Finite Group Databasebeta

← C59×#23592 #28435701

Subgroups — C59×#23592 #28435701

Order 15104.

1086 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3461C211yesmaximal, characteristic, complemented
2C4×AGL(1,ℤ/8)×C59C212yesmaximal, complemented
3C59×#2326C211yesmaximal, characteristic, complemented
4C59×#2060C211yesmaximal, characteristic, complemented
5C59×#2039C211yesmaximal, characteristic, complemented
6C59×#2038C212yesmaximal, complemented
7C59×#2027C211yesmaximal, characteristic, complemented
8C59×#2006C211yesmaximal, characteristic
9C59×#2003C212yesmaximal, complemented
10C59×#1759C212yesmaximal, complemented
11C4×#101×C59C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C59C2212yes—
13C59×#544C2211yescharacteristic
14C59×#542—22no—
15C59×#541—22no—
16C59×#539—22no—
17C59×#537C2211yescharacteristic
18C59×#536C2212yes—
19C59×#533—22no—
20C59×#529—22no—
21C59×#529—22no—
22C2×#124×C59C2211yescharacteristic
23C4×#41×C59C2212yes—
24C2×C4×Q8×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×#101×C59C2212yes—
36C4×M16×C59C2212yes—
37C59×#391C2211yescharacteristic
38C59×#390C2211yescharacteristic
39C59×#388—22no—
40C59×#385—22no—
41C59×#384C2211yescharacteristic
42C59×#376C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2211yescharacteristic
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59C2312yes—
48C22×Q8×C59C2311yescharacteristic
49C22×D4×C59C2311yescharacteristic
50C23×C4×C59—21no—
51AGL(1,ℤ/8)×C59—24no—
52AGL(1,ℤ/8)×C59—24no—
53C2×SD16×C59—22no—
54C2×D8×C59—22no—
55C2×M16×C59C2312yes—
56C2×M16×C59C2312yes—
57C4⋊Q8×C59—22no—
58C4⋊Q8×C59—22no—
59#126×C59—24no—
60#126×C59—24no—
61#125×C59—22no—
62#125×C59—24no—
63#125×C59—22no—
64#124×C59—22no—
65#124×C59—22no—
66#124×C59—22no—
67#124×C59—22no—
68#123×C59—24no—
69#123×C59—24no—
70C22⋊Q8×C59—24no—
71C22⋊Q8×C59—22no—
72C22⋊Q8×C59—22no—
73C22⋊Q8×C59—22no—
74C22⋊Q8×C59—22no—
75#121×C59—22no—
76#121×C59—22no—
77C4×Q8×C59—22no—
78C4×Q8×C59—22no—
79C4×Q8×C59—22no—
80C4×Q8×C59D412yes—
81C4×Q8×C59D412yes—
82C4×D4×C59—22no—
83C4×D4×C59—22no—
84C4×D4×C59—22no—
85C4×D4×C59—22no—
86C4×D4×C59—22no—
87C4×D4×C59D412yes—
88C4×D4×C59D412yes—
89#117×C59—24no—
90#117×C59—22no—
91#117×C59C2312yes—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59D412yes—
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—22no—
99C2×C42×C59—22no—
100C2×C42×C59D412yes—
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#101×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—
119#41×C59—22no—
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—22no—
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59C2×D412yes—
129C2×D4×C59—22no—
130C2×D4×C59—22no—
131C2×D4×C59—22no—
132C2×D4×C59—22no—
133C2×D4×C59—21no—
134C2×D4×C59C2×D412yes—
135C2×D4×C59#4112yes—
136C22×C4×C59—21no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—42no—
143C22×C4×C59—82no—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149SD16×C59—28no—
150D8×C59—28no—
151M16×C59—22no—
152M16×C59—22no—
153M16×C59—22no—
154M16×C59#4114yes—
155C2×C8×C59—22no—
156C2×C8×C59—22no—
157C4⋊C4×C59—24no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—82no—
168C4⋊C4×C59C2×D412yes—
169C4⋊C4×C59C2×D412yes—
170C22⋊C4×C59—28no—
171C22⋊C4×C59—24no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—41no—
178C22⋊C4×C59—41no—
179C22⋊C4×C59—82no—
180C42×C59—22no—
181C42×C59—22no—
182C42×C59—22no—
183C42×C59—22no—
184C42×C59—22no—
185C42×C59—21no—
186C42×C59—22no—
187C42×C59C2×D412yes—
188C42×C59C2×D412yes—
189C42×C59#4112yes—
190C23×C59—22no—
191C23×C59—22no—
192C23×C59—42no—
193C23×C59—42no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59C22×D411yescharacteristic
197Q8×C59—22no—
198Q8×C59—22no—
199Q8×C59—42no—
200Q8×C59—42no—
201Q8×C59—82no—
202D4×C59—22no—
203D4×C59—24no—
204D4×C59—22no—
205D4×C59—42no—
206D4×C59—42no—
207D4×C59—42no—
208D4×C59—82no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—44no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—41no—
226C2×C4×C59—42no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—82no—
231C2×C4×C59—84no—
232C2×C4×C59#12011yescharacteristic
233C2×C4×C59#12011yescharacteristic
234C2×C4×C59#12012yes—
235C2×C4×C59C22×D412yes—
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4112yes—
238C2×C4×C59C2×#4111yescharacteristic
239C8×C59—24no—
240C8×C59—42no—
241#23592C5911yesmaximal, characteristic, direct product factor, Sylow
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—44no—
245C22×C59—42no—
246C22×C59—42no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—82no—
250C22×C59—81no—
251C22×C59—81no—
252C22×C59C2×#12011yescharacteristic
253C22×C59#52511yescharacteristic
254C22×C59#53211yescharacteristic
255C4×C59—21no—
256C4×C59—21no—
257C4×C59—22no—
258C4×C59—42no—
259C4×C59—42no—
260C4×C59—42no—
261C4×C59—42no—
262C4×C59—42no—
263C4×C59—42no—
264C4×C59—81no—
265C4×C59—81no—
266C4×C59#52512yes—
267C4×C59#53212yes—
268#3461C2×C5911yescharacteristic, complemented
269C4×AGL(1,ℤ/8)C2×C5912yescomplemented
270#2326C2×C5911yescharacteristic, complemented
271#2060C2×C5911yescharacteristic, complemented
272#2039C2×C5911yescharacteristic, complemented
273#2038C2×C5912yescomplemented
274#2027C2×C5911yescharacteristic, complemented
275#2006C2×C5911yescharacteristic
276#2003C2×C5912yescomplemented
277#1759C2×C5912yescomplemented
278C4×#101C2×C5911yescharacteristic, complemented
279C2×C59—21no—
280C2×C59—21no—
281C2×C59—82no—
282C2×C59—82no—
283C2×C59#238811yescharacteristic
284C2×C59#302011yescharacteristic
285C2×C59#302111yescharacteristic
286C2×AGL(1,ℤ/8)C22×C5912yes—
287#544C22×C5911yescharacteristic
288#542—22no—
289#541—22no—
290#539—22no—
291#537C22×C5911yescharacteristic
292#536C22×C5912yes—
293#533—22no—
294#529—22no—
295#529—22no—
296C2×#124C22×C5911yescharacteristic
297C4×#41C22×C5912yes—
298C2×C4×Q8C22×C5911yescharacteristic
299C2×C4×D4C22×C5911yescharacteristic
300#442—22no—
301#440—22no—
302C4×SD16—22no—
303C4×D8—22no—
304#428C22×C5912yes—
305C2×#104C22×C5912yes—
306C2×#104C22×C5912yes—
307#418C22×C5912yes—
308#417C22×C5912yes—
309C2×#101C22×C5912yes—
310C4×M16C22×C5912yes—
311#391C22×C5911yescharacteristic
312#390C22×C5911yescharacteristic
313#388—22no—
314#385—22no—
315#384C22×C5911yescharacteristic
316#376C22×C5911yescharacteristic
317#343C22×C5911yescharacteristic
318#343C22×C5911yescharacteristic
319#339C22×C5912yes—
320#339C22×C5912yes—
321C59#2359211yescharacteristic, direct product factor, Sylow
322C2×#41C23×C5912yes—
323C22×Q8C23×C5911yescharacteristic
324C22×D4C23×C5911yescharacteristic
325C23×C4—21no—
326AGL(1,ℤ/8)—24no—
327AGL(1,ℤ/8)—24no—
328C2×SD16—22no—
329C2×D8—22no—
330C2×M16C23×C5912yes—
331C2×M16C23×C5912yes—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334#126—24no—
335#126—24no—
336#125—22no—
337#125—24no—
338#125—22no—
339#124—22no—
340#124—22no—
341#124—22no—
342#124—22no—
343#123—24no—
344#123—24no—
345C22⋊Q8—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350#121—22no—
351#121—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8D4×C5912yes—
356C4×Q8D4×C5912yes—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4D4×C5912yes—
363C4×D4D4×C5912yes—
364#117—24no—
365#117—22no—
366#117C23×C5912yes—
367C2×C4⋊C4—22no—
368C2×C4⋊C4D4×C5912yes—
369C2×C4⋊C4C23×C5911yescharacteristic
370C2×C4⋊C4C23×C5911yescharacteristic
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C42—22no—
375C2×C42D4×C5912yes—
376C2×C42C23×C5911yescharacteristic
377C2×C42C23×C5911yescharacteristic
378C2×C42C23×C5911yescharacteristic
379#107—22no—
380#106—22no—
381#104—24no—
382#104—24no—
383#103—22no—
384#102—22no—
385#102—22no—
386#102—22no—
387#101—24no—
388#97—22no—
389C4×C8—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393C24—21no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8C2×D4×C5912yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4C2×D4×C5912yes—
410C2×D4#41×C5912yes—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—82no—
419C22×C4C2×D4×C5912yes—
420C22×C4C2×D4×C5912yes—
421C22×C4C2×D4×C5911yescharacteristic
422C22×C4C2×D4×C5911yescharacteristic
423C22×C4C24×C5911yescharacteristic
424SD16—28no—
425D8—28no—
426M16—22no—
427M16—22no—
428M16—22no—
429M16#41×C5914yes—
430C2×C8—22no—
431C2×C8—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—82no—
443C4⋊C4C2×D4×C5912yes—
444C4⋊C4C2×D4×C5912yes—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—82no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—22no—
462C42C2×D4×C5912yes—
463C42C2×D4×C5912yes—
464C42#41×C5912yes—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—41no—
470C23—41no—
471C23C22×D4×C5911yescharacteristic
472Q8—22no—
473Q8—22no—
474Q8—42no—
475Q8—42no—
476Q8—82no—
477D4—22no—
478D4—24no—
479D4—22no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—82no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4#120×C5911yescharacteristic
508C2×C4#120×C5911yescharacteristic
509C2×C4#120×C5912yes—
510C2×C4C22×D4×C5912yes—
511C2×C4C2×#41×C5911yescharacteristic
512C2×C4C2×#41×C5912yes—
513C2×C4C2×#41×C5911yescharacteristic
514C8—24no—
515C8—42no—
516C22—21no—
517C22—21no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—81no—
525C22—81no—
526C22C2×#120×C5911yescharacteristic
527C22C59×#52511yescharacteristic
528C22C59×#53211yescharacteristic
529C4—21no—
530C4—21no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4C59×#52512yes—
541C4C59×#53212yes—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—82no—
546C2C59×#238811yescharacteristic
547C2C59×#302011yescharacteristic
548C2C59×#302111yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#2006C2
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×#3461C2semidirect
2. C4×AGL(1,ℤ/8)×C59C2semidirect
3. C59×#2326C2semidirect
4. C59×#2060C2semidirect
5. C59×#2039C2semidirect
6. C59×#2038C2semidirect
7. C59×#2027C2semidirect
9. C59×#2003C2semidirect
10. C59×#1759C2semidirect
11. C4×#101×C59C2semidirect
241. #23592C59direct (N × H)
268. #3461C2×C59semidirect
269. C4×AGL(1,ℤ/8)C2×C59semidirect
270. #2326C2×C59semidirect
271. #2060C2×C59semidirect
272. #2039C2×C59semidirect
273. #2038C2×C59semidirect
274. #2027C2×C59semidirect
276. #2003C2×C59semidirect
277. #1759C2×C59semidirect
278. C4×#101C2×C59semidirect
321. C59#23592direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34612
C4×AGL(1,ℤ/8)×C592
C59×#23262
C59×#20602
C59×#20392
C59×#20382
C59×#20272
C59×#20062
C59×#20032
C59×#17592
C4×#101×C592
#2359259