Finite Group Databasebeta

← C59×#18421 #28430530

Subgroups — C59×#18421 #28430530

Order 15104.

994 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3536C211yesmaximal, characteristic
2C59×#3195C211yesmaximal, characteristic, complemented
3C59×#3190C211yesmaximal, characteristic, complemented
4C59×#2639C211yesmaximal, characteristic, complemented
5C59×#1612C212yesmaximal, complemented
6C59×#1608C212yesmaximal, complemented
7C59×#1606C212yesmaximal, complemented
8C59×#1602C212yesmaximal, complemented
9C59×#1504C211yesmaximal, characteristic, complemented
10C2×C59×#328C211yesmaximal, characteristic, complemented
11C59×#1471C211yesmaximal, characteristic, complemented
12C59×#562—22no—
13C59×#554—21no—
14C59×#548—21no—
15C59×#547—21no—
16C59×#546—22no—
17C59×#546—21no—
18C59×#543C2212yes—
19C59×#542—21no—
20C59×#539—21no—
21C59×#538C2212yes—
22C59×#537—21no—
23C59×#537—21no—
24C59×#525C2211yescharacteristic
25C2×C22⋊Q8×C59C2211yescharacteristic
26C2×C22⋊Q8×C59C2211yescharacteristic
27C2×#121×C59C2211yescharacteristic
28C59×#485—21no—
29C59×#483—21no—
30C59×#483C2211yescharacteristic
31C59×#483C2212yes—
32C59×#480C2211yescharacteristic
33C59×#470—22no—
34C59×#467—21no—
35C59×#465—21no—
36C59×#452—21no—
37C59×#451—21no—
38C59×#451C2211yescharacteristic
39C59×#451C2212yes—
40C59×#448C2211yescharacteristic
41C59×#407C2211yescharacteristic
42C2×C22⋊C8×C59C2211yescharacteristic
43C59×#400—21no—
44C59×#397C2212yes—
45C59×#394—21no—
46C59×#393C2212yes—
47C59×#386—22no—
48C2×#95×C59C2211yescharacteristic
49C59×#328C2212yes—
50C59×#328C2212yes—
51C59×#328C2212yes—
52C59×#327C2212yes—
53C59×#323C2212yes—
54C59×#323C2212yes—
55C2×#41×C59—22no—
56C2×#41×C59—21no—
57C2×#41×C59D411yescharacteristic
58C22×Q8×C59D411yescharacteristic
59C22×Q8×C59C2311yescharacteristic
60C22×D4×C59D412yes—
61C22×D4×C59D411yescharacteristic
62C23×C4×C59—21no—
63C23×C4×C59C2311yescharacteristic
64C2×Q16×C59—22no—
65C2×Q16×C59—21no—
66C2×SD16×C59—21no—
67C2×M16×C59—21no—
68C22×C8×C59—21no—
69C4⋊Q8×C59—21no—
70#126×C59—22no—
71#125×C59—22no—
72#124×C59—21no—
73#124×C59—22no—
74#124×C59—22no—
75#123×C59—22no—
76#123×C59—22no—
77#123×C59—22no—
78#123×C59—21no—
79#123×C59—22no—
80C22⋊Q8×C59—21no—
81C22⋊Q8×C59—22no—
82C22⋊Q8×C59—22no—
83C22⋊Q8×C59—22no—
84C22⋊Q8×C59—21no—
85C22⋊Q8×C59—21no—
86C22⋊Q8×C59—22no—
87C22⋊Q8×C59—22no—
88C22⋊Q8×C59—22no—
89C22⋊Q8×C59—21no—
90C22⋊Q8×C59—21no—
91C22⋊Q8×C59—21no—
92C22⋊Q8×C59—21no—
93C22⋊Q8×C59D411yescharacteristic
94C22⋊Q8×C59D412yes—
95C22⋊Q8×C59C2312yes—
96C22⋊Q8×C59C2311yescharacteristic
97#121×C59—21no—
98#121×C59—22no—
99#121×C59—22no—
100#121×C59—22no—
101#121×C59—22no—
102#121×C59—22no—
103#121×C59—21no—
104#121×C59—21no—
105#121×C59—21no—
106#121×C59—21no—
107#121×C59D411yescharacteristic
108#121×C59C2311yescharacteristic
109#120×C59—21no—
110#120×C59—22no—
111C4×Q8×C59—21no—
112C4×D4×C59—22no—
113C4×D4×C59—21no—
114C4×D4×C59—21no—
115C4×D4×C59—21no—
116C4×D4×C59—22no—
117#117×C59D411yescharacteristic
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59—21no—
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59D411yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C22⋊C4×C59—21no—
125C2×C22⋊C4×C59—22no—
126C2×C22⋊C4×C59—21no—
127C2×C22⋊C4×C59—22no—
128C2×C22⋊C4×C59—22no—
129C2×C22⋊C4×C59C2312yes—
130#107×C59—21no—
131#106×C59—21no—
132#106×C59—22no—
133#103×C59—22no—
134#103×C59—21no—
135#103×C59—21no—
136#103×C59—21no—
137#103×C59—22no—
138#102×C59—21no—
139C22⋊C8×C59—21no—
140C22⋊C8×C59—21no—
141C22⋊C8×C59C2311yescharacteristic
142C22⋊C8×C59C2311yescharacteristic
143C22⋊C8×C59C2312yes—
144#95×C59—21no—
145#95×C59—22no—
146#95×C59—22no—
147#95×C59C2312yes—
148C24×C59C2×D411yescharacteristic
149C24×C59C2×D412yes—
150#41×C59—81no—
151#41×C59—84no—
152#41×C59—82no—
153C2×Q8×C59—22no—
154C2×Q8×C59—21no—
155C2×Q8×C59—21no—
156C2×Q8×C59—21no—
157C2×Q8×C59—81no—
158C2×Q8×C59—81no—
159C2×Q8×C59C2×D411yescharacteristic
160C2×Q8×C59C2×D412yes—
161C2×D4×C59—21no—
162C2×D4×C59—22no—
163C2×D4×C59—21no—
164C2×D4×C59—22no—
165C2×D4×C59—22no—
166C2×D4×C59—21no—
167C2×D4×C59—22no—
168C2×D4×C59—21no—
169C2×D4×C59—21no—
170C2×D4×C59—44no—
171C2×D4×C59—81no—
172C2×D4×C59C2×D411yescharacteristic
173C22×C4×C59—21no—
174C22×C4×C59—22no—
175C22×C4×C59—22no—
176C22×C4×C59—22no—
177C22×C4×C59—21no—
178C22×C4×C59—21no—
179C22×C4×C59—22no—
180C22×C4×C59—21no—
181C22×C4×C59—21no—
182C22×C4×C59—42no—
183C22×C4×C59—81no—
184C22×C4×C59—81no—
185C22×C4×C59—81no—
186C22×C4×C59C2×D412yes—
187C22×C4×C59C2×D411yescharacteristic
188C22×C4×C59C2×D411yescharacteristic
189C22×C4×C59C2×D411yescharacteristic
190C22×C4×C59C2×D411yescharacteristic
191C22×C4×C59C2×D411yescharacteristic
192C22×C4×C59C2411yescharacteristic
193Q16×C59—81no—
194Q16×C59—82no—
195SD16×C59—81no—
196M16×C59—81no—
197C2×C8×C59—21no—
198C2×C8×C59—21no—
199C2×C8×C59—22no—
200C2×C8×C59—81no—
201C4⋊C4×C59—22no—
202C4⋊C4×C59—22no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59—22no—
205C4⋊C4×C59—21no—
206C4⋊C4×C59—21no—
207C4⋊C4×C59—22no—
208C4⋊C4×C59—21no—
209C4⋊C4×C59—21no—
210C4⋊C4×C59—22no—
211C4⋊C4×C59—21no—
212C4⋊C4×C59—21no—
213C4⋊C4×C59—21no—
214C4⋊C4×C59—41no—
215C4⋊C4×C59—41no—
216C4⋊C4×C59C2×D412yes—
217C4⋊C4×C59C2×D411yescharacteristic
218C4⋊C4×C59C2×D411yescharacteristic
219C22⋊C4×C59—22no—
220C22⋊C4×C59—22no—
221C22⋊C4×C59—22no—
222C22⋊C4×C59—22no—
223C22⋊C4×C59—22no—
224C22⋊C4×C59—22no—
225C22⋊C4×C59—21no—
226C22⋊C4×C59—21no—
227C22⋊C4×C59—22no—
228C22⋊C4×C59—21no—
229C22⋊C4×C59—21no—
230C22⋊C4×C59—22no—
231C22⋊C4×C59—22no—
232C22⋊C4×C59—21no—
233C22⋊C4×C59—41no—
234C22⋊C4×C59—41no—
235C22⋊C4×C59—82no—
236C22⋊C4×C59—82no—
237C42×C59—22no—
238C42×C59—21no—
239C23×C59—21no—
240C23×C59—21no—
241C23×C59—22no—
242C23×C59—22no—
243C23×C59—21no—
244C23×C59—22no—
245C23×C59—22no—
246C23×C59—41no—
247C23×C59—42no—
248C23×C59—42no—
249C23×C59#12011yescharacteristic
250C23×C59#12011yescharacteristic
251C23×C59C22×D411yescharacteristic
252Q8×C59—42no—
253Q8×C59—41no—
254Q8×C59—42no—
255Q8×C59—41no—
256Q8×C59—41no—
257Q8×C59—42no—
258Q8×C59—82no—
259D4×C59—41no—
260D4×C59—42no—
261D4×C59—41no—
262D4×C59—41no—
263D4×C59—44no—
264D4×C59—42no—
265D4×C59—81no—
266D4×C59—82no—
267D4×C59—81no—
268D4×C59—82no—
269D4×C59—82no—
270C2×C4×C59—21no—
271C2×C4×C59—22no—
272C2×C4×C59—21no—
273C2×C4×C59—22no—
274C2×C4×C59—21no—
275C2×C4×C59—22no—
276C2×C4×C59—41no—
277C2×C4×C59—41no—
278C2×C4×C59—41no—
279C2×C4×C59—41no—
280C2×C4×C59—41no—
281C2×C4×C59—41no—
282C2×C4×C59—42no—
283C2×C4×C59—42no—
284C2×C4×C59—42no—
285C2×C4×C59—42no—
286C2×C4×C59—42no—
287C2×C4×C59—42no—
288C2×C4×C59—41no—
289C2×C4×C59—41no—
290C2×C4×C59—41no—
291C2×C4×C59—42no—
292C2×C4×C59—41no—
293C2×C4×C59—81no—
294C2×C4×C59—82no—
295C2×C4×C59—82no—
296C2×C4×C59—82no—
297C2×C4×C59—84no—
298C2×C4×C59#12011yescharacteristic
299C2×C4×C59#12011yescharacteristic
300C2×C4×C59C22×D411yescharacteristic
301C2×C4×C59C22×D411yescharacteristic
302C8×C59—41no—
303C8×C59—41no—
304C8×C59—42no—
305#18421C5911yesmaximal, characteristic, direct product factor, Sylow
306C22×C59—21no—
307C22×C59—21no—
308C22×C59—21no—
309C22×C59—21no—
310C22×C59—21no—
311C22×C59—21no—
312C22×C59—21no—
313C22×C59—42no—
314C22×C59—42no—
315C22×C59—41no—
316C22×C59—42no—
317C22×C59—41no—
318C22×C59—42no—
319C22×C59—41no—
320C22×C59—42no—
321C22×C59—82no—
322C22×C59—82no—
323C22×C59#45512yes—
324C22×C59C2×#12011yescharacteristic
325C22×C59#57711yescharacteristic
326C22×C59#57811yescharacteristic
327C4×C59—21no—
328C4×C59—21no—
329C4×C59—21no—
330C4×C59—21no—
331C4×C59—41no—
332C4×C59—42no—
333C4×C59—42no—
334C4×C59—41no—
335C4×C59—41no—
336C4×C59—42no—
337C4×C59—82no—
338C4×C59—81no—
339C4×C59—81no—
340#3536C2×C5911yescharacteristic
341#3195C2×C5911yescharacteristic, complemented
342#3190C2×C5911yescharacteristic, complemented
343#2639C2×C5911yescharacteristic, complemented
344#1612C2×C5912yescomplemented
345#1608C2×C5912yescomplemented
346#1606C2×C5912yescomplemented
347#1602C2×C5912yescomplemented
348#1504C2×C5911yescharacteristic, complemented
349C2×#328C2×C5911yescharacteristic, complemented
350#1471C2×C5911yescharacteristic, complemented
351C2×C59—21no—
352C2×C59—21no—
353C2×C59—22no—
354C2×C59—41no—
355C2×C59—81no—
356C2×C59—82no—
357C2×C59#301311yescharacteristic
358C2×C59C2×#45511yescharacteristic
359C2×C59#302611yescharacteristic
360#562—22no—
361#554—21no—
362#548—21no—
363#547—21no—
364#546—22no—
365#546—21no—
366#543C22×C5912yes—
367#542—21no—
368#539—21no—
369#538C22×C5912yes—
370#537—21no—
371#537—21no—
372#525C22×C5911yescharacteristic
373C2×C22⋊Q8C22×C5911yescharacteristic
374C2×C22⋊Q8C22×C5911yescharacteristic
375C2×#121C22×C5911yescharacteristic
376#485—21no—
377#483—21no—
378#483C22×C5911yescharacteristic
379#483C22×C5912yes—
380#480C22×C5911yescharacteristic
381#470—22no—
382#467—21no—
383#465—21no—
384#452—21no—
385#451—21no—
386#451C22×C5911yescharacteristic
387#451C22×C5912yes—
388#448C22×C5911yescharacteristic
389#407C22×C5911yescharacteristic
390C2×C22⋊C8C22×C5911yescharacteristic
391#400—21no—
392#397C22×C5912yes—
393#394—21no—
394#393C22×C5912yes—
395#386—22no—
396C2×#95C22×C5911yescharacteristic
397#328C22×C5912yes—
398#328C22×C5912yes—
399#328C22×C5912yes—
400#327C22×C5912yes—
401#323C22×C5912yes—
402#323C22×C5912yes—
403C59#1842111yescharacteristic, direct product factor, Sylow
404C2×#41—22no—
405C2×#41—21no—
406C2×#41D4×C5911yescharacteristic
407C22×Q8D4×C5911yescharacteristic
408C22×Q8C23×C5911yescharacteristic
409C22×D4D4×C5912yes—
410C22×D4D4×C5911yescharacteristic
411C23×C4—21no—
412C23×C4C23×C5911yescharacteristic
413C2×Q16—22no—
414C2×Q16—21no—
415C2×SD16—21no—
416C2×M16—21no—
417C22×C8—21no—
418C4⋊Q8—21no—
419#126—22no—
420#125—22no—
421#124—21no—
422#124—22no—
423#124—22no—
424#123—22no—
425#123—22no—
426#123—22no—
427#123—21no—
428#123—22no—
429C22⋊Q8—21no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—22no—
433C22⋊Q8—21no—
434C22⋊Q8—21no—
435C22⋊Q8—22no—
436C22⋊Q8—22no—
437C22⋊Q8—22no—
438C22⋊Q8—21no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8—21no—
442C22⋊Q8D4×C5912yes—
443C22⋊Q8D4×C5911yescharacteristic
444C22⋊Q8C23×C5912yes—
445C22⋊Q8C23×C5911yescharacteristic
446#121—21no—
447#121—22no—
448#121—22no—
449#121—22no—
450#121—22no—
451#121—22no—
452#121—21no—
453#121—21no—
454#121—21no—
455#121—21no—
456#121D4×C5911yescharacteristic
457#121C23×C5911yescharacteristic
458#120—21no—
459#120—22no—
460C4×Q8—21no—
461C4×D4—22no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—21no—
465C4×D4—22no—
466#117D4×C5911yescharacteristic
467C2×C4⋊C4—21no—
468C2×C4⋊C4—21no—
469C2×C4⋊C4—21no—
470C2×C4⋊C4D4×C5911yescharacteristic
471C2×C4⋊C4D4×C5911yescharacteristic
472C2×C4⋊C4C23×C5911yescharacteristic
473C2×C22⋊C4—21no—
474C2×C22⋊C4—22no—
475C2×C22⋊C4—21no—
476C2×C22⋊C4—22no—
477C2×C22⋊C4—22no—
478C2×C22⋊C4C23×C5912yes—
479#107—21no—
480#106—21no—
481#106—22no—
482#103—22no—
483#103—21no—
484#103—21no—
485#103—21no—
486#103—22no—
487#102—21no—
488C22⋊C8—21no—
489C22⋊C8—21no—
490C22⋊C8C23×C5912yes—
491C22⋊C8C23×C5911yescharacteristic
492C22⋊C8C23×C5911yescharacteristic
493#95—21no—
494#95—22no—
495#95—22no—
496#95C23×C5912yes—
497C24C2×D4×C5911yescharacteristic
498C24C2×D4×C5912yes—
499#41—81no—
500#41—84no—
501#41—82no—
502C2×Q8—22no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—81no—
507C2×Q8—81no—
508C2×Q8C2×D4×C5911yescharacteristic
509C2×Q8C2×D4×C5912yes—
510C2×D4—21no—
511C2×D4—22no—
512C2×D4—21no—
513C2×D4—22no—
514C2×D4—22no—
515C2×D4—21no—
516C2×D4—22no—
517C2×D4—21no—
518C2×D4—21no—
519C2×D4—44no—
520C2×D4—81no—
521C2×D4C2×D4×C5911yescharacteristic
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4—81no—
533C22×C4—81no—
534C22×C4—81no—
535C22×C4C2×D4×C5912yes—
536C22×C4C2×D4×C5911yescharacteristic
537C22×C4C2×D4×C5911yescharacteristic
538C22×C4C2×D4×C5911yescharacteristic
539C22×C4C2×D4×C5911yescharacteristic
540C22×C4C2×D4×C5911yescharacteristic
541C22×C4C24×C5911yescharacteristic
542Q16—81no—
543Q16—82no—
544SD16—81no—
545M16—81no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—22no—
549C2×C8—81no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—41no—
564C4⋊C4—41no—
565C4⋊C4C2×D4×C5912yes—
566C4⋊C4C2×D4×C5911yescharacteristic
567C4⋊C4C2×D4×C5911yescharacteristic
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—21no—
575C22⋊C4—21no—
576C22⋊C4—22no—
577C22⋊C4—21no—
578C22⋊C4—21no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—21no—
582C22⋊C4—41no—
583C22⋊C4—41no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C42—22no—
587C42—21no—
588C23—22no—
589C23—21no—
590C23—21no—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—22no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23#120×C5911yescharacteristic
599C23#120×C5911yescharacteristic
600C23C22×D4×C5911yescharacteristic
601Q8—42no—
602Q8—41no—
603Q8—42no—
604Q8—41no—
605Q8—41no—
606Q8—42no—
607Q8—82no—
608D4—41no—
609D4—42no—
610D4—44no—
611D4—42no—
612D4—41no—
613D4—41no—
614D4—81no—
615D4—82no—
616D4—81no—
617D4—82no—
618D4—82no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—81no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—84no—
647C2×C4#120×C5911yescharacteristic
648C2×C4#120×C5911yescharacteristic
649C2×C4C22×D4×C5911yescharacteristic
650C2×C4C22×D4×C5911yescharacteristic
651C8—41no—
652C8—41no—
653C8—42no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—42no—
662C22—42no—
663C22—41no—
664C22—42no—
665C22—41no—
666C22—42no—
667C22—41no—
668C22—42no—
669C22—82no—
670C22—82no—
671C22C59×#45512yes—
672C22C2×#120×C5911yescharacteristic
673C22C59×#57711yescharacteristic
674C22C59×#57811yescharacteristic
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—41no—
683C4—41no—
684C4—42no—
685C4—82no—
686C4—81no—
687C4—81no—
688C2—21no—
689C2—21no—
690C2—22no—
691C2—41no—
692C2—82no—
693C2—81no—
694C2C59×#301311yescharacteristic
695C2C2×C59×#45511yescharacteristic
696C2C59×#302611yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3536C2
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×#3195C2semidirect
3. C59×#3190C2semidirect
4. C59×#2639C2semidirect
5. C59×#1612C2semidirect
6. C59×#1608C2semidirect
7. C59×#1606C2semidirect
8. C59×#1602C2semidirect
9. C59×#1504C2semidirect
10. C2×C59×#328C2semidirect
11. C59×#1471C2semidirect
305. #18421C59direct (N × H)
341. #3195C2×C59semidirect
342. #3190C2×C59semidirect
343. #2639C2×C59semidirect
344. #1612C2×C59semidirect
345. #1608C2×C59semidirect
346. #1606C2×C59semidirect
347. #1602C2×C59semidirect
348. #1504C2×C59semidirect
349. C2×#328C2×C59semidirect
350. #1471C2×C59semidirect
403. C59#18421direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35362
C59×#31952
C59×#31902
C59×#26392
C59×#16122
C59×#16082
C59×#16062
C59×#16022
C59×#15042
C2×C59×#3282
C59×#14712
#1842159