Finite Group Databasebeta

← C59×#23588 #28435697

Subgroups — C59×#23588 #28435697

Order 15104.

1106 conjugacy classes of subgroups.

The subgroup lattice has 554 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
3C4×#124×C59C211yesmaximal, characteristic, complemented
4C59×#2061C211yesmaximal, characteristic, complemented
5C59×#2040C211yesmaximal, characteristic, complemented
6C59×#2038C212yesmaximal, complemented
7C59×#2027C211yesmaximal, characteristic, complemented
8C59×#2006C211yesmaximal, characteristic
9C59×#2003C212yesmaximal, complemented
10C4×#104×C59C212yesmaximal, 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×#382C2211yescharacteristic
41C4×C22⋊C4×C59—22no—
42C43×C59C2211yescharacteristic
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×C59D412yes—
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—21no—
98C2×C42×C59—22no—
99C2×C42×C59—21no—
100C2×C42×C59—21no—
101C2×C42×C59D412yes—
102C2×C42×C59C2311yescharacteristic
103C2×C42×C59C2311yescharacteristic
104C2×C42×C59C2311yescharacteristic
105#107×C59—22no—
106#106×C59—22no—
107#104×C59—24no—
108#104×C59—24no—
109#103×C59—22no—
110#102×C59—22no—
111#102×C59—22no—
112#102×C59—22no—
113#101×C59—24no—
114#97×C59—22no—
115C4×C8×C59—22no—
116#95×C59—22no—
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—21no—
138C22×C4×C59—22no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—21no—
142C22×C4×C59—44no—
143C22×C4×C59—41no—
144C22×C4×C59—41no—
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D412yes—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150SD16×C59—28no—
151D8×C59—28no—
152M16×C59—22no—
153M16×C59—22no—
154M16×C59—22no—
155M16×C59#4114yes—
156C2×C8×C59—22no—
157C2×C8×C59—22no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—24no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
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—44no—
178C22⋊C4×C59—42no—
179C42×C59—22no—
180C42×C59—22no—
181C42×C59—22no—
182C42×C59—22no—
183C42×C59—22no—
184C42×C59—21no—
185C42×C59—22no—
186C42×C59—42no—
187C42×C59—42no—
188C42×C59C2×D412yes—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—42no—
194C23×C59—42no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59C22×D411yescharacteristic
198Q8×C59—22no—
199Q8×C59—22no—
200Q8×C59—42no—
201Q8×C59—42no—
202Q8×C59—82no—
203D4×C59—22no—
204D4×C59—24no—
205D4×C59—22no—
206D4×C59—42no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—82no—
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—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—44no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—82no—
232C2×C4×C59—82no—
233C2×C4×C59—82no—
234C2×C4×C59#12011yescharacteristic
235C2×C4×C59#12012yes—
236C2×C4×C59#12011yescharacteristic
237C2×C4×C59C22×D412yes—
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4112yes—
240C2×C4×C59C2×#4111yescharacteristic
241C8×C59—24no—
242C8×C59—42no—
243#23588C5911yesmaximal, characteristic, direct product factor, Sylow
244C22×C59—21no—
245C22×C59—21no—
246C22×C59—44no—
247C22×C59—42no—
248C22×C59—42no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—82no—
252C22×C59—81no—
253C22×C59—81no—
254C22×C59C2×#12011yescharacteristic
255C22×C59#52511yescharacteristic
256C22×C59#53211yescharacteristic
257C4×C59—21no—
258C4×C59—22no—
259C4×C59—42no—
260C4×C59—42no—
261C4×C59—42no—
262C4×C59—42no—
263C4×C59—42no—
264C4×C59—42no—
265C4×C59—42no—
266C4×C59—81no—
267C4×C59#45412yes—
268C4×C59#52512yes—
269C4×C59#53212yes—
270#3461C2×C5911yescharacteristic, complemented
271C4×AGL(1,ℤ/8)C2×C5912yescomplemented
272C4×#124C2×C5911yescharacteristic, complemented
273#2061C2×C5911yescharacteristic, complemented
274#2040C2×C5911yescharacteristic, complemented
275#2038C2×C5912yescomplemented
276#2027C2×C5911yescharacteristic, complemented
277#2006C2×C5911yescharacteristic
278#2003C2×C5912yescomplemented
279C4×#104C2×C5912yescomplemented
280C4×#101C2×C5911yescharacteristic, complemented
281C2×C59—21no—
282C2×C59—21no—
283C2×C59—82no—
284C2×C59—82no—
285C2×C59#238811yescharacteristic
286C2×C59C2×#45411yescharacteristic
287C2×C59#301811yescharacteristic
288C2×AGL(1,ℤ/8)C22×C5912yes—
289#544C22×C5911yescharacteristic
290#542—22no—
291#541—22no—
292#539—22no—
293#537C22×C5911yescharacteristic
294#536C22×C5912yes—
295#533—22no—
296#529—22no—
297#529—22no—
298C2×#124C22×C5911yescharacteristic
299C4×#41C22×C5912yes—
300C2×C4×Q8C22×C5911yescharacteristic
301C2×C4×D4C22×C5911yescharacteristic
302#442—22no—
303#440—22no—
304C4×SD16—22no—
305C4×D8—22no—
306#428C22×C5912yes—
307C2×#104C22×C5912yes—
308C2×#104C22×C5912yes—
309#418C22×C5912yes—
310#417C22×C5912yes—
311C2×#101C22×C5912yes—
312C4×M16C22×C5912yes—
313#391C22×C5911yescharacteristic
314#390C22×C5911yescharacteristic
315#388—22no—
316#382C22×C5911yescharacteristic
317C4×C22⋊C4—22no—
318C43C22×C5911yescharacteristic
319#343C22×C5911yescharacteristic
320#343C22×C5911yescharacteristic
321#339C22×C5912yes—
322#339C22×C5912yes—
323C59#2358811yescharacteristic, direct product factor, Sylow
324C2×#41C23×C5912yes—
325C22×Q8C23×C5911yescharacteristic
326C22×D4C23×C5911yescharacteristic
327C23×C4—21no—
328AGL(1,ℤ/8)—24no—
329AGL(1,ℤ/8)—24no—
330C2×SD16—22no—
331C2×D8—22no—
332C2×M16C23×C5912yes—
333C2×M16C23×C5912yes—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336#126—24no—
337#126—24no—
338#125—22no—
339#125—24no—
340#125—22no—
341#124—22no—
342#124—22no—
343#124—22no—
344#124—22no—
345#123—24no—
346#123—24no—
347C22⋊Q8—24no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352#121—22no—
353#121—22no—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8D4×C5912yes—
358C4×Q8D4×C5912yes—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4D4×C5912yes—
365C4×D4D4×C5912yes—
366#117—24no—
367#117—22no—
368#117C23×C5912yes—
369C2×C4⋊C4D4×C5912yes—
370C2×C4⋊C4C23×C5911yescharacteristic
371C2×C4⋊C4C23×C5911yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42D4×C5912yes—
379C2×C42C23×C5911yescharacteristic
380C2×C42C23×C5911yescharacteristic
381C2×C42C23×C5911yescharacteristic
382#107—22no—
383#106—22no—
384#104—24no—
385#104—24no—
386#103—22no—
387#102—22no—
388#102—22no—
389#102—22no—
390#101—24no—
391#97—22no—
392C4×C8—22no—
393#95—22no—
394#95—22no—
395C24—21no—
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8C2×D4×C5912yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4C2×D4×C5912yes—
412C2×D4#41×C5912yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—44no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4C2×D4×C5912yes—
423C22×C4C2×D4×C5911yescharacteristic
424C22×C4C2×D4×C5912yes—
425C22×C4C2×D4×C5911yescharacteristic
426C22×C4C24×C5911yescharacteristic
427SD16—28no—
428D8—28no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16#41×C5914yes—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4C2×D4×C5912yes—
446C4⋊C4C2×D4×C5912yes—
447C22⋊C4—28no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42—22no—
463C42—42no—
464C42—42no—
465C42C2×D4×C5912yes—
466C42C2×D4×C5912yes—
467C42#41×C5912yes—
468C23—22no—
469C23—22no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23C22×D4×C5911yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—42no—
478Q8—42no—
479Q8—82no—
480D4—22no—
481D4—24no—
482D4—22no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#120×C5911yescharacteristic
512C2×C4#120×C5912yes—
513C2×C4#120×C5911yescharacteristic
514C2×C4C22×D4×C5912yes—
515C2×C4C2×#41×C5911yescharacteristic
516C2×C4C2×#41×C5912yes—
517C2×C4C2×#41×C5911yescharacteristic
518C8—24no—
519C8—42no—
520C22—21no—
521C22—21no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22C2×#120×C5911yescharacteristic
531C22C59×#52511yescharacteristic
532C22C59×#53211yescharacteristic
533C4—21no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—81no—
543C4C59×#45412yes—
544C4C59×#52512yes—
545C4C59×#53212yes—
546C2—21no—
547C2—21no—
548C2—82no—
549C2—82no—
550C2C59×#238811yescharacteristic
551C2C2×C59×#45411yescharacteristic
552C2C59×#301811yescharacteristic
subgroupG/H
centerC2×C4×C59#120
commutator, Frattini, cosocleC22×C4C24×C59
socleC22×C59C2×#120
prime radicalC59×#2006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×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. C4×#124×C59C2semidirect
4. C59×#2061C2semidirect
5. C59×#2040C2semidirect
6. C59×#2038C2semidirect
7. C59×#2027C2semidirect
9. C59×#2003C2semidirect
10. C4×#104×C59C2semidirect
11. C4×#101×C59C2semidirect
243. #23588C59direct (N × H)
270. #3461C2×C59semidirect
271. C4×AGL(1,ℤ/8)C2×C59semidirect
272. C4×#124C2×C59semidirect
273. #2061C2×C59semidirect
274. #2040C2×C59semidirect
275. #2038C2×C59semidirect
276. #2027C2×C59semidirect
278. #2003C2×C59semidirect
279. C4×#104C2×C59semidirect
280. C4×#101C2×C59semidirect
323. C59#23588direct (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
C4×#124×C592
C59×#20612
C59×#20402
C59×#20382
C59×#20272
C59×#20062
C59×#20032
C4×#104×C592
C4×#101×C592
#2358859