Finite Group Databasebeta

← C59×#18206 #28430315

Subgroups — C59×#18206 #28430315

Order 15104.

1028 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
1C59×#3505C211yesmaximal, characteristic
2C2×C59×#483C211yesmaximal, characteristic, complemented
3C2×C59×#480C211yesmaximal, characteristic, complemented
4C59×#2486C211yesmaximal, characteristic, complemented
5C59×#1612C214yesmaximal, complemented
6C59×#1609C214yesmaximal, complemented
7C59×#1520C211yesmaximal, characteristic, complemented
8C59×#1519C211yesmaximal, characteristic, complemented
9C59×#1459C211yesmaximal, characteristic, complemented
10C59×#553—22no—
11C59×#553—21no—
12C59×#552—21no—
13C59×#551—22no—
14C59×#548—21no—
15C59×#546—21no—
16C59×#539C2212yes—
17C59×#538C2212yes—
18C59×#534—21no—
19C59×#529—21no—
20C59×#525—21no—
21C2×#123×C59—21no—
22C2×C22⋊Q8×C59C2211yescharacteristic
23C2×#121×C59C2211yescharacteristic
24C59×#518—21no—
25C2×C4×D4×C59C2211yescharacteristic
26C22×C4⋊C4×C59C2211yescharacteristic
27C59×#483—22no—
28C59×#483C2214yescomplemented
29C59×#480—22no—
30C59×#480C2214yes—
31C2×#107×C59—21no—
32C2×#106×C59—21no—
33C2×#103×C59—21no—
34C2×#102×C59—21no—
35C2×C22⋊C8×C59C2211yescharacteristic
36C2×C22⋊C8×C59C2211yescharacteristic
37C59×#398C2212yescomplemented
38C59×#397C2212yescomplemented
39C59×#389—21no—
40C59×#381—21no—
41C59×#380—21no—
42C59×#380C2211yescharacteristic
43C59×#328C2214yescomplemented
44C59×#327C2214yes—
45C59×#324C2212yescomplemented
46C59×#323C2212yescomplemented
47C2×#41×C59—21no—
48C22×Q8×C59D411yescharacteristic
49C22×D4×C59—21no—
50C22×D4×C59D411yescharacteristic
51C22×D4×C59D411yescharacteristic
52C23×C4×C59—21no—
53C23×C4×C59D411yescharacteristic
54C23×C4×C59C2311yescharacteristic
55C22×C8×C59—21no—
56C22×C8×C59—21no—
57#126×C59—22no—
58#126×C59—22no—
59#126×C59—22no—
60#125×C59—21no—
61#125×C59—22no—
62#124×C59—21no—
63#124×C59—22no—
64#123×C59—22no—
65#123×C59—22no—
66#123×C59—22no—
67#123×C59—22no—
68#123×C59—22no—
69#123×C59—44no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—21no—
73C22⋊Q8×C59—21no—
74C22⋊Q8×C59D412yes—
75C22⋊Q8×C59C2312yes—
76#121×C59—21no—
77#121×C59—22no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—21no—
81#121×C59—21no—
82#121×C59D412yes—
83#121×C59C2312yes—
84#120×C59—22no—
85#120×C59—22no—
86C4×Q8×C59—21no—
87C4×D4×C59—22no—
88C4×D4×C59—22no—
89C4×D4×C59—22no—
90C4×D4×C59—21no—
91C4×D4×C59—21no—
92C4×D4×C59—21no—
93C4×D4×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59—22no—
96C4×D4×C59—22no—
97#117×C59—22no—
98#117×C59—21no—
99#117×C59—22no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—22no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59C2312yes—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—22no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—22no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C22⋊C4×C59C2311yescharacteristic
119C2×C42×C59D411yescharacteristic
120#107×C59—24no—
121#106×C59—24no—
122#103×C59—24no—
123#102×C59—24no—
124C22⋊C8×C59—21no—
125C22⋊C8×C59—21no—
126C22⋊C8×C59C2312yes—
127C22⋊C8×C59C2312yes—
128#95×C59—22no—
129#95×C59—22no—
130#95×C59C2312yes—
131C24×C59—21no—
132C24×C59C2×D411yescharacteristic
133C24×C59C2×D411yescharacteristic
134#41×C59—82no—
135C2×Q8×C59—21no—
136C2×Q8×C59—42no—
137C2×Q8×C59C2×D412yes—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—22no—
141C2×D4×C59—22no—
142C2×D4×C59—22no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—22no—
146C2×D4×C59—21no—
147C2×D4×C59—42no—
148C2×D4×C59—42no—
149C2×D4×C59—42no—
150C2×D4×C59C2×D412yes—
151C22×C4×C59—21no—
152C22×C4×C59—22no—
153C22×C4×C59—22no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—22no—
159C22×C4×C59—22no—
160C22×C4×C59—42no—
161C22×C4×C59—42no—
162C22×C4×C59—41no—
163C22×C4×C59—41no—
164C22×C4×C59—42no—
165C22×C4×C59—42no—
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D411yescharacteristic
171C22×C4×C59C2×D411yescharacteristic
172C22×C4×C59C2×D411yescharacteristic
173C22×C4×C59C2×D411yescharacteristic
174C22×C4×C59C2411yescharacteristic
175C2×C8×C59—22no—
176C2×C8×C59—22no—
177C2×C8×C59—42no—
178C2×C8×C59—42no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—22no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—42no—
192C4⋊C4×C59—44no—
193C4⋊C4×C59—41no—
194C4⋊C4×C59—41no—
195C4⋊C4×C59C2×D412yes—
196C4⋊C4×C59C2×D412yes—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—22no—
199C22⋊C4×C59—22no—
200C22⋊C4×C59—22no—
201C22⋊C4×C59—22no—
202C22⋊C4×C59—22no—
203C22⋊C4×C59—21no—
204C22⋊C4×C59—22no—
205C22⋊C4×C59—21no—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—22no—
209C22⋊C4×C59—21no—
210C22⋊C4×C59—21no—
211C22⋊C4×C59—44no—
212C22⋊C4×C59—42no—
213C42×C59—22no—
214C42×C59—22no—
215C42×C59—21no—
216C42×C59—21no—
217C23×C59—21no—
218C23×C59—22no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—42no—
226C23×C59—81no—
227C23×C59—81no—
228C23×C59#12011yescharacteristic
229C23×C59#12011yescharacteristic
230C23×C59#13512yes—
231C23×C59AGL(1,ℤ/8)12yes—
232C23×C59C22×D411yescharacteristic
233Q8×C59—42no—
234Q8×C59—42no—
235D4×C59—42no—
236D4×C59—42no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—42no—
240D4×C59—44no—
241D4×C59—42no—
242D4×C59—82no—
243D4×C59—82no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—22no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—22no—
252C2×C4×C59—22no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—42no—
259C2×C4×C59—42no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
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—81no—
269C2×C4×C59—81no—
270C2×C4×C59—81no—
271C2×C4×C59—82no—
272C2×C4×C59—82no—
273C2×C4×C59—82no—
274C2×C4×C59—82no—
275C2×C4×C59#12011yescharacteristic
276C2×C4×C59#12011yescharacteristic
277C2×C4×C59C22×D411yescharacteristic
278C2×C4×C59C22×D411yescharacteristic
279C8×C59—42no—
280C8×C59—42no—
281#18206C5911yesmaximal, characteristic, direct product factor, Sylow
282C22×C59—21no—
283C22×C59—21no—
284C22×C59—21no—
285C22×C59—21no—
286C22×C59—21no—
287C22×C59—21no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—41no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—42no—
297C22×C59—81no—
298C22×C59—162no—
299C22×C59#44912yes—
300C22×C59#44912yes—
301C22×C59C2×#12011yescharacteristic
302C22×C59C2×#13511yescharacteristic
303C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
304C4×C59—21no—
305C4×C59—21no—
306C4×C59—21no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—42no—
313C4×C59—82no—
314C4×C59—82no—
315C4×C59#45812yes—
316#3505C2×C5911yescharacteristic
317C2×#483C2×C5911yescharacteristic, complemented
318C2×#480C2×C5911yescharacteristic, complemented
319#2486C2×C5911yescharacteristic, complemented
320#1612C2×C5914yescomplemented
321#1609C2×C5914yescomplemented
322#1520C2×C5911yescharacteristic, complemented
323#1519C2×C5911yescharacteristic, complemented
324#1459C2×C5911yescharacteristic, complemented
325C2×C59—21no—
326C2×C59—21no—
327C2×C59—22no—
328C2×C59—22no—
329C2×C59—81no—
330C2×C59—82no—
331C2×C59C2×#44911yescharacteristic
332C2×C59#302211yescharacteristic
333C2×C59C2×#45811yescharacteristic
334#553—22no—
335#553—21no—
336#552—21no—
337#551—22no—
338#548—21no—
339#546—21no—
340#539C22×C5912yes—
341#538C22×C5912yes—
342#534—21no—
343#529—21no—
344#525—21no—
345C2×#123—21no—
346C2×C22⋊Q8C22×C5911yescharacteristic
347C2×#121C22×C5911yescharacteristic
348#518—21no—
349C2×C4×D4C22×C5911yescharacteristic
350C22×C4⋊C4C22×C5911yescharacteristic
351#483—22no—
352#483C22×C5914yescomplemented
353#480—22no—
354#480C22×C5914yes—
355C2×#107—21no—
356C2×#106—21no—
357C2×#103—21no—
358C2×#102—21no—
359C2×C22⋊C8C22×C5911yescharacteristic
360C2×C22⋊C8C22×C5911yescharacteristic
361#398C22×C5912yescomplemented
362#397C22×C5912yescomplemented
363#389—21no—
364#381—21no—
365#380—21no—
366#380C22×C5911yescharacteristic
367#328C22×C5914yescomplemented
368#327C22×C5914yes—
369#324C22×C5912yescomplemented
370#323C22×C5912yescomplemented
371C59#1820611yescharacteristic, direct product factor, Sylow
372C2×#41—21no—
373C22×Q8D4×C5911yescharacteristic
374C22×D4—21no—
375C22×D4D4×C5911yescharacteristic
376C22×D4D4×C5911yescharacteristic
377C23×C4—21no—
378C23×C4D4×C5911yescharacteristic
379C23×C4C23×C5911yescharacteristic
380C22×C8—21no—
381C22×C8—21no—
382#126—22no—
383#126—22no—
384#126—22no—
385#125—21no—
386#125—22no—
387#124—21no—
388#124—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—44no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8D4×C5912yes—
400C22⋊Q8C23×C5912yes—
401#121—21no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121—21no—
407#121D4×C5912yes—
408#121C23×C5912yes—
409#120—22no—
410#120—22no—
411C4×Q8—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—22no—
421C4×D4—22no—
422#117—22no—
423#117—21no—
424#117—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4D4×C5911yescharacteristic
434C2×C4⋊C4D4×C5911yescharacteristic
435C2×C4⋊C4C23×C5912yes—
436C2×C4⋊C4C23×C5911yescharacteristic
437C2×C22⋊C4—22no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4D4×C5911yescharacteristic
443C2×C22⋊C4C23×C5911yescharacteristic
444C2×C42D4×C5911yescharacteristic
445#107—24no—
446#106—24no—
447#103—24no—
448#102—24no—
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8C23×C5912yes—
452C22⋊C8C23×C5912yes—
453#95—22no—
454#95—22no—
455#95C23×C5912yes—
456C24—21no—
457C24C2×D4×C5911yescharacteristic
458C24C2×D4×C5911yescharacteristic
459#41—82no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C2×D4×C5912yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C5912yes—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4C2×D4×C5911yescharacteristic
492C22×C4C2×D4×C5911yescharacteristic
493C22×C4C2×D4×C5911yescharacteristic
494C22×C4C2×D4×C5911yescharacteristic
495C22×C4C2×D4×C5911yescharacteristic
496C22×C4C2×D4×C5911yescharacteristic
497C22×C4C2×D4×C5911yescharacteristic
498C22×C4C2×D4×C5911yescharacteristic
499C22×C4C24×C5911yescharacteristic
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C2×C8—42no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C5912yes—
521C4⋊C4C2×D4×C5912yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23#120×C5911yescharacteristic
554C23#120×C5911yescharacteristic
555C23#135×C5912yes—
556C23AGL(1,ℤ/8)×C5912yes—
557C23C22×D4×C5911yescharacteristic
558Q8—42no—
559Q8—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—44no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—82no—
568D4—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#120×C5911yescharacteristic
601C2×C4#120×C5911yescharacteristic
602C2×C4C22×D4×C5911yescharacteristic
603C2×C4C22×D4×C5911yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—81no—
622C22—162no—
623C22C59×#44912yes—
624C22C59×#44912yes—
625C22C2×#120×C5911yescharacteristic
626C22C2×#135×C5911yescharacteristic
627C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4C59×#45812yes—
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—81no—
645C2—82no—
646C2C2×C59×#44911yescharacteristic
647C2C59×#302211yescharacteristic
648C2C2×C59×#45811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3505C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C59×#483C2semidirect
3. C2×C59×#480C2semidirect
4. C59×#2486C2semidirect
5. C59×#1612C2semidirect
6. C59×#1609C2semidirect
7. C59×#1520C2semidirect
8. C59×#1519C2semidirect
9. C59×#1459C2semidirect
28. C59×#483C22semidirect
37. C59×#398C22semidirect
38. C59×#397C22semidirect
43. C59×#328C22semidirect
45. C59×#324C22semidirect
46. C59×#323C22semidirect
281. #18206C59direct (N × H)
317. C2×#483C2×C59semidirect
318. C2×#480C2×C59semidirect
319. #2486C2×C59semidirect
320. #1612C2×C59semidirect
321. #1609C2×C59semidirect
322. #1520C2×C59semidirect
323. #1519C2×C59semidirect
324. #1459C2×C59semidirect
352. #483C22×C59semidirect
361. #398C22×C59semidirect
362. #397C22×C59semidirect
367. #328C22×C59semidirect
369. #324C22×C59semidirect
370. #323C22×C59semidirect
371. C59#18206direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35052
C2×C59×#4832
C2×C59×#4802
C59×#24862
C59×#16122
C59×#16092
C59×#15202
C59×#15192
C59×#14592
#1820659