Finite Group Databasebeta

← C59×#18247 #28430356

Subgroups — C59×#18247 #28430356

Order 15104.

852 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3563C211yesmaximal, characteristic
2C59×#3117C211yesmaximal, characteristic
3C59×#3104C211yesmaximal, characteristic, complemented
4C59×#2634C211yesmaximal, characteristic, complemented
5C59×#1617C212yesmaximal, complemented
6C59×#1615C212yesmaximal
7C59×#1605C212yesmaximal, complemented
8C59×#1603C212yesmaximal
9C59×#1521C211yesmaximal, characteristic
10C59×#1515C211yesmaximal, characteristic, complemented
11C59×#1470C211yesmaximal, characteristic, complemented
12C59×#564—21no—
13C59×#563—21no—
14C59×#557—21no—
15C59×#554—21no—
16C59×#552—21no—
17C59×#552—22no—
18C59×#552—21no—
19C59×#544—21no—
20C59×#544C2212yes—
21C59×#541—21no—
22C59×#541—22no—
23C59×#539C2212yes—
24C59×#533C2211yescharacteristic
25C59×#529C2211yescharacteristic
26C2×#125×C59C2211yescharacteristic
27C2×C4×Q8×C59C2211yescharacteristic
28C59×#484C2212yes—
29C59×#484C2211yescharacteristic
30C59×#481C2211yescharacteristic
31C59×#479—21no—
32C59×#477—21no—
33C59×#464—21no—
34C59×#462—21no—
35C59×#451C2212yes—
36C59×#451C2211yescharacteristic
37C59×#448C2211yescharacteristic
38C59×#441—21no—
39C59×#440—21no—
40C4×Q16×C59—22no—
41C59×#432C2211yescharacteristic
42C59×#431C2211yescharacteristic
43C59×#400—21no—
44C59×#400C2212yes—
45C59×#398—21no—
46C59×#395C2212yes—
47C59×#387—22no—
48C59×#383C2211yescharacteristic
49C59×#328C2212yes—
50C59×#328C2212yes—
51C59×#328C2212yes—
52C59×#327C2212yes—
53C59×#324C2212yes—
54C59×#324C2212yes—
55C22×Q8×C59C2311yescharacteristic
56C2×Q16×C59—22no—
57C2×Q16×C59—21no—
58C2×SD16×C59—21no—
59C4⋊Q8×C59—21no—
60C4⋊Q8×C59—22no—
61C4⋊Q8×C59—22no—
62C4⋊Q8×C59D411yescharacteristic
63C4⋊Q8×C59D411yescharacteristic
64#126×C59—22no—
65#126×C59—22no—
66#126×C59—22no—
67#126×C59—22no—
68#126×C59—21no—
69#125×C59—22no—
70#125×C59—22no—
71#125×C59—22no—
72#125×C59—22no—
73#125×C59—21no—
74#125×C59—22no—
75#125×C59—22no—
76#125×C59—22no—
77#125×C59—22no—
78#125×C59—21no—
79#125×C59—21no—
80#125×C59—21no—
81#125×C59—21no—
82#125×C59—21no—
83#125×C59D411yescharacteristic
84#124×C59D411yescharacteristic
85#123×C59—22no—
86#123×C59—21no—
87#123×C59—22no—
88C22⋊Q8×C59—21no—
89C22⋊Q8×C59—21no—
90C22⋊Q8×C59—22no—
91C22⋊Q8×C59—22no—
92C22⋊Q8×C59—22no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—22no—
95C22⋊Q8×C59C2311yescharacteristic
96C22⋊Q8×C59C2312yes—
97#121×C59C2311yescharacteristic
98C4×Q8×C59—21no—
99C4×Q8×C59—22no—
100C4×Q8×C59—21no—
101C4×Q8×C59—21no—
102C4×Q8×C59—21no—
103C4×Q8×C59—21no—
104C4×Q8×C59—21no—
105C4×Q8×C59—21no—
106C4×Q8×C59D412yes—
107C4×Q8×C59D411yescharacteristic
108C4×D4×C59—21no—
109C4×D4×C59—22no—
110C4×D4×C59D411yescharacteristic
111#117×C59—22no—
112#117×C59—21no—
113#117×C59—21no—
114#117×C59—22no—
115#117×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—22no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59D412yes—
120C2×C4⋊C4×C59D411yescharacteristic
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2312yes—
124C2×C42×C59—21no—
125C2×C42×C59C2311yescharacteristic
126#107×C59—22no—
127#107×C59—21no—
128#106×C59—21no—
129C4⋊C8×C59—21no—
130C4⋊C8×C59—21no—
131#103×C59—22no—
132#103×C59—21no—
133#103×C59—22no—
134#103×C59—21no—
135#103×C59—21no—
136#102×C59—21no—
137C22⋊C8×C59C2312yes—
138C22⋊C8×C59C2311yescharacteristic
139C22⋊C8×C59C2311yescharacteristic
140#97×C59—21no—
141C4×C8×C59—21no—
142#95×C59—21no—
143#95×C59—21no—
144#95×C59—22no—
145#95×C59—22no—
146#95×C59—22no—
147#95×C59C2312yes—
148C2×Q8×C59—22no—
149C2×Q8×C59—21no—
150C2×Q8×C59—21no—
151C2×Q8×C59—21no—
152C2×Q8×C59—42no—
153C2×Q8×C59C2×D412yes—
154C2×Q8×C59C2×D411yescharacteristic
155C2×D4×C59—21no—
156C2×D4×C59C2×D411yescharacteristic
157C22×C4×C59—21no—
158C22×C4×C59—22no—
159C22×C4×C59—21no—
160C22×C4×C59—22no—
161C22×C4×C59—21no—
162C22×C4×C59C2×D412yes—
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2×D412yes—
165C22×C4×C59C2×D411yescharacteristic
166C22×C4×C59C2411yescharacteristic
167Q16×C59—42no—
168Q16×C59—44no—
169SD16×C59—42no—
170C2×C8×C59—22no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—22no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—21no—
190C4⋊C4×C59—21no—
191C4⋊C4×C59—21no—
192C4⋊C4×C59—21no—
193C4⋊C4×C59—22no—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—22no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—21no—
199C4⋊C4×C59—21no—
200C4⋊C4×C59—44no—
201C4⋊C4×C59—41no—
202C4⋊C4×C59—41no—
203C4⋊C4×C59C2×D411yescharacteristic
204C4⋊C4×C59C2×D411yescharacteristic
205C4⋊C4×C59C2×D412yes—
206C22⋊C4×C59—22no—
207C22⋊C4×C59—22no—
208C22⋊C4×C59—22no—
209C22⋊C4×C59—22no—
210C22⋊C4×C59—21no—
211C22⋊C4×C59—21no—
212C22⋊C4×C59—21no—
213C22⋊C4×C59—21no—
214C22⋊C4×C59—21no—
215C42×C59—22no—
216C42×C59—22no—
217C42×C59—22no—
218C42×C59—22no—
219C42×C59—21no—
220C42×C59—41no—
221C42×C59—41no—
222C42×C59C2×D411yescharacteristic
223C42×C59C2×D411yescharacteristic
224C42×C59C2×D411yescharacteristic
225C42×C59C2×D411yescharacteristic
226C23×C59—21no—
227C23×C59C22×D411yescharacteristic
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59—41no—
231Q8×C59—42no—
232Q8×C59—42no—
233Q8×C59—41no—
234Q8×C59—41no—
235Q8×C59—82no—
236D4×C59—41no—
237D4×C59—42no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—22no—
244C2×C4×C59—21no—
245C2×C4×C59—22no—
246C2×C4×C59—22no—
247C2×C4×C59—24no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—42no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—42no—
257C2×C4×C59—41no—
258C2×C4×C59—42no—
259C2×C4×C59—82no—
260C2×C4×C59#12011yescharacteristic
261C2×C4×C59#12011yescharacteristic
262C2×C4×C59#12011yescharacteristic
263C2×C4×C59#12011yescharacteristic
264C2×C4×C59C22×D411yescharacteristic
265C2×C4×C59C22×D411yescharacteristic
266C8×C59—41no—
267C8×C59—41no—
268C8×C59—42no—
269#18247C5911yesmaximal, characteristic, direct product factor, Sylow
270C22×C59—21no—
271C22×C59—21no—
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59C2×#12011yescharacteristic
276C22×C59#57812yes—
277C4×C59—21no—
278C4×C59—21no—
279C4×C59—21no—
280C4×C59—21no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—42no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—42no—
290C4×C59—41no—
291C4×C59—82no—
292C4×C59—81no—
293#3563C2×C5911yescharacteristic
294#3117C2×C5911yescharacteristic
295#3104C2×C5911yescharacteristic, complemented
296#2634C2×C5911yescharacteristic, complemented
297#1617C2×C5912yescomplemented
298#1615C2×C5912yes—
299#1605C2×C5912yescomplemented
300#1603C2×C5912yes—
301#1521C2×C5911yescharacteristic
302#1515C2×C5911yescharacteristic, complemented
303#1470C2×C5911yescharacteristic, complemented
304C2×C59—22no—
305C2×C59—81no—
306C2×C59#301111yescharacteristic
307C2×C59#302011yescharacteristic
308C2×C59#302511yescharacteristic
309#564—21no—
310#563—21no—
311#557—21no—
312#554—21no—
313#552—21no—
314#552—22no—
315#552—21no—
316#544—21no—
317#544C22×C5912yes—
318#541—21no—
319#541—22no—
320#539C22×C5912yes—
321#533C22×C5911yescharacteristic
322#529C22×C5911yescharacteristic
323C2×#125C22×C5911yescharacteristic
324C2×C4×Q8C22×C5911yescharacteristic
325#484C22×C5912yes—
326#484C22×C5911yescharacteristic
327#481C22×C5911yescharacteristic
328#479—21no—
329#477—21no—
330#464—21no—
331#462—21no—
332#451C22×C5912yes—
333#451C22×C5911yescharacteristic
334#448C22×C5911yescharacteristic
335#441—21no—
336#440—21no—
337C4×Q16—22no—
338#432C22×C5911yescharacteristic
339#431C22×C5911yescharacteristic
340#400—21no—
341#400C22×C5912yes—
342#398—21no—
343#395C22×C5912yes—
344#387—22no—
345#383C22×C5911yescharacteristic
346#328C22×C5912yes—
347#328C22×C5912yes—
348#328C22×C5912yes—
349#327C22×C5912yes—
350#324C22×C5912yes—
351#324C22×C5912yes—
352C59#1824711yescharacteristic, direct product factor, Sylow
353C22×Q8C23×C5911yescharacteristic
354C2×Q16—22no—
355C2×Q16—21no—
356C2×SD16—21no—
357C4⋊Q8—21no—
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360C4⋊Q8D4×C5911yescharacteristic
361C4⋊Q8D4×C5911yescharacteristic
362#126—22no—
363#126—22no—
364#126—22no—
365#126—22no—
366#126—21no—
367#125—22no—
368#125—22no—
369#125—22no—
370#125—22no—
371#125—21no—
372#125—22no—
373#125—22no—
374#125—22no—
375#125—22no—
376#125—21no—
377#125—21no—
378#125—21no—
379#125—21no—
380#125—21no—
381#125D4×C5911yescharacteristic
382#124D4×C5911yescharacteristic
383#123—22no—
384#123—21no—
385#123—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8C23×C5911yescharacteristic
394C22⋊Q8C23×C5912yes—
395#121C23×C5911yescharacteristic
396C4×Q8—21no—
397C4×Q8—22no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8D4×C5911yescharacteristic
405C4×Q8D4×C5912yes—
406C4×D4—21no—
407C4×D4—22no—
408C4×D4D4×C5911yescharacteristic
409#117—22no—
410#117—21no—
411#117—21no—
412#117—22no—
413#117—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4D4×C5912yes—
418C2×C4⋊C4D4×C5911yescharacteristic
419C2×C4⋊C4D4×C5911yescharacteristic
420C2×C4⋊C4C23×C5911yescharacteristic
421C2×C4⋊C4C23×C5912yes—
422C2×C42—21no—
423C2×C42C23×C5911yescharacteristic
424#107—22no—
425#107—21no—
426#106—21no—
427C4⋊C8—21no—
428C4⋊C8—21no—
429#103—22no—
430#103—21no—
431#103—22no—
432#103—21no—
433#103—21no—
434#102—21no—
435C22⋊C8C23×C5912yes—
436C22⋊C8C23×C5911yescharacteristic
437C22⋊C8C23×C5911yescharacteristic
438#97—21no—
439C4×C8—21no—
440#95—21no—
441#95—21no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95C23×C5912yes—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—42no—
451C2×Q8C2×D4×C5912yes—
452C2×Q8C2×D4×C5911yescharacteristic
453C2×D4—21no—
454C2×D4C2×D4×C5911yescharacteristic
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4C2×D4×C5912yes—
461C22×C4C2×D4×C5911yescharacteristic
462C22×C4C2×D4×C5912yes—
463C22×C4C2×D4×C5911yescharacteristic
464C22×C4C24×C5911yescharacteristic
465Q16—42no—
466Q16—44no—
467SD16—42no—
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4C2×D4×C5911yescharacteristic
502C4⋊C4C2×D4×C5911yescharacteristic
503C4⋊C4C2×D4×C5912yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—41no—
519C42—41no—
520C42C2×D4×C5911yescharacteristic
521C42C2×D4×C5911yescharacteristic
522C42C2×D4×C5911yescharacteristic
523C42C2×D4×C5911yescharacteristic
524C23—21no—
525C23C22×D4×C5911yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—42no—
530Q8—42no—
531Q8—41no—
532Q8—41no—
533Q8—82no—
534D4—41no—
535D4—42no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4#120×C5911yescharacteristic
559C2×C4#120×C5911yescharacteristic
560C2×C4#120×C5911yescharacteristic
561C2×C4#120×C5911yescharacteristic
562C2×C4C22×D4×C5911yescharacteristic
563C2×C4C22×D4×C5911yescharacteristic
564C8—41no—
565C8—41no—
566C8—42no—
567C22—21no—
568C22—21no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22C2×#120×C5911yescharacteristic
573C22C59×#57812yes—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4—81no—
590C2—22no—
591C2—81no—
592C2C59×#301111yescharacteristic
593C2C59×#302011yescharacteristic
594C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radical#121×C59C23
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
3. C59×#3104C2semidirect
4. C59×#2634C2semidirect
5. C59×#1617C2semidirect
7. C59×#1605C2semidirect
10. C59×#1515C2semidirect
11. C59×#1470C2semidirect
269. #18247C59direct (N × H)
295. #3104C2×C59semidirect
296. #2634C2×C59semidirect
297. #1617C2×C59semidirect
299. #1605C2×C59semidirect
302. #1515C2×C59semidirect
303. #1470C2×C59semidirect
352. C59#18247direct (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×#31172
C59×#31042
C59×#26342
C59×#16172
C59×#16152
C59×#16052
C59×#16032
C59×#15212
C59×#15152
C59×#14702
#1824759