Finite Group Databasebeta

← C59×#13877 #28425986

Subgroups — C59×#13877 #28425986

Order 15104.

4046 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4×C59C211yesmaximal, characteristic
2C23×C22⋊C4×C59C211yesmaximal, characteristic
3C2×C59×#385C212yesmaximal, complemented
4C2×C59×#380C212yesmaximal, complemented
5C2×C59×#379C211yesmaximal, characteristic
6C59×#1439C214yesmaximal, complemented
7C59×C22⋊#95C212yesmaximal, complemented
8C59×C22⋊#95C212yesmaximal, complemented
9C26×C59C2211yescharacteristic
10C24×C4×C59C2212yes—
11C24×C4×C59C2212yes—
12C22×C4⋊C4×C59C2214yescomplemented
13C22×C22⋊C4×C59—24no—
14C22×C22⋊C4×C59—28no—
15C22×C22⋊C4×C59—28no—
16C22×C22⋊C4×C59C418yescomplemented
17C22×C22⋊C4×C59C2211yescharacteristic
18C22×C22⋊C4×C59C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C22×C22⋊C4×C59C2212yes—
21C22×C22⋊C4×C59C2211yescharacteristic
22C22×C22⋊C4×C59C2212yes—
23C22×C22⋊C4×C59C2212yes—
24C22×C22⋊C4×C59C2212yes—
25C22×C22⋊C4×C59C2211yescharacteristic
26C59×#385—28no—
27C59×#380—28no—
28C59×#379—24no—
29C2×#95×C59C2214yescomplemented
30C2×#95×C59C2214yescomplemented
31C2×#95×C59C2214yescomplemented
32C25×C59—24no—
33C25×C59—24no—
34C25×C59—24no—
35C25×C59—24no—
36C25×C59—44no—
37C25×C59C2×C414yes—
38C25×C59D414yes—
39C25×C59Q814yes—
40C25×C59C2311yescharacteristic
41C25×C59C2311yescharacteristic
42C25×C59C2311yescharacteristic
43C23×C4×C59—24no—
44C23×C4×C59—24no—
45C23×C4×C59—24no—
46C23×C4×C59—28no—
47C23×C4×C59—28no—
48C23×C4×C59—24no—
49C23×C4×C59—24no—
50C23×C4×C59—44no—
51C23×C4×C59—44no—
52C23×C4×C59C2×C418yescomplemented
53C23×C4×C59C2312yes—
54C23×C4×C59C2312yes—
55C23×C4×C59C2312yes—
56C23×C4×C59C2314yes—
57C23×C4×C59C2312yes—
58C2×C4⋊C4×C59—28no—
59C2×C4⋊C4×C59—28no—
60C2×C4⋊C4×C59—44no—
61C2×C22⋊C4×C59—232no—
62C2×C22⋊C4×C59—24no—
63C2×C22⋊C4×C59—24no—
64C2×C22⋊C4×C59—24no—
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—28no—
67C2×C22⋊C4×C59—28no—
68C2×C22⋊C4×C59—232no—
69C2×C22⋊C4×C59—24no—
70C2×C22⋊C4×C59—22no—
71C2×C22⋊C4×C59—24no—
72C2×C22⋊C4×C59—22no—
73C2×C22⋊C4×C59—24no—
74C2×C22⋊C4×C59—24no—
75C2×C22⋊C4×C59—24no—
76C2×C22⋊C4×C59—24no—
77C2×C22⋊C4×C59—44no—
78C2×C22⋊C4×C59C2×C414yes—
79C2×C22⋊C4×C59C2×C414yes—
80C2×C22⋊C4×C59C2×C418yescomplemented
81C2×C22⋊C4×C59D418yes—
82C2×C22⋊C4×C59D418yes—
83#95×C59—28no—
84#95×C59—28no—
85#95×C59—44no—
86C24×C59—22no—
87C24×C59—22no—
88C24×C59—22no—
89C24×C59—22no—
90C24×C59—24no—
91C24×C59—24no—
92C24×C59—28no—
93C24×C59—28no—
94C24×C59—216no—
95C24×C59—28no—
96C24×C59—216no—
97C24×C59—216no—
98C24×C59—216no—
99C24×C59—28no—
100C24×C59—22no—
101C24×C59—22no—
102C24×C59—22no—
103C24×C59—22no—
104C24×C59—24no—
105C24×C59—48no—
106C24×C59—48no—
107C24×C59—48no—
108C24×C59—416no—
109C24×C59—416no—
110C24×C59—416no—
111C24×C59—416no—
112C24×C59C4⋊C4116yescomplemented
113C24×C59C22×C412yes—
114C24×C59C22×C412yes—
115C24×C59C22×C412yes—
116C24×C59C2×D412yes—
117C24×C59C2×D414yes—
118C24×C59C2×D412yes—
119C24×C59C2×D412yes—
120C24×C59C2×D414yes—
121C24×C59C2×Q812yes—
122C24×C59C2×Q812yes—
123C24×C59C2×Q812yes—
124C24×C59#4114yes—
125C24×C59#4114yes—
126C24×C59C2411yescharacteristic
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—24no—
130C22×C4×C59—216no—
131C22×C4×C59—216no—
132C22×C4×C59—22no—
133C22×C4×C59—24no—
134C22×C4×C59—24no—
135C22×C4×C59—24no—
136C22×C4×C59—24no—
137C22×C4×C59—22no—
138C22×C4×C59—24no—
139C22×C4×C59—24no—
140C22×C4×C59—24no—
141C22×C4×C59—24no—
142C22×C4×C59—216no—
143C22×C4×C59—22no—
144C22×C4×C59—24no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—22no—
150C22×C4×C59—24no—
151C22×C4×C59—24no—
152C22×C4×C59—24no—
153C22×C4×C59—48no—
154C22×C4×C59—416no—
155C22×C4×C59—416no—
156C22×C4×C59—416no—
157C22×C4×C59—416no—
158C22×C4×C59—48no—
159C22×C4×C59—48no—
160C22×C4×C59C22×C414yes—
161C22×C4×C59C22×C414yes—
162C22×C4×C59C2×D418yes—
163C22×C4×C59C2×D418yes—
164C4⋊C4×C59—416no—
165C22⋊C4×C59—216no—
166C22⋊C4×C59—28no—
167C22⋊C4×C59—216no—
168C22⋊C4×C59—216no—
169C22⋊C4×C59—48no—
170C22⋊C4×C59—44no—
171C22⋊C4×C59—48no—
172C22⋊C4×C59—44no—
173C22⋊C4×C59C22⋊C4116yescomplemented
174C23×C59—22no—
175C23×C59—22no—
176C23×C59—24no—
177C23×C59—24no—
178C23×C59—24no—
179C23×C59—22no—
180C23×C59—24no—
181C23×C59—24no—
182C23×C59—24no—
183C23×C59—22no—
184C23×C59—24no—
185C23×C59—24no—
186C23×C59—24no—
187C23×C59—24no—
188C23×C59—28no—
189C23×C59—28no—
190C23×C59—28no—
191C23×C59—28no—
192C23×C59—28no—
193C23×C59—28no—
194C23×C59—28no—
195C23×C59—28no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—24no—
199C23×C59—24no—
200C23×C59—22no—
201C23×C59—24no—
202C23×C59—24no—
203C23×C59—24no—
204C23×C59—22no—
205C23×C59—24no—
206C23×C59—24no—
207C23×C59—232no—
208C23×C59—232no—
209C23×C59—416no—
210C23×C59—432no—
211C23×C59—432no—
212C23×C59—432no—
213C23×C59—432no—
214C23×C59—464no—
215C23×C59—44no—
216C23×C59—44no—
217C23×C59—44no—
218C23×C59—44no—
219C23×C59C2×C22⋊C414yes—
220C23×C59C2×C4⋊C414yes—
221C23×C59C2×C4⋊C414yes—
222C23×C59C2×C4⋊C414yes—
223C23×C59#11714yes—
224C23×C59C4×D418yes—
225C23×C59#12014yes—
226C23×C59#12118yes—
227C23×C59C22⋊Q818yes—
228C23×C59C22⋊Q818yes—
229C23×C59#12314yes—
230C23×C59C23×C411yescharacteristic
231C23×C59C22×D412yes—
232C23×C59C22×D411yescharacteristic
233C23×C59C22×D412yes—
234C23×C59C22×Q811yescharacteristic
235C23×C59C2×#4112yes—
236C23×C59C2×#4112yes—
237C23×C59H(5,𝔽2)14yes—
238C2×C4×C59—24no—
239C2×C4×C59—24no—
240C2×C4×C59—28no—
241C2×C4×C59—28no—
242C2×C4×C59—24no—
243C2×C4×C59—24no—
244C2×C4×C59—24no—
245C2×C4×C59—28no—
246C2×C4×C59—28no—
247C2×C4×C59—28no—
248C2×C4×C59—432no—
249C2×C4×C59—432no—
250C2×C4×C59—44no—
251C2×C4×C59—44no—
252C2×C4×C59—44no—
253C2×C4×C59—44no—
254C2×C4×C59—44no—
255C2×C4×C59—44no—
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—44no—
263C2×C4×C59—44no—
264C2×C4×C59—44no—
265C2×C4×C59—44no—
266C2×C4×C59C2×C22⋊C418yescomplemented
267#13877C5911yesmaximal, characteristic, direct product factor, Sylow
268C22×C59—24no—
269C22×C59—24no—
270C22×C59—28no—
271C22×C59—28no—
272C22×C59—28no—
273C22×C59—28no—
274C22×C59—28no—
275C22×C59—28no—
276C22×C59—28no—
277C22×C59—28no—
278C22×C59—216no—
279C22×C59—216no—
280C22×C59—24no—
281C22×C59—24no—
282C22×C59—24no—
283C22×C59—24no—
284C22×C59—24no—
285C22×C59—464no—
286C22×C59—44no—
287C22×C59—44no—
288C22×C59—44no—
289C22×C59—44no—
290C22×C59—48no—
291C22×C59#38018yescomplemented
292C22×C59#38518yescomplemented
293C22×C59C22×C22⋊C411yescharacteristic
294C22×C59C22×C4⋊C411yescharacteristic
295C22×C59C2×#11711yescharacteristic
296C22×C59C2×C4×D412yes—
297C22×C59#51812yes—
298C22×C59C2×#12011yescharacteristic
299C22×C59C2×#12112yes—
300C22×C59C2×C22⋊Q812yes—
301C22×C59C2×C22⋊Q812yes—
302C22×C59C2×#12311yescharacteristic
303C22×C59#53412yes—
304C22×C59#54312yes—
305C22×C59D4214yes—
306C22×C59#54614yes—
307C22×C59#54614yes—
308C22×C59#55114yes—
309C4×C59—28no—
310C4×C59—44no—
311C4×C59—48no—
312C4×C59—82no—
313C4×C59—82no—
314C23×C22⋊C4C2×C5911yescharacteristic
315C23×C22⋊C4C2×C5911yescharacteristic
316C2×#385C2×C5912yescomplemented
317C2×#380C2×C5912yescomplemented
318C2×#379C2×C5911yescharacteristic
319#1439C2×C5914yescomplemented
320C22⋊#95C2×C5912yescomplemented
321C22⋊#95C2×C5912yescomplemented
322C2×C59—28no—
323C2×C59—28no—
324C2×C59—44no—
325C2×C59C2×#38011yescharacteristic
326C2×C59C2×#38511yescharacteristic
327C2×C59#231511yescharacteristic
328C2×C59D4×C22⋊C412yes—
329C2×C59#234212yes—
330C2×C59#241012yes—
331C2×C59#241212yes—
332C2×C59#243714yes—
333C26C22×C5911yescharacteristic
334C24×C4C22×C5912yes—
335C24×C4C22×C5912yes—
336C22×C4⋊C4C22×C5914yescomplemented
337C22×C22⋊C4—24no—
338C22×C22⋊C4—28no—
339C22×C22⋊C4—28no—
340C22×C22⋊C4C4×C5918yescomplemented
341C22×C22⋊C4C22×C5912yes—
342C22×C22⋊C4C22×C5911yescharacteristic
343C22×C22⋊C4C22×C5912yes—
344C22×C22⋊C4C22×C5911yescharacteristic
345C22×C22⋊C4C22×C5911yescharacteristic
346C22×C22⋊C4C22×C5912yes—
347C22×C22⋊C4C22×C5912yes—
348C22×C22⋊C4C22×C5912yes—
349C22×C22⋊C4C22×C5911yescharacteristic
350#385—28no—
351#380—28no—
352#379—24no—
353C2×#95C22×C5914yescomplemented
354C2×#95C22×C5914yescomplemented
355C2×#95C22×C5914yescomplemented
356C59#1387711yescharacteristic, direct product factor, Sylow
357C25—24no—
358C25—24no—
359C25—24no—
360C25—24no—
361C25—44no—
362C25C2×C4×C5914yes—
363C25D4×C5914yes—
364C25Q8×C5914yes—
365C25C23×C5911yescharacteristic
366C25C23×C5911yescharacteristic
367C25C23×C5911yescharacteristic
368C23×C4—24no—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—28no—
372C23×C4—28no—
373C23×C4—24no—
374C23×C4—24no—
375C23×C4—44no—
376C23×C4—44no—
377C23×C4C2×C4×C5918yescomplemented
378C23×C4C23×C5912yes—
379C23×C4C23×C5912yes—
380C23×C4C23×C5912yes—
381C23×C4C23×C5914yes—
382C23×C4C23×C5912yes—
383C2×C4⋊C4—28no—
384C2×C4⋊C4—28no—
385C2×C4⋊C4—44no—
386C2×C22⋊C4—232no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—28no—
392C2×C22⋊C4—28no—
393C2×C22⋊C4—232no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—44no—
403C2×C22⋊C4C2×C4×C5914yes—
404C2×C22⋊C4C2×C4×C5914yes—
405C2×C22⋊C4C2×C4×C5918yescomplemented
406C2×C22⋊C4D4×C5918yes—
407C2×C22⋊C4D4×C5918yes—
408#95—28no—
409#95—28no—
410#95—44no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—24no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—216no—
423C24—216no—
424C24—28no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—24no—
430C24—48no—
431C24—48no—
432C24—48no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—416no—
437C24C4⋊C4×C59116yescomplemented
438C24C22×C4×C5912yes—
439C24C22×C4×C5912yes—
440C24C22×C4×C5912yes—
441C24C2×D4×C5912yes—
442C24C2×D4×C5914yes—
443C24C2×D4×C5912yes—
444C24C2×D4×C5912yes—
445C24C2×D4×C5914yes—
446C24C2×Q8×C5912yes—
447C24C2×Q8×C5912yes—
448C24C2×Q8×C5912yes—
449C24#41×C5914yes—
450C24#41×C5914yes—
451C24C24×C5911yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4C22×C4×C5914yes—
486C22×C4C22×C4×C5914yes—
487C22×C4C2×D4×C5918yes—
488C22×C4C2×D4×C5918yes—
489C4⋊C4—416no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4C22⋊C4×C59116yescomplemented
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—232no—
522C23—232no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—416no—
535C23—432no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—464no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C2×C22⋊C4×C5914yes—
545C23C2×C4⋊C4×C5914yes—
546C23C2×C4⋊C4×C5914yes—
547C23C2×C4⋊C4×C5914yes—
548C23#117×C5914yes—
549C23C4×D4×C5918yes—
550C23#120×C5914yes—
551C23#121×C5918yes—
552C23C22⋊Q8×C5918yes—
553C23C22⋊Q8×C5918yes—
554C23#123×C5914yes—
555C23C23×C4×C5911yescharacteristic
556C23C22×D4×C5912yes—
557C23C22×D4×C5911yescharacteristic
558C23C22×D4×C5912yes—
559C23C22×Q8×C5911yescharacteristic
560C23C2×#41×C5912yes—
561C23C2×#41×C5912yes—
562C23H(5,𝔽2)×C5914yes—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—432no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C2×C22⋊C4×C5918yescomplemented
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—216no—
605C22—216no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—464no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—48no—
615C22C59×#38018yescomplemented
616C22C59×#38518yescomplemented
617C22C22×C22⋊C4×C5911yescharacteristic
618C22C22×C4⋊C4×C5911yescharacteristic
619C22C2×#117×C5911yescharacteristic
620C22C2×C4×D4×C5912yes—
621C22C59×#51812yes—
622C22C2×#120×C5911yescharacteristic
623C22C2×#121×C5912yes—
624C22C2×C22⋊Q8×C5912yes—
625C22C2×C22⋊Q8×C5912yes—
626C22C2×#123×C5911yescharacteristic
627C22C59×#53412yes—
628C22C59×#54312yes—
629C22D42×C5914yes—
630C22C59×#54614yes—
631C22C59×#54614yes—
632C22C59×#55114yes—
633C4—28no—
634C4—44no—
635C4—48no—
636C4—82no—
637C4—82no—
638C2—28no—
639C2—28no—
640C2—44no—
641C2C2×C59×#38011yescharacteristic
642C2C2×C59×#38511yescharacteristic
643C2C59×#231511yescharacteristic
644C2D4×C22⋊C4×C5912yes—
645C2C59×#234212yes—
646C2C59×#241012yes—
647C2C59×#241212yes—
648C2C59×#243714yes—
subgroupG/H
center, socleC24×C59C24
commutatorC23C23×C4×C59
Frattini, cosocleC24C24×C59
prime radicalC26×C59C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C59 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C59×#385C2semidirect
4. C2×C59×#380C2semidirect
6. C59×#1439C2semidirect
7. C59×C22⋊#95C2semidirect
8. C59×C22⋊#95C2semidirect
12. C22×C4⋊C4×C59C22semidirect
16. C22×C22⋊C4×C59C4semidirect
29. C2×#95×C59C22semidirect
30. C2×#95×C59C22semidirect
31. C2×#95×C59C22semidirect
52. C23×C4×C59C2×C4semidirect
80. C2×C22⋊C4×C59C2×C4semidirect
112. C24×C59C4⋊C4semidirect
173. C22⋊C4×C59C22⋊C4semidirect
266. C2×C4×C59C2×C22⋊C4semidirect
267. #13877C59direct (N × H)
291. C22×C59#380semidirect
292. C22×C59#385semidirect
316. C2×#385C2×C59semidirect
317. C2×#380C2×C59semidirect
319. #1439C2×C59semidirect
320. C22⋊#95C2×C59semidirect
321. C22⋊#95C2×C59semidirect
336. C22×C4⋊C4C22×C59semidirect
340. C22×C22⋊C4C4×C59semidirect
353. C2×#95C22×C59semidirect
354. C2×#95C22×C59semidirect
355. C2×#95C22×C59semidirect
356. C59#13877direct (N × H)
377. C23×C4C2×C4×C59semidirect
405. C2×C22⋊C4C2×C4×C59semidirect
437. C24C4⋊C4×C59semidirect
498. C22⋊C4C22⋊C4×C59semidirect
591. C2×C4C2×C22⋊C4×C59semidirect
615. C22C59×#380semidirect
616. C22C59×#385semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C4×C592
C2×C59×#3852
C2×C59×#3802
C2×C59×#3792
C59×#14392
C59×C22⋊#952
#1387759