Finite Group Databasebeta

← C59×#18146 #28430255

Subgroups — C59×#18146 #28430255

Order 15104.

1186 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3460C211yesmaximal, characteristic
2C59×#3004C212yesmaximal, complemented
3C59×#2409C211yesmaximal, characteristic, complemented
4C59×#1606C214yesmaximal, complemented
5C59×#1605C212yesmaximal, complemented
6C59×#1604C212yesmaximal, complemented
7C59×#1513C212yesmaximal, complemented
8C59×#1465C211yesmaximal, characteristic, complemented
9C23×Q8×C59C2211yescharacteristic
10C2×#137×C59—22no—
11C59×#544C2211yescharacteristic
12C59×#542—22no—
13C59×#540—22no—
14C59×#538—21no—
15C59×#538C2212yes—
16C59×#537C2211yescharacteristic
17C59×#535—21no—
18C59×#532—22no—
19C59×#529—22no—
20C59×#525—21no—
21C59×#525C2212yes—
22C2×C22⋊Q8×C59—21no—
23C4×#41×C59—21no—
24C2×C4×D4×C59C2211yescharacteristic
25C59×#451—22no—
26C59×#451C2214yes—
27C59×#448—22no—
28C59×#448C2214yes—
29C59×#419—22no—
30C59×#407C2212yes—
31C59×#395C2212yes—
32C59×#393C2212yes—
33C59×#391—21no—
34C59×#386—21no—
35C59×#385—21no—
36C59×#380C2211yescharacteristic
37C59×#328C2214yes—
38C59×#327C2214yes—
39C59×#324C2212yes—
40C59×#323C2212yes—
41C2×#41×C59—21no—
42C2×#41×C59—21no—
43C2×#41×C59D412yes—
44C22×Q8×C59—21no—
45C22×Q8×C59—22no—
46C22×Q8×C59—22no—
47C22×Q8×C59—44no—
48C22×Q8×C59C2312yes—
49C22×D4×C59D411yescharacteristic
50C23×C4×C59—21no—
51C23×C4×C59D411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53#137×C59—48no—
54C2×Q16×C59—24no—
55C2×SD16×C59—24no—
56C2×M16×C59—22no—
57C4⋊Q8×C59—22no—
58#127×C59—22no—
59#126×C59—22no—
60#126×C59—22no—
61#125×C59—22no—
62#124×C59—22no—
63#124×C59—22no—
64#124×C59—22no—
65#123×C59—22no—
66#123×C59—22no—
67#123×C59—22no—
68#123×C59—22no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—44no—
73C22⋊Q8×C59D412yes—
74C22⋊Q8×C59C2312yes—
75#121×C59—21no—
76#121×C59—22no—
77#121×C59—22no—
78#121×C59—22no—
79#121×C59—22no—
80#121×C59—22no—
81#121×C59—21no—
82#121×C59D412yes—
83#121×C59C2312yes—
84#120×C59—22no—
85#120×C59—22no—
86C4×Q8×C59—21no—
87C4×Q8×C59—21no—
88C4×D4×C59—21no—
89C4×D4×C59—21no—
90C4×D4×C59—22no—
91C4×D4×C59—22no—
92C4×D4×C59—22no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C4×D4×C59—22no—
96#117×C59—22no—
97#117×C59—21no—
98#117×C59—21no—
99#117×C59D412yes—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—22no—
105C2×C22⋊C4×C59D411yescharacteristic
106C2×C22⋊C4×C59C2311yescharacteristic
107C2×C42×C59—22no—
108C2×C42×C59D411yescharacteristic
109#103×C59—24no—
110#103×C59—24no—
111C22⋊C8×C59—22no—
112C22⋊C8×C59C2314yes—
113#95×C59—22no—
114#95×C59—22no—
115#95×C59C2312yes—
116C24×C59C2×D411yescharacteristic
117C24×C59C2×D411yescharacteristic
118#41×C59—42no—
119#41×C59—44no—
120#41×C59—42no—
121#41×C59—82no—
122C2×Q8×C59—21no—
123C2×Q8×C59—24no—
124C2×Q8×C59—21no—
125C2×Q8×C59—28no—
126C2×Q8×C59—24no—
127C2×Q8×C59—44no—
128C2×Q8×C59—42no—
129C2×Q8×C59—44no—
130C2×Q8×C59—48no—
131C2×Q8×C59—41no—
132C2×Q8×C59—41no—
133C2×Q8×C59C2×D412yes—
134C2×D4×C59—21no—
135C2×D4×C59—22no—
136C2×D4×C59—22no—
137C2×D4×C59—22no—
138C2×D4×C59—22no—
139C2×D4×C59—21no—
140C2×D4×C59—22no—
141C2×D4×C59—22no—
142C2×D4×C59—22no—
143C2×D4×C59—41no—
144C2×D4×C59C2×D412yes—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—22no—
148C22×C4×C59—22no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—22no—
152C22×C4×C59—42no—
153C22×C4×C59—44no—
154C22×C4×C59—42no—
155C22×C4×C59—42no—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D412yes—
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D412yes—
162C22×C4×C59C2411yescharacteristic
163Q16×C59—48no—
164SD16×C59—48no—
165M16×C59—44no—
166C2×C8×C59—24no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59—44no—
178C4⋊C4×C59C2×D412yes—
179C4⋊C4×C59C2×D412yes—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—22no—
186C22⋊C4×C59—22no—
187C22⋊C4×C59—22no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—44no—
192C22⋊C4×C59—42no—
193C42×C59—22no—
194C42×C59—22no—
195C42×C59—21no—
196C42×C59—22no—
197C42×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—22no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59#12012yes—
209C23×C59C22×D411yescharacteristic
210Q8×C59—28no—
211Q8×C59—42no—
212Q8×C59—44no—
213Q8×C59—44no—
214Q8×C59—42no—
215Q8×C59—44no—
216Q8×C59—81no—
217D4×C59—44no—
218D4×C59—42no—
219D4×C59—42no—
220D4×C59—42no—
221D4×C59—42no—
222D4×C59—82no—
223D4×C59—82no—
224D4×C59—82no—
225D4×C59—81no—
226D4×C59—82no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—24no—
231C2×C4×C59—24no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—21no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—44no—
240C2×C4×C59—42no—
241C2×C4×C59—48no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—44no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—81no—
251C2×C4×C59—82no—
252C2×C4×C59#12011yescharacteristic
253C2×C4×C59#12011yescharacteristic
254C2×C4×C59#13712yescomplemented
255C2×C4×C59C22×D412yes—
256C8×C59—44no—
257#18146C5911yesmaximal, characteristic, direct product factor, Sylow
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—41no—
268C22×C59—42no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—42no—
272C22×C59—81no—
273C22×C59C2×#12011yescharacteristic
274C22×C59C2×#13711yescharacteristic
275C22×C59#57511yescharacteristic
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—21no—
279C4×C59—42no—
280C4×C59—41no—
281C4×C59—42no—
282C4×C59—44no—
283C4×C59—41no—
284C4×C59—42no—
285C4×C59—82no—
286C4×C59B(4,𝔽2)12yes—
287#3460C2×C5911yescharacteristic
288#3004C2×C5912yescomplemented
289#2409C2×C5911yescharacteristic, complemented
290#1606C2×C5914yescomplemented
291#1605C2×C5912yescomplemented
292#1604C2×C5912yescomplemented
293#1513C2×C5912yescomplemented
294#1465C2×C5911yescharacteristic, complemented
295C2×C59—21no—
296C2×C59—21no—
297C2×C59—42no—
298C2×C59—82no—
299C2×C59—81no—
300C2×C59#300711yescharacteristic
301C2×C59#302211yescharacteristic
302C2×C59C2×B(4,𝔽2)11yescharacteristic
303C23×Q8C22×C5911yescharacteristic
304C2×#137—22no—
305#544C22×C5911yescharacteristic
306#542—22no—
307#540—22no—
308#538—21no—
309#538C22×C5912yes—
310#537C22×C5911yescharacteristic
311#535—21no—
312#532—22no—
313#529—22no—
314#525—21no—
315#525C22×C5912yes—
316C2×C22⋊Q8—21no—
317C4×#41—21no—
318C2×C4×D4C22×C5911yescharacteristic
319#451—22no—
320#451C22×C5914yes—
321#448—22no—
322#448C22×C5914yes—
323#419—22no—
324#407C22×C5912yes—
325#395C22×C5912yes—
326#393C22×C5912yes—
327#391—21no—
328#386—21no—
329#385—21no—
330#380C22×C5911yescharacteristic
331#328C22×C5914yes—
332#327C22×C5914yes—
333#324C22×C5912yes—
334#323C22×C5912yes—
335C59#1814611yescharacteristic, direct product factor, Sylow
336C2×#41—21no—
337C2×#41—21no—
338C2×#41D4×C5912yes—
339C22×Q8—21no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—44no—
343C22×Q8C23×C5912yes—
344C22×D4D4×C5911yescharacteristic
345C23×C4—21no—
346C23×C4D4×C5911yescharacteristic
347C23×C4C23×C5911yescharacteristic
348#137—48no—
349C2×Q16—24no—
350C2×SD16—24no—
351C2×M16—22no—
352C4⋊Q8—22no—
353#127—22no—
354#126—22no—
355#126—22no—
356#125—22no—
357#124—22no—
358#124—22no—
359#124—22no—
360#123—22no—
361#123—22no—
362#123—22no—
363#123—22no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—44no—
368C22⋊Q8D4×C5912yes—
369C22⋊Q8C23×C5912yes—
370#121—21no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121D4×C5912yes—
378#121C23×C5912yes—
379#120—22no—
380#120—22no—
381C4×Q8—21no—
382C4×Q8—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391#117—22no—
392#117—21no—
393#117—21no—
394#117D4×C5912yes—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4C23×C5911yescharacteristic
398C2×C22⋊C4—22no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4D4×C5911yescharacteristic
401C2×C22⋊C4C23×C5911yescharacteristic
402C2×C42—22no—
403C2×C42D4×C5911yescharacteristic
404#103—24no—
405#103—24no—
406C22⋊C8—22no—
407C22⋊C8C23×C5914yes—
408#95—22no—
409#95—22no—
410#95C23×C5912yes—
411C24C2×D4×C5911yescharacteristic
412C24C2×D4×C5911yescharacteristic
413#41—42no—
414#41—44no—
415#41—42no—
416#41—82no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—21no—
420C2×Q8—28no—
421C2×Q8—24no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8C2×D4×C5912yes—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—41no—
439C2×D4C2×D4×C5912yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4C2×D4×C5911yescharacteristic
452C22×C4C2×D4×C5912yes—
453C22×C4C2×D4×C5911yescharacteristic
454C22×C4C2×D4×C5911yescharacteristic
455C22×C4C2×D4×C5911yescharacteristic
456C22×C4C2×D4×C5912yes—
457C22×C4C24×C5911yescharacteristic
458Q16—48no—
459SD16—48no—
460M16—44no—
461C2×C8—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4C2×D4×C5912yes—
474C4⋊C4C2×D4×C5912yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—22no—
492C42—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23#120×C5912yes—
504C23C22×D4×C5911yescharacteristic
505Q8—28no—
506Q8—42no—
507Q8—44no—
508Q8—44no—
509Q8—42no—
510Q8—44no—
511Q8—81no—
512D4—44no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—81no—
521D4—82no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4#120×C5911yescharacteristic
548C2×C4#120×C5911yescharacteristic
549C2×C4#137×C5912yescomplemented
550C2×C4C22×D4×C5912yes—
551C8—44no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—81no—
567C22C2×#120×C5911yescharacteristic
568C22C2×#137×C5911yescharacteristic
569C22C59×#57511yescharacteristic
570C4—22no—
571C4—22no—
572C4—21no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—44no—
577C4—41no—
578C4—42no—
579C4—82no—
580C4C59×B(4,𝔽2)12yes—
581C2—21no—
582C2—21no—
583C2—42no—
584C2—81no—
585C2—82no—
586C2C59×#300711yescharacteristic
587C2C59×#302211yescharacteristic
588C2C2×C59×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3460C2
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×#3004C2semidirect
3. C59×#2409C2semidirect
4. C59×#1606C2semidirect
5. C59×#1605C2semidirect
6. C59×#1604C2semidirect
7. C59×#1513C2semidirect
8. C59×#1465C2semidirect
254. C2×C4×C59#137semidirect
257. #18146C59direct (N × H)
288. #3004C2×C59semidirect
289. #2409C2×C59semidirect
290. #1606C2×C59semidirect
291. #1605C2×C59semidirect
292. #1604C2×C59semidirect
293. #1513C2×C59semidirect
294. #1465C2×C59semidirect
335. C59#18146direct (N × H)
549. C2×C4#137×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34602
C59×#30042
C59×#24092
C59×#16062
C59×#16052
C59×#16042
C59×#15132
C59×#14652
#1814659