Finite Group Databasebeta

← C59×#18304 #28430413

Subgroups — C59×#18304 #28430413

Order 15104.

1054 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3455C211yesmaximal, characteristic, complemented
2C59×#3208C211yesmaximal, characteristic, complemented
3C59×#3205C211yesmaximal, characteristic, complemented
4C59×#2619C211yesmaximal, characteristic, complemented
5C59×#1600C214yesmaximal, complemented
6C59×#1597C214yesmaximal, complemented
7C59×#1518C211yesmaximal, characteristic, complemented
8C59×#1517C211yesmaximal, characteristic, complemented
9C59×#1458C211yesmaximal, characteristic, complemented
10C59×#542—21no—
11C59×#542—21no—
12C59×#541—21no—
13C59×#540—21no—
14C59×#539—21no—
15C59×#539C2212yes—
16C59×#538—21no—
17C59×#538C2212yes—
18C59×#533C2211yescharacteristic
19C59×#532C2211yescharacteristic
20C2×#127×C59C2211yescharacteristic
21C59×#529—22no—
22C59×#529—22no—
23C59×#525—21no—
24C2×#121×C59—21no—
25C4×#41×C59—21no—
26C2×#117×C59C2211yescharacteristic
27C59×#493—21no—
28C59×#492—21no—
29C59×#486—21no—
30C59×#486—21no—
31C59×#460—22no—
32C59×#459—22no—
33C59×#450C2214yes—
34C59×#447C2214yes—
35C59×#431C2211yescharacteristic
36C59×#431C2211yescharacteristic
37C59×#394—21no—
38C59×#394C2212yes—
39C59×#392—21no—
40C59×#392C2212yes—
41C59×#390—21no—
42C59×#382C2211yescharacteristic
43C59×#328C2214yes—
44C59×#327C2214yes—
45C59×#324C2212yes—
46C59×#323C2212yes—
47C2×#41×C59—21no—
48C22×D4×C59—22no—
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C22×D4×C59C2312yes—
52C23×C4×C59D411yescharacteristic
53C2×SD16×C59—24no—
54C2×D8×C59—24no—
55C4⋊Q8×C59D411yescharacteristic, complemented
56C4⋊Q8×C59D411yescharacteristic, complemented
57C4⋊Q8×C59D411yescharacteristic, complemented
58#127×C59—21no—
59#127×C59—21no—
60#127×C59—21no—
61#127×C59—21no—
62#127×C59D411yescharacteristic, complemented
63#126×C59—22no—
64#126×C59—22no—
65#126×C59—22no—
66#126×C59—22no—
67#125×C59—22no—
68#125×C59—22no—
69#125×C59—21no—
70#124×C59—22no—
71#124×C59—22no—
72#124×C59—21no—
73#123×C59—22no—
74#123×C59—22no—
75#123×C59—22no—
76#123×C59—22no—
77#123×C59—22no—
78C22⋊Q8×C59—21no—
79C22⋊Q8×C59—21no—
80C22⋊Q8×C59—22no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—21no—
83C22⋊Q8×C59C2312yes—
84#121×C59—21no—
85#121×C59—21no—
86#121×C59—22no—
87#121×C59—22no—
88#121×C59—21no—
89#121×C59—44no—
90#121×C59C2312yes—
91#120×C59—22no—
92C4×Q8×C59—21no—
93C4×Q8×C59—21no—
94C4×Q8×C59—21no—
95C4×Q8×C59D412yescomplemented
96C4×D4×C59—21no—
97C4×D4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59—21no—
103C4×D4×C59D412yescomplemented
104#117×C59—22no—
105#117×C59—21no—
106#117×C59—21no—
107#117×C59—21no—
108#117×C59—21no—
109#117×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59D411yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—22no—
114C2×C22⋊C4×C59—22no—
115C2×C22⋊C4×C59—22no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59C2311yescharacteristic
118C2×C42×C59—22no—
119C2×C42×C59D411yescharacteristic
120C2×C42×C59C2311yescharacteristic
121C4⋊C8×C59—21no—
122C4⋊C8×C59—21no—
123#102×C59—24no—
124#102×C59—24no—
125C22⋊C8×C59C2312yes—
126C22⋊C8×C59C2312yes—
127C4×C8×C59—21no—
128C4×C8×C59—21no—
129#95×C59—21no—
130#95×C59C2312yes—
131C24×C59—22no—
132C24×C59C2×D411yescharacteristic
133#41×C59—42no—
134#41×C59—42no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59C2×D412yes—
138C2×D4×C59—22no—
139C2×D4×C59—22no—
140C2×D4×C59—22no—
141C2×D4×C59—21no—
142C2×D4×C59—21no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—44no—
146C2×D4×C59—41no—
147C2×D4×C59—41no—
148C2×D4×C59—41no—
149C2×D4×C59—41no—
150C2×D4×C59—42no—
151C2×D4×C59—42no—
152C2×D4×C59—82no—
153C2×D4×C59—82no—
154C2×D4×C59—82no—
155C2×D4×C59—82no—
156C2×D4×C59C2×D412yes—
157C22×C4×C59—22no—
158C22×C4×C59—22no—
159C22×C4×C59—22no—
160C22×C4×C59—21no—
161C22×C4×C59—21no—
162C22×C4×C59—22no—
163C22×C4×C59—22no—
164C22×C4×C59—22no—
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2×D411yescharacteristic
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2411yescharacteristic
171SD16×C59—84no—
172D8×C59—84no—
173C2×C8×C59—22no—
174C2×C8×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—22no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—42no—
190C4⋊C4×C59C2×D412yes—
191C4⋊C4×C59C2×D412yes—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—22no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—22no—
199C22⋊C4×C59—21no—
200C22⋊C4×C59—21no—
201C22⋊C4×C59—22no—
202C22⋊C4×C59—21no—
203C22⋊C4×C59—21no—
204C22⋊C4×C59—22no—
205C22⋊C4×C59—44no—
206C22⋊C4×C59—82no—
207C22⋊C4×C59—82no—
208C42×C59—22no—
209C42×C59—22no—
210C42×C59—21no—
211C42×C59—22no—
212C42×C59—21no—
213C42×C59C2×D411yescharacteristic
214C42×C59C2×D411yescharacteristic
215C42×C59C2×D411yescharacteristic
216C42×C59C2×D411yescharacteristic
217C23×C59—21no—
218C23×C59—22no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—42no—
224C23×C59—42no—
225C23×C59—42no—
226C23×C59—42no—
227C23×C59—42no—
228C23×C59—42no—
229C23×C59—42no—
230C23×C59C22×D411yescharacteristic
231Q8×C59—42no—
232Q8×C59—42no—
233Q8×C59—81no—
234D4×C59—42no—
235D4×C59—44no—
236D4×C59—44no—
237D4×C59—42no—
238D4×C59—42no—
239D4×C59—42no—
240D4×C59—81no—
241D4×C59—82no—
242D4×C59—82no—
243D4×C59—82no—
244D4×C59—82no—
245D4×C59—88no—
246D4×C59—82no—
247D4×C59—82no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—22no—
252C2×C4×C59—22no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—42no—
259C2×C4×C59—42no—
260C2×C4×C59—44no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—42no—
264C2×C4×C59—42no—
265C2×C4×C59—42no—
266C2×C4×C59—82no—
267C2×C4×C59—82no—
268C2×C4×C59#12011yescharacteristic
269C2×C4×C59#12011yescharacteristic
270C2×C4×C59#12011yescharacteristic
271C2×C4×C59#12011yescharacteristic
272C2×C4×C59#13512yes—
273C2×C4×C59AGL(1,ℤ/8)12yes—
274C2×C4×C59C22×D411yescharacteristic
275C2×C4×C59C22×D411yescharacteristic
276C8×C59—42no—
277C8×C59—42no—
278#18304C5911yesmaximal, characteristic, direct product factor, Sylow
279C22×C59—21no—
280C22×C59—21no—
281C22×C59—42no—
282C22×C59—41no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59—41no—
286C22×C59—42no—
287C22×C59—41no—
288C22×C59—81no—
289C22×C59—82no—
290C22×C59—82no—
291C22×C59—82no—
292C22×C59—82no—
293C22×C59—82no—
294C22×C59—82no—
295C22×C59—82no—
296C22×C59C2×#12011yescharacteristic
297C22×C59C2×#13511yescharacteristic
298C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C59—21no—
300C4×C59—21no—
301C4×C59—21no—
302C4×C59—21no—
303C4×C59—41no—
304C4×C59—42no—
305C4×C59—41no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—82no—
310C4×C59#44912yes—
311C4×C59#44912yes—
312#3455C2×C5911yescharacteristic, complemented
313#3208C2×C5911yescharacteristic, complemented
314#3205C2×C5911yescharacteristic, complemented
315#2619C2×C5911yescharacteristic, complemented
316#1600C2×C5914yescomplemented
317#1597C2×C5914yescomplemented
318#1518C2×C5911yescharacteristic, complemented
319#1517C2×C5911yescharacteristic, complemented
320#1458C2×C5911yescharacteristic, complemented
321C2×C59—21no—
322C2×C59—21no—
323C2×C59—82no—
324C2×C59—81no—
325C2×C59—162no—
326C2×C59C2×#44911yescharacteristic
327C2×C59#302111yescharacteristic
328C2×C59#302711yescharacteristic
329#542—21no—
330#542—21no—
331#541—21no—
332#540—21no—
333#539—21no—
334#539C22×C5912yes—
335#538—21no—
336#538C22×C5912yes—
337#533C22×C5911yescharacteristic
338#532C22×C5911yescharacteristic
339C2×#127C22×C5911yescharacteristic
340#529—22no—
341#529—22no—
342#525—21no—
343C2×#121—21no—
344C4×#41—21no—
345C2×#117C22×C5911yescharacteristic
346#493—21no—
347#492—21no—
348#486—21no—
349#486—21no—
350#460—22no—
351#459—22no—
352#450C22×C5914yes—
353#447C22×C5914yes—
354#431C22×C5911yescharacteristic
355#431C22×C5911yescharacteristic
356#394—21no—
357#394C22×C5912yes—
358#392—21no—
359#392C22×C5912yes—
360#390—21no—
361#382C22×C5911yescharacteristic
362#328C22×C5914yes—
363#327C22×C5914yes—
364#324C22×C5912yes—
365#323C22×C5912yes—
366C59#1830411yescharacteristic, direct product factor, Sylow
367C2×#41—21no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—21no—
371C22×D4C23×C5912yes—
372C23×C4D4×C5911yescharacteristic
373C2×SD16—24no—
374C2×D8—24no—
375C4⋊Q8D4×C5911yescharacteristic, complemented
376C4⋊Q8D4×C5911yescharacteristic, complemented
377C4⋊Q8D4×C5911yescharacteristic, complemented
378#127—21no—
379#127—21no—
380#127—21no—
381#127—21no—
382#127D4×C5911yescharacteristic, complemented
383#126—22no—
384#126—22no—
385#126—22no—
386#126—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#124—22no—
391#124—22no—
392#124—21no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8—21no—
403C22⋊Q8C23×C5912yes—
404#121—21no—
405#121—21no—
406#121—22no—
407#121—22no—
408#121—21no—
409#121—44no—
410#121C23×C5912yes—
411#120—22no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8D4×C5912yescomplemented
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—21no—
423C4×D4D4×C5912yescomplemented
424#117—22no—
425#117—21no—
426#117—21no—
427#117—21no—
428#117—21no—
429#117—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4D4×C5911yescharacteristic
432C2×C4⋊C4C23×C5911yescharacteristic
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—22no—
436C2×C22⋊C4D4×C5911yescharacteristic
437C2×C22⋊C4C23×C5911yescharacteristic
438C2×C42—22no—
439C2×C42D4×C5911yescharacteristic
440C2×C42C23×C5911yescharacteristic
441C4⋊C8—21no—
442C4⋊C8—21no—
443#102—24no—
444#102—24no—
445C22⋊C8C23×C5912yes—
446C22⋊C8C23×C5912yes—
447C4×C8—21no—
448C4×C8—21no—
449#95—21no—
450#95C23×C5912yes—
451C24—22no—
452C24C2×D4×C5911yescharacteristic
453#41—42no—
454#41—42no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8C2×D4×C5912yes—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—82no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4—82no—
476C2×D4C2×D4×C5912yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4C2×D4×C5911yescharacteristic
486C22×C4C2×D4×C5911yescharacteristic
487C22×C4C2×D4×C5911yescharacteristic
488C22×C4C2×D4×C5911yescharacteristic
489C22×C4C2×D4×C5911yescharacteristic
490C22×C4C24×C5911yescharacteristic
491SD16—84no—
492D8—84no—
493C2×C8—22no—
494C2×C8—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—42no—
510C4⋊C4C2×D4×C5912yes—
511C4⋊C4C2×D4×C5912yes—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—44no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—22no—
532C42—21no—
533C42C2×D4×C5911yescharacteristic
534C42C2×D4×C5911yescharacteristic
535C42C2×D4×C5911yescharacteristic
536C42C2×D4×C5911yescharacteristic
537C23—21no—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C22×D4×C5911yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—81no—
554D4—42no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—81no—
561D4—82no—
562D4—82no—
563D4—82no—
564D4—82no—
565D4—88no—
566D4—82no—
567D4—82no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4#120×C5911yescharacteristic
589C2×C4#120×C5911yescharacteristic
590C2×C4#120×C5911yescharacteristic
591C2×C4#120×C5911yescharacteristic
592C2×C4#135×C5912yes—
593C2×C4AGL(1,ℤ/8)×C5912yes—
594C2×C4C22×D4×C5911yescharacteristic
595C2×C4C22×D4×C5911yescharacteristic
596C8—42no—
597C8—42no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—42no—
606C22—41no—
607C22—81no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C2×#120×C5911yescharacteristic
616C22C2×#135×C5911yescharacteristic
617C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—82no—
629C4C59×#44912yes—
630C4C59×#44912yes—
631C2—21no—
632C2—21no—
633C2—81no—
634C2—82no—
635C2—162no—
636C2C2×C59×#44911yescharacteristic
637C2C59×#302111yescharacteristic
638C2C59×#302711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3455C2semidirect
2. C59×#3208C2semidirect
3. C59×#3205C2semidirect
4. C59×#2619C2semidirect
5. C59×#1600C2semidirect
6. C59×#1597C2semidirect
7. C59×#1518C2semidirect
8. C59×#1517C2semidirect
9. C59×#1458C2semidirect
55. C4⋊Q8×C59D4semidirect
56. C4⋊Q8×C59D4semidirect
57. C4⋊Q8×C59D4semidirect
62. #127×C59D4semidirect
95. C4×Q8×C59D4semidirect
103. C4×D4×C59D4semidirect
278. #18304C59direct (N × H)
312. #3455C2×C59semidirect
313. #3208C2×C59semidirect
314. #3205C2×C59semidirect
315. #2619C2×C59semidirect
316. #1600C2×C59semidirect
317. #1597C2×C59semidirect
318. #1518C2×C59semidirect
319. #1517C2×C59semidirect
320. #1458C2×C59semidirect
366. C59#18304direct (N × H)
375. C4⋊Q8D4×C59semidirect
376. C4⋊Q8D4×C59semidirect
377. C4⋊Q8D4×C59semidirect
382. #127D4×C59semidirect
415. C4×Q8D4×C59semidirect
423. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34552
C59×#32082
C59×#32052
C59×#26192
C59×#16002
C59×#15972
C59×#15182
C59×#15172
C59×#14582
#1830459