Finite Group Databasebeta

← C59×#16675 #28428784

Subgroups — C59×#16675 #28428784

Order 15104.

1454 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#397C211yesmaximal, characteristic
2C2×C59×#385C212yesmaximal
3C2×C59×#381C211yesmaximal, characteristic
4C59×#1444C212yesmaximal, complemented
5C59×#1443C212yesmaximal, complemented
6C59×#1442C212yesmaximal, complemented
7C59×#1441C212yesmaximal, complemented
8C59×#1440C212yesmaximal
9C59×#1439C211yesmaximal, characteristic
10C24×C4×C59C2211yescharacteristic
11C22×C4⋊C4×C59C2211yescharacteristic
12C22×C4⋊C4×C59C2212yes—
13C22×C4⋊C4×C59C2212yes—
14C22×C4⋊C4×C59C2211yescharacteristic
15C22×C22⋊C4×C59C2212yes—
16C22×C22⋊C4×C59C2212yes—
17C22×C22⋊C4×C59C2211yescharacteristic
18C22×C22⋊C4×C59C2211yescharacteristic
19C59×#397—24no—
20C59×#385—28no—
21C59×#381—24no—
22C2×#95×C59C2212yes—
23C2×#95×C59C2212yes—
24C2×#95×C59C2212yes—
25C2×#95×C59C2212yes—
26C2×#95×C59C2212yes—
27C2×#95×C59C2211yescharacteristic
28C2×#95×C59C2212yes—
29C2×#95×C59C2212yes—
30C2×#95×C59C2212yes—
31C2×#95×C59C2211yescharacteristic
32C2×#95×C59C2212yes—
33C2×#95×C59C2212yes—
34C25×C59C2311yescharacteristic
35C23×C4×C59—22no—
36C23×C4×C59—22no—
37C23×C4×C59—41no—
38C23×C4×C59—41no—
39C23×C4×C59—41no—
40C23×C4×C59—41no—
41C23×C4×C59D414yes—
42C23×C4×C59C2312yes—
43C23×C4×C59C2312yes—
44C23×C4×C59C2312yes—
45C23×C4×C59C2311yescharacteristic
46C23×C4×C59C2311yescharacteristic
47C23×C4×C59C2312yes—
48C23×C4×C59C2312yes—
49C23×C4×C59C2312yes—
50C2×C4⋊C4×C59—24no—
51C2×C4⋊C4×C59—24no—
52C2×C4⋊C4×C59—22no—
53C2×C4⋊C4×C59—24no—
54C2×C4⋊C4×C59—41no—
55C2×C4⋊C4×C59—42no—
56C2×C4⋊C4×C59—42no—
57C2×C4⋊C4×C59—42no—
58C2×C4⋊C4×C59—41no—
59C2×C4⋊C4×C59—41no—
60C2×C4⋊C4×C59—42no—
61C2×C22⋊C4×C59—24no—
62C2×C22⋊C4×C59—24no—
63C2×C22⋊C4×C59—24no—
64C2×C22⋊C4×C59—22no—
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—24no—
67C2×C22⋊C4×C59—24no—
68C2×C22⋊C4×C59—41no—
69C2×C22⋊C4×C59—41no—
70C2×C22⋊C4×C59—41no—
71C2×C22⋊C4×C59—41no—
72C2×C22⋊C4×C59D414yes—
73#95×C59—24no—
74#95×C59—24no—
75#95×C59—24no—
76#95×C59—24no—
77#95×C59—22no—
78#95×C59—24no—
79#95×C59—24no—
80#95×C59—24no—
81#95×C59—24no—
82#95×C59—41no—
83#95×C59—42no—
84#95×C59—42no—
85C24×C59—22no—
86C24×C59—22no—
87C24×C59—21no—
88C24×C59—21no—
89C24×C59—21no—
90C24×C59—21no—
91C24×C59—22no—
92C24×C59—22no—
93C24×C59C2×D412yes—
94C24×C59C2×D412yes—
95C24×C59#4112yes—
96C24×C59C2411yescharacteristic
97C22×C4×C59—28no—
98C22×C4×C59—22no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—21no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
114C22×C4×C59—21no—
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—22no—
123C22×C4×C59—22no—
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—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—44no—
146C22×C4×C59—44no—
147C22×C4×C59—44no—
148C22×C4×C59—44no—
149C22×C4×C59—44no—
150C22×C4×C59—44no—
151C22×C4×C59C2×D412yes—
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D414yes—
154C22×C4×C59#4114yes—
155C22×C4×C59#4114yes—
156C4⋊C4×C59—44no—
157C4⋊C4×C59—44no—
158C4⋊C4×C59—48no—
159C4⋊C4×C59—48no—
160C22⋊C4×C59—216no—
161C22⋊C4×C59—44no—
162C22⋊C4×C59—44no—
163C22⋊C4×C59—48no—
164C23×C59—24no—
165C23×C59—22no—
166C23×C59—22no—
167C23×C59—22no—
168C23×C59—22no—
169C23×C59—24no—
170C23×C59—22no—
171C23×C59—22no—
172C23×C59—24no—
173C23×C59—24no—
174C23×C59—24no—
175C23×C59—24no—
176C23×C59—42no—
177C23×C59—42no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—41no—
181C23×C59—42no—
182C23×C59—41no—
183C23×C59—41no—
184C23×C59—42no—
185C23×C59—41no—
186C23×C59#12114yes—
187C23×C59C22×D411yescharacteristic
188C23×C59C22×D411yescharacteristic
189C23×C59C2×#4111yescharacteristic
190C23×C59C2×#4112yes—
191C23×C59C2×#4112yes—
192C23×C59H(5,𝔽2)11yescharacteristic
193C23×C59H(5,𝔽2)11yescharacteristic
194C23×C59H(5,𝔽2)11yescharacteristic
195C23×C59H(5,𝔽2)11yescharacteristic
196C23×C592−1+412yes—
197C23×C592−1+412yes—
198C2×C4×C59—24no—
199C2×C4×C59—24no—
200C2×C4×C59—24no—
201C2×C4×C59—22no—
202C2×C4×C59—22no—
203C2×C4×C59—24no—
204C2×C4×C59—24no—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—24no—
208C2×C4×C59—24no—
209C2×C4×C59—24no—
210C2×C4×C59—24no—
211C2×C4×C59—416no—
212C2×C4×C59—42no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
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—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251#16675C5911yesmaximal, characteristic, direct product factor, Sylow
252C22×C59—24no—
253C22×C59—28no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—44no—
257C22×C59—42no—
258C22×C59—44no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—44no—
262C22×C59—42no—
263C22×C59C2×#12111yescharacteristic
264C22×C59#52512yes—
265C22×C59#53411yescharacteristic
266C22×C59#53411yescharacteristic
267C22×C59#53411yescharacteristic
268C22×C59#53511yescharacteristic
269C22×C59#53612yes—
270C22×C59#53811yescharacteristic
271C22×C59#53811yescharacteristic
272C22×C59#53912yes—
273C22×C59#53912yes—
274C22×C59#54212yes—
275C22×C59#54212yes—
276C22×C59#54712yes—
277C22×C59#54712yes—
278C22×C59#55111yescharacteristic
279C22×C59#55212yes—
280C22×C59#55212yes—
281C22×C59#55312yes—
282C22×C59#55911yescharacteristic
283C22×C59#56311yescharacteristic
284C22×C59#56312yes—
285C22×C59#56311yescharacteristic
286C4×C59—44no—
287C4×C59—44no—
288C4×C59—44no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—82no—
292C4×C59—82no—
293C4×C59—82no—
294C2×#397C2×C5911yescharacteristic
295C2×#385C2×C5912yes—
296C2×#381C2×C5911yescharacteristic
297#1444C2×C5912yescomplemented
298#1443C2×C5912yescomplemented
299#1442C2×C5912yescomplemented
300#1441C2×C5912yescomplemented
301#1440C2×C5912yes—
302#1439C2×C5911yescharacteristic
303C2×C59—44no—
304C2×C59#241312yes—
305C2×C59#260011yescharacteristic
306C2×C59#261711yescharacteristic
307C2×C59#263812yes—
308C2×C59#271711yescharacteristic
309C2×C59#271812yes—
310C2×C59#272411yescharacteristic
311C2×C59#276411yescharacteristic
312C2×C59#276512yes—
313C2×C59#277111yescharacteristic
314C2×C59#281711yescharacteristic
315C24×C4C22×C5911yescharacteristic
316C22×C4⋊C4C22×C5911yescharacteristic
317C22×C4⋊C4C22×C5911yescharacteristic
318C22×C4⋊C4C22×C5912yes—
319C22×C4⋊C4C22×C5912yes—
320C22×C22⋊C4C22×C5912yes—
321C22×C22⋊C4C22×C5912yes—
322C22×C22⋊C4C22×C5911yescharacteristic
323C22×C22⋊C4C22×C5911yescharacteristic
324#397—24no—
325#385—28no—
326#381—24no—
327C2×#95C22×C5912yes—
328C2×#95C22×C5911yescharacteristic
329C2×#95C22×C5912yes—
330C2×#95C22×C5912yes—
331C2×#95C22×C5912yes—
332C2×#95C22×C5912yes—
333C2×#95C22×C5912yes—
334C2×#95C22×C5912yes—
335C2×#95C22×C5912yes—
336C2×#95C22×C5911yescharacteristic
337C2×#95C22×C5912yes—
338C2×#95C22×C5912yes—
339C59#1667511yescharacteristic, direct product factor, Sylow
340C25C23×C5911yescharacteristic
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—41no—
344C23×C4—41no—
345C23×C4—41no—
346C23×C4—41no—
347C23×C4D4×C5914yes—
348C23×C4C23×C5912yes—
349C23×C4C23×C5912yes—
350C23×C4C23×C5912yes—
351C23×C4C23×C5911yescharacteristic
352C23×C4C23×C5911yescharacteristic
353C23×C4C23×C5912yes—
354C23×C4C23×C5912yes—
355C23×C4C23×C5912yes—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—42no—
362C2×C4⋊C4—42no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—42no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4D4×C5914yes—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—41no—
389#95—42no—
390#95—42no—
391C24—22no—
392C24—22no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—22no—
398C24—22no—
399C24C2×D4×C5912yes—
400C24C2×D4×C5912yes—
401C24#41×C5912yes—
402C24C24×C5911yescharacteristic
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
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—22no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4C2×D4×C5912yes—
458C22×C4C2×D4×C5912yes—
459C22×C4C2×D4×C5914yes—
460C22×C4#41×C5914yes—
461C22×C4#41×C5914yes—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C22⋊C4—216no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23#121×C5914yes—
493C23C22×D4×C5911yescharacteristic
494C23C22×D4×C5911yescharacteristic
495C23C2×#41×C5911yescharacteristic
496C23C2×#41×C5912yes—
497C23C2×#41×C5912yes—
498C23H(5,𝔽2)×C5911yescharacteristic
499C23H(5,𝔽2)×C5911yescharacteristic
500C23H(5,𝔽2)×C5911yescharacteristic
501C23H(5,𝔽2)×C5911yescharacteristic
502C232−1+4×C5912yes—
503C232−1+4×C5912yes—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—416no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C22—24no—
558C22—28no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22C2×#121×C5911yescharacteristic
569C22C59×#52512yes—
570C22C59×#53411yescharacteristic
571C22C59×#53411yescharacteristic
572C22C59×#53411yescharacteristic
573C22C59×#53511yescharacteristic
574C22C59×#53612yes—
575C22C59×#53811yescharacteristic
576C22C59×#53811yescharacteristic
577C22C59×#53912yes—
578C22C59×#53912yes—
579C22C59×#54212yes—
580C22C59×#54212yes—
581C22C59×#54712yes—
582C22C59×#54712yes—
583C22C59×#55111yescharacteristic
584C22C59×#55212yes—
585C22C59×#55212yes—
586C22C59×#55312yes—
587C22C59×#55911yescharacteristic
588C22C59×#56311yescharacteristic
589C22C59×#56312yes—
590C22C59×#56311yescharacteristic
591C4—44no—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—42no—
596C4—82no—
597C4—82no—
598C4—82no—
599C2—44no—
600C2C59×#241312yes—
601C2C59×#260011yescharacteristic
602C2C59×#261711yescharacteristic
603C2C59×#263812yes—
604C2C59×#271711yescharacteristic
605C2C59×#271812yes—
606C2C59×#272411yescharacteristic
607C2C59×#276411yescharacteristic
608C2C59×#276512yes—
609C2C59×#277111yescharacteristic
610C2C59×#281711yescharacteristic
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×#1444C2semidirect
5. C59×#1443C2semidirect
6. C59×#1442C2semidirect
7. C59×#1441C2semidirect
251. #16675C59direct (N × H)
297. #1444C2×C59semidirect
298. #1443C2×C59semidirect
299. #1442C2×C59semidirect
300. #1441C2×C59semidirect
339. C59#16675direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3972
C2×C59×#3852
C2×C59×#3812
C59×#14442
C59×#14432
C59×#14422
C59×#14412
C59×#14402
C59×#14392
#1667559