Finite Group Databasebeta

← C59×#18258 #28430367

Subgroups — C59×#18258 #28430367

Order 15104.

1276 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3538C211yesmaximal, characteristic, complemented
2C59×#3003C211yesmaximal, characteristic
3C2×C59×#450C211yesmaximal, characteristic, complemented
4C59×#2435C211yesmaximal, characteristic, complemented
5C59×#1599C214yesmaximal, complemented
6C59×#1597C214yesmaximal, complemented
7C59×#1508C211yesmaximal, characteristic, complemented
8C59×#1501C211yesmaximal, characteristic, complemented
9C59×#1472C211yesmaximal, characteristic, complemented
10C23×D4×C59C2211yescharacteristic
11C2×AGL(1,ℤ/8)×C59—21no—
12C22×SD16×C59—21no—
13C59×#563—22no—
14C59×#552—21no—
15C59×#552—21no—
16C59×#551—22no—
17D4×Q8×C59—21no—
18D4×Q8×C59—21no—
19C59×#544—21no—
20C59×#544C2212yescomplemented
21C59×#541—21no—
22C59×#539—21no—
23C59×#539C2212yescomplemented
24C59×#536—21no—
25C2×#124×C59—21no—
26C59×#525C2211yescharacteristic
27C2×C22⋊Q8×C59C2211yescharacteristic
28C2×C22⋊Q8×C59C2211yescharacteristic
29C59×#450—22no—
30C59×#450—21no—
31C59×#450C2214yes—
32C59×#450C2212yes—
33C59×#447—21no—
34C59×#447C2212yes—
35C59×#418—21no—
36C2×#102×C59—21no—
37C59×#407C2211yescharacteristic
38C2×C22⋊C8×C59C2211yescharacteristic
39C59×#394C2214yes—
40C59×#390—21no—
41C59×#379—22no—
42C4×C22⋊C4×C59C2211yescharacteristic
43C59×#328C2212yescomplemented
44C59×#328C2214yescomplemented
45C59×#327C2212yescomplemented
46C59×#324C2214yescomplemented
47C25×C59—21no—
48C2×#41×C59D411yescharacteristic
49C22×Q8×C59—21no—
50C22×Q8×C59D411yescharacteristic
51C22×Q8×C59D411yescharacteristic
52C22×D4×C59—21no—
53C22×D4×C59—21no—
54C22×D4×C59—21no—
55C22×D4×C59—21no—
56C22×D4×C59—21no—
57C22×D4×C59—44no—
58C22×D4×C59C2312yes—
59C23×C4×C59C2311yescharacteristic
60AGL(1,ℤ/8)×C59—44no—
61C2×SD16×C59—24no—
62C2×SD16×C59—22no—
63C2×SD16×C59—44no—
64C2×D8×C59—22no—
65C2×M16×C59—21no—
66C22×C8×C59—21no—
67C4⋊Q8×C59—21no—
68C4⋊Q8×C59—22no—
69C4⋊Q8×C59—22no—
70C4⋊Q8×C59—22no—
71#126×C59—22no—
72#126×C59—22no—
73#126×C59—22no—
74#126×C59—22no—
75#125×C59—21no—
76#125×C59—22no—
77#125×C59—22no—
78#125×C59—22no—
79#125×C59—22no—
80#124×C59—22no—
81#124×C59—44no—
82#123×C59—22no—
83#123×C59—22no—
84#123×C59—22no—
85#123×C59—21no—
86#123×C59—22no—
87C22⋊Q8×C59—21no—
88C22⋊Q8×C59—21no—
89C22⋊Q8×C59—22no—
90C22⋊Q8×C59—22no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—21no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59—21no—
96C22⋊Q8×C59—21no—
97C22⋊Q8×C59—21no—
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59D412yes—
100C22⋊Q8×C59D411yescharacteristic
101C22⋊Q8×C59C2312yes—
102C22⋊Q8×C59C2311yescharacteristic
103#121×C59D411yescharacteristic
104#121×C59C2311yescharacteristic
105#120×C59—21no—
106C4×Q8×C59—21no—
107C4×Q8×C59—21no—
108C4×D4×C59—22no—
109C4×D4×C59—21no—
110C4×D4×C59—21no—
111C4×D4×C59—22no—
112#117×C59—21no—
113#117×C59—22no—
114#117×C59—22no—
115#117×C59D411yescharacteristic
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59D411yescharacteristic
118C2×C4⋊C4×C59D411yescharacteristic
119C2×C4⋊C4×C59C2312yes—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—24no—
123C2×C22⋊C4×C59—24no—
124C2×C22⋊C4×C59D412yes—
125C2×C42×C59—21no—
126#102×C59—24no—
127#102×C59—22no—
128#102×C59—22no—
129C22⋊C8×C59—21no—
130C22⋊C8×C59—21no—
131C22⋊C8×C59C2312yes—
132C22⋊C8×C59C2312yes—
133#95×C59C2312yes—
134C24×C59—22no—
135C24×C59—44no—
136C24×C59—44no—
137C24×C59—41no—
138C24×C59—41no—
139C24×C59—82no—
140C24×C59C2×D411yescharacteristic
141#41×C59—42no—
142C2×Q8×C59—22no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—42no—
147C2×Q8×C59—42no—
148C2×Q8×C59—82no—
149C2×Q8×C59C2×D412yes—
150C2×Q8×C59C2×D411yescharacteristic
151C2×D4×C59—24no—
152C2×D4×C59—24no—
153C2×D4×C59—24no—
154C2×D4×C59—21no—
155C2×D4×C59—21no—
156C2×D4×C59—21no—
157C2×D4×C59—22no—
158C2×D4×C59—22no—
159C2×D4×C59—42no—
160C2×D4×C59—42no—
161C2×D4×C59—42no—
162C2×D4×C59—44no—
163C2×D4×C59—44no—
164C2×D4×C59—44no—
165C2×D4×C59—41no—
166C2×D4×C59—41no—
167C2×D4×C59C2×D411yescharacteristic
168C22×C4×C59—21no—
169C22×C4×C59—22no—
170C22×C4×C59—21no—
171C22×C4×C59—21no—
172C22×C4×C59—22no—
173C22×C4×C59—42no—
174C22×C4×C59C2×D412yes—
175C22×C4×C59C2×D411yescharacteristic
176C22×C4×C59C2×D411yescharacteristic
177C22×C4×C59C2×D411yescharacteristic
178C22×C4×C59C2×D412yes—
179C22×C4×C59C2×D411yescharacteristic
180C22×C4×C59C2×D411yescharacteristic
181C22×C4×C59C2411yescharacteristic
182SD16×C59—48no—
183SD16×C59—44no—
184D8×C59—44no—
185M16×C59—42no—
186C2×C8×C59—22no—
187C2×C8×C59—22no—
188C2×C8×C59—42no—
189C4⋊C4×C59—22no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—22no—
192C4⋊C4×C59—22no—
193C4⋊C4×C59—22no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—22no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—21no—
200C4⋊C4×C59—21no—
201C4⋊C4×C59—21no—
202C4⋊C4×C59—22no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59—22no—
205C4⋊C4×C59—21no—
206C4⋊C4×C59C2×D412yes—
207C4⋊C4×C59C2×D411yescharacteristic
208C4⋊C4×C59C2×D411yescharacteristic
209C22⋊C4×C59—22no—
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—22no—
213C22⋊C4×C59—22no—
214C22⋊C4×C59—22no—
215C22⋊C4×C59—21no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—22no—
218C22⋊C4×C59—21no—
219C22⋊C4×C59—21no—
220C22⋊C4×C59—21no—
221C22⋊C4×C59—42no—
222C22⋊C4×C59—48no—
223C22⋊C4×C59—84no—
224C42×C59—22no—
225C42×C59—22no—
226C42×C59—22no—
227C42×C59—21no—
228C42×C59—42no—
229C23×C59—22no—
230C23×C59—22no—
231C23×C59—24no—
232C23×C59—21no—
233C23×C59—41no—
234C23×C59—44no—
235C23×C59—44no—
236C23×C59—42no—
237C23×C59—44no—
238C23×C59—42no—
239C23×C59—44no—
240C23×C59—82no—
241C23×C59—82no—
242C23×C59—82no—
243C23×C59—816no—
244C23×C59—82no—
245C23×C59#12011yescharacteristic
246C23×C59#12011yescharacteristic
247C23×C59C22×D411yescharacteristic
248Q8×C59—42no—
249Q8×C59—41no—
250Q8×C59—42no—
251Q8×C59—42no—
252Q8×C59—81no—
253Q8×C59—82no—
254D4×C59—24no—
255D4×C59—24no—
256D4×C59—41no—
257D4×C59—42no—
258D4×C59—42no—
259D4×C59—44no—
260D4×C59—42no—
261D4×C59—42no—
262D4×C59—44no—
263D4×C59—81no—
264C2×C4×C59—22no—
265C2×C4×C59—22no—
266C2×C4×C59—22no—
267C2×C4×C59—21no—
268C2×C4×C59—21no—
269C2×C4×C59—22no—
270C2×C4×C59—22no—
271C2×C4×C59—21no—
272C2×C4×C59—22no—
273C2×C4×C59—22no—
274C2×C4×C59—24no—
275C2×C4×C59—41no—
276C2×C4×C59—41no—
277C2×C4×C59—41no—
278C2×C4×C59—42no—
279C2×C4×C59—42no—
280C2×C4×C59—42no—
281C2×C4×C59—42no—
282C2×C4×C59—42no—
283C2×C4×C59—42no—
284C2×C4×C59—81no—
285C2×C4×C59—82no—
286C2×C4×C59#12011yescharacteristic
287C2×C4×C59#12011yescharacteristic
288C2×C4×C59C22×D411yescharacteristic
289C2×C4×C59C22×D411yescharacteristic
290C8×C59—42no—
291C8×C59—42no—
292#18258C5911yesmaximal, characteristic, direct product factor, Sylow
293C22×C59—21no—
294C22×C59—21no—
295C22×C59—21no—
296C22×C59—21no—
297C22×C59—21no—
298C22×C59—21no—
299C22×C59—21no—
300C22×C59—44no—
301C22×C59—48no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59—44no—
306C22×C59—84no—
307C22×C59—88no—
308C22×C59—88no—
309C22×C59—82no—
310C22×C59—82no—
311C22×C59#45412yes—
312C22×C59C2×#12011yescharacteristic
313C22×C59#57712yes—
314C4×C59—21no—
315C4×C59—21no—
316C4×C59—21no—
317C4×C59—42no—
318C4×C59—41no—
319C4×C59—42no—
320C4×C59—41no—
321C4×C59—41no—
322C4×C59—42no—
323C4×C59—42no—
324C4×C59—82no—
325C4×C59B(4,𝔽2)12yes—
326#3538C2×C5911yescharacteristic, complemented
327#3003C2×C5911yescharacteristic
328C2×#450C2×C5911yescharacteristic, complemented
329#2435C2×C5911yescharacteristic, complemented
330#1599C2×C5914yescomplemented
331#1597C2×C5914yescomplemented
332#1508C2×C5911yescharacteristic, complemented
333#1501C2×C5911yescharacteristic, complemented
334#1472C2×C5911yescharacteristic, complemented
335C2×C59—22no—
336C2×C59—22no—
337C2×C59—41no—
338C2×C59—81no—
339C2×C59—84no—
340C2×C59#301011yescharacteristic
341C2×C59C2×#45411yescharacteristic
342C2×C59C2×B(4,𝔽2)11yescharacteristic
343C23×D4C22×C5911yescharacteristic
344C2×AGL(1,ℤ/8)—21no—
345C22×SD16—21no—
346#563—22no—
347#552—21no—
348#552—21no—
349#551—22no—
350D4×Q8—21no—
351D4×Q8—21no—
352#544—21no—
353#544C22×C5912yescomplemented
354#541—21no—
355#539—21no—
356#539C22×C5912yescomplemented
357#536—21no—
358C2×#124—21no—
359#525C22×C5911yescharacteristic
360C2×C22⋊Q8C22×C5911yescharacteristic
361C2×C22⋊Q8C22×C5911yescharacteristic
362#450—22no—
363#450—21no—
364#450C22×C5914yes—
365#450C22×C5912yes—
366#447—21no—
367#447C22×C5912yes—
368#418—21no—
369C2×#102—21no—
370#407C22×C5911yescharacteristic
371C2×C22⋊C8C22×C5911yescharacteristic
372#394C22×C5914yes—
373#390—21no—
374#379—22no—
375C4×C22⋊C4C22×C5911yescharacteristic
376#328C22×C5914yescomplemented
377#328C22×C5912yescomplemented
378#327C22×C5912yescomplemented
379#324C22×C5914yescomplemented
380C59#1825811yescharacteristic, direct product factor, Sylow
381C25—21no—
382C2×#41D4×C5911yescharacteristic
383C22×Q8—21no—
384C22×Q8D4×C5911yescharacteristic
385C22×Q8D4×C5911yescharacteristic
386C22×D4—21no—
387C22×D4—21no—
388C22×D4—21no—
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—44no—
392C22×D4C23×C5912yes—
393C23×C4C23×C5911yescharacteristic
394AGL(1,ℤ/8)—44no—
395C2×SD16—24no—
396C2×SD16—22no—
397C2×SD16—44no—
398C2×D8—22no—
399C2×M16—21no—
400C22×C8—21no—
401C4⋊Q8—21no—
402C4⋊Q8—22no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#126—22no—
406#126—22no—
407#126—22no—
408#126—22no—
409#125—21no—
410#125—22no—
411#125—22no—
412#125—22no—
413#125—22no—
414#124—22no—
415#124—44no—
416#123—22no—
417#123—22no—
418#123—22no—
419#123—21no—
420#123—22no—
421C22⋊Q8—21no—
422C22⋊Q8—21no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—21no—
431C22⋊Q8—21no—
432C22⋊Q8—22no—
433C22⋊Q8D4×C5912yes—
434C22⋊Q8D4×C5911yescharacteristic
435C22⋊Q8C23×C5912yes—
436C22⋊Q8C23×C5911yescharacteristic
437#121D4×C5911yescharacteristic
438#121C23×C5911yescharacteristic
439#120—21no—
440C4×Q8—21no—
441C4×Q8—21no—
442C4×D4—22no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4—22no—
446#117—21no—
447#117—22no—
448#117—22no—
449#117D4×C5911yescharacteristic
450C2×C4⋊C4—21no—
451C2×C4⋊C4D4×C5911yescharacteristic
452C2×C4⋊C4D4×C5911yescharacteristic
453C2×C4⋊C4C23×C5912yes—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—24no—
457C2×C22⋊C4—24no—
458C2×C22⋊C4D4×C5912yes—
459C2×C42—21no—
460#102—24no—
461#102—22no—
462#102—22no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8C23×C5912yes—
466C22⋊C8C23×C5912yes—
467#95C23×C5912yes—
468C24—22no—
469C24—44no—
470C24—44no—
471C24—41no—
472C24—41no—
473C24—82no—
474C24C2×D4×C5911yescharacteristic
475#41—42no—
476C2×Q8—22no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—82no—
483C2×Q8C2×D4×C5912yes—
484C2×Q8C2×D4×C5911yescharacteristic
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—24no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4—44no—
499C2×D4—41no—
500C2×D4—41no—
501C2×D4C2×D4×C5911yescharacteristic
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—42no—
508C22×C4C2×D4×C5912yes—
509C22×C4C2×D4×C5911yescharacteristic
510C22×C4C2×D4×C5911yescharacteristic
511C22×C4C2×D4×C5911yescharacteristic
512C22×C4C2×D4×C5912yes—
513C22×C4C2×D4×C5911yescharacteristic
514C22×C4C2×D4×C5911yescharacteristic
515C22×C4C24×C5911yescharacteristic
516SD16—48no—
517SD16—44no—
518D8—44no—
519M16—42no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—42no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4C2×D4×C5912yes—
541C4⋊C4C2×D4×C5911yescharacteristic
542C4⋊C4C2×D4×C5911yescharacteristic
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—48no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—42no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—21no—
567C23—41no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—816no—
578C23—82no—
579C23#120×C5911yescharacteristic
580C23#120×C5911yescharacteristic
581C23C22×D4×C5911yescharacteristic
582Q8—42no—
583Q8—41no—
584Q8—42no—
585Q8—42no—
586Q8—81no—
587Q8—82no—
588D4—24no—
589D4—24no—
590D4—41no—
591D4—42no—
592D4—42no—
593D4—44no—
594D4—42no—
595D4—42no—
596D4—44no—
597D4—81no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4#120×C5911yescharacteristic
621C2×C4#120×C5911yescharacteristic
622C2×C4C22×D4×C5911yescharacteristic
623C2×C4C22×D4×C5911yescharacteristic
624C8—42no—
625C8—42no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—44no—
634C22—48no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—44no—
639C22—84no—
640C22—88no—
641C22—88no—
642C22—82no—
643C22—82no—
644C22C59×#45412yes—
645C22C2×#120×C5911yescharacteristic
646C22C59×#57712yes—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4C59×B(4,𝔽2)12yes—
659C2—22no—
660C2—22no—
661C2—41no—
662C2—84no—
663C2—81no—
664C2C59×#301011yescharacteristic
665C2C2×C59×#45411yescharacteristic
666C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3003C2
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
1. C59×#3538C2semidirect
3. C2×C59×#450C2semidirect
4. C59×#2435C2semidirect
5. C59×#1599C2semidirect
6. C59×#1597C2semidirect
7. C59×#1508C2semidirect
8. C59×#1501C2semidirect
9. C59×#1472C2semidirect
20. C59×#544C22semidirect
23. C59×#539C22semidirect
43. C59×#328C22semidirect
44. C59×#328C22semidirect
45. C59×#327C22semidirect
46. C59×#324C22semidirect
292. #18258C59direct (N × H)
326. #3538C2×C59semidirect
328. C2×#450C2×C59semidirect
329. #2435C2×C59semidirect
330. #1599C2×C59semidirect
331. #1597C2×C59semidirect
332. #1508C2×C59semidirect
333. #1501C2×C59semidirect
334. #1472C2×C59semidirect
353. #544C22×C59semidirect
356. #539C22×C59semidirect
376. #328C22×C59semidirect
377. #328C22×C59semidirect
378. #327C22×C59semidirect
379. #324C22×C59semidirect
380. C59#18258direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35382
C59×#30032
C2×C59×#4502
C59×#24352
C59×#15992
C59×#15972
C59×#15082
C59×#15012
C59×#14722
#1825859