Finite Group Databasebeta

← C59×#16364 #28428473

Subgroups — C59×#16364 #28428473

Order 15104.

1706 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8×C59C211yesmaximal, characteristic
2C2×C59×#397C212yesmaximal
3C2×C59×#381C211yesmaximal, characteristic
4C59×#1446C211yesmaximal, characteristic, complemented
5C59×#1446C211yesmaximal, characteristic, complemented
6C59×#1444C211yesmaximal, characteristic, complemented
7C59×#1442C212yesmaximal, complemented
8C59×#1442C211yesmaximal, characteristic, complemented
9C59×#1441C212yesmaximal, complemented
10C59×#1440C212yesmaximal
11C59×C22⋊#95C211yesmaximal, characteristic
12C23×Q8×C59C2211yescharacteristic
13C24×C4×C59C2211yescharacteristic
14C2×C22⋊Q8×C59—24no—
15C2×C22⋊Q8×C59—28no—
16C22×C4⋊C4×C59C2212yes—
17C22×C4⋊C4×C59C2211yescharacteristic
18C22×C4⋊C4×C59C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C22×C22⋊C4×C59C2212yes—
21C22×C22⋊C4×C59C2212yes—
22C22×C22⋊C4×C59C2211yescharacteristic
23C59×#397—28no—
24C59×#381—24no—
25C2×#95×C59C2211yescharacteristic
26C2×#95×C59C2211yescharacteristic
27C2×#95×C59C2211yescharacteristic
28C2×#95×C59C2211yescharacteristic
29C2×#95×C59C2212yes—
30C2×#95×C59C2212yes—
31C2×#95×C59C2212yes—
32C2×#95×C59C2211yescharacteristic
33C2×#95×C59C2211yescharacteristic
34C2×#95×C59C2212yes—
35C2×#95×C59C2212yes—
36C2×#95×C59C2212yes—
37C2×#95×C59C2212yes—
38C2×#95×C59C2212yes—
39C25×C59C2311yescharacteristic
40C22×Q8×C59—24no—
41C22×Q8×C59—22no—
42C22×Q8×C59—22no—
43C22×Q8×C59—22no—
44C22×Q8×C59—42no—
45C23×C4×C59—24no—
46C23×C4×C59—22no—
47C23×C4×C59—22no—
48C23×C4×C59—42no—
49C23×C4×C59D414yes—
50C23×C4×C59C2311yescharacteristic
51C23×C4×C59C2311yescharacteristic
52C23×C4×C59C2312yes—
53C23×C4×C59C2311yescharacteristic
54C23×C4×C59C2311yescharacteristic
55C23×C4×C59C2312yes—
56C23×C4×C59C2311yescharacteristic
57C23×C4×C59C2312yes—
58C23×C4×C59C2312yes—
59C23×C4×C59C2311yescharacteristic
60C22⋊Q8×C59—232no—
61C2×C4⋊C4×C59—24no—
62C2×C4⋊C4×C59—22no—
63C2×C4⋊C4×C59—24no—
64C2×C4⋊C4×C59—24no—
65C2×C4⋊C4×C59—24no—
66C2×C4⋊C4×C59—24no—
67C2×C4⋊C4×C59—24no—
68C2×C4⋊C4×C59—42no—
69C2×C22⋊C4×C59—24no—
70C2×C22⋊C4×C59—24no—
71C2×C22⋊C4×C59—22no—
72C2×C22⋊C4×C59—22no—
73C2×C22⋊C4×C59—24no—
74C2×C22⋊C4×C59—24no—
75C2×C22⋊C4×C59—24no—
76C2×C22⋊C4×C59—42no—
77C2×C22⋊C4×C59—42no—
78C2×C22⋊C4×C59D418yes—
79#95×C59—22no—
80#95×C59—22no—
81#95×C59—22no—
82#95×C59—22no—
83#95×C59—24no—
84#95×C59—24no—
85#95×C59—24no—
86#95×C59—22no—
87#95×C59—24no—
88#95×C59—22no—
89#95×C59—42no—
90#95×C59—42no—
91#95×C59—42no—
92#95×C59—42no—
93C24×C59—22no—
94C24×C59—21no—
95C24×C59—21no—
96C24×C59—21no—
97C24×C59—22no—
98C24×C59—21no—
99C24×C59—21no—
100C24×C59—22no—
101C24×C59—21no—
102C24×C59C2×D412yes—
103C24×C59C2×D414yes—
104C24×C59C2411yescharacteristic
105C2×Q8×C59—28no—
106C2×Q8×C59—48no—
107C2×Q8×C59—48no—
108C2×Q8×C59—44no—
109C2×Q8×C59—44no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—28no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—28no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—21no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—22no—
151C22×C4×C59—22no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—22no—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—22no—
159C22×C4×C59—22no—
160C22×C4×C59—22no—
161C22×C4×C59—21no—
162C22×C4×C59—21no—
163C22×C4×C59—44no—
164C22×C4×C59—48no—
165C22×C4×C59—48no—
166C22×C4×C59C2×D414yes—
167C22×C4×C59C2×D412yes—
168C22×C4×C59C2×D414yes—
169C22×C4×C59C2×D412yes—
170C22×C4×C59#4114yes—
171C22×C4×C59#4112yes—
172C22×C4×C59#4112yes—
173C4⋊C4×C59—28no—
174C4⋊C4×C59—216no—
175C4⋊C4×C59—48no—
176C22⋊C4×C59—216no—
177C22⋊C4×C59—48no—
178C22⋊C4×C59—44no—
179C22⋊C4×C59—44no—
180C23×C59—22no—
181C23×C59—22no—
182C23×C59—22no—
183C23×C59—22no—
184C23×C59—24no—
185C23×C59—24no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59—24no—
189C23×C59—24no—
190C23×C59—24no—
191C23×C59—24no—
192C23×C59—42no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59#12014yes—
203C23×C59C22×D411yescharacteristic
204C23×C59C22×D412yes—
205C23×C59C2×#4111yescharacteristic
206C23×C59C2×#4111yescharacteristic
207C23×C59C2×#4112yes—
208C23×C59H(5,𝔽2)12yes—
209C23×C59H(5,𝔽2)11yescharacteristic
210C23×C592−1+411yescharacteristic
211C23×C592−1+411yescharacteristic
212C23×C592−1+412yes—
213C23×C592−1+411yescharacteristic
214Q8×C59—416no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—24no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—24no—
225C2×C4×C59—24no—
226C2×C4×C59—24no—
227C2×C4×C59—24no—
228C2×C4×C59—24no—
229C2×C4×C59—24no—
230C2×C4×C59—24no—
231C2×C4×C59—416no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—42no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—42no—
262C2×C4×C59—42no—
263C2×C4×C59—42no—
264C2×C4×C59—42no—
265C2×C4×C59—42no—
266C2×C4×C59—42no—
267C2×C4×C59—42no—
268C2×C4×C59—42no—
269C2×C4×C59—42no—
270C2×C4×C59—42no—
271C2×C4×C59—42no—
272C2×C4×C59—42no—
273C2×C4×C59#12314yes—
274#16364C5911yesmaximal, characteristic, direct product factor, Sylow
275C22×C59—24no—
276C22×C59—24no—
277C22×C59—24no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—44no—
281C22×C59—44no—
282C22×C59—44no—
283C22×C59—44no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59C2×#12011yescharacteristic
287C22×C59C2×#12311yescharacteristic
288C22×C59#53211yescharacteristic
289C22×C59#53411yescharacteristic
290C22×C59#53611yescharacteristic
291C22×C59#53612yes—
292C22×C59#53612yes—
293C22×C59#53611yescharacteristic
294C22×C59#53711yescharacteristic
295C22×C59#53912yes—
296C22×C59#54111yescharacteristic
297C22×C59#54111yescharacteristic
298C22×C59#54211yescharacteristic
299C22×C59#54211yescharacteristic
300C22×C59#54212yes—
301C22×C59#54612yes—
302C22×C59#54612yes—
303C22×C59#54612yes—
304C22×C59#54712yes—
305C22×C59#55212yes—
306C22×C59#55612yes—
307C22×C59#56212yes—
308C22×C59#56312yes—
309C4×C59—24no—
310C4×C59—42no—
311C4×C59—44no—
312C4×C59—44no—
313C4×C59—44no—
314C4×C59—81no—
315C4×C59—81no—
316C4×C59—81no—
317C4×C59—82no—
318C4×C59—81no—
319C22×C22⋊Q8C2×C5911yescharacteristic
320C2×#397C2×C5912yes—
321C2×#381C2×C5911yescharacteristic
322#1446C2×C5911yescharacteristic, complemented
323#1446C2×C5911yescharacteristic, complemented
324#1444C2×C5911yescharacteristic, complemented
325#1442C2×C5912yescomplemented
326#1442C2×C5911yescharacteristic, complemented
327#1441C2×C5912yescomplemented
328#1440C2×C5912yes—
329C22⋊#95C2×C5911yescharacteristic
330C2×C59—44no—
331C2×C59#240511yescharacteristic
332C2×C59#243011yescharacteristic
333C2×C59#249911yescharacteristic
334C2×C59#261411yescharacteristic
335C2×C59#262511yescharacteristic
336C2×C59#266311yescharacteristic
337C2×C59#268712yes—
338C2×C59#268912yes—
339C2×C59#269812yes—
340C2×C59#270012yes—
341C2×C59#281411yescharacteristic
342C23×Q8C22×C5911yescharacteristic
343C24×C4C22×C5911yescharacteristic
344C2×C22⋊Q8—24no—
345C2×C22⋊Q8—28no—
346C22×C4⋊C4C22×C5912yes—
347C22×C4⋊C4C22×C5911yescharacteristic
348C22×C4⋊C4C22×C5912yes—
349C22×C22⋊C4C22×C5911yescharacteristic
350C22×C22⋊C4C22×C5912yes—
351C22×C22⋊C4C22×C5912yes—
352C22×C22⋊C4C22×C5911yescharacteristic
353#397—28no—
354#381—24no—
355C2×#95C22×C5912yes—
356C2×#95C22×C5912yes—
357C2×#95C22×C5912yes—
358C2×#95C22×C5911yescharacteristic
359C2×#95C22×C5911yescharacteristic
360C2×#95C22×C5912yes—
361C2×#95C22×C5911yescharacteristic
362C2×#95C22×C5911yescharacteristic
363C2×#95C22×C5911yescharacteristic
364C2×#95C22×C5911yescharacteristic
365C2×#95C22×C5912yes—
366C2×#95C22×C5912yes—
367C2×#95C22×C5912yes—
368C2×#95C22×C5912yes—
369C59#1636411yescharacteristic, direct product factor, Sylow
370C25C23×C5911yescharacteristic
371C22×Q8—24no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—42no—
376C23×C4—24no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—42no—
380C23×C4D4×C5914yes—
381C23×C4C23×C5911yescharacteristic
382C23×C4C23×C5911yescharacteristic
383C23×C4C23×C5911yescharacteristic
384C23×C4C23×C5912yes—
385C23×C4C23×C5911yescharacteristic
386C23×C4C23×C5912yes—
387C23×C4C23×C5912yes—
388C23×C4C23×C5911yescharacteristic
389C23×C4C23×C5912yes—
390C23×C4C23×C5911yescharacteristic
391C22⋊Q8—232no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—42no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—24no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4D4×C5918yes—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—42no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24C2×D4×C5912yes—
434C24C2×D4×C5914yes—
435C24C24×C5911yescharacteristic
436C2×Q8—28no—
437C2×Q8—48no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4C2×D4×C5914yes—
498C22×C4C2×D4×C5912yes—
499C22×C4C2×D4×C5914yes—
500C22×C4C2×D4×C5912yes—
501C22×C4#41×C5914yes—
502C22×C4#41×C5912yes—
503C22×C4#41×C5912yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—48no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#120×C5914yes—
534C23C22×D4×C5911yescharacteristic
535C23C22×D4×C5912yes—
536C23C2×#41×C5911yescharacteristic
537C23C2×#41×C5911yescharacteristic
538C23C2×#41×C5912yes—
539C23H(5,𝔽2)×C5912yes—
540C23H(5,𝔽2)×C5911yescharacteristic
541C232−1+4×C5911yescharacteristic
542C232−1+4×C5911yescharacteristic
543C232−1+4×C5912yes—
544C232−1+4×C5911yescharacteristic
545Q8—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4#123×C5914yes—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22C2×#120×C5911yescharacteristic
617C22C2×#123×C5911yescharacteristic
618C22C59×#53211yescharacteristic
619C22C59×#53411yescharacteristic
620C22C59×#53611yescharacteristic
621C22C59×#53611yescharacteristic
622C22C59×#53612yes—
623C22C59×#53612yes—
624C22C59×#53711yescharacteristic
625C22C59×#53912yes—
626C22C59×#54111yescharacteristic
627C22C59×#54111yescharacteristic
628C22C59×#54211yescharacteristic
629C22C59×#54211yescharacteristic
630C22C59×#54212yes—
631C22C59×#54612yes—
632C22C59×#54612yes—
633C22C59×#54612yes—
634C22C59×#54712yes—
635C22C59×#55212yes—
636C22C59×#55612yes—
637C22C59×#56212yes—
638C22C59×#56312yes—
639C4—24no—
640C4—42no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—82no—
649C2—44no—
650C2C59×#240511yescharacteristic
651C2C59×#243011yescharacteristic
652C2C59×#249911yescharacteristic
653C2C59×#261411yescharacteristic
654C2C59×#262511yescharacteristic
655C2C59×#266311yescharacteristic
656C2C59×#268712yes—
657C2C59×#268912yes—
658C2C59×#269812yes—
659C2C59×#270012yes—
660C2C59×#281411yescharacteristic
subgroupG/H
center, socleC24×C59C24
commutator, Frattini, cosocleC24C24×C59
prime radicalC25×C59C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C59 ◁ G

Complements

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

NH ≅ G/Ntype
4. C59×#1446C2semidirect
5. C59×#1446C2semidirect
6. C59×#1444C2semidirect
7. C59×#1442C2semidirect
8. C59×#1442C2semidirect
9. C59×#1441C2semidirect
274. #16364C59direct (N × H)
322. #1446C2×C59semidirect
323. #1446C2×C59semidirect
324. #1444C2×C59semidirect
325. #1442C2×C59semidirect
326. #1442C2×C59semidirect
327. #1441C2×C59semidirect
369. C59#16364direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C592
C2×C59×#3972
C2×C59×#3812
C59×#14462
C59×#14442
C59×#14422
C59×#14412
C59×#14402
C59×C22⋊#952
#1636459