Finite Group Databasebeta

← C59×#18278 #28430387

Subgroups — C59×#18278 #28430387

Order 15104.

910 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3563C211yesmaximal, characteristic, complemented
2C59×#3110C211yesmaximal, characteristic, complemented
3C59×#3105C211yesmaximal, characteristic
4C59×#2562C211yesmaximal, characteristic, complemented
5C59×#1617C212yesmaximal, complemented
6C59×#1615C212yesmaximal, complemented
7C59×#1599C212yesmaximal, complemented
8C59×#1597C212yesmaximal
9C59×#1509C211yesmaximal, characteristic, complemented
10C59×#1503C211yesmaximal, characteristic, complemented
11C59×#1470C211yesmaximal, characteristic, complemented
12C59×#564—21no—
13C59×#563—21no—
14C59×#557—21no—
15C59×#554—21no—
16C59×#552—21no—
17C59×#552—21no—
18C59×#552—22no—
19C59×#544—21no—
20C59×#544C2212yes—
21C59×#541—21no—
22C59×#541—22no—
23C59×#539C2212yes—
24C59×#533C2211yescharacteristic
25C59×#529C2211yescharacteristic
26C2×#125×C59C2211yescharacteristic
27C2×C4×D4×C59C2211yescharacteristic
28C59×#484C2211yescharacteristic
29C59×#484C2212yes—
30C59×#481C2211yescharacteristic
31C59×#478—21no—
32C59×#474—21no—
33C59×#463—21no—
34C59×#461—21no—
35C59×#450C2211yescharacteristic
36C59×#450C2212yes—
37C59×#447C2211yescharacteristic
38C59×#440—22no—
39C4×SD16×C59—21no—
40C4×D8×C59—21no—
41C59×#432C2211yescharacteristic
42C59×#431C2211yescharacteristic
43C59×#400C2212yes—
44C59×#394C2212yes—
45C59×#388—22no—
46C59×#380—21no—
47C59×#380—21no—
48C4×C4⋊C4×C59C2211yescharacteristic
49C59×#328C2212yes—
50C59×#328C2212yes—
51C59×#328C2212yes—
52C59×#327C2212yes—
53C59×#324C2212yes—
54C59×#324C2212yes—
55C22×D4×C59C2311yescharacteristic
56C23×C4×C59—21no—
57C2×SD16×C59—21no—
58C2×SD16×C59—22no—
59C2×D8×C59—21no—
60C4⋊Q8×C59—21no—
61C4⋊Q8×C59—22no—
62C4⋊Q8×C59—22no—
63C4⋊Q8×C59D411yescharacteristic
64C4⋊Q8×C59D411yescharacteristic
65#126×C59—22no—
66#126×C59—22no—
67#126×C59—22no—
68#126×C59—22no—
69#126×C59—21no—
70#125×C59—22no—
71#125×C59—22no—
72#125×C59—22no—
73#125×C59—22no—
74#125×C59—21no—
75#125×C59—22no—
76#125×C59—22no—
77#125×C59—22no—
78#125×C59—22no—
79#125×C59—21no—
80#125×C59—21no—
81#125×C59—21no—
82#125×C59—21no—
83#125×C59—21no—
84#125×C59D411yescharacteristic
85#124×C59D411yescharacteristic
86#123×C59—22no—
87#123×C59—21no—
88#123×C59—22no—
89C22⋊Q8×C59—21no—
90C22⋊Q8×C59—21no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—22no—
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59—22no—
96C22⋊Q8×C59C2311yescharacteristic
97C22⋊Q8×C59C2312yes—
98#121×C59C2311yescharacteristic
99C4×Q8×C59—21no—
100C4×Q8×C59—22no—
101C4×Q8×C59—21no—
102C4×Q8×C59—21no—
103C4×Q8×C59D411yescharacteristic
104C4×Q8×C59D412yes—
105C4×D4×C59—21no—
106C4×D4×C59—22no—
107C4×D4×C59—21no—
108C4×D4×C59—21no—
109C4×D4×C59—21no—
110C4×D4×C59—21no—
111C4×D4×C59D411yescharacteristic
112#117×C59—22no—
113#117×C59—21no—
114#117×C59—21no—
115#117×C59—22no—
116#117×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59D411yescharacteristic
120C2×C4⋊C4×C59D411yescharacteristic
121C2×C4⋊C4×C59D412yes—
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2312yes—
124C2×C22⋊C4×C59—22no—
125C2×C22⋊C4×C59—21no—
126C2×C22⋊C4×C59—22no—
127C2×C42×C59—21no—
128C2×C42×C59C2311yescharacteristic
129#107×C59—21no—
130#107×C59—22no—
131#106×C59—21no—
132C4⋊C8×C59—21no—
133C4⋊C8×C59—21no—
134#103×C59—21no—
135#103×C59—22no—
136#102×C59—21no—
137#102×C59—21no—
138#102×C59—21no—
139#102×C59—22no—
140C22⋊C8×C59C2311yescharacteristic
141C22⋊C8×C59C2311yescharacteristic
142C22⋊C8×C59C2312yes—
143#97×C59—21no—
144C4×C8×C59—21no—
145#95×C59—22no—
146#95×C59—22no—
147#95×C59C2312yes—
148C24×C59—21no—
149C2×Q8×C59—22no—
150C2×Q8×C59—21no—
151C2×Q8×C59C2×D412yes—
152C2×Q8×C59C2×D411yescharacteristic
153C2×D4×C59—21no—
154C2×D4×C59—21no—
155C2×D4×C59—21no—
156C2×D4×C59—42no—
157C2×D4×C59C2×D411yescharacteristic
158C22×C4×C59—22no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59—22no—
162C22×C4×C59—42no—
163C22×C4×C59—42no—
164C22×C4×C59—41no—
165C22×C4×C59—41no—
166C22×C4×C59C2×D412yes—
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D412yes—
170C22×C4×C59C2411yescharacteristic
171SD16×C59—42no—
172SD16×C59—44no—
173D8×C59—42no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—22no—
183C4⋊C4×C59—22no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—22no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—22no—
191C4⋊C4×C59—21no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—21no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—22no—
199C4⋊C4×C59—22no—
200C4⋊C4×C59—21no—
201C4⋊C4×C59—22no—
202C4⋊C4×C59—21no—
203C4⋊C4×C59—21no—
204C4⋊C4×C59—42no—
205C4⋊C4×C59C2×D411yescharacteristic
206C4⋊C4×C59C2×D411yescharacteristic
207C4⋊C4×C59C2×D412yes—
208C22⋊C4×C59—22no—
209C22⋊C4×C59—22no—
210C22⋊C4×C59—22no—
211C22⋊C4×C59—22no—
212C22⋊C4×C59—21no—
213C22⋊C4×C59—21no—
214C22⋊C4×C59—21no—
215C22⋊C4×C59—21no—
216C22⋊C4×C59—21no—
217C22⋊C4×C59—44no—
218C22⋊C4×C59—41no—
219C22⋊C4×C59—41no—
220C22⋊C4×C59—82no—
221C42×C59—22no—
222C42×C59—22no—
223C42×C59—22no—
224C42×C59—22no—
225C42×C59—21no—
226C42×C59—81no—
227C42×C59C2×D411yescharacteristic
228C42×C59C2×D411yescharacteristic
229C42×C59C2×D411yescharacteristic
230C42×C59C2×D411yescharacteristic
231C23×C59—22no—
232C23×C59—21no—
233C23×C59—41no—
234C23×C59—41no—
235C23×C59—41no—
236C23×C59—41no—
237C23×C59—41no—
238C23×C59—41no—
239C23×C59C22×D411yescharacteristic
240Q8×C59—41no—
241Q8×C59—42no—
242Q8×C59—42no—
243Q8×C59—82no—
244D4×C59—22no—
245D4×C59—22no—
246D4×C59—41no—
247D4×C59—41no—
248D4×C59—41no—
249D4×C59—41no—
250D4×C59—41no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—22no—
254C2×C4×C59—22no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—22no—
258C2×C4×C59—21no—
259C2×C4×C59—22no—
260C2×C4×C59—22no—
261C2×C4×C59—22no—
262C2×C4×C59—22no—
263C2×C4×C59—41no—
264C2×C4×C59—42no—
265C2×C4×C59—42no—
266C2×C4×C59—42no—
267C2×C4×C59—42no—
268C2×C4×C59—41no—
269C2×C4×C59—41no—
270C2×C4×C59—41no—
271C2×C4×C59—42no—
272C2×C4×C59—82no—
273C2×C4×C59—82no—
274C2×C4×C59—82no—
275C2×C4×C59#12011yescharacteristic
276C2×C4×C59#12011yescharacteristic
277C2×C4×C59#12011yescharacteristic
278C2×C4×C59#12011yescharacteristic
279C2×C4×C59C22×D411yescharacteristic
280C2×C4×C59C22×D411yescharacteristic
281C8×C59—41no—
282C8×C59—41no—
283C8×C59—42no—
284#18278C5911yesmaximal, characteristic, direct product factor, Sylow
285C22×C59—21no—
286C22×C59—21no—
287C22×C59—42no—
288C22×C59—42no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—42no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59C2×#12011yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57811yescharacteristic
300C4×C59—21no—
301C4×C59—21no—
302C4×C59—21no—
303C4×C59—21no—
304C4×C59—21no—
305C4×C59—42no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—42no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—82no—
313C4×C59—81no—
314C4×C59#45412yes—
315#3563C2×C5911yescharacteristic, complemented
316#3110C2×C5911yescharacteristic, complemented
317#3105C2×C5911yescharacteristic
318#2562C2×C5911yescharacteristic, complemented
319#1617C2×C5912yescomplemented
320#1615C2×C5912yescomplemented
321#1599C2×C5912yescomplemented
322#1597C2×C5912yes—
323#1509C2×C5911yescharacteristic, complemented
324#1503C2×C5911yescharacteristic, complemented
325#1470C2×C5911yescharacteristic, complemented
326C2×C59—21no—
327C2×C59—21no—
328C2×C59—81no—
329C2×C59—82no—
330C2×C59#301311yescharacteristic
331C2×C59C2×#45411yescharacteristic
332C2×C59#302511yescharacteristic
333#564—21no—
334#563—21no—
335#557—21no—
336#554—21no—
337#552—21no—
338#552—21no—
339#552—22no—
340#544—21no—
341#544C22×C5912yes—
342#541—21no—
343#541—22no—
344#539C22×C5912yes—
345#533C22×C5911yescharacteristic
346#529C22×C5911yescharacteristic
347C2×#125C22×C5911yescharacteristic
348C2×C4×D4C22×C5911yescharacteristic
349#484C22×C5911yescharacteristic
350#484C22×C5912yes—
351#481C22×C5911yescharacteristic
352#478—21no—
353#474—21no—
354#463—21no—
355#461—21no—
356#450C22×C5911yescharacteristic
357#450C22×C5912yes—
358#447C22×C5911yescharacteristic
359#440—22no—
360C4×SD16—21no—
361C4×D8—21no—
362#432C22×C5911yescharacteristic
363#431C22×C5911yescharacteristic
364#400C22×C5912yes—
365#394C22×C5912yes—
366#388—22no—
367#380—21no—
368#380—21no—
369C4×C4⋊C4C22×C5911yescharacteristic
370#328C22×C5912yes—
371#328C22×C5912yes—
372#328C22×C5912yes—
373#327C22×C5912yes—
374#324C22×C5912yes—
375#324C22×C5912yes—
376C59#1827811yescharacteristic, direct product factor, Sylow
377C22×D4C23×C5911yescharacteristic
378C23×C4—21no—
379C2×SD16—21no—
380C2×SD16—22no—
381C2×D8—21no—
382C4⋊Q8—21no—
383C4⋊Q8—22no—
384C4⋊Q8—22no—
385C4⋊Q8D4×C5911yescharacteristic
386C4⋊Q8D4×C5911yescharacteristic
387#126—22no—
388#126—22no—
389#126—22no—
390#126—22no—
391#126—21no—
392#125—22no—
393#125—22no—
394#125—22no—
395#125—22no—
396#125—21no—
397#125—22no—
398#125—22no—
399#125—22no—
400#125—22no—
401#125—21no—
402#125—21no—
403#125—21no—
404#125—21no—
405#125—21no—
406#125D4×C5911yescharacteristic
407#124D4×C5911yescharacteristic
408#123—22no—
409#123—21no—
410#123—22no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417C22⋊Q8—22no—
418C22⋊Q8C23×C5911yescharacteristic
419C22⋊Q8C23×C5912yes—
420#121C23×C5911yescharacteristic
421C4×Q8—21no—
422C4×Q8—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C5911yescharacteristic
426C4×Q8D4×C5912yes—
427C4×D4—21no—
428C4×D4—22no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4D4×C5911yescharacteristic
434#117—22no—
435#117—21no—
436#117—21no—
437#117—22no—
438#117—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4D4×C5911yescharacteristic
442C2×C4⋊C4D4×C5912yes—
443C2×C4⋊C4D4×C5911yescharacteristic
444C2×C4⋊C4C23×C5911yescharacteristic
445C2×C4⋊C4C23×C5912yes—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—22no—
448C2×C22⋊C4—22no—
449C2×C42—21no—
450C2×C42C23×C5911yescharacteristic
451#107—21no—
452#107—22no—
453#106—21no—
454C4⋊C8—21no—
455C4⋊C8—21no—
456#103—21no—
457#103—22no—
458#102—21no—
459#102—21no—
460#102—21no—
461#102—22no—
462C22⋊C8C23×C5912yes—
463C22⋊C8C23×C5911yescharacteristic
464C22⋊C8C23×C5911yescharacteristic
465#97—21no—
466C4×C8—21no—
467#95—22no—
468#95—22no—
469#95C23×C5912yes—
470C24—21no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8C2×D4×C5912yes—
474C2×Q8C2×D4×C5911yescharacteristic
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4C2×D4×C5911yescharacteristic
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4C2×D4×C5912yes—
489C22×C4C2×D4×C5911yescharacteristic
490C22×C4C2×D4×C5911yescharacteristic
491C22×C4C2×D4×C5912yes—
492C22×C4C24×C5911yescharacteristic
493SD16—42no—
494SD16—44no—
495D8—42no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4C2×D4×C5911yescharacteristic
528C4⋊C4C2×D4×C5911yescharacteristic
529C4⋊C4C2×D4×C5912yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—44no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—82no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—81no—
549C42C2×D4×C5911yescharacteristic
550C42C2×D4×C5911yescharacteristic
551C42C2×D4×C5911yescharacteristic
552C42C2×D4×C5911yescharacteristic
553C23—22no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×D4×C5911yescharacteristic
562Q8—41no—
563Q8—42no—
564Q8—42no—
565Q8—82no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#120×C5911yescharacteristic
598C2×C4#120×C5911yescharacteristic
599C2×C4#120×C5911yescharacteristic
600C2×C4#120×C5911yescharacteristic
601C2×C4C22×D4×C5911yescharacteristic
602C2×C4C22×D4×C5911yescharacteristic
603C8—41no—
604C8—41no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#120×C5911yescharacteristic
619C22C59×#57711yescharacteristic
620C22C59×#57811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C59×#45412yes—
636C2—21no—
637C2—21no—
638C2—82no—
639C2—81no—
640C2C59×#301311yescharacteristic
641C2C2×C59×#45411yescharacteristic
642C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#447C22
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×#3563C2semidirect
2. C59×#3110C2semidirect
4. C59×#2562C2semidirect
5. C59×#1617C2semidirect
6. C59×#1615C2semidirect
7. C59×#1599C2semidirect
9. C59×#1509C2semidirect
10. C59×#1503C2semidirect
11. C59×#1470C2semidirect
284. #18278C59direct (N × H)
315. #3563C2×C59semidirect
316. #3110C2×C59semidirect
318. #2562C2×C59semidirect
319. #1617C2×C59semidirect
320. #1615C2×C59semidirect
321. #1599C2×C59semidirect
323. #1509C2×C59semidirect
324. #1503C2×C59semidirect
325. #1470C2×C59semidirect
376. C59#18278direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35632
C59×#31102
C59×#31052
C59×#25622
C59×#16172
C59×#16152
C59×#15992
C59×#15972
C59×#15092
C59×#15032
C59×#14702
#1827859