Finite Group Databasebeta

← C59×#23591 #28435700

Subgroups — C59×#23591 #28435700

Order 15104.

1182 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×#3460C211yesmaximal, characteristic, complemented
2C4×AGL(1,ℤ/8)×C59C212yesmaximal, complemented
3C59×#2328C211yesmaximal, characteristic, complemented
4C59×#2061C211yesmaximal, characteristic, complemented
5C59×#2040C211yesmaximal, characteristic, complemented
6C59×#2037C212yesmaximal, complemented
7C59×#2027C211yesmaximal, characteristic, complemented
8C59×#2006C211yesmaximal, characteristic, complemented
9C59×#2002C212yesmaximal, complemented
10C59×#1759C212yesmaximal, complemented
11C4×#100×C59C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C59C2212yes—
13C59×#544C2211yescharacteristic
14C59×#542—22no—
15C59×#540—22no—
16C59×#538—22no—
17C59×#537C2211yescharacteristic
18C59×#535C2212yes—
19C59×#532—22no—
20C2×#127×C59C2211yescharacteristic
21C59×#529—22no—
22C59×#525—22no—
23C4×#41×C59C2212yes—
24C2×C4×D4×C59C2211yescharacteristic
25C2×C4×D4×C59C2211yescharacteristic
26C59×#442—22no—
27C59×#440—22no—
28C4×SD16×C59—22no—
29C4×D8×C59—22no—
30C59×#428C2212yes—
31C2×#104×C59C2212yes—
32C2×#104×C59C2212yes—
33C59×#418C2212yes—
34C59×#417C2212yes—
35C2×#100×C59C2212yes—
36C4×M16×C59C2212yes—
37C59×#390C2211yescharacteristic
38C59×#390C2211yescharacteristic
39C59×#386—22no—
40C59×#386—22no—
41C59×#382C2211yescharacteristic
42C59×#376C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2211yescharacteristic
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59—22no—
48C2×#41×C59C2312yes—
49C22×D4×C59—22no—
50C22×D4×C59C2311yescharacteristic
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59—21no—
53C23×C4×C59D412yes—
54AGL(1,ℤ/8)×C59—24no—
55AGL(1,ℤ/8)×C59—24no—
56C2×SD16×C59—22no—
57C2×D8×C59—22no—
58C2×M16×C59C2312yes—
59C2×M16×C59C2312yes—
60C4⋊Q8×C59—22no—
61#127×C59—22no—
62#127×C59—22no—
63#127×C59—22no—
64#126×C59—24no—
65#125×C59—22no—
66#124×C59—24no—
67#124×C59—22no—
68#123×C59—24no—
69#123×C59—24no—
70C22⋊Q8×C59—22no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—22no—
73#121×C59—22no—
74#121×C59—24no—
75#121×C59—22no—
76#121×C59—22no—
77#121×C59—22no—
78#121×C59—22no—
79#120×C59—24no—
80C4×Q8×C59D412yescomplemented
81C4×Q8×C59D412yescomplemented
82C4×D4×C59—24no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59—22no—
88C4×D4×C59—22no—
89C4×D4×C59D412yescomplemented
90C4×D4×C59D412yescomplemented
91#117×C59—22no—
92#117×C59—22no—
93#117×C59C2312yes—
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—22no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59D412yes—
100C2×C42×C59—22no—
101C2×C42×C59C2311yescharacteristic
102C2×C42×C59C2311yescharacteristic
103C2×C42×C59C2311yescharacteristic
104#107×C59—22no—
105#106×C59—22no—
106#104×C59—24no—
107#104×C59—24no—
108#103×C59—22no—
109#102×C59—22no—
110#102×C59—22no—
111#102×C59—22no—
112#100×C59—24no—
113#97×C59—22no—
114C4×C8×C59—22no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—22no—
118C24×C59—21no—
119C24×C59C2×D412yes—
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123#41×C59—22no—
124#41×C59—84no—
125C2×Q8×C59—22no—
126C2×Q8×C59C2×D412yes—
127C2×D4×C59—22no—
128C2×D4×C59—24no—
129C2×D4×C59—24no—
130C2×D4×C59—22no—
131C2×D4×C59—28no—
132C2×D4×C59—22no—
133C2×D4×C59—22no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—82no—
140C2×D4×C59—82no—
141C2×D4×C59C2×D412yes—
142C2×D4×C59#4112yescomplemented
143C22×C4×C59—21no—
144C22×C4×C59—22no—
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—42no—
152C22×C4×C59—82no—
153C22×C4×C59—82no—
154C22×C4×C59C2×D412yes—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158SD16×C59—28no—
159D8×C59—28no—
160M16×C59—22no—
161M16×C59—22no—
162M16×C59—22no—
163M16×C59#4114yescomplemented
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59C2×D412yes—
174C4⋊C4×C59C2×D412yes—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—24no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—22no—
181C22⋊C4×C59—22no—
182C22⋊C4×C59—41no—
183C22⋊C4×C59—41no—
184C22⋊C4×C59—82no—
185C22⋊C4×C59—82no—
186C42×C59—22no—
187C42×C59—22no—
188C42×C59—22no—
189C42×C59—21no—
190C42×C59—22no—
191C42×C59C2×D412yes—
192C42×C59C2×D412yes—
193C42×C59#4112yescomplemented
194C23×C59—22no—
195C23×C59—22no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59C22×D411yescharacteristic
206Q8×C59—22no—
207Q8×C59—22no—
208Q8×C59—82no—
209D4×C59—22no—
210D4×C59—24no—
211D4×C59—22no—
212D4×C59—42no—
213D4×C59—42no—
214D4×C59—42no—
215D4×C59—42no—
216D4×C59—42no—
217D4×C59—84no—
218D4×C59—88no—
219D4×C59—82no—
220D4×C59—84no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—44no—
229C2×C4×C59—44no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—42no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—82no—
241C2×C4×C59—82no—
242C2×C4×C59—84no—
243C2×C4×C59#12011yescharacteristic
244C2×C4×C59#12012yes—
245C2×C4×C59#12011yescharacteristic
246C2×C4×C59C22×D412yes—
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4112yes—
249C2×C4×C59C2×#4111yescharacteristic
250C8×C59—24no—
251C8×C59—42no—
252#23591C5911yesmaximal, characteristic, direct product factor, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—44no—
256C22×C59—44no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—82no—
265C22×C59—82no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59C2×#12011yescharacteristic
269C22×C59#52511yescharacteristic
270C22×C59#53211yescharacteristic
271C4×C59—21no—
272C4×C59—21no—
273C4×C59—22no—
274C4×C59—42no—
275C4×C59—42no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59—42no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59#52512yes—
282C4×C59#53212yes—
283#3460C2×C5911yescharacteristic, complemented
284C4×AGL(1,ℤ/8)C2×C5912yescomplemented
285#2328C2×C5911yescharacteristic, complemented
286#2061C2×C5911yescharacteristic, complemented
287#2040C2×C5911yescharacteristic, complemented
288#2037C2×C5912yescomplemented
289#2027C2×C5911yescharacteristic, complemented
290#2006C2×C5911yescharacteristic, complemented
291#2002C2×C5912yescomplemented
292#1759C2×C5912yescomplemented
293C4×#100C2×C5911yescharacteristic, complemented
294C2×C59—21no—
295C2×C59—21no—
296C2×C59—82no—
297C2×C59—82no—
298C2×C59—82no—
299C2×C59#238811yescharacteristic
300C2×C59#301911yescharacteristic
301C2×C59#302111yescharacteristic
302C2×AGL(1,ℤ/8)C22×C5912yes—
303#544C22×C5911yescharacteristic
304#542—22no—
305#540—22no—
306#538—22no—
307#537C22×C5911yescharacteristic
308#535C22×C5912yes—
309#532—22no—
310C2×#127C22×C5911yescharacteristic
311#529—22no—
312#525—22no—
313C4×#41C22×C5912yes—
314C2×C4×D4C22×C5911yescharacteristic
315C2×C4×D4C22×C5911yescharacteristic
316#442—22no—
317#440—22no—
318C4×SD16—22no—
319C4×D8—22no—
320#428C22×C5912yes—
321C2×#104C22×C5912yes—
322C2×#104C22×C5912yes—
323#418C22×C5912yes—
324#417C22×C5912yes—
325C2×#100C22×C5912yes—
326C4×M16C22×C5912yes—
327#390C22×C5911yescharacteristic
328#390C22×C5911yescharacteristic
329#386—22no—
330#386—22no—
331#382C22×C5911yescharacteristic
332#376C22×C5911yescharacteristic
333#343C22×C5911yescharacteristic
334#343C22×C5911yescharacteristic
335#339C22×C5912yes—
336#339C22×C5912yes—
337C59#2359111yescharacteristic, direct product factor, Sylow
338C2×#41—22no—
339C2×#41C23×C5912yes—
340C22×D4—22no—
341C22×D4C23×C5911yescharacteristic
342C22×D4C23×C5911yescharacteristic
343C23×C4—21no—
344C23×C4D4×C5912yes—
345AGL(1,ℤ/8)—24no—
346AGL(1,ℤ/8)—24no—
347C2×SD16—22no—
348C2×D8—22no—
349C2×M16C23×C5912yes—
350C2×M16C23×C5912yes—
351C4⋊Q8—22no—
352#127—22no—
353#127—22no—
354#127—22no—
355#126—24no—
356#125—22no—
357#124—24no—
358#124—22no—
359#123—24no—
360#123—24no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364#121—22no—
365#121—24no—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—22no—
370#120—24no—
371C4×Q8D4×C5912yescomplemented
372C4×Q8D4×C5912yescomplemented
373C4×D4—24no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4D4×C5912yescomplemented
381C4×D4D4×C5912yescomplemented
382#117—22no—
383#117—22no—
384#117C23×C5912yes—
385C2×C4⋊C4C23×C5911yescharacteristic
386C2×C4⋊C4C23×C5911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4D4×C5912yes—
391C2×C42—22no—
392C2×C42C23×C5911yescharacteristic
393C2×C42C23×C5911yescharacteristic
394C2×C42C23×C5911yescharacteristic
395#107—22no—
396#106—22no—
397#104—24no—
398#104—24no—
399#103—22no—
400#102—22no—
401#102—22no—
402#102—22no—
403#100—24no—
404#97—22no—
405C4×C8—22no—
406#95—21no—
407#95—21no—
408#95—22no—
409C24—21no—
410C24C2×D4×C5912yes—
411#41—22no—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—84no—
416C2×Q8—22no—
417C2×Q8C2×D4×C5912yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—28no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4C2×D4×C5912yes—
433C2×D4#41×C5912yescomplemented
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×D4×C5912yes—
446C22×C4C2×D4×C5911yescharacteristic
447C22×C4C2×D4×C5911yescharacteristic
448C22×C4C24×C5911yescharacteristic
449SD16—28no—
450D8—28no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16#41×C5914yescomplemented
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4C2×D4×C5912yes—
465C4⋊C4C2×D4×C5912yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—22no—
482C42C2×D4×C5912yes—
483C42C2×D4×C5912yes—
484C42#41×C5912yescomplemented
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23C22×D4×C5911yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—82no—
500D4—22no—
501D4—24no—
502D4—22no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—84no—
509D4—88no—
510D4—82no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—84no—
534C2×C4#120×C5911yescharacteristic
535C2×C4#120×C5912yes—
536C2×C4#120×C5911yescharacteristic
537C2×C4C22×D4×C5912yes—
538C2×C4C2×#41×C5911yescharacteristic
539C2×C4C2×#41×C5912yes—
540C2×C4C2×#41×C5911yescharacteristic
541C8—24no—
542C8—42no—
543C22—21no—
544C22—21no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22C2×#120×C5911yescharacteristic
559C22C59×#52511yescharacteristic
560C22C59×#53211yescharacteristic
561C4—21no—
562C4—21no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4C59×#52512yes—
572C4C59×#53212yes—
573C2—21no—
574C2—21no—
575C2—82no—
576C2—82no—
577C2—82no—
578C2C59×#238811yescharacteristic
579C2C59×#301911yescharacteristic
580C2C59×#302111yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C59×#3460C2semidirect
2. C4×AGL(1,ℤ/8)×C59C2semidirect
3. C59×#2328C2semidirect
4. C59×#2061C2semidirect
5. C59×#2040C2semidirect
6. C59×#2037C2semidirect
7. C59×#2027C2semidirect
8. C59×#2006C2semidirect
9. C59×#2002C2semidirect
10. C59×#1759C2semidirect
11. C4×#100×C59C2semidirect
80. C4×Q8×C59D4semidirect
81. C4×Q8×C59D4semidirect
89. C4×D4×C59D4semidirect
90. C4×D4×C59D4semidirect
142. C2×D4×C59#41semidirect
163. M16×C59#41semidirect
193. C42×C59#41semidirect
252. #23591C59direct (N × H)
283. #3460C2×C59semidirect
284. C4×AGL(1,ℤ/8)C2×C59semidirect
285. #2328C2×C59semidirect
286. #2061C2×C59semidirect
287. #2040C2×C59semidirect
288. #2037C2×C59semidirect
289. #2027C2×C59semidirect
290. #2006C2×C59semidirect
291. #2002C2×C59semidirect
292. #1759C2×C59semidirect
293. C4×#100C2×C59semidirect
337. C59#23591direct (N × H)
371. C4×Q8D4×C59semidirect
372. C4×Q8D4×C59semidirect
380. C4×D4D4×C59semidirect
381. C4×D4D4×C59semidirect
433. C2×D4#41×C59semidirect
454. M16#41×C59semidirect
484. C42#41×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#34602
C4×AGL(1,ℤ/8)×C592
C59×#23282
C59×#20612
C59×#20402
C59×#20372
C59×#20272
C59×#20062
C59×#20022
C59×#17592
C4×#100×C592
#2359159