Finite Group Databasebeta

← C59×#14777 #28426886

Subgroups — C59×#14777 #28426886

Order 15104.

2506 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C4×C59C211yesmaximal, characteristic, complemented
2C2×C59×#399C212yesmaximal, complemented
3C2×C59×#386C211yesmaximal, characteristic
4C2×C59×#386C212yesmaximal
5C2×C59×#385C212yesmaximal, complemented
6C59×#1445C212yesmaximal, complemented
7C59×#1444C212yesmaximal, complemented
8C59×#1439C212yesmaximal, complemented
9C59×C22⋊#95C211yesmaximal, characteristic, complemented
10C23×D4×C59C2211yescharacteristic
11C24×C4×C59C2211yescharacteristic
12C24×C4×C59C2212yes—
13C22×C4⋊C4×C59—24no—
14C22×C4⋊C4×C59—28no—
15C22×C4⋊C4×C59C2212yes—
16C22×C4⋊C4×C59C2214yes—
17C22×C4⋊C4×C59C2212yes—
18C22×C22⋊C4×C59C2211yescharacteristic
19C22×C22⋊C4×C59C2211yescharacteristic
20C22×C22⋊C4×C59C2211yescharacteristic
21C22×C22⋊C4×C59C2214yes—
22C22×C22⋊C4×C59C2212yes—
23C59×#399—28no—
24C59×#386—24no—
25C59×#386—28no—
26C59×#385—28no—
27C2×#95×C59C2214yes—
28C2×#95×C59C2211yescharacteristic
29C2×#95×C59C2214yes—
30C2×#95×C59C2211yescharacteristic
31C2×#95×C59C2212yes—
32C2×#95×C59C2212yes—
33C25×C59C2311yescharacteristic
34C25×C59C2311yescharacteristic
35C22×D4×C59—24no—
36C22×D4×C59—22no—
37C22×D4×C59—22no—
38C22×D4×C59—24no—
39C22×D4×C59Q814yes—
40C23×C4×C59—24no—
41C23×C4×C59—24no—
42C23×C4×C59—24no—
43C23×C4×C59—24no—
44C23×C4×C59—28no—
45C23×C4×C59—24no—
46C23×C4×C59—24no—
47C23×C4×C59—44no—
48C23×C4×C59D414yes—
49C23×C4×C59C2311yescharacteristic
50C23×C4×C59C2311yescharacteristic
51C23×C4×C59C2314yes—
52C23×C4×C59C2314yes—
53C23×C4×C59C2311yescharacteristic
54C23×C4×C59C2311yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C2×C4⋊C4×C59—232no—
57C2×C4⋊C4×C59—28no—
58C2×C4⋊C4×C59—24no—
59C2×C4⋊C4×C59—24no—
60C2×C4⋊C4×C59—24no—
61C2×C4⋊C4×C59—28no—
62C2×C4⋊C4×C59—28no—
63C2×C4⋊C4×C59—44no—
64C2×C4⋊C4×C59D418yescomplemented
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—24no—
67C2×C22⋊C4×C59—24no—
68C2×C22⋊C4×C59—22no—
69C2×C22⋊C4×C59—28no—
70C2×C22⋊C4×C59—22no—
71C2×C22⋊C4×C59—22no—
72C2×C22⋊C4×C59—28no—
73C2×C22⋊C4×C59—24no—
74C2×C22⋊C4×C59—24no—
75C2×C22⋊C4×C59—44no—
76C2×C22⋊C4×C59D418yes—
77#95×C59—28no—
78#95×C59—28no—
79#95×C59—22no—
80#95×C59—22no—
81#95×C59—24no—
82#95×C59—24no—
83C24×C59—22no—
84C24×C59—22no—
85C24×C59—22no—
86C24×C59—22no—
87C24×C59—22no—
88C24×C59—21no—
89C24×C59—22no—
90C24×C59—22no—
91C24×C59—22no—
92C24×C59—21no—
93C24×C59—22no—
94C24×C59—22no—
95C24×C59C2×D414yes—
96C24×C59C2×D412yes—
97C24×C59C2×Q812yes—
98C24×C59C2×Q812yes—
99C24×C59#4112yes—
100C24×C59#4114yes—
101C24×C59C2411yescharacteristic
102C2×D4×C59—28no—
103C2×D4×C59—216no—
104C2×D4×C59—48no—
105C2×D4×C59—44no—
106C2×D4×C59—44no—
107C22×C4×C59—232no—
108C22×C4×C59—28no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—28no—
114C22×C4×C59—28no—
115C22×C4×C59—216no—
116C22×C4×C59—21no—
117C22×C4×C59—22no—
118C22×C4×C59—24no—
119C22×C4×C59—24no—
120C22×C4×C59—24no—
121C22×C4×C59—24no—
122C22×C4×C59—24no—
123C22×C4×C59—24no—
124C22×C4×C59—24no—
125C22×C4×C59—24no—
126C22×C4×C59—22no—
127C22×C4×C59—21no—
128C22×C4×C59—24no—
129C22×C4×C59—24no—
130C22×C4×C59—24no—
131C22×C4×C59—24no—
132C22×C4×C59—22no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—416no—
141C22×C4×C59—416no—
142C22×C4×C59—48no—
143C22×C4×C59—48no—
144C22×C4×C59C2×D418yes—
145C22×C4×C59C2×D418yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D414yes—
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2×Q812yes—
150C22×C4×C59#4114yes—
151C4⋊C4×C59—216no—
152C4⋊C4×C59—216no—
153C4⋊C4×C59—416no—
154C22⋊C4×C59—216no—
155C22⋊C4×C59—416no—
156C22⋊C4×C59—44no—
157C22⋊C4×C59—44no—
158C22⋊C4×C59—44no—
159C23×C59—22no—
160C23×C59—22no—
161C23×C59—28no—
162C23×C59—22no—
163C23×C59—22no—
164C23×C59—28no—
165C23×C59—28no—
166C23×C59—28no—
167C23×C59—24no—
168C23×C59—24no—
169C23×C59—24no—
170C23×C59—24no—
171C23×C59—22no—
172C23×C59—24no—
173C23×C59—24no—
174C23×C59—24no—
175C23×C59—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—42no—
179C23×C59—44no—
180C23×C59—41no—
181C23×C59—41no—
182C23×C59—41no—
183C23×C59—44no—
184C23×C59—44no—
185C23×C59—44no—
186C23×C59—42no—
187C23×C59—42no—
188C23×C59C22⋊Q814yes—
189C23×C59C22⋊Q818yes—
190C23×C59#12318yes—
191C23×C59#12314yes—
192C23×C59#12714yes—
193C23×C59C22×D411yescharacteristic
194C23×C59C22×D412yes—
195C23×C59C22×D412yes—
196C23×C59C22×Q811yescharacteristic
197C23×C59C2×#4112yes—
198C23×C59C2×#4111yescharacteristic
199C23×C59C2×#4112yes—
200C23×C59H(5,𝔽2)12yes—
201C23×C592−1+412yes—
202D4×C59—416no—
203C2×C4×C59—232no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
208C2×C4×C59—28no—
209C2×C4×C59—28no—
210C2×C4×C59—28no—
211C2×C4×C59—22no—
212C2×C4×C59—24no—
213C2×C4×C59—28no—
214C2×C4×C59—28no—
215C2×C4×C59—28no—
216C2×C4×C59—28no—
217C2×C4×C59—28no—
218C2×C4×C59—432no—
219C2×C4×C59—41no—
220C2×C4×C59—42no—
221C2×C4×C59—44no—
222C2×C4×C59—44no—
223C2×C4×C59—41no—
224C2×C4×C59—44no—
225C2×C4×C59—44no—
226C2×C4×C59—44no—
227C2×C4×C59—44no—
228C2×C4×C59—44no—
229C2×C4×C59—44no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—44no—
237C2×C4×C59—44no—
238C2×C4×C59#12014yes—
239C2×C4×C59#12314yes—
240#14777C5911yesmaximal, characteristic, direct product factor, Sylow
241C22×C59—24no—
242C22×C59—24no—
243C22×C59—24no—
244C22×C59—24no—
245C22×C59—216no—
246C22×C59—216no—
247C22×C59—24no—
248C22×C59—24no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—48no—
252C22×C59—42no—
253C22×C59—81no—
254C22×C59—81no—
255C22×C59—81no—
256C22×C59—84no—
257C22×C59—81no—
258C22×C59#39918yescomplemented
259C22×C59C2×#12011yescharacteristic
260C22×C59C2×C22⋊Q811yescharacteristic
261C22×C59C2×C22⋊Q812yes—
262C22×C59C2×#12311yescharacteristic
263C22×C59C2×#12311yescharacteristic
264C22×C59C2×#12312yes—
265C22×C59#52514yes—
266C22×C59C2×#12711yescharacteristic
267C22×C59#53511yescharacteristic
268C22×C59#53611yescharacteristic
269C22×C59#53914yes—
270C22×C59D4214yes—
271C22×C59#54714yes—
272C22×C59#54812yes—
273C22×C59D4×Q812yes—
274C22×C59#55312yes—
275C22×C59#55412yes—
276C4×C59—24no—
277C4×C59—24no—
278C4×C59—42no—
279C4×C59—48no—
280C4×C59—48no—
281C4×C59—81no—
282C4×C59—81no—
283C23×C4⋊C4C2×C5911yescharacteristic, complemented
284C2×#399C2×C5912yescomplemented
285C2×#386C2×C5911yescharacteristic
286C2×#386C2×C5912yes—
287C2×#385C2×C5912yescomplemented
288#1445C2×C5912yescomplemented
289#1444C2×C5912yescomplemented
290#1439C2×C5912yescomplemented
291C22⋊#95C2×C5911yescharacteristic, complemented
292C2×C59—28no—
293C2×C59—82no—
294C2×C59C2×#39911yescharacteristic
295C2×C59#240911yescharacteristic
296C2×C59#243311yescharacteristic
297C2×C59#244314yes—
298C2×C59#244914yes—
299C2×C59#248711yescharacteristic
300C2×C59#250111yescharacteristic
301C2×C59#262311yescharacteristic
302C2×C59#284911yescharacteristic
303C23×D4C22×C5911yescharacteristic
304C24×C4C22×C5911yescharacteristic
305C24×C4C22×C5912yes—
306C22×C4⋊C4—24no—
307C22×C4⋊C4—28no—
308C22×C4⋊C4C22×C5912yes—
309C22×C4⋊C4C22×C5914yes—
310C22×C4⋊C4C22×C5912yes—
311C22×C22⋊C4C22×C5911yescharacteristic
312C22×C22⋊C4C22×C5911yescharacteristic
313C22×C22⋊C4C22×C5911yescharacteristic
314C22×C22⋊C4C22×C5914yes—
315C22×C22⋊C4C22×C5912yes—
316#399—28no—
317#386—24no—
318#386—28no—
319#385—28no—
320C2×#95C22×C5912yes—
321C2×#95C22×C5912yes—
322C2×#95C22×C5914yes—
323C2×#95C22×C5911yescharacteristic
324C2×#95C22×C5914yes—
325C2×#95C22×C5911yescharacteristic
326C59#1477711yescharacteristic, direct product factor, Sylow
327C25C23×C5911yescharacteristic
328C25C23×C5911yescharacteristic
329C22×D4—24no—
330C22×D4—22no—
331C22×D4—22no—
332C22×D4—24no—
333C22×D4Q8×C5914yes—
334C23×C4—24no—
335C23×C4—24no—
336C23×C4—24no—
337C23×C4—24no—
338C23×C4—28no—
339C23×C4—24no—
340C23×C4—24no—
341C23×C4—44no—
342C23×C4D4×C5914yes—
343C23×C4C23×C5911yescharacteristic
344C23×C4C23×C5911yescharacteristic
345C23×C4C23×C5914yes—
346C23×C4C23×C5914yes—
347C23×C4C23×C5911yescharacteristic
348C23×C4C23×C5911yescharacteristic
349C23×C4C23×C5911yescharacteristic
350C2×C4⋊C4—232no—
351C2×C4⋊C4—28no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—28no—
356C2×C4⋊C4—28no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4D4×C5918yescomplemented
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—28no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—28no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4D4×C5918yes—
371#95—28no—
372#95—28no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—22no—
389C24C2×D4×C5914yes—
390C24C2×D4×C5912yes—
391C24C2×Q8×C5912yes—
392C24C2×Q8×C5912yes—
393C24#41×C5912yes—
394C24#41×C5914yes—
395C24C24×C5911yescharacteristic
396C2×D4—28no—
397C2×D4—216no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—44no—
401C22×C4—232no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C2×D4×C5918yes—
439C22×C4C2×D4×C5918yes—
440C22×C4C2×D4×C5912yes—
441C22×C4C2×D4×C5914yes—
442C22×C4C2×D4×C5912yes—
443C22×C4C2×Q8×C5912yes—
444C22×C4#41×C5914yes—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—22no—
457C23—22no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23C22⋊Q8×C5914yes—
483C23C22⋊Q8×C5918yes—
484C23#123×C5918yes—
485C23#123×C5914yes—
486C23#127×C5914yes—
487C23C22×D4×C5911yescharacteristic
488C23C22×D4×C5912yes—
489C23C22×D4×C5912yes—
490C23C22×Q8×C5911yescharacteristic
491C23C2×#41×C5912yes—
492C23C2×#41×C5911yescharacteristic
493C23C2×#41×C5912yes—
494C23H(5,𝔽2)×C5912yes—
495C232−1+4×C5912yes—
496D4—416no—
497C2×C4—232no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—432no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4#120×C5914yes—
533C2×C4#123×C5914yes—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—216no—
541C22—216no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—48no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—84no—
551C22C59×#39918yescomplemented
552C22C2×#120×C5911yescharacteristic
553C22C2×C22⋊Q8×C5911yescharacteristic
554C22C2×C22⋊Q8×C5912yes—
555C22C2×#123×C5911yescharacteristic
556C22C2×#123×C5911yescharacteristic
557C22C2×#123×C5912yes—
558C22C59×#52514yes—
559C22C2×#127×C5911yescharacteristic
560C22C59×#53511yescharacteristic
561C22C59×#53611yescharacteristic
562C22C59×#53914yes—
563C22D42×C5914yes—
564C22C59×#54714yes—
565C22C59×#54812yes—
566C22D4×Q8×C5912yes—
567C22C59×#55312yes—
568C22C59×#55412yes—
569C4—24no—
570C4—24no—
571C4—42no—
572C4—48no—
573C4—48no—
574C4—81no—
575C4—81no—
576C2—28no—
577C2—82no—
578C2C2×C59×#39911yescharacteristic
579C2C59×#240911yescharacteristic
580C2C59×#243311yescharacteristic
581C2C59×#244314yes—
582C2C59×#244914yes—
583C2C59×#248711yescharacteristic
584C2C59×#250111yescharacteristic
585C2C59×#262311yescharacteristic
586C2C59×#284911yescharacteristic
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. C23×C4⋊C4×C59C2semidirect
2. C2×C59×#399C2semidirect
5. C2×C59×#385C2semidirect
6. C59×#1445C2semidirect
7. C59×#1444C2semidirect
8. C59×#1439C2semidirect
9. C59×C22⋊#95C2semidirect
64. C2×C4⋊C4×C59D4semidirect
240. #14777C59direct (N × H)
258. C22×C59#399semidirect
283. C23×C4⋊C4C2×C59semidirect
284. C2×#399C2×C59semidirect
287. C2×#385C2×C59semidirect
288. #1445C2×C59semidirect
289. #1444C2×C59semidirect
290. #1439C2×C59semidirect
291. C22⋊#95C2×C59semidirect
326. C59#14777direct (N × H)
358. C2×C4⋊C4D4×C59semidirect
551. C22C59×#399semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C4×C592
C2×C59×#3992
C2×C59×#3862
C2×C59×#3852
C59×#14452
C59×#14442
C59×#14392
C59×C22⋊#952
#1477759