Finite Group Databasebeta

← C59×#17140 #28429249

Subgroups — C59×#17140 #28429249

Order 15104.

1466 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#398C211yesmaximal, characteristic, complemented
2C2×C59×#394C211yesmaximal, characteristic
3C59×#1445C211yesmaximal, characteristic, complemented
4C59×#1443C212yesmaximal, complemented
5C59×#1440C212yesmaximal, complemented
6C59×#1439C212yesmaximal, complemented
7C59×#1439C212yesmaximal, complemented
8C59×#1439C212yesmaximal, complemented
9C59×#1438C212yesmaximal
10C23×D4×C59C2211yescharacteristic
11C22×C4⋊C4×C59C2211yescharacteristic
12C22×C4⋊C4×C59C2212yes—
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—
20C59×#398—24no—
21C59×#394—24no—
22C2×#95×C59C2211yescharacteristic
23C2×#95×C59C2212yes—
24C2×#95×C59C2211yescharacteristic
25C2×#95×C59C2212yes—
26C2×#95×C59C2212yes—
27C2×#95×C59C2212yes—
28C2×#95×C59C2212yes—
29C2×#95×C59C2211yescharacteristic
30C2×#95×C59C2211yescharacteristic
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×C59C2311yescharacteristic
46C23×C4×C59C2312yes—
47C2×C4⋊C4×C59—24no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59—41no—
51C2×C4⋊C4×C59—42no—
52C2×C4⋊C4×C59—42no—
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—42no—
63C2×C22⋊C4×C59—42no—
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—
74#95×C59—22no—
75#95×C59—22no—
76#95×C59—24no—
77#95×C59—24no—
78#95×C59—42no—
79#95×C59—42no—
80#95×C59—41no—
81#95×C59—41no—
82#95×C59—42no—
83#95×C59—42no—
84#95×C59—42no—
85C24×C59—22no—
86C24×C59—22no—
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#4114yes—
100C24×C59C2411yescharacteristic
101C2×D4×C59—48no—
102C2×D4×C59—44no—
103C2×D4×C59—44no—
104C2×D4×C59—48no—
105C2×D4×C59—44no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—22no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
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—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×C59C2×D412yes—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D412yes—
155C22×C4×C59#4114yes—
156C4⋊C4×C59—44no—
157C4⋊C4×C59—44no—
158C4⋊C4×C59—48no—
159C22⋊C4×C59—48no—
160C22⋊C4×C59—48no—
161C22⋊C4×C59—48no—
162C22⋊C4×C59—48no—
163C22⋊C4×C59—48no—
164C22⋊C4×C59—48no—
165C23×C59—24no—
166C23×C59—24no—
167C23×C59—24no—
168C23×C59—24no—
169C23×C59—24no—
170C23×C59—24no—
171C23×C59—24no—
172C23×C59—42no—
173C23×C59—42no—
174C23×C59—42no—
175C23×C59—42no—
176C23×C59—42no—
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×C59C22×D411yescharacteristic
201C23×C59C2×#4112yes—
202C23×C59C2×#4112yes—
203C23×C59H(5,𝔽2)11yescharacteristic
204C23×C59H(5,𝔽2)11yescharacteristic
205C23×C59H(5,𝔽2)12yes—
206C23×C59H(5,𝔽2)12yes—
207C23×C59H(5,𝔽2)12yes—
208C23×C592−1+412yes—
209D4×C59—416no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—24no—
216C2×C4×C59—24no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
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—
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—
260#17140C5911yesmaximal, characteristic, direct product factor, Sylow
261C22×C59—44no—
262C22×C59—44no—
263C22×C59—44no—
264C22×C59—44no—
265C22×C59—44no—
266C22×C59—44no—
267C22×C59—44no—
268C22×C59—82no—
269C22×C59—82no—
270C22×C59—82no—
271C22×C59—82no—
272C22×C59—82no—
273C22×C59—82no—
274C22×C59—82no—
275C22×C59—82no—
276C22×C59#53411yescharacteristic
277C22×C59#53511yescharacteristic
278C22×C59#53511yescharacteristic
279C22×C59#53812yes—
280C22×C59#53812yes—
281C22×C59#53912yes—
282C22×C59#53912yes—
283C22×C59#54012yes—
284C22×C59#54212yes—
285C22×C59#54612yes—
286C22×C59#54812yes—
287C22×C59#55112yes—
288C22×C59#55111yescharacteristic
289C22×C59#55312yes—
290C22×C59#55511yescharacteristic
291C22×C59#56012yes—
292C22×C59#56011yescharacteristic
293C22×C59#56012yes—
294C22×C59#56212yes—
295C22×C59#56212yes—
296C22×C59#56311yescharacteristic
297C4×C59—44no—
298C4×C59—42no—
299C4×C59—42no—
300C4×C59—42no—
301C4×C59—82no—
302C4×C59—82no—
303C4×C59—82no—
304C4×C59—82no—
305C2×#398C2×C5911yescharacteristic, complemented
306C2×#394C2×C5911yescharacteristic
307#1445C2×C5911yescharacteristic, complemented
308#1443C2×C5912yescomplemented
309#1440C2×C5912yescomplemented
310#1439C2×C5912yescomplemented
311#1439C2×C5912yescomplemented
312#1439C2×C5912yescomplemented
313#1438C2×C5912yes—
314C2×C59—84no—
315C2×C59#269612yes—
316C2×C59#269711yescharacteristic
317C2×C59#271112yes—
318C2×C59#273111yescharacteristic
319C2×C59#274612yes—
320C2×C59#275612yes—
321C2×C59#281511yescharacteristic
322C2×C59#282712yes—
323C2×C59#282912yes—
324C23×D4C22×C5911yescharacteristic
325C22×C4⋊C4C22×C5911yescharacteristic
326C22×C4⋊C4C22×C5912yes—
327C22×C4⋊C4C22×C5911yescharacteristic
328C22×C22⋊C4C22×C5912yes—
329C22×C22⋊C4C22×C5912yes—
330C22×C22⋊C4C22×C5912yes—
331C22×C22⋊C4C22×C5912yes—
332C22×C22⋊C4C22×C5912yes—
333C22×C22⋊C4C22×C5912yes—
334#398—24no—
335#394—24no—
336C2×#95C22×C5911yescharacteristic
337C2×#95C22×C5912yes—
338C2×#95C22×C5911yescharacteristic
339C2×#95C22×C5912yes—
340C2×#95C22×C5912yes—
341C2×#95C22×C5912yes—
342C2×#95C22×C5912yes—
343C2×#95C22×C5911yescharacteristic
344C2×#95C22×C5911yescharacteristic
345C2×#95C22×C5912yes—
346C2×#95C22×C5912yes—
347C59#1714011yescharacteristic, direct product factor, Sylow
348C25C23×C5912yes—
349C22×D4—24no—
350C22×D4—22no—
351C22×D4—42no—
352C22×D4—41no—
353C22×D4—41no—
354C23×C4C23×C5912yes—
355C23×C4C23×C5912yes—
356C23×C4C23×C5912yes—
357C23×C4C23×C5912yes—
358C23×C4C23×C5911yescharacteristic
359C23×C4C23×C5911yescharacteristic
360C23×C4C23×C5911yescharacteristic
361C23×C4C23×C5912yes—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—42no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4D4×C5914yes—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C22⋊C4—42no—
382C2×C22⋊C4—42no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—42no—
388C2×C22⋊C4—42no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—42no—
394#95—42no—
395#95—41no—
396#95—41no—
397#95—42no—
398#95—42no—
399#95—42no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24#41×C5914yes—
415C24C24×C5911yescharacteristic
416C2×D4—48no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—44no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
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—22no—
452C22×C4—22no—
453C22×C4—22no—
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×C4C2×D4×C5912yes—
468C22×C4C2×D4×C5912yes—
469C22×C4C2×D4×C5912yes—
470C22×C4#41×C5914yes—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C22×D4×C5911yescharacteristic
516C23C2×#41×C5912yes—
517C23C2×#41×C5912yes—
518C23H(5,𝔽2)×C5911yescharacteristic
519C23H(5,𝔽2)×C5911yescharacteristic
520C23H(5,𝔽2)×C5912yes—
521C23H(5,𝔽2)×C5912yes—
522C23H(5,𝔽2)×C5912yes—
523C232−1+4×C5912yes—
524D4—416no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
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—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
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—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22C59×#53411yescharacteristic
591C22C59×#53511yescharacteristic
592C22C59×#53511yescharacteristic
593C22C59×#53812yes—
594C22C59×#53812yes—
595C22C59×#53912yes—
596C22C59×#53912yes—
597C22C59×#54012yes—
598C22C59×#54212yes—
599C22C59×#54612yes—
600C22C59×#54812yes—
601C22C59×#55111yescharacteristic
602C22C59×#55112yes—
603C22C59×#55312yes—
604C22C59×#55511yescharacteristic
605C22C59×#56012yes—
606C22C59×#56012yes—
607C22C59×#56011yescharacteristic
608C22C59×#56212yes—
609C22C59×#56212yes—
610C22C59×#56311yescharacteristic
611C4—42no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C2—84no—
620C2C59×#269612yes—
621C2C59×#269711yescharacteristic
622C2C59×#271112yes—
623C2C59×#273111yescharacteristic
624C2C59×#274612yes—
625C2C59×#275612yes—
626C2C59×#281511yescharacteristic
627C2C59×#282712yes—
628C2C59×#282912yes—
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×#398C2semidirect
3. C59×#1445C2semidirect
4. C59×#1443C2semidirect
5. C59×#1440C2semidirect
6. C59×#1439C2semidirect
7. C59×#1439C2semidirect
8. C59×#1439C2semidirect
260. #17140C59direct (N × H)
305. C2×#398C2×C59semidirect
307. #1445C2×C59semidirect
308. #1443C2×C59semidirect
309. #1440C2×C59semidirect
310. #1439C2×C59semidirect
311. #1439C2×C59semidirect
312. #1439C2×C59semidirect
347. C59#17140direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#3982
C2×C59×#3942
C59×#14452
C59×#14432
C59×#14402
C59×#14392
C59×#14382
#1714059