Finite Group Databasebeta

← C59×#18356 #28430465

Subgroups — C59×#18356 #28430465

Order 15104.

1054 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3491C211yesmaximal, characteristic, complemented
2C59×#3217C212yesmaximal, complemented
3C59×#2619C211yesmaximal, characteristic, complemented
4C59×#1597C214yesmaximal, complemented
5C59×#1596C212yesmaximal, complemented
6C59×#1595C212yesmaximal, complemented
7C59×#1512C212yesmaximal, complemented
8C59×#1464C211yesmaximal, characteristic, complemented
9C59×#552—22no—
10C59×#552—22no—
11C59×#551—22no—
12C59×#551—22no—
13C59×#543C2211yescharacteristic
14C59×#539—21no—
15C59×#539C2212yes—
16C59×#534C2211yescharacteristic
17C2×#127×C59C2211yescharacteristic, complemented
18C59×#529—21no—
19C59×#529C2212yes—
20C2×#123×C59—21no—
21C2×#121×C59—21no—
22C59×#518—21no—
23C2×#117×C59C2211yescharacteristic
24C59×#496—22no—
25C59×#490—22no—
26C59×#460—22no—
27C59×#459—22no—
28C59×#450C2214yescomplemented
29C59×#447C2214yes—
30C59×#432C2212yescomplemented
31C59×#394—21no—
32C59×#394C2212yescomplemented
33C59×#392—21no—
34C59×#392C2212yes—
35C59×#390—21no—
36C59×#382C2211yescharacteristic, complemented
37C59×#328C2214yescomplemented
38C59×#327C2214yes—
39C59×#324C2212yescomplemented
40C59×#323C2212yes—
41C22×D4×C59—21no—
42C22×D4×C59—22no—
43C22×D4×C59—21no—
44C22×D4×C59—21no—
45C22×D4×C59C2312yes—
46C23×C4×C59D411yescharacteristic
47C2×SD16×C59—24no—
48C2×D8×C59—24no—
49#127×C59—22no—
50#127×C59—22no—
51#126×C59—22no—
52#126×C59—22no—
53#126×C59—22no—
54#126×C59—22no—
55#125×C59—21no—
56#125×C59—22no—
57#125×C59—22no—
58#125×C59D412yescomplemented
59#124×C59—21no—
60#124×C59D412yescomplemented
61#123×C59—22no—
62#123×C59—22no—
63#123×C59—22no—
64#123×C59—22no—
65#123×C59—22no—
66#123×C59—22no—
67#123×C59—22no—
68C22⋊Q8×C59—22no—
69C22⋊Q8×C59—22no—
70C22⋊Q8×C59—22no—
71C22⋊Q8×C59—21no—
72C22⋊Q8×C59—22no—
73C22⋊Q8×C59C2312yes—
74#121×C59—21no—
75#121×C59—44no—
76#121×C59C2312yes—
77#120×C59—22no—
78C4×Q8×C59—21no—
79C4×Q8×C59D412yescomplemented
80C4×D4×C59—22no—
81C4×D4×C59—22no—
82C4×D4×C59—21no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59D412yescomplemented
86#117×C59—22no—
87#117×C59—22no—
88#117×C59—22no—
89#117×C59—22no—
90#117×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59D411yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—22no—
101C2×C22⋊C4×C59—22no—
102C2×C22⋊C4×C59—22no—
103C2×C22⋊C4×C59D411yescharacteristic
104C2×C22⋊C4×C59C2311yescharacteristic
105C2×C42×C59D411yescharacteristic
106C2×C42×C59C2311yescharacteristic
107C4⋊C8×C59—22no—
108#102×C59—24no—
109#102×C59—24no—
110C22⋊C8×C59C2314yes—
111#97×C59—22no—
112#95×C59—21no—
113#95×C59C2312yes—
114C24×C59—21no—
115C24×C59—22no—
116C24×C59C2×D411yescharacteristic
117C2×Q8×C59—21no—
118C2×Q8×C59C2×D412yes—
119C2×D4×C59—22no—
120C2×D4×C59—22no—
121C2×D4×C59—21no—
122C2×D4×C59—24no—
123C2×D4×C59—42no—
124C2×D4×C59—42no—
125C2×D4×C59—42no—
126C2×D4×C59—42no—
127C2×D4×C59—44no—
128C2×D4×C59—41no—
129C2×D4×C59—41no—
130C2×D4×C59—42no—
131C2×D4×C59—82no—
132C2×D4×C59—82no—
133C2×D4×C59—82no—
134C2×D4×C59—82no—
135C2×D4×C59C2×D412yes—
136C22×C4×C59—21no—
137C22×C4×C59—22no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—22no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—41no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2411yescharacteristic
152SD16×C59—84no—
153D8×C59—84no—
154C2×C8×C59—24no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59C2×D412yes—
169C4⋊C4×C59C2×D412yes—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—44no—
183C22⋊C4×C59—82no—
184C22⋊C4×C59—82no—
185C42×C59—22no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—22no—
189C42×C59C2×D412yes—
190C42×C59C2×D412yes—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—42no—
204C23×C59—81no—
205C23×C59—81no—
206C23×C59AGL(1,ℤ/8)12yes—
207C23×C59C22×D411yescharacteristic
208Q8×C59—42no—
209Q8×C59—81no—
210D4×C59—42no—
211D4×C59—44no—
212D4×C59—44no—
213D4×C59—44no—
214D4×C59—42no—
215D4×C59—42no—
216D4×C59—82no—
217D4×C59—82no—
218D4×C59—84no—
219D4×C59—81no—
220D4×C59—88no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—82no—
242C2×C4×C59—81no—
243C2×C4×C59—82no—
244C2×C4×C59—81no—
245C2×C4×C59#12011yescharacteristic
246C2×C4×C59#12011yescharacteristic
247C2×C4×C59#12012yes—
248C2×C4×C59C22×D412yes—
249C8×C59—44no—
250#18356C5911yesmaximal, characteristic, direct product factor, Sylow
251C22×C59—21no—
252C22×C59—21no—
253C22×C59—22no—
254C22×C59—42no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—81no—
261C22×C59—82no—
262C22×C59—82no—
263C22×C59—82no—
264C22×C59—82no—
265C22×C59—82no—
266C22×C59—82no—
267C22×C59—82no—
268C22×C59—162no—
269C22×C59C2×#12011yescharacteristic
270C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C59#57511yescharacteristic
272C4×C59—21no—
273C4×C59—21no—
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—41no—
279C4×C59—42no—
280C4×C59—41no—
281C4×C59—42no—
282C4×C59—82no—
283#3491C2×C5911yescharacteristic, complemented
284#3217C2×C5912yescomplemented
285#2619C2×C5911yescharacteristic, complemented
286#1597C2×C5914yescomplemented
287#1596C2×C5912yescomplemented
288#1595C2×C5912yescomplemented
289#1512C2×C5912yescomplemented
290#1464C2×C5911yescharacteristic, complemented
291C2×C59—21no—
292C2×C59—21no—
293C2×C59—42no—
294C2×C59—82no—
295C2×C59—162no—
296C2×C59#300611yescharacteristic
297C2×C59#302111yescharacteristic
298C2×C59#302711yescharacteristic
299#552—22no—
300#552—22no—
301#551—22no—
302#551—22no—
303#543C22×C5911yescharacteristic
304#539—21no—
305#539C22×C5912yes—
306#534C22×C5911yescharacteristic
307C2×#127C22×C5911yescharacteristic, complemented
308#529—21no—
309#529C22×C5912yes—
310C2×#123—21no—
311C2×#121—21no—
312#518—21no—
313C2×#117C22×C5911yescharacteristic
314#496—22no—
315#490—22no—
316#460—22no—
317#459—22no—
318#450C22×C5914yescomplemented
319#447C22×C5914yes—
320#432C22×C5912yescomplemented
321#394—21no—
322#394C22×C5912yescomplemented
323#392—21no—
324#392C22×C5912yes—
325#390—21no—
326#382C22×C5911yescharacteristic, complemented
327#328C22×C5914yescomplemented
328#327C22×C5914yes—
329#324C22×C5912yescomplemented
330#323C22×C5912yes—
331C59#1835611yescharacteristic, direct product factor, Sylow
332C22×D4—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C23×C5912yes—
337C23×C4D4×C5911yescharacteristic
338C2×SD16—24no—
339C2×D8—24no—
340#127—22no—
341#127—22no—
342#126—22no—
343#126—22no—
344#126—22no—
345#126—22no—
346#125—21no—
347#125—22no—
348#125—22no—
349#125D4×C5912yescomplemented
350#124—21no—
351#124D4×C5912yescomplemented
352#123—22no—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8C23×C5912yes—
365#121—21no—
366#121—44no—
367#121C23×C5912yes—
368#120—22no—
369C4×Q8—21no—
370C4×Q8D4×C5912yescomplemented
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4D4×C5912yescomplemented
377#117—22no—
378#117—22no—
379#117—22no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C5911yescharacteristic
386C2×C4⋊C4C23×C5911yescharacteristic
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4D4×C5911yescharacteristic
395C2×C22⋊C4C23×C5911yescharacteristic
396C2×C42D4×C5911yescharacteristic
397C2×C42C23×C5911yescharacteristic
398C4⋊C8—22no—
399#102—24no—
400#102—24no—
401C22⋊C8C23×C5914yes—
402#97—22no—
403#95—21no—
404#95C23×C5912yes—
405C24—21no—
406C24—22no—
407C24C2×D4×C5911yescharacteristic
408C2×Q8—21no—
409C2×Q8C2×D4×C5912yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—24no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4C2×D4×C5912yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4C2×D4×C5911yescharacteristic
438C22×C4C2×D4×C5911yescharacteristic
439C22×C4C2×D4×C5911yescharacteristic
440C22×C4C2×D4×C5911yescharacteristic
441C22×C4C2×D4×C5911yescharacteristic
442C22×C4C24×C5911yescharacteristic
443SD16—84no—
444D8—84no—
445C2×C8—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4C2×D4×C5912yes—
460C4⋊C4C2×D4×C5912yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42C2×D4×C5912yes—
481C42C2×D4×C5912yes—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23AGL(1,ℤ/8)×C5912yes—
498C23C22×D4×C5911yescharacteristic
499Q8—42no—
500Q8—81no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—81no—
511D4—88no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4#120×C5911yescharacteristic
537C2×C4#120×C5911yescharacteristic
538C2×C4#120×C5912yes—
539C2×C4C22×D4×C5912yes—
540C8—44no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—162no—
559C22C2×#120×C5911yescharacteristic
560C22C2×AGL(1,ℤ/8)×C5911yescharacteristic
561C22C59×#57511yescharacteristic
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—82no—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—82no—
577C2—162no—
578C2C59×#300611yescharacteristic
579C2C59×#302111yescharacteristic
580C2C59×#302711yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
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×#3491C2semidirect
2. C59×#3217C2semidirect
3. C59×#2619C2semidirect
4. C59×#1597C2semidirect
5. C59×#1596C2semidirect
6. C59×#1595C2semidirect
7. C59×#1512C2semidirect
8. C59×#1464C2semidirect
17. C2×#127×C59C22semidirect
28. C59×#450C22semidirect
30. C59×#432C22semidirect
32. C59×#394C22semidirect
36. C59×#382C22semidirect
37. C59×#328C22semidirect
39. C59×#324C22semidirect
58. #125×C59D4semidirect
60. #124×C59D4semidirect
79. C4×Q8×C59D4semidirect
85. C4×D4×C59D4semidirect
250. #18356C59direct (N × H)
283. #3491C2×C59semidirect
284. #3217C2×C59semidirect
285. #2619C2×C59semidirect
286. #1597C2×C59semidirect
287. #1596C2×C59semidirect
288. #1595C2×C59semidirect
289. #1512C2×C59semidirect
290. #1464C2×C59semidirect
307. C2×#127C22×C59semidirect
318. #450C22×C59semidirect
320. #432C22×C59semidirect
322. #394C22×C59semidirect
326. #382C22×C59semidirect
327. #328C22×C59semidirect
329. #324C22×C59semidirect
331. C59#18356direct (N × H)
349. #125D4×C59semidirect
351. #124D4×C59semidirect
370. C4×Q8D4×C59semidirect
376. C4×D4D4×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34912
C59×#32172
C59×#26192
C59×#15972
C59×#15962
C59×#15952
C59×#15122
C59×#14642
#1835659