Finite Group Databasebeta

← C59×#18297 #28430406

Subgroups — C59×#18297 #28430406

Order 15104.

1084 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C59×#538C211yesmaximal, characteristic
2C59×#3189C211yesmaximal, characteristic, complemented
3C59×#3187C211yesmaximal, characteristic, complemented
4C59×#2626C211yesmaximal, characteristic, complemented
5C59×#1618C214yesmaximal, complemented
6C59×#1606C214yesmaximal, complemented
7C59×#1520C211yesmaximal, characteristic, complemented
8C59×#1519C211yesmaximal, characteristic, complemented
9C2×C59×#323C211yesmaximal, characteristic, complemented
10C59×#538—22no—
11C59×#538—22no—
12C59×#538—22no—
13C59×#538C2214yes—
14C2×#126×C59—21no—
15C2×#124×C59—21no—
16C2×#123×C59—21no—
17C2×C22⋊Q8×C59C2211yescharacteristic
18C2×C22⋊Q8×C59C2211yescharacteristic
19C2×#121×C59—21no—
20C2×#121×C59C2211yescharacteristic
21C2×#120×C59—21no—
22C2×C4×D4×C59—21no—
23C22×C22⋊C4×C59C2211yescharacteristic
24C59×#484—21no—
25C59×#484C2212yescomplemented
26C59×#483—21no—
27C59×#481C2212yescomplemented
28C59×#467—22no—
29C59×#465—22no—
30C59×#451—21no—
31C59×#451C2212yescomplemented
32C59×#450—21no—
33C59×#448C2212yescomplemented
34C2×C22⋊C8×C59C2211yescharacteristic
35C2×C22⋊C8×C59C2211yescharacteristic
36C59×#399C2212yescomplemented
37C59×#396—22no—
38C59×#393C2212yescomplemented
39C59×#385—21no—
40C59×#380C2211yescharacteristic
41C59×#379—21no—
42C59×#328C2214yescomplemented
43C59×#327C2214yescomplemented
44C59×#323C2214yescomplemented
45C25×C59D411yescharacteristic
46C22×Q8×C59D411yescharacteristic
47C22×Q8×C59C2311yescharacteristic
48C22×D4×C59—21no—
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C22×D4×C59D411yescharacteristic
52C23×C4×C59—21no—
53C23×C4×C59D411yescharacteristic
54C23×C4×C59C2311yescharacteristic
55C2×Q16×C59—22no—
56C2×SD16×C59—22no—
57C22×C8×C59—21no—
58C22×C8×C59—21no—
59#126×C59—24no—
60#126×C59—24no—
61#124×C59—24no—
62#124×C59—24no—
63#123×C59—24no—
64#123×C59—24no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C22⋊Q8×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59D412yes—
72C22⋊Q8×C59C2312yes—
73#121×C59—24no—
74#121×C59—24no—
75#121×C59—21no—
76#121×C59—21no—
77#121×C59D412yes—
78#121×C59C2312yes—
79#120×C59—24no—
80#120×C59—24no—
81C4×D4×C59—24no—
82C4×D4×C59—24no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—21no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59D411yescharacteristic
89C2×C4⋊C4×C59D411yescharacteristic
90C2×C4⋊C4×C59C2311yescharacteristic
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59—21no—
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59—22no—
104C2×C22⋊C4×C59D412yes—
105C2×C22⋊C4×C59C2312yes—
106C2×C42×C59—21no—
107#107×C59—22no—
108#106×C59—22no—
109#103×C59—22no—
110#103×C59—22no—
111#103×C59—22no—
112#102×C59—22no—
113C22⋊C8×C59—21no—
114C22⋊C8×C59—21no—
115C22⋊C8×C59C2312yes—
116C22⋊C8×C59C2312yes—
117#95×C59—22no—
118#95×C59C2312yes—
119C24×C59—21no—
120C24×C59—21no—
121C24×C59—21no—
122C24×C59—22no—
123C24×C59—22no—
124C24×C59—81no—
125C24×C59—81no—
126C24×C59C2×D412yes—
127C24×C59C2×D411yescharacteristic
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—81no—
132C2×Q8×C59—81no—
133C2×Q8×C59C2×D412yes—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—24no—
137C2×D4×C59—24no—
138C2×D4×C59—21no—
139C2×D4×C59—42no—
140C2×D4×C59—42no—
141C2×D4×C59—81no—
142C2×D4×C59—82no—
143C2×D4×C59—82no—
144C2×D4×C59C2×D412yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—24no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—22no—
153C22×C4×C59—21no—
154C22×C4×C59—81no—
155C22×C4×C59—81no—
156C22×C4×C59—82no—
157C22×C4×C59C2×D412yes—
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2411yescharacteristic
164Q16×C59—82no—
165SD16×C59—82no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—81no—
169C2×C8×C59—81no—
170C4⋊C4×C59—24no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—41no—
178C4⋊C4×C59—41no—
179C4⋊C4×C59—81no—
180C4⋊C4×C59—82no—
181C4⋊C4×C59C2×D412yes—
182C4⋊C4×C59C2×D412yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—24no—
185C22⋊C4×C59—24no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—41no—
195C22⋊C4×C59—41no—
196C22⋊C4×C59—42no—
197C22⋊C4×C59—82no—
198C22⋊C4×C59—82no—
199C42×C59—24no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—42no—
207C23×C59—42no—
208C23×C59—42no—
209C23×C59—42no—
210C23×C59—42no—
211C23×C59—42no—
212C23×C59—42no—
213C23×C59—42no—
214C23×C59—42no—
215C23×C59—81no—
216C23×C59—81no—
217C23×C59—84no—
218C23×C59—84no—
219C23×C59#12011yescharacteristic
220C23×C59#12011yescharacteristic
221C23×C59#13512yes—
222C23×C59#13712yes—
223C23×C59C22×D411yescharacteristic
224Q8×C59—42no—
225Q8×C59—42no—
226Q8×C59—42no—
227Q8×C59—81no—
228D4×C59—42no—
229D4×C59—42no—
230D4×C59—42no—
231D4×C59—44no—
232D4×C59—81no—
233D4×C59—84no—
234D4×C59—84no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
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—41no—
255C2×C4×C59—41no—
256C2×C4×C59—42no—
257C2×C4×C59—81no—
258C2×C4×C59—84no—
259C2×C4×C59#12011yescharacteristic
260C2×C4×C59#12011yescharacteristic
261C2×C4×C59C22×D411yescharacteristic
262C2×C4×C59C22×D411yescharacteristic
263C8×C59—42no—
264C8×C59—42no—
265#18297C5911yesmaximal, characteristic, direct product factor, Sylow
266C22×C59—21no—
267C22×C59—21no—
268C22×C59—21no—
269C22×C59—21no—
270C22×C59—21no—
271C22×C59—21no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—44no—
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—42no—
281C22×C59—82no—
282C22×C59—84no—
283C22×C59—84no—
284C22×C59—84no—
285C22×C59#45212yes—
286C22×C59#45212yes—
287C22×C59C2×#12011yescharacteristic
288C22×C59C2×#13511yescharacteristic
289C22×C59C2×#13711yescharacteristic
290C4×C59—21no—
291C4×C59—21no—
292C4×C59—21no—
293C4×C59—21no—
294C4×C59—42no—
295C4×C59—42no—
296C4×C59—42no—
297C4×C59—42no—
298C4×C59—82no—
299C4×C59—81no—
300C4×C59—81no—
301C2×#538C2×C5911yescharacteristic
302#3189C2×C5911yescharacteristic, complemented
303#3187C2×C5911yescharacteristic, complemented
304#2626C2×C5911yescharacteristic, complemented
305#1618C2×C5914yescomplemented
306#1606C2×C5914yescomplemented
307#1520C2×C5911yescharacteristic, complemented
308#1519C2×C5911yescharacteristic, complemented
309C2×#323C2×C5911yescharacteristic, complemented
310C2×C59—21no—
311C2×C59—21no—
312C2×C59—22no—
313C2×C59—22no—
314C2×C59—82no—
315C2×C59—82no—
316C2×C59C2×#45211yescharacteristic
317C2×C59#301911yescharacteristic
318C2×C59#302611yescharacteristic
319#538—22no—
320#538—22no—
321#538—22no—
322#538C22×C5914yes—
323C2×#126—21no—
324C2×#124—21no—
325C2×#123—21no—
326C2×C22⋊Q8C22×C5911yescharacteristic
327C2×C22⋊Q8C22×C5911yescharacteristic
328C2×#121—21no—
329C2×#121C22×C5911yescharacteristic
330C2×#120—21no—
331C2×C4×D4—21no—
332C22×C22⋊C4C22×C5911yescharacteristic
333#484—21no—
334#484C22×C5912yescomplemented
335#483—21no—
336#481C22×C5912yescomplemented
337#467—22no—
338#465—22no—
339#451—21no—
340#451C22×C5912yescomplemented
341#450—21no—
342#448C22×C5912yescomplemented
343C2×C22⋊C8C22×C5911yescharacteristic
344C2×C22⋊C8C22×C5911yescharacteristic
345#399C22×C5912yescomplemented
346#396—22no—
347#393C22×C5912yescomplemented
348#385—21no—
349#380C22×C5911yescharacteristic
350#379—21no—
351#328C22×C5914yescomplemented
352#327C22×C5914yescomplemented
353#323C22×C5914yescomplemented
354C59#1829711yescharacteristic, direct product factor, Sylow
355C25D4×C5911yescharacteristic
356C22×Q8D4×C5911yescharacteristic
357C22×Q8C23×C5911yescharacteristic
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4D4×C5911yescharacteristic
362C23×C4—21no—
363C23×C4D4×C5911yescharacteristic
364C23×C4C23×C5911yescharacteristic
365C2×Q16—22no—
366C2×SD16—22no—
367C22×C8—21no—
368C22×C8—21no—
369#126—24no—
370#126—24no—
371#124—24no—
372#124—24no—
373#123—24no—
374#123—24no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8D4×C5912yes—
382C22⋊Q8C23×C5912yes—
383#121—24no—
384#121—24no—
385#121—21no—
386#121—21no—
387#121D4×C5912yes—
388#121C23×C5912yes—
389#120—24no—
390#120—24no—
391C4×D4—24no—
392C4×D4—24no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4D4×C5911yescharacteristic
399C2×C4⋊C4D4×C5911yescharacteristic
400C2×C4⋊C4C23×C5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4D4×C5912yes—
415C2×C22⋊C4C23×C5912yes—
416C2×C42—21no—
417#107—22no—
418#106—22no—
419#103—22no—
420#103—22no—
421#103—22no—
422#102—22no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8C23×C5912yes—
426C22⋊C8C23×C5912yes—
427#95—22no—
428#95C23×C5912yes—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—81no—
435C24—81no—
436C24C2×D4×C5912yes—
437C24C2×D4×C5911yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8C2×D4×C5912yes—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4C2×D4×C5912yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4—82no—
467C22×C4C2×D4×C5912yes—
468C22×C4C2×D4×C5911yescharacteristic
469C22×C4C2×D4×C5911yescharacteristic
470C22×C4C2×D4×C5911yescharacteristic
471C22×C4C2×D4×C5911yescharacteristic
472C22×C4C2×D4×C5911yescharacteristic
473C22×C4C24×C5911yescharacteristic
474Q16—82no—
475SD16—82no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—81no—
479C2×C8—81no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4C2×D4×C5912yes—
492C4⋊C4C2×D4×C5912yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—81no—
527C23—84no—
528C23—84no—
529C23#120×C5911yescharacteristic
530C23#120×C5911yescharacteristic
531C23#135×C5912yes—
532C23#137×C5912yes—
533C23C22×D4×C5911yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—81no—
538D4—42no—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—81no—
543D4—84no—
544D4—84no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
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—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—81no—
568C2×C4—84no—
569C2×C4#120×C5911yescharacteristic
570C2×C4#120×C5911yescharacteristic
571C2×C4C22×D4×C5911yescharacteristic
572C2×C4C22×D4×C5911yescharacteristic
573C8—42no—
574C8—42no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22C59×#45212yes—
595C22C59×#45212yes—
596C22C2×#120×C5911yescharacteristic
597C22C2×#135×C5911yescharacteristic
598C22C2×#137×C5911yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—82no—
615C2—82no—
616C2C2×C59×#45211yescharacteristic
617C2C59×#301911yescharacteristic
618C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC2×C59×#538C2
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. C59×#3189C2semidirect
3. C59×#3187C2semidirect
4. C59×#2626C2semidirect
5. C59×#1618C2semidirect
6. C59×#1606C2semidirect
7. C59×#1520C2semidirect
8. C59×#1519C2semidirect
9. C2×C59×#323C2semidirect
25. C59×#484C22semidirect
27. C59×#481C22semidirect
31. C59×#451C22semidirect
33. C59×#448C22semidirect
36. C59×#399C22semidirect
38. C59×#393C22semidirect
42. C59×#328C22semidirect
43. C59×#327C22semidirect
44. C59×#323C22semidirect
265. #18297C59direct (N × H)
302. #3189C2×C59semidirect
303. #3187C2×C59semidirect
304. #2626C2×C59semidirect
305. #1618C2×C59semidirect
306. #1606C2×C59semidirect
307. #1520C2×C59semidirect
308. #1519C2×C59semidirect
309. C2×#323C2×C59semidirect
334. #484C22×C59semidirect
336. #481C22×C59semidirect
340. #451C22×C59semidirect
342. #448C22×C59semidirect
345. #399C22×C59semidirect
347. #393C22×C59semidirect
351. #328C22×C59semidirect
352. #327C22×C59semidirect
353. #323C22×C59semidirect
354. C59#18297direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C59×#5382
C59×#31892
C59×#31872
C59×#26262
C59×#16182
C59×#16062
C59×#15202
C59×#15192
C2×C59×#3232
#1829759