Finite Group Databasebeta

← C59×#18161 #28430270

Subgroups — C59×#18161 #28430270

Order 15104.

908 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3461C211yesmaximal, characteristic
2C59×#3117C212yesmaximal, complemented
3C59×#2581C211yesmaximal, characteristic, complemented
4C59×#1617C211yesmaximal, characteristic, complemented
5C59×#1616C211yesmaximal, characteristic
6C59×#1615C212yesmaximal, complemented
7C59×#1605C211yesmaximal, characteristic, complemented
8C59×#1604C211yesmaximal, characteristic
9C59×#1603C212yesmaximal, complemented
10C59×#1512C212yesmaximal, complemented
11C59×#1467C211yesmaximal, characteristic, complemented
12C59×#544C2211yescharacteristic
13C59×#542—22no—
14C59×#541—22no—
15C59×#539—21no—
16C59×#539C2212yes—
17C59×#537C2211yescharacteristic
18C59×#536—21no—
19C59×#533—22no—
20C59×#529—22no—
21C59×#529—21no—
22C59×#529C2212yes—
23C4×#41×C59—21no—
24C2×C4×Q8×C59C2211yescharacteristic
25C2×C4×Q8×C59C2211yescharacteristic
26C59×#484C2212yes—
27C59×#481C2212yes—
28C59×#479—22no—
29C59×#464—22no—
30C59×#451C2212yes—
31C59×#448C2212yes—
32C59×#441—22no—
33C59×#440—22no—
34C59×#432C2212yes—
35C59×#400C2211yescharacteristic
36C59×#400C2211yescharacteristic
37C59×#395C2211yescharacteristic
38C59×#395C2211yescharacteristic
39C59×#391—21no—
40C59×#389—21no—
41C59×#387—21no—
42C59×#387—21no—
43C59×#382C2211yescharacteristic
44C59×#328C2212yes—
45C59×#328C2212yes—
46C59×#327C2212yes—
47C59×#327C2212yes—
48C59×#324C2212yes—
49C59×#324C2211yescharacteristic
50C59×#324C2211yescharacteristic
51C2×#41×C59—21no—
52C22×Q8×C59D411yescharacteristic
53C22×Q8×C59C2311yescharacteristic
54C2×Q16×C59—22no—
55C2×SD16×C59—22no—
56C4⋊Q8×C59—22no—
57C4⋊Q8×C59—22no—
58#126×C59—22no—
59#126×C59—22no—
60#126×C59—22no—
61#126×C59—22no—
62#125×C59—21no—
63#125×C59—22no—
64#125×C59—22no—
65#125×C59—22no—
66#125×C59D412yes—
67#124×C59—22no—
68#124×C59—21no—
69#124×C59D412yes—
70#123×C59—22no—
71#123×C59—22no—
72#123×C59—22no—
73#123×C59—22no—
74C22⋊Q8×C59—22no—
75C22⋊Q8×C59—22no—
76C22⋊Q8×C59—22no—
77C22⋊Q8×C59—22no—
78C22⋊Q8×C59—21no—
79C22⋊Q8×C59—22no—
80C22⋊Q8×C59C2312yes—
81#121×C59—22no—
82#121×C59—21no—
83#121×C59C2312yes—
84C4×Q8×C59—21no—
85C4×Q8×C59—21no—
86C4×Q8×C59—21no—
87C4×Q8×C59—22no—
88C4×Q8×C59—22no—
89C4×Q8×C59—22no—
90C4×Q8×C59—22no—
91C4×Q8×C59D412yes—
92C4×D4×C59—21no—
93C4×D4×C59—21no—
94C4×D4×C59—21no—
95C4×D4×C59—22no—
96C4×D4×C59—22no—
97C4×D4×C59D412yes—
98#117×C59—22no—
99#117×C59—22no—
100#117×C59—21no—
101#117×C59—22no—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59D411yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C42×C59—22no—
111C2×C42×C59—21no—
112C2×C42×C59D411yescharacteristic
113C2×C42×C59D411yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#107×C59—22no—
116#106×C59—22no—
117C4⋊C8×C59—22no—
118#103×C59—22no—
119#103×C59—22no—
120#103×C59—22no—
121#102×C59—22no—
122C22⋊C8×C59C2312yes—
123C22⋊C8×C59C2312yes—
124#97×C59—22no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134#41×C59—42no—
135#41×C59—42no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—22no—
139C2×Q8×C59—22no—
140C2×Q8×C59—22no—
141C2×Q8×C59—41no—
142C2×Q8×C59—42no—
143C2×Q8×C59C2×D412yes—
144C2×D4×C59—21no—
145C2×D4×C59—22no—
146C2×D4×C59—21no—
147C2×D4×C59C2×D412yes—
148C22×C4×C59—22no—
149C22×C4×C59—21no—
150C22×C4×C59—22no—
151C22×C4×C59—22no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2411yescharacteristic
162Q16×C59—44no—
163SD16×C59—44no—
164C2×C8×C59—22no—
165C2×C8×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—22no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—22no—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—42no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59—42no—
185C4⋊C4×C59—41no—
186C4⋊C4×C59—41no—
187C4⋊C4×C59C2×D412yes—
188C4⋊C4×C59C2×D412yes—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—22no—
192C22⋊C4×C59—22no—
193C22⋊C4×C59—22no—
194C22⋊C4×C59—22no—
195C22⋊C4×C59—22no—
196C22⋊C4×C59—22no—
197C22⋊C4×C59—22no—
198C42×C59—22no—
199C42×C59—22no—
200C42×C59—21no—
201C42×C59—22no—
202C42×C59—21no—
203C42×C59—21no—
204C42×C59—21no—
205C42×C59—42no—
206C42×C59C2×D412yes—
207C42×C59C2×D412yes—
208C23×C59—22no—
209C23×C59C22×D411yescharacteristic
210Q8×C59—24no—
211Q8×C59—44no—
212Q8×C59—42no—
213Q8×C59—42no—
214Q8×C59—42no—
215Q8×C59—81no—
216D4×C59—42no—
217D4×C59—42no—
218D4×C59—42no—
219D4×C59—42no—
220D4×C59—81no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—22no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—44no—
237C2×C4×C59—44no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—82no—
251C2×C4×C59#12011yescharacteristic
252C2×C4×C59#12011yescharacteristic
253C2×C4×C59#12012yes—
254C2×C4×C59#13712yes—
255C2×C4×C59C22×D412yes—
256C8×C59—42no—
257C8×C59—42no—
258#18161C5911yesmaximal, characteristic, direct product factor, Sylow
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59C2×#12011yescharacteristic
265C22×C59C2×#13711yescharacteristic
266C22×C59#57511yescharacteristic
267C4×C59—22no—
268C4×C59—22no—
269C4×C59—21no—
270C4×C59—21no—
271C4×C59—22no—
272C4×C59—41no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59—42no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282#3461C2×C5911yescharacteristic
283#3117C2×C5912yescomplemented
284#2581C2×C5911yescharacteristic, complemented
285#1617C2×C5911yescharacteristic, complemented
286#1616C2×C5911yescharacteristic
287#1615C2×C5912yescomplemented
288#1605C2×C5911yescharacteristic, complemented
289#1604C2×C5911yescharacteristic
290#1603C2×C5912yescomplemented
291#1512C2×C5912yescomplemented
292#1467C2×C5911yescharacteristic, complemented
293C2×C59—21no—
294C2×C59—21no—
295C2×C59—82no—
296C2×C59#300711yescharacteristic
297C2×C59#302011yescharacteristic
298C2×C59#302511yescharacteristic
299#544C22×C5911yescharacteristic
300#542—22no—
301#541—22no—
302#539—21no—
303#539C22×C5912yes—
304#537C22×C5911yescharacteristic
305#536—21no—
306#533—22no—
307#529—22no—
308#529—21no—
309#529C22×C5912yes—
310C4×#41—21no—
311C2×C4×Q8C22×C5911yescharacteristic
312C2×C4×Q8C22×C5911yescharacteristic
313#484C22×C5912yes—
314#481C22×C5912yes—
315#479—22no—
316#464—22no—
317#451C22×C5912yes—
318#448C22×C5912yes—
319#441—22no—
320#440—22no—
321#432C22×C5912yes—
322#400C22×C5911yescharacteristic
323#400C22×C5911yescharacteristic
324#395C22×C5911yescharacteristic
325#395C22×C5911yescharacteristic
326#391—21no—
327#389—21no—
328#387—21no—
329#387—21no—
330#382C22×C5911yescharacteristic
331#328C22×C5912yes—
332#328C22×C5912yes—
333#327C22×C5912yes—
334#327C22×C5912yes—
335#324C22×C5912yes—
336#324C22×C5911yescharacteristic
337#324C22×C5911yescharacteristic
338C59#1816111yescharacteristic, direct product factor, Sylow
339C2×#41—21no—
340C22×Q8D4×C5911yescharacteristic
341C22×Q8C23×C5911yescharacteristic
342C2×Q16—22no—
343C2×SD16—22no—
344C4⋊Q8—22no—
345C4⋊Q8—22no—
346#126—22no—
347#126—22no—
348#126—22no—
349#126—22no—
350#125—21no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125D4×C5912yes—
355#124—22no—
356#124—21no—
357#124D4×C5912yes—
358#123—22no—
359#123—22no—
360#123—22no—
361#123—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8C23×C5912yes—
369#121—22no—
370#121—21no—
371#121C23×C5912yes—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8D4×C5912yes—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—22no—
384C4×D4—22no—
385C4×D4D4×C5912yes—
386#117—22no—
387#117—22no—
388#117—21no—
389#117—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4D4×C5911yescharacteristic
395C2×C4⋊C4C23×C5911yescharacteristic
396C2×C4⋊C4C23×C5911yescharacteristic
397C2×C4⋊C4C23×C5911yescharacteristic
398C2×C42—22no—
399C2×C42—21no—
400C2×C42D4×C5911yescharacteristic
401C2×C42D4×C5911yescharacteristic
402C2×C42C23×C5911yescharacteristic
403#107—22no—
404#106—22no—
405C4⋊C8—22no—
406#103—22no—
407#103—22no—
408#103—22no—
409#102—22no—
410C22⋊C8C23×C5912yes—
411C22⋊C8C23×C5912yes—
412#97—22no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95C23×C5911yescharacteristic
421#95C23×C5911yescharacteristic
422#41—42no—
423#41—42no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—42no—
431C2×Q8C2×D4×C5912yes—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4C2×D4×C5912yes—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4C2×D4×C5911yescharacteristic
444C22×C4C2×D4×C5911yescharacteristic
445C22×C4C2×D4×C5911yescharacteristic
446C22×C4C2×D4×C5911yescharacteristic
447C22×C4C2×D4×C5911yescharacteristic
448C22×C4C2×D4×C5911yescharacteristic
449C22×C4C24×C5911yescharacteristic
450Q16—44no—
451SD16—44no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4C2×D4×C5912yes—
476C4⋊C4C2×D4×C5912yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42C2×D4×C5912yes—
495C42C2×D4×C5912yes—
496C23—22no—
497C23C22×D4×C5911yescharacteristic
498Q8—24no—
499Q8—44no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—81no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—42no—
508D4—81no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—82no—
539C2×C4#120×C5911yescharacteristic
540C2×C4#120×C5911yescharacteristic
541C2×C4#120×C5912yes—
542C2×C4#137×C5912yes—
543C2×C4C22×D4×C5912yes—
544C8—42no—
545C8—42no—
546C22—21no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22C2×#120×C5911yescharacteristic
552C22C2×#137×C5911yescharacteristic
553C22C59×#57511yescharacteristic
554C4—22no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—41no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—81no—
569C2—21no—
570C2—21no—
571C2—82no—
572C2C59×#300711yescharacteristic
573C2C59×#302011yescharacteristic
574C2C59×#302511yescharacteristic
subgroupG/H
center, socleC22×C59C2×#120
commutator, Frattini, cosocleC22×C4C24×C59
prime radicalC59×#537C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3117C2semidirect
3. C59×#2581C2semidirect
4. C59×#1617C2semidirect
6. C59×#1615C2semidirect
7. C59×#1605C2semidirect
9. C59×#1603C2semidirect
10. C59×#1512C2semidirect
11. C59×#1467C2semidirect
258. #18161C59direct (N × H)
283. #3117C2×C59semidirect
284. #2581C2×C59semidirect
285. #1617C2×C59semidirect
287. #1615C2×C59semidirect
288. #1605C2×C59semidirect
290. #1603C2×C59semidirect
291. #1512C2×C59semidirect
292. #1467C2×C59semidirect
338. C59#18161direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#34612
C59×#31172
C59×#25812
C59×#16172
C59×#16162
C59×#16152
C59×#16052
C59×#16042
C59×#16032
C59×#15122
C59×#14672
#1816159