Finite Group Databasebeta

← C59×#23589 #28435698

Subgroups — C59×#23589 #28435698

Order 15104.

1138 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3460C211yesmaximal, characteristic
2C4×#137×C59C212yesmaximal, complemented
3C4×#124×C59C211yesmaximal, characteristic, complemented
4C59×#2061C211yesmaximal, characteristic, complemented
5C59×#2040C211yesmaximal, characteristic, complemented
6C59×#2037C212yesmaximal, complemented
7C59×#2028C211yesmaximal, characteristic, complemented
8C59×#2007C211yesmaximal, characteristic, complemented
9C59×#2004C212yesmaximal, complemented
10C4×#104×C59C212yesmaximal, complemented
11C4×#100×C59C211yesmaximal, characteristic, complemented
12C2×#137×C59C2212yes—
13C59×#544C2211yescharacteristic
14C59×#542—22no—
15C59×#540—22no—
16C59×#538—22no—
17C59×#537C2211yescharacteristic
18C59×#535C2212yes—
19C59×#532—22no—
20C59×#529—22no—
21C2×#124×C59C2211yescharacteristic
22C59×#525—22no—
23C4×#41×C59C2212yes—
24C2×C4×Q8×C59C2211yescharacteristic
25C2×C4×D4×C59C2211yescharacteristic
26C59×#441—22no—
27C59×#440—22no—
28C4×Q16×C59—22no—
29C4×SD16×C59—22no—
30C59×#428C2212yes—
31C2×#104×C59C2212yes—
32C2×#104×C59C2212yes—
33C59×#419C2212yes—
34C59×#417C2212yes—
35C2×#100×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×C59—22no—
48C2×#41×C59C2312yes—
49C22×Q8×C59C2311yescharacteristic
50C22×D4×C59C2311yescharacteristic
51C23×C4×C59D412yes—
52#137×C59—24no—
53#137×C59—24no—
54C2×Q16×C59—22no—
55C2×SD16×C59—22no—
56C2×M16×C59C2312yes—
57C2×M16×C59C2312yes—
58C4⋊Q8×C59—22no—
59#127×C59—22no—
60#126×C59—24no—
61#125×C59—22no—
62#124×C59—24no—
63#124×C59—22no—
64#124×C59—22no—
65#124×C59—22no—
66#123×C59—24no—
67#123×C59—24no—
68C22⋊Q8×C59—22no—
69C22⋊Q8×C59—22no—
70C22⋊Q8×C59—22no—
71#121×C59—22no—
72#121×C59—24no—
73#121×C59—22no—
74#121×C59—22no—
75#121×C59—22no—
76#121×C59—22no—
77#120×C59—24no—
78C4×Q8×C59—22no—
79C4×Q8×C59—22no—
80C4×Q8×C59D412yes—
81C4×Q8×C59D412yes—
82C4×D4×C59—24no—
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—22no—
90#117×C59—22no—
91#117×C59C2312yes—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59D412yes—
98C2×C42×C59—22no—
99C2×C42×C59—21no—
100C2×C42×C59—21no—
101C2×C42×C59—21no—
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#103×C59—22no—
111#103×C59—22no—
112#102×C59—22no—
113#100×C59—24no—
114#97×C59—22no—
115C4×C8×C59—22no—
116#95×C59—22no—
117#95×C59—22no—
118C24×C59C2×D412yes—
119#41×C59—22no—
120#41×C59—22no—
121#41×C59—22no—
122#41×C59—22no—
123#41×C59—84no—
124C2×Q8×C59—22no—
125C2×Q8×C59—21no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59C2×D412yes—
129C2×Q8×C59#4112yescomplemented
130C2×D4×C59—22no—
131C2×D4×C59—24no—
132C2×D4×C59—24no—
133C2×D4×C59—22no—
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59C2×D412yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—21no—
148C22×C4×C59—44no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D412yes—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153Q16×C59—28no—
154SD16×C59—28no—
155M16×C59—22no—
156M16×C59—22no—
157M16×C59—22no—
158M16×C59#4114yescomplemented
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59C2×D412yes—
170C4⋊C4×C59C2×D412yes—
171C22⋊C4×C59—28no—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—24no—
174C22⋊C4×C59—24no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—24no—
177C22⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—44no—
180C42×C59—22no—
181C42×C59—22no—
182C42×C59—22no—
183C42×C59—21no—
184C42×C59—22no—
185C42×C59—42no—
186C42×C59—42no—
187C42×C59—41no—
188C42×C59—41no—
189C42×C59C2×D412yes—
190C42×C59C2×D412yes—
191C42×C59#4112yescomplemented
192C23×C59—22no—
193C23×C59—22no—
194C23×C59—22no—
195C23×C59—22no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59C22×D411yescharacteristic
199Q8×C59—24no—
200Q8×C59—22no—
201Q8×C59—22no—
202Q8×C59—42no—
203Q8×C59—82no—
204D4×C59—22no—
205D4×C59—22no—
206D4×C59—42no—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—42no—
210D4×C59—84no—
211D4×C59—82no—
212D4×C59—84no—
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—22no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—44no—
231C2×C4×C59—44no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—41no—
235C2×C4×C59—82no—
236C2×C4×C59—82no—
237C2×C4×C59#12012yes—
238C2×C4×C59#12011yescharacteristic
239C2×C4×C59#12011yescharacteristic
240C2×C4×C59C22×D412yes—
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4112yes—
243C2×C4×C59C2×#4111yescharacteristic
244C8×C59—24no—
245C8×C59—42no—
246#23589C5911yesmaximal, characteristic, direct product factor, Sylow
247C22×C59—21no—
248C22×C59—21no—
249C22×C59—44no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—82no—
257C22×C59C2×#12011yescharacteristic
258C22×C59#52511yescharacteristic
259C22×C59#53211yescharacteristic
260C4×C59—21no—
261C4×C59—22no—
262C4×C59—42no—
263C4×C59—42no—
264C4×C59—42no—
265C4×C59—42no—
266C4×C59—42no—
267C4×C59—42no—
268C4×C59—42no—
269C4×C59—81no—
270C4×C59#45512yes—
271C4×C59#52512yes—
272C4×C59#53212yes—
273#3460C2×C5911yescharacteristic
274C4×#137C2×C5912yescomplemented
275C4×#124C2×C5911yescharacteristic, complemented
276#2061C2×C5911yescharacteristic, complemented
277#2040C2×C5911yescharacteristic, complemented
278#2037C2×C5912yescomplemented
279#2028C2×C5911yescharacteristic, complemented
280#2007C2×C5911yescharacteristic, complemented
281#2004C2×C5912yescomplemented
282C4×#104C2×C5912yescomplemented
283C4×#100C2×C5911yescharacteristic, complemented
284C2×C59—21no—
285C2×C59—21no—
286C2×C59—82no—
287C2×C59—82no—
288C2×C59#238811yescharacteristic
289C2×C59C2×#45511yescharacteristic
290C2×C59#301811yescharacteristic
291C2×#137C22×C5912yes—
292#544C22×C5911yescharacteristic
293#542—22no—
294#540—22no—
295#538—22no—
296#537C22×C5911yescharacteristic
297#535C22×C5912yes—
298#532—22no—
299#529—22no—
300C2×#124C22×C5911yescharacteristic
301#525—22no—
302C4×#41C22×C5912yes—
303C2×C4×Q8C22×C5911yescharacteristic
304C2×C4×D4C22×C5911yescharacteristic
305#441—22no—
306#440—22no—
307C4×Q16—22no—
308C4×SD16—22no—
309#428C22×C5912yes—
310C2×#104C22×C5912yes—
311C2×#104C22×C5912yes—
312#419C22×C5912yes—
313#417C22×C5912yes—
314C2×#100C22×C5912yes—
315C4×M16C22×C5912yes—
316#391C22×C5911yescharacteristic
317#390C22×C5911yescharacteristic
318#388—22no—
319#382C22×C5911yescharacteristic
320C4×C22⋊C4—22no—
321C43C22×C5911yescharacteristic
322#343C22×C5911yescharacteristic
323#343C22×C5911yescharacteristic
324#339C22×C5912yes—
325#339C22×C5912yes—
326C59#2358911yescharacteristic, direct product factor, Sylow
327C2×#41—22no—
328C2×#41C23×C5912yes—
329C22×Q8C23×C5911yescharacteristic
330C22×D4C23×C5911yescharacteristic
331C23×C4D4×C5912yes—
332#137—24no—
333#137—24no—
334C2×Q16—22no—
335C2×SD16—22no—
336C2×M16C23×C5912yes—
337C2×M16C23×C5912yes—
338C4⋊Q8—22no—
339#127—22no—
340#126—24no—
341#125—22no—
342#124—24no—
343#124—22no—
344#124—22no—
345#124—22no—
346#123—24no—
347#123—24no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351#121—22no—
352#121—24no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#120—24no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8D4×C5912yes—
361C4×Q8D4×C5912yes—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4D4×C5912yes—
368C4×D4D4×C5912yes—
369#117—22no—
370#117—22no—
371#117C23×C5912yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4C23×C5911yescharacteristic
374C2×C4⋊C4C23×C5911yescharacteristic
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4D4×C5912yes—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C23×C5911yescharacteristic
383C2×C42C23×C5911yescharacteristic
384C2×C42C23×C5911yescharacteristic
385#107—22no—
386#106—22no—
387#104—24no—
388#104—24no—
389#103—22no—
390#103—22no—
391#103—22no—
392#102—22no—
393#100—24no—
394#97—22no—
395C4×C8—22no—
396#95—22no—
397#95—22no—
398C24C2×D4×C5912yes—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—84no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×D4×C5912yes—
409C2×Q8#41×C5912yescomplemented
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×D4×C5912yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—44no—
429C22×C4C2×D4×C5911yescharacteristic
430C22×C4C2×D4×C5912yes—
431C22×C4C2×D4×C5911yescharacteristic
432C22×C4C24×C5911yescharacteristic
433Q16—28no—
434SD16—28no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16#41×C5914yescomplemented
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4C2×D4×C5912yes—
450C4⋊C4C2×D4×C5912yes—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—44no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—21no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—41no—
468C42—41no—
469C42C2×D4×C5912yes—
470C42C2×D4×C5912yes—
471C42#41×C5912yescomplemented
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23C22×D4×C5911yescharacteristic
479Q8—24no—
480Q8—22no—
481Q8—22no—
482Q8—42no—
483Q8—82no—
484D4—22no—
485D4—22no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—84no—
491D4—82no—
492D4—84no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#120×C5912yes—
518C2×C4#120×C5911yescharacteristic
519C2×C4#120×C5911yescharacteristic
520C2×C4C22×D4×C5912yes—
521C2×C4C2×#41×C5911yescharacteristic
522C2×C4C2×#41×C5912yes—
523C2×C4C2×#41×C5911yescharacteristic
524C8—24no—
525C8—42no—
526C22—21no—
527C22—21no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—82no—
536C22C2×#120×C5911yescharacteristic
537C22C59×#52511yescharacteristic
538C22C59×#53211yescharacteristic
539C4—21no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4C59×#45512yes—
550C4C59×#52512yes—
551C4C59×#53212yes—
552C2—21no—
553C2—21no—
554C2—82no—
555C2—82no—
556C2C59×#238811yescharacteristic
557C2C2×C59×#45511yescharacteristic
558C2C59×#301811yescharacteristic
subgroupG/H
centerC2×C4×C59#120
commutator, Frattini, cosocleC22×C4C24×C59
socleC22×C59C2×#120
prime radicalC59×#3460C2
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
2. C4×#137×C59C2semidirect
3. C4×#124×C59C2semidirect
4. C59×#2061C2semidirect
5. C59×#2040C2semidirect
6. C59×#2037C2semidirect
7. C59×#2028C2semidirect
8. C59×#2007C2semidirect
9. C59×#2004C2semidirect
10. C4×#104×C59C2semidirect
11. C4×#100×C59C2semidirect
129. C2×Q8×C59#41semidirect
158. M16×C59#41semidirect
191. C42×C59#41semidirect
246. #23589C59direct (N × H)
274. C4×#137C2×C59semidirect
275. C4×#124C2×C59semidirect
276. #2061C2×C59semidirect
277. #2040C2×C59semidirect
278. #2037C2×C59semidirect
279. #2028C2×C59semidirect
280. #2007C2×C59semidirect
281. #2004C2×C59semidirect
282. C4×#104C2×C59semidirect
283. C4×#100C2×C59semidirect
326. C59#23589direct (N × H)
409. C2×Q8#41×C59semidirect
438. M16#41×C59semidirect
471. 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×#137×C592
C4×#124×C592
C59×#20612
C59×#20402
C59×#20372
C59×#20282
C59×#20072
C59×#20042
C4×#104×C592
C4×#100×C592
#2358959