C2×C59#300911yescharacteristic
307C2×C59#330712yes—
308C2×C59#332012yes—
309C22×SD16C22×C5912yes—
310#553—24no—
311#553—24no—
312C2×C4⋊Q8C22×C5911yescharacteristic
313C2×#126—21no—
314C2×#125C22×C5911yescharacteristic
315C2×#123—21no—
316C2×C22⋊Q8C22×C5912yes—
317C2×C22⋊Q8C22×C5912yes—
318C2×#121—21no—
319C2×#121C22×C5911yescharacteristic
320C2×#121C22×C5911yescharacteristic
321C2×C4×D4—21no—
322C2×C4×D4C22×C5911yescharacteristic
323C2×C4×D4C22×C5911yescharacteristic
324C2×#117C22×C5911yescharacteristic
325C22×C4⋊C4C22×C5911yescharacteristic
326#491—24no—
327#461—24no—
328C2×C4⋊C8C22×C5911yescharacteristic
329C2×#103C22×C5912yes—
330C2×#103C22×C5912yes—
331C2×#103C22×C5912yes—
332C2×#102C22×C5912yes—
333C2×C22⋊C8C22×C5912yes—
334C2×C22⋊C8C22×C5912yes—
335C2×#97C22×C5911yescharacteristic
336#399—21no—
337#399—21no—
338#390—21no—
339#384—21no—
340C4×C4⋊C4C22×C5911yescharacteristic
341#340C22×C5912yes—
342#340C22×C5912yes—
343#340C22×C5912yes—
344C59#2558211yescharacteristic, direct product factor, Sylow
345C22×Q8C23×C5912yes—
346C22×D4—21no—
347C22×D4—21no—
348C22×D4D4×C5911yescharacteristic
349C22×D4C23×C5911yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4C23×C5911yescharacteristic
353C23×C4C23×C5911yescharacteristic
354C2×SD16—24no—
355C2×SD16—42no—
356C2×SD16—42no—
357C22×C8C23×C5912yes—
358C22×C8C23×C5912yes—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361#126—28no—
362#125—22no—
363#125D4×C5914yes—
364#123—28no—
365C22⋊Q8—24no—
366C22⋊Q8—24no—
367C22⋊Q8—42no—
368C22⋊Q8—42no—
369#121—28no—
370#121—22no—
371#121—22no—
372#121—22no—
373#121—22no—
374C4×D4—28no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4D4×C5914yes—
379#117—22no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4D4×C5911yescharacteristic
391C2×C4⋊C4C23×C5911yescharacteristic
392C2×C4⋊C4C23×C5911yescharacteristic
393C2×C4⋊C4C23×C5911yescharacteristic
394C2×C4⋊C4C23×C5912yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4D4×C5911yescharacteristic
402C2×C22⋊C4D4×C5911yescharacteristic
403C2×C42—21no—
404C2×C42—21no—
405C2×C42C23×C5911yescharacteristic
406C4⋊C8—22no—
407#103—24no—
408#103—24no—
409#103—24no—
410#102—24no—
411C22⋊C8—42no—
412C22⋊C8—42no—
413#97—22no—
414#95—21no—
415C24—21no—
416C24C2×D4×C5911yescharacteristic
417C24C2×D4×C5911yescharacteristic
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4C2×D4×C5912yes—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C2×D4×C5911yescharacteristic
454C22×C4C2×D4×C5911yescharacteristic
455C22×C4C2×D4×C5911yescharacteristic
456C22×C4C2×D4×C5911yescharacteristic
457C22×C4C24×C5911yescharacteristic
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4C2×D4×C5912yes—
478C4⋊C4C2×D4×C5912yes—
479C4⋊C4C2×D4×C5912yes—
480C4⋊C4#41×C5914yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C42—24no—
490C42—42no—
491C42C2×D4×C5912yes—
492C42C2×D4×C5912yes—
493C23—21no—
494C23—22no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23#137×C5912yes—
508C23C22×D4×C5911yescharacteristic
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513D4—22no—
514D4—22no—
515D4—44no—
516D4—48no—
517D4—42no—
518D4—84no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4#120×C5912yes—
548C2×C4#120×C5912yes—
549C2×C4C22×D4×C5911yescharacteristic
550C2×C4C22×D4×C5911yescharacteristic
551C2×C4C2×#41×C5912yes—
552C2×C4H(5,𝔽2)×C5912yes—
553C8—42no—
554C8—42no—
555C22—22no—
556C22—24no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—42no—
568C22—82no—
569C22—82no—
570C22C2×#120×C5911yescharacteristic
571C22C59×#54612yes—
572C22C2×#137×C5911yescharacteristic
573C22C59×#57511yescharacteristic
574C22C59×#57712yes—
575C4—22no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—41no—
582C4—41no—
583C4—82no—
584C4C59×#53211yescharacteristic
585C4C59×#53511yescharacteristic
586C4C59×#55012yes—
587C2—42no—
588C2—81no—
589C2—82no—
590C2C59×#243111yescharacteristic
591C2C59×#300711yescharacteristic
592C2C59×#300911yescharacteristic
593C2C59×#330712yes—
594C2C59×#332012yes—
subgroupG/H
center, socleC23×C59C22×D4
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC2×C59×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C59 ◁ C2×C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#491C2semidirect
3. C2×C59×#461C2semidirect
4. C59×#2542C2semidirect
5. C59×#2000C2semidirect
6. C59×#2000C2semidirect
7. C59×#1989C2semidirect
8. C59×#1955C2semidirect
9. C59×#1873C2semidirect
10. C59×#1872C2semidirect
11. C59×#1762C2semidirect
257. #25582C59direct (N × H)
291. C2×#491C2×C59semidirect
292. C2×#461C2×C59semidirect
293. #2542C2×C59semidirect
294. #2000C2×C59semidirect
295. #2000C2×C59semidirect
296. #1989C2×C59semidirect
297. #1955C2×C59semidirect
298. #1873C2×C59semidirect
299. #1872C2×C59semidirect
300. #1762C2×C59semidirect
344. C59#25582direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5532
C2×C59×#4912
C2×C59×#4612
C59×#25422
C59×#20002
C59×#19892
C59×#19552
C59×#18732
C59×#18722
C59×#17622
#2558259