Finite Group Databasebeta

← C59×#24078 #28436187

Subgroups — C59×#24078 #28436187

Order 15104.

1038 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3493C211yesmaximal, characteristic
2C59×#3093C212yesmaximal, complemented
3C59×#2285C211yesmaximal, characteristic, complemented
4C59×#2061C211yesmaximal, characteristic, complemented
5C59×#2060C211yesmaximal, characteristic, complemented
6C59×#2040C211yesmaximal, characteristic, complemented
7C59×#2039C211yesmaximal, characteristic, complemented
8C59×#1896C212yesmaximal, complemented
9C59×#1860C212yesmaximal, complemented
10C59×#1859C212yesmaximal, complemented
11C59×#1822C211yesmaximal, characteristic, complemented
12C59×#556—22no—
13C59×#555—22no—
14C59×#553—22no—
15C59×#552—22no—
16C59×#544C2211yescharacteristic
17C59×#538—22no—
18C59×#537C2211yescharacteristic
19C59×#529—22no—
20C59×#525C2212yes—
21C59×#520C2212yes—
22C2×#117×C59C2211yescharacteristic
23C2×#117×C59C2211yescharacteristic
24C22×C42×C59C2211yescharacteristic
25C59×#485—22no—
26C59×#485—22no—
27C59×#482—22no—
28C59×#482—22no—
29C59×#428C2212yes—
30C59×#428C2212yes—
31C2×#104×C59C2212yes—
32C2×#104×C59C2212yes—
33C59×#417C2212yes—
34C59×#417C2212yes—
35C59×#407C2212yes—
36C59×#384C2211yescharacteristic
37C59×#384C2211yescharacteristic
38C59×#382C2211yescharacteristic
39C59×#382C2211yescharacteristic
40C59×#380—22no—
41C4×C4⋊C4×C59—22no—
42C59×#343C2211yescharacteristic
43C59×#343C2211yescharacteristic
44C59×#343C2212yes—
45C59×#339C2212yes—
46C59×#339C2212yes—
47C2×#41×C59C2312yes—
48C23×C4×C59—21no—
49C23×C4×C59C2311yescharacteristic
50C2×M16×C59C2312yes—
51C2×M16×C59C2312yes—
52C4⋊Q8×C59—22no—
53#126×C59—24no—
54#126×C59—24no—
55#125×C59—22no—
56#125×C59—22no—
57#124×C59—22no—
58#124×C59—24no—
59#123×C59—24no—
60#123×C59—22no—
61C22⋊Q8×C59—22no—
62C22⋊Q8×C59—22no—
63C22⋊Q8×C59D412yes—
64C22⋊Q8×C59D412yes—
65#121×C59—24no—
66#121×C59—22no—
67#121×C59—22no—
68#121×C59D412yes—
69#121×C59D412yes—
70#120×C59—22no—
71C4×Q8×C59—22no—
72C4×Q8×C59—22no—
73C4×D4×C59—22no—
74C4×D4×C59—22no—
75C4×D4×C59—24no—
76C4×D4×C59—22no—
77C4×D4×C59—22no—
78#117×C59—24no—
79#117×C59—22no—
80#117×C59—22no—
81#117×C59—22no—
82#117×C59—22no—
83#117×C59—22no—
84#117×C59C2312yes—
85C2×C4⋊C4×C59—22no—
86C2×C4⋊C4×C59—22no—
87C2×C4⋊C4×C59—22no—
88C2×C4⋊C4×C59C2311yescharacteristic
89C2×C4⋊C4×C59C2311yescharacteristic
90C2×C4⋊C4×C59C2311yescharacteristic
91C2×C4⋊C4×C59C2311yescharacteristic
92C2×C22⋊C4×C59—21no—
93C2×C22⋊C4×C59D412yes—
94C2×C42×C59—21no—
95C2×C42×C59—22no—
96C2×C42×C59—22no—
97C2×C42×C59—21no—
98C2×C42×C59D412yes—
99C2×C42×C59C2311yescharacteristic
100C2×C42×C59C2311yescharacteristic
101#107×C59—22no—
102#107×C59—22no—
103#106×C59—22no—
104#106×C59—22no—
105#104×C59—24no—
106#104×C59—24no—
107#103×C59—22no—
108#103×C59—22no—
109#102×C59—22no—
110#102×C59—22no—
111C22⋊C8×C59—22no—
112C22⋊C8×C59—22no—
113#95×C59—22no—
114#95×C59—22no—
115C24×C59C2×D411yescharacteristic
116#41×C59—22no—
117#41×C59—22no—
118#41×C59—22no—
119#41×C59—22no—
120C2×Q8×C59—22no—
121C2×Q8×C59C2×D412yes—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59—22no—
125C2×D4×C59—22no—
126C2×D4×C59C2×D412yes—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—22no—
131C22×C4×C59—24no—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—22no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D412yes—
141C22×C4×C59C2×D412yes—
142C22×C4×C59C2×D412yes—
143C22×C4×C59C2×D412yes—
144C22×C4×C59C2411yescharacteristic
145M16×C59—22no—
146M16×C59—22no—
147M16×C59—22no—
148M16×C59—22no—
149C2×C8×C59—22no—
150C2×C8×C59—22no—
151C4⋊C4×C59—24no—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—22no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—22no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—44no—
162C4⋊C4×C59—44no—
163C4⋊C4×C59C2×D412yes—
164C4⋊C4×C59C2×D412yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—42no—
174C42×C59—24no—
175C42×C59—24no—
176C42×C59—22no—
177C42×C59—22no—
178C42×C59—22no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—41no—
182C42×C59—41no—
183C42×C59#4112yes—
184C42×C59#4112yes—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—22no—
188C23×C59—41no—
189C23×C59—41no—
190C23×C59#12012yes—
191C23×C59C22×D411yescharacteristic
192Q8×C59—22no—
193Q8×C59—22no—
194Q8×C59—82no—
195D4×C59—22no—
196D4×C59—22no—
197D4×C59—42no—
198D4×C59—42no—
199D4×C59—82no—
200D4×C59—82no—
201D4×C59—82no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—21no—
205C2×C4×C59—22no—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
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—48no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59#12011yescharacteristic
230C2×C4×C59#12011yescharacteristic
231C2×C4×C59C22×D412yes—
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C592−1+412yes—
235C8×C59—42no—
236C8×C59—42no—
237#24078C5911yesmaximal, characteristic, direct product factor, Sylow
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—22no—
241C22×C59—22no—
242C22×C59—22no—
243C22×C59—42no—
244C22×C59—42no—
245C22×C59—42no—
246C22×C59—42no—
247C22×C59—42no—
248C22×C59C2×#12011yescharacteristic
249C22×C59#52511yescharacteristic
250C22×C59#53611yescharacteristic
251C4×C59—24no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—42no—
255C4×C59—42no—
256C4×C59—42no—
257C4×C59—42no—
258C4×C59—42no—
259C4×C59—42no—
260C4×C59—81no—
261C4×C59—81no—
262C4×C59#54712yes—
263C4×C59#54712yes—
264#3493C2×C5911yescharacteristic
265#3093C2×C5912yescomplemented
266#2285C2×C5911yescharacteristic, complemented
267#2061C2×C5911yescharacteristic, complemented
268#2060C2×C5911yescharacteristic, complemented
269#2040C2×C5911yescharacteristic, complemented
270#2039C2×C5911yescharacteristic, complemented
271#1896C2×C5912yescomplemented
272#1860C2×C5912yescomplemented
273#1859C2×C5912yescomplemented
274#1822C2×C5911yescharacteristic, complemented
275C2×C59—21no—
276C2×C59—21no—
277C2×C59—42no—
278C2×C59—82no—
279C2×C59#240611yescharacteristic
280C2×C59#301811yescharacteristic
281C2×C59#301811yescharacteristic
282#556—22no—
283#555—22no—
284#553—22no—
285#552—22no—
286#544C22×C5911yescharacteristic
287#538—22no—
288#537C22×C5911yescharacteristic
289#529—22no—
290#525C22×C5912yes—
291#520C22×C5912yes—
292C2×#117C22×C5911yescharacteristic
293C2×#117C22×C5911yescharacteristic
294C22×C42C22×C5911yescharacteristic
295#485—22no—
296#485—22no—
297#482—22no—
298#482—22no—
299#428C22×C5912yes—
300#428C22×C5912yes—
301C2×#104C22×C5912yes—
302C2×#104C22×C5912yes—
303#417C22×C5912yes—
304#417C22×C5912yes—
305#407C22×C5912yes—
306#384C22×C5911yescharacteristic
307#384C22×C5911yescharacteristic
308#382C22×C5911yescharacteristic
309#382C22×C5911yescharacteristic
310#380—22no—
311C4×C4⋊C4—22no—
312#343C22×C5911yescharacteristic
313#343C22×C5911yescharacteristic
314#343C22×C5912yes—
315#339C22×C5912yes—
316#339C22×C5912yes—
317C59#2407811yescharacteristic, direct product factor, Sylow
318C2×#41C23×C5912yes—
319C23×C4—21no—
320C23×C4C23×C5911yescharacteristic
321C2×M16C23×C5912yes—
322C2×M16C23×C5912yes—
323C4⋊Q8—22no—
324#126—24no—
325#126—24no—
326#125—22no—
327#125—22no—
328#124—22no—
329#124—24no—
330#123—24no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8D4×C5912yes—
335C22⋊Q8D4×C5912yes—
336#121—24no—
337#121—22no—
338#121—22no—
339#121D4×C5912yes—
340#121D4×C5912yes—
341#120—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—24no—
347C4×D4—22no—
348C4×D4—22no—
349#117—24no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117C23×C5912yes—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C23×C5911yescharacteristic
360C2×C4⋊C4C23×C5911yescharacteristic
361C2×C4⋊C4C23×C5911yescharacteristic
362C2×C4⋊C4C23×C5911yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4D4×C5912yes—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42D4×C5912yes—
370C2×C42C23×C5911yescharacteristic
371C2×C42C23×C5911yescharacteristic
372#107—22no—
373#107—22no—
374#106—22no—
375#106—22no—
376#104—24no—
377#104—24no—
378#103—22no—
379#103—22no—
380#102—22no—
381#102—22no—
382C22⋊C8—22no—
383C22⋊C8—22no—
384#95—22no—
385#95—22no—
386C24C2×D4×C5911yescharacteristic
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391C2×Q8—22no—
392C2×Q8C2×D4×C5912yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4C2×D4×C5912yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×D4×C5911yescharacteristic
411C22×C4C2×D4×C5912yes—
412C22×C4C2×D4×C5912yes—
413C22×C4C2×D4×C5912yes—
414C22×C4C2×D4×C5912yes—
415C22×C4C24×C5911yescharacteristic
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420C2×C8—22no—
421C2×C8—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4C2×D4×C5912yes—
435C4⋊C4C2×D4×C5912yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—41no—
453C42—41no—
454C42#41×C5912yes—
455C42#41×C5912yes—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23#120×C5912yes—
462C23C22×D4×C5911yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—82no—
466D4—22no—
467D4—22no—
468D4—42no—
469D4—42no—
470D4—82no—
471D4—82no—
472D4—82no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4#120×C5911yescharacteristic
501C2×C4#120×C5911yescharacteristic
502C2×C4C22×D4×C5912yes—
503C2×C4C2×#41×C5911yescharacteristic
504C2×C4C2×#41×C5911yescharacteristic
505C2×C42−1+4×C5912yes—
506C8—42no—
507C8—42no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22C2×#120×C5911yescharacteristic
519C22C59×#52511yescharacteristic
520C22C59×#53611yescharacteristic
521C4—24no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—81no—
531C4—81no—
532C4C59×#54712yes—
533C4C59×#54712yes—
534C2—21no—
535C2—21no—
536C2—42no—
537C2—82no—
538C2C59×#240611yescharacteristic
539C2C59×#301811yescharacteristic
540C2C59×#301811yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#3493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C23×C4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3093C2semidirect
3. C59×#2285C2semidirect
4. C59×#2061C2semidirect
5. C59×#2060C2semidirect
6. C59×#2040C2semidirect
7. C59×#2039C2semidirect
8. C59×#1896C2semidirect
9. C59×#1860C2semidirect
10. C59×#1859C2semidirect
11. C59×#1822C2semidirect
237. #24078C59direct (N × H)
265. #3093C2×C59semidirect
266. #2285C2×C59semidirect
267. #2061C2×C59semidirect
268. #2060C2×C59semidirect
269. #2040C2×C59semidirect
270. #2039C2×C59semidirect
271. #1896C2×C59semidirect
272. #1860C2×C59semidirect
273. #1859C2×C59semidirect
274. #1822C2×C59semidirect
317. C59#24078direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34932
C59×#30932
C59×#22852
C59×#20612
C59×#20602
C59×#20402
C59×#20392
C59×#18962
C59×#18602
C59×#18592
C59×#18222
#2407859