Finite Group Databasebeta

← C59×#18217 #28430326

Subgroups — C59×#18217 #28430326

Order 15104.

932 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3493C211yesmaximal, characteristic
2C59×#3108C212yesmaximal, complemented
3C59×#2599C211yesmaximal, characteristic, complemented
4C59×#1612C212yesmaximal, complemented
5C59×#1611C211yesmaximal, characteristic, complemented
6C59×#1610C211yesmaximal, characteristic, complemented
7C59×#1606C212yesmaximal, complemented
8C59×#1605C211yesmaximal, characteristic, complemented
9C59×#1604C211yesmaximal, characteristic, complemented
10C59×#1521C212yesmaximal, complemented
11C59×#1458C211yesmaximal, characteristic, complemented
12C59×#556—22no—
13C59×#555—22no—
14C59×#553—22no—
15C59×#552—22no—
16C59×#544C2211yescharacteristic
17C59×#538—21no—
18C59×#538C2212yes—
19C59×#537C2211yescharacteristic
20C59×#529—21no—
21C59×#529C2212yes—
22C59×#525—21no—
23C59×#520—21no—
24C2×C4×Q8×C59C2211yescharacteristic
25C2×#117×C59C2211yescharacteristic
26C59×#483C2212yes—
27C59×#480C2212yes—
28C59×#479—22no—
29C59×#464—22no—
30C59×#451C2212yes—
31C59×#448C2212yes—
32C4×Q16×C59—22no—
33C4×SD16×C59—22no—
34C59×#431C2212yes—
35C59×#398C2211yescharacteristic
36C59×#397C2211yescharacteristic
37C59×#395C2211yescharacteristic
38C59×#393C2211yescharacteristic
39C59×#388—21no—
40C59×#387—21no—
41C59×#387—21no—
42C59×#385—21no—
43C59×#383C2211yescharacteristic
44C59×#328C2212yes—
45C59×#328C2212yes—
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2211yescharacteristic
49C59×#324C2211yescharacteristic
50C59×#323C2212yes—
51C2×#41×C59—21no—
52C22×Q8×C59C2311yescharacteristic
53C23×C4×C59D411yescharacteristic
54C2×Q16×C59—22no—
55C2×SD16×C59—22no—
56C4⋊Q8×C59—22no—
57#126×C59—22no—
58#126×C59—22no—
59#126×C59—22no—
60#126×C59—22no—
61#125×C59—21no—
62#125×C59—22no—
63#125×C59D412yes—
64#124×C59—21no—
65#124×C59—22no—
66#124×C59—22no—
67#124×C59D412yes—
68#123×C59—22no—
69#123×C59—22no—
70#123×C59—22no—
71C22⋊Q8×C59—21no—
72C22⋊Q8×C59—21no—
73C22⋊Q8×C59—21no—
74C22⋊Q8×C59—22no—
75C22⋊Q8×C59C2312yes—
76#121×C59—21no—
77#121×C59—21no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—22no—
81#121×C59—21no—
82#121×C59C2312yes—
83#120×C59—22no—
84C4×Q8×C59—22no—
85C4×Q8×C59—21no—
86C4×Q8×C59—22no—
87C4×Q8×C59—22no—
88C4×Q8×C59D412yes—
89C4×D4×C59—22no—
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92C4×D4×C59—21no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C4×D4×C59D412yes—
96#117×C59—22no—
97#117×C59—22no—
98#117×C59—21no—
99#117×C59—22no—
100#117×C59—22no—
101#117×C59—22no—
102C2×C4⋊C4×C59—22no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C22⋊C4×C59C2311yescharacteristic
113C2×C42×C59—21no—
114C2×C42×C59D411yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#107×C59—22no—
117#106×C59—22no—
118C4⋊C8×C59—22no—
119#103×C59—22no—
120#103×C59—22no—
121#103×C59—22no—
122#102×C59—22no—
123C22⋊C8×C59C2312yes—
124C22⋊C8×C59C2312yes—
125C4×C8×C59—22no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134C24×C59C2×D411yescharacteristic
135#41×C59—42no—
136#41×C59—42no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—22no—
140C2×Q8×C59—42no—
141C2×Q8×C59C2×D412yes—
142C2×D4×C59—21no—
143C2×D4×C59—22no—
144C2×D4×C59—22no—
145C2×D4×C59—22no—
146C2×D4×C59—21no—
147C2×D4×C59C2×D412yes—
148C22×C4×C59—22no—
149C22×C4×C59—21no—
150C22×C4×C59—22no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—22no—
154C22×C4×C59—22no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—41no—
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59C2411yescharacteristic
165Q16×C59—44no—
166SD16×C59—44no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
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—22no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—41no—
184C4⋊C4×C59—41no—
185C4⋊C4×C59—42no—
186C4⋊C4×C59C2×D412yes—
187C4⋊C4×C59C2×D412yes—
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—22no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—22no—
198C22⋊C4×C59—22no—
199C22⋊C4×C59—42no—
200C22⋊C4×C59—42no—
201C42×C59—22no—
202C42×C59—22no—
203C42×C59—21no—
204C42×C59—21no—
205C42×C59—22no—
206C42×C59—42no—
207C42×C59C2×D412yes—
208C42×C59C2×D412yes—
209C23×C59—21no—
210C23×C59—22no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59C22×D411yescharacteristic
216Q8×C59—24no—
217Q8×C59—42no—
218Q8×C59—42no—
219Q8×C59—42no—
220Q8×C59—81no—
221D4×C59—42no—
222D4×C59—42no—
223D4×C59—42no—
224D4×C59—42no—
225D4×C59—82no—
226D4×C59—81no—
227D4×C59—82no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—44no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—82no—
257C2×C4×C59—82no—
258C2×C4×C59#12012yes—
259C2×C4×C59#12011yescharacteristic
260C2×C4×C59#12011yescharacteristic
261C2×C4×C59#13512yes—
262C2×C4×C59C22×D412yes—
263C8×C59—42no—
264C8×C59—42no—
265#18217C5911yesmaximal, characteristic, direct product factor, Sylow
266C22×C59—21no—
267C22×C59—21no—
268C22×C59—42no—
269C22×C59—41no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—41no—
273C22×C59—42no—
274C22×C59—41no—
275C22×C59—81no—
276C22×C59C2×#12011yescharacteristic
277C22×C59C2×#13511yescharacteristic
278C22×C59#57511yescharacteristic
279C4×C59—22no—
280C4×C59—21no—
281C4×C59—21no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—42no—
285C4×C59—41no—
286C4×C59—42no—
287C4×C59—42no—
288C4×C59—41no—
289C4×C59—42no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59—81no—
293#3493C2×C5911yescharacteristic
294#3108C2×C5912yescomplemented
295#2599C2×C5911yescharacteristic, complemented
296#1612C2×C5912yescomplemented
297#1611C2×C5911yescharacteristic, complemented
298#1610C2×C5911yescharacteristic, complemented
299#1606C2×C5912yescomplemented
300#1605C2×C5911yescharacteristic, complemented
301#1604C2×C5911yescharacteristic, complemented
302#1521C2×C5912yescomplemented
303#1458C2×C5911yescharacteristic, complemented
304C2×C59—21no—
305C2×C59—21no—
306C2×C59—82no—
307C2×C59—81no—
308C2×C59#300211yescharacteristic
309C2×C59#302211yescharacteristic
310C2×C59#302511yescharacteristic
311#556—22no—
312#555—22no—
313#553—22no—
314#552—22no—
315#544C22×C5911yescharacteristic
316#538—21no—
317#538C22×C5912yes—
318#537C22×C5911yescharacteristic
319#529—21no—
320#529C22×C5912yes—
321#525—21no—
322#520—21no—
323C2×C4×Q8C22×C5911yescharacteristic
324C2×#117C22×C5911yescharacteristic
325#483C22×C5912yes—
326#480C22×C5912yes—
327#479—22no—
328#464—22no—
329#451C22×C5912yes—
330#448C22×C5912yes—
331C4×Q16—22no—
332C4×SD16—22no—
333#431C22×C5912yes—
334#398C22×C5911yescharacteristic
335#397C22×C5911yescharacteristic
336#395C22×C5911yescharacteristic
337#393C22×C5911yescharacteristic
338#388—21no—
339#387—21no—
340#387—21no—
341#385—21no—
342#383C22×C5911yescharacteristic
343#328C22×C5912yes—
344#328C22×C5912yes—
345#327C22×C5912yes—
346#327C22×C5912yes—
347#324C22×C5911yescharacteristic
348#324C22×C5911yescharacteristic
349#323C22×C5912yes—
350C59#1821711yescharacteristic, direct product factor, Sylow
351C2×#41—21no—
352C22×Q8C23×C5911yescharacteristic
353C23×C4D4×C5911yescharacteristic
354C2×Q16—22no—
355C2×SD16—22no—
356C4⋊Q8—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#126—22no—
361#125—21no—
362#125—22no—
363#125D4×C5912yes—
364#124—21no—
365#124—22no—
366#124—22no—
367#124D4×C5912yes—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—21no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8C23×C5912yes—
376#121—21no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121—22no—
381#121—21no—
382#121C23×C5912yes—
383#120—22no—
384C4×Q8—22no—
385C4×Q8—21no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8D4×C5912yes—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—21no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4D4×C5912yes—
396#117—22no—
397#117—22no—
398#117—21no—
399#117—22no—
400#117—22no—
401#117—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4D4×C5911yescharacteristic
407C2×C4⋊C4C23×C5911yescharacteristic
408C2×C4⋊C4C23×C5911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4D4×C5911yescharacteristic
412C2×C22⋊C4C23×C5911yescharacteristic
413C2×C42—21no—
414C2×C42D4×C5911yescharacteristic
415C2×C42C23×C5911yescharacteristic
416#107—22no—
417#106—22no—
418C4⋊C8—22no—
419#103—22no—
420#103—22no—
421#103—22no—
422#102—22no—
423C22⋊C8C23×C5912yes—
424C22⋊C8C23×C5912yes—
425C4×C8—22no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95C23×C5911yescharacteristic
433#95C23×C5911yescharacteristic
434C24C2×D4×C5911yescharacteristic
435#41—42no—
436#41—42no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8C2×D4×C5912yes—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4C2×D4×C5912yes—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—41no—
459C22×C4C2×D4×C5911yescharacteristic
460C22×C4C2×D4×C5911yescharacteristic
461C22×C4C2×D4×C5911yescharacteristic
462C22×C4C2×D4×C5911yescharacteristic
463C22×C4C2×D4×C5911yescharacteristic
464C22×C4C24×C5911yescharacteristic
465Q16—44no—
466SD16—44no—
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4C2×D4×C5912yes—
487C4⋊C4C2×D4×C5912yes—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—42no—
507C42C2×D4×C5912yes—
508C42C2×D4×C5912yes—
509C23—21no—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—41no—
514C23—41no—
515C23C22×D4×C5911yescharacteristic
516Q8—24no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—81no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—82no—
526D4—81no—
527D4—82no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#120×C5912yes—
559C2×C4#120×C5911yescharacteristic
560C2×C4#120×C5911yescharacteristic
561C2×C4#135×C5912yes—
562C2×C4C22×D4×C5912yes—
563C8—42no—
564C8—42no—
565C22—21no—
566C22—21no—
567C22—42no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—81no—
575C22C2×#120×C5911yescharacteristic
576C22C2×#135×C5911yescharacteristic
577C22C59×#57511yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—81no—
592C2—21no—
593C2—21no—
594C2—81no—
595C2—82no—
596C2C59×#300211yescharacteristic
597C2C59×#302211yescharacteristic
598C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3493C2
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
2. C59×#3108C2semidirect
3. C59×#2599C2semidirect
4. C59×#1612C2semidirect
5. C59×#1611C2semidirect
6. C59×#1610C2semidirect
7. C59×#1606C2semidirect
8. C59×#1605C2semidirect
9. C59×#1604C2semidirect
10. C59×#1521C2semidirect
11. C59×#1458C2semidirect
265. #18217C59direct (N × H)
294. #3108C2×C59semidirect
295. #2599C2×C59semidirect
296. #1612C2×C59semidirect
297. #1611C2×C59semidirect
298. #1610C2×C59semidirect
299. #1606C2×C59semidirect
300. #1605C2×C59semidirect
301. #1604C2×C59semidirect
302. #1521C2×C59semidirect
303. #1458C2×C59semidirect
350. C59#18217direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34932
C59×#31082
C59×#25992
C59×#16122
C59×#16112
C59×#16102
C59×#16062
C59×#16052
C59×#16042
C59×#15212
C59×#14582
#1821759