Finite Group Databasebeta

← C59×#16932 #28429041

Subgroups — C59×#16932 #28429041

Order 15104.

1498 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#400C211yesmaximal, characteristic, complemented
2C2×C59×#394C211yesmaximal, characteristic
3C59×#1443C212yesmaximal, complemented
4C59×#1440C212yesmaximal, complemented
5C59×#1439C212yesmaximal, complemented
6C59×#1439C212yesmaximal, complemented
7C59×#1438C212yesmaximal
8C59×#1437C212yesmaximal, complemented
9C59×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C59C2211yescharacteristic
11C22×C4⋊C4×C59C2211yescharacteristic
12C22×C4⋊C4×C59C2211yescharacteristic
13C22×C4⋊C4×C59C2211yescharacteristic
14C22×C22⋊C4×C59C2212yes—
15C22×C22⋊C4×C59C2212yes—
16C22×C22⋊C4×C59C2212yes—
17C22×C22⋊C4×C59C2212yes—
18C22×C22⋊C4×C59C2212yes—
19C22×C22⋊C4×C59C2212yes—
20C22×C42×C59C2211yescharacteristic
21C59×#400—24no—
22C59×#394—24no—
23C2×#95×C59C2211yescharacteristic
24C2×#95×C59C2212yes—
25C2×#95×C59C2212yes—
26C2×#95×C59C2211yescharacteristic
27C2×#95×C59C2212yes—
28C2×#95×C59C2212yes—
29C2×#95×C59C2212yes—
30C2×#95×C59C2212yes—
31C2×#95×C59C2212yes—
32C2×#95×C59C2212yes—
33C25×C59C2312yes—
34C22×D4×C59—24no—
35C22×D4×C59—22no—
36C22×D4×C59—42no—
37C22×D4×C59—41no—
38C22×D4×C59—41no—
39C23×C4×C59C2312yes—
40C23×C4×C59C2312yes—
41C23×C4×C59C2312yes—
42C23×C4×C59C2312yes—
43C23×C4×C59C2311yescharacteristic
44C23×C4×C59C2311yescharacteristic
45C23×C4×C59C2312yes—
46C23×C4×C59C2311yescharacteristic
47C2×C4⋊C4×C59—22no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59—41no—
51C2×C4⋊C4×C59—41no—
52C2×C4⋊C4×C59—41no—
53C2×C4⋊C4×C59—41no—
54C2×C4⋊C4×C59—41no—
55C2×C4⋊C4×C59D414yes—
56C2×C22⋊C4×C59—24no—
57C2×C22⋊C4×C59—24no—
58C2×C22⋊C4×C59—24no—
59C2×C22⋊C4×C59—24no—
60C2×C22⋊C4×C59—24no—
61C2×C22⋊C4×C59—24no—
62C2×C22⋊C4×C59—24no—
63C2×C22⋊C4×C59—24no—
64C2×C22⋊C4×C59—42no—
65C2×C22⋊C4×C59—42no—
66C2×C22⋊C4×C59—42no—
67C2×C22⋊C4×C59—42no—
68C2×C22⋊C4×C59—42no—
69C2×C22⋊C4×C59—42no—
70C2×C22⋊C4×C59—42no—
71C2×C22⋊C4×C59—42no—
72C2×C22⋊C4×C59—42no—
73C2×C22⋊C4×C59—42no—
74C2×C42×C59—22no—
75C2×C42×C59—41no—
76C2×C42×C59—41no—
77#95×C59—22no—
78#95×C59—22no—
79#95×C59—24no—
80#95×C59—24no—
81#95×C59—24no—
82#95×C59—42no—
83#95×C59—42no—
84#95×C59—42no—
85#95×C59—42no—
86#95×C59—42no—
87C24×C59—22no—
88C24×C59—22no—
89C24×C59—22no—
90C24×C59—22no—
91C24×C59—22no—
92C24×C59—22no—
93C24×C59—22no—
94C24×C59—22no—
95C24×C59—22no—
96C24×C59—22no—
97C24×C59—22no—
98C24×C59—22no—
99C24×C59—22no—
100C24×C59—22no—
101C24×C59#4114yes—
102C24×C59C2411yescharacteristic
103C2×D4×C59—48no—
104C2×D4×C59—44no—
105C2×D4×C59—44no—
106C2×D4×C59—48no—
107C2×D4×C59—44no—
108C22×C4×C59—21no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—21no—
112C22×C4×C59—22no—
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—21no—
125C22×C4×C59—21no—
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—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
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—22no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—22no—
151C22×C4×C59—22no—
152C22×C4×C59—22no—
153C22×C4×C59—22no—
154C22×C4×C59C2×D412yes—
155C22×C4×C59C2×D412yes—
156C22×C4×C59C2×D412yes—
157C22×C4×C59#4114yes—
158C22×C4×C59#4112yes—
159C22×C4×C59#4112yes—
160C4⋊C4×C59—44no—
161C4⋊C4×C59—44no—
162C4⋊C4×C59—44no—
163C22⋊C4×C59—48no—
164C22⋊C4×C59—48no—
165C22⋊C4×C59—48no—
166C22⋊C4×C59—48no—
167C22⋊C4×C59—48no—
168C22⋊C4×C59—48no—
169C42×C59—44no—
170C23×C59—24no—
171C23×C59—24no—
172C23×C59—24no—
173C23×C59—24no—
174C23×C59—24no—
175C23×C59—24no—
176C23×C59—24no—
177C23×C59—42no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59—42no—
183C23×C59—42no—
184C23×C59—42no—
185C23×C59—42no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59—42no—
189C23×C59—42no—
190C23×C59—42no—
191C23×C59—42no—
192C23×C59—42no—
193C23×C59—42no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—42no—
204C23×C59—42no—
205C23×C59C22×D411yescharacteristic
206C23×C59C2×#4111yescharacteristic
207C23×C59C2×#4111yescharacteristic
208C23×C59C2×#4112yes—
209C23×C59C2×#4112yes—
210C23×C59H(5,𝔽2)11yescharacteristic
211C23×C59H(5,𝔽2)11yescharacteristic
212C23×C59H(5,𝔽2)11yescharacteristic
213C23×C59H(5,𝔽2)12yes—
214C23×C59H(5,𝔽2)12yes—
215C23×C592−1+411yescharacteristic
216D4×C59—416no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
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—24no—
226C2×C4×C59—24no—
227C2×C4×C59—24no—
228C2×C4×C59—24no—
229C2×C4×C59—24no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—42no—
237C2×C4×C59—41no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
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—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—42no—
259C2×C4×C59—42no—
260C2×C4×C59—42no—
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#12414yes—
269#16932C5911yesmaximal, characteristic, direct product factor, Sylow
270C22×C59—44no—
271C22×C59—44no—
272C22×C59—44no—
273C22×C59—44no—
274C22×C59—44no—
275C22×C59—44no—
276C22×C59—44no—
277C22×C59—82no—
278C22×C59—82no—
279C22×C59—82no—
280C22×C59—82no—
281C22×C59—82no—
282C22×C59—82no—
283C22×C59—82no—
284C22×C59—82no—
285C22×C59C2×#12411yescharacteristic
286C22×C59#52912yes—
287C22×C59#53411yescharacteristic
288C22×C59#53711yescharacteristic
289C22×C59#53811yescharacteristic
290C22×C59#53811yescharacteristic
291C22×C59#53812yes—
292C22×C59#53812yes—
293C22×C59#53911yescharacteristic
294C22×C59#53911yescharacteristic
295C22×C59#53912yes—
296C22×C59#54012yes—
297C22×C59#54212yes—
298C22×C59#54612yes—
299C22×C59#55012yes—
300C22×C59#55111yescharacteristic
301C22×C59#55112yes—
302C22×C59#55312yes—
303C22×C59#55312yes—
304C22×C59#55511yescharacteristic
305C22×C59#56011yescharacteristic
306C22×C59#56012yes—
307C22×C59H(3,𝔽4)11yescharacteristic
308C4×C59—24no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59—44no—
312C4×C59—82no—
313C4×C59—82no—
314C4×C59—82no—
315C4×C59—82no—
316C2×#400C2×C5911yescharacteristic, complemented
317C2×#394C2×C5911yescharacteristic
318#1443C2×C5912yescomplemented
319#1440C2×C5912yescomplemented
320#1439C2×C5912yescomplemented
321#1439C2×C5912yescomplemented
322#1438C2×C5912yes—
323#1437C2×C5912yescomplemented
324C4⋊#95C2×C5911yescharacteristic, complemented
325C2×C59—84no—
326C2×C59#249012yes—
327C2×C59#257211yescharacteristic
328C2×C59#263911yescharacteristic
329C2×C59#264912yes—
330C2×C59#269711yescharacteristic
331C2×C59#270512yes—
332C2×C59#273311yescharacteristic
333C2×C59#277812yes—
334C2×C59#278211yescharacteristic
335C2×C59#279511yescharacteristic
336C2×C59#282811yescharacteristic
337C23×D4C22×C5911yescharacteristic
338C22×C4⋊C4C22×C5911yescharacteristic
339C22×C4⋊C4C22×C5911yescharacteristic
340C22×C4⋊C4C22×C5911yescharacteristic
341C22×C22⋊C4C22×C5912yes—
342C22×C22⋊C4C22×C5912yes—
343C22×C22⋊C4C22×C5912yes—
344C22×C22⋊C4C22×C5912yes—
345C22×C22⋊C4C22×C5912yes—
346C22×C22⋊C4C22×C5912yes—
347C22×C42C22×C5911yescharacteristic
348#400—24no—
349#394—24no—
350C2×#95C22×C5911yescharacteristic
351C2×#95C22×C5912yes—
352C2×#95C22×C5912yes—
353C2×#95C22×C5911yescharacteristic
354C2×#95C22×C5912yes—
355C2×#95C22×C5912yes—
356C2×#95C22×C5912yes—
357C2×#95C22×C5912yes—
358C2×#95C22×C5912yes—
359C2×#95C22×C5912yes—
360C59#1693211yescharacteristic, direct product factor, Sylow
361C25C23×C5912yes—
362C22×D4—24no—
363C22×D4—22no—
364C22×D4—42no—
365C22×D4—41no—
366C22×D4—41no—
367C23×C4C23×C5912yes—
368C23×C4C23×C5912yes—
369C23×C4C23×C5912yes—
370C23×C4C23×C5912yes—
371C23×C4C23×C5911yescharacteristic
372C23×C4C23×C5911yescharacteristic
373C23×C4C23×C5912yes—
374C23×C4C23×C5911yescharacteristic
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—41no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4D4×C5914yes—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—42no—
393C2×C22⋊C4—42no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—42no—
402C2×C42—22no—
403C2×C42—41no—
404C2×C42—41no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—42no—
411#95—42no—
412#95—42no—
413#95—42no—
414#95—42no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24#41×C5914yes—
430C24C24×C5911yescharacteristic
431C2×D4—48no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—44no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
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—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4C2×D4×C5912yes—
483C22×C4C2×D4×C5912yes—
484C22×C4C2×D4×C5912yes—
485C22×C4#41×C5914yes—
486C22×C4#41×C5912yes—
487C22×C4#41×C5912yes—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C42—44no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23C22×D4×C5911yescharacteristic
534C23C2×#41×C5911yescharacteristic
535C23C2×#41×C5911yescharacteristic
536C23C2×#41×C5912yes—
537C23C2×#41×C5912yes—
538C23H(5,𝔽2)×C5911yescharacteristic
539C23H(5,𝔽2)×C5911yescharacteristic
540C23H(5,𝔽2)×C5911yescharacteristic
541C23H(5,𝔽2)×C5912yes—
542C23H(5,𝔽2)×C5912yes—
543C232−1+4×C5911yescharacteristic
544D4—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
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—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4#124×C5914yes—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22C2×#124×C5911yescharacteristic
613C22C59×#52912yes—
614C22C59×#53411yescharacteristic
615C22C59×#53711yescharacteristic
616C22C59×#53811yescharacteristic
617C22C59×#53812yes—
618C22C59×#53812yes—
619C22C59×#53811yescharacteristic
620C22C59×#53911yescharacteristic
621C22C59×#53912yes—
622C22C59×#53911yescharacteristic
623C22C59×#54012yes—
624C22C59×#54212yes—
625C22C59×#54612yes—
626C22C59×#55012yes—
627C22C59×#55112yes—
628C22C59×#55111yescharacteristic
629C22C59×#55312yes—
630C22C59×#55312yes—
631C22C59×#55511yescharacteristic
632C22C59×#56012yes—
633C22C59×#56011yescharacteristic
634C22C59×H(3,𝔽4)11yescharacteristic
635C4—24no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C2—84no—
644C2C59×#249012yes—
645C2C59×#257211yescharacteristic
646C2C59×#263911yescharacteristic
647C2C59×#264912yes—
648C2C59×#269711yescharacteristic
649C2C59×#270512yes—
650C2C59×#273311yescharacteristic
651C2C59×#277812yes—
652C2C59×#278211yescharacteristic
653C2C59×#279511yescharacteristic
654C2C59×#282811yescharacteristic
subgroupG/H
center, socleC24×C59C24
commutator, Frattini, cosocleC24C24×C59
prime radicalC23×D4×C59C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C59×#400C2semidirect
3. C59×#1443C2semidirect
4. C59×#1440C2semidirect
5. C59×#1439C2semidirect
6. C59×#1439C2semidirect
8. C59×#1437C2semidirect
9. C59×C4⋊#95C2semidirect
269. #16932C59direct (N × H)
316. C2×#400C2×C59semidirect
318. #1443C2×C59semidirect
319. #1440C2×C59semidirect
320. #1439C2×C59semidirect
321. #1439C2×C59semidirect
323. #1437C2×C59semidirect
324. C4⋊#95C2×C59semidirect
360. C59#16932direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#4002
C2×C59×#3942
C59×#14432
C59×#14402
C59×#14392
C59×#14382
C59×#14372
C59×C4⋊#952
#1693259