Finite Group Databasebeta

← #28559357

Subgroups — #28559357

Order 15104.

600 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434652C211yesmaximal, characteristic
2#21433192C211yesmaximal, characteristic, complemented
3#21433173C211yesmaximal, characteristic
4#21422998C211yesmaximal, characteristic, complemented
5#21422996C211yesmaximal, characteristic, complemented
6#21422988C211yesmaximal, characteristic, complemented
7#21422911C211yesmaximal, characteristic, complemented
8#21422905C211yesmaximal, characteristic, complemented
9SD16×C59⋊C8C211yesmaximal, characteristic, complemented
10#21422895C211yesmaximal, characteristic, complemented
11#21422856C211yesmaximal, characteristic, complemented
12#21422567C211yesmaximal, characteristic, complemented
13#21422563C211yesmaximal, characteristic
14#21422378C211yesmaximal, characteristic, complemented
15C59×#1590C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17#16837949C2211yescharacteristic
18#16837799C411yescharacteristic
19#16837773C411yescharacteristic
20#16837751C411yescharacteristic
21#16837747C411yescharacteristic
22#16837634C411yescharacteristic
23#16837632C411yescharacteristic
24#16837630C411yescharacteristic
25#16837628C411yescharacteristic
26#16837532C2211yescharacteristic
27#16837530—21no—
28#16837529—21no—
29Q8×C59⋊C8C2211yescharacteristic
30#16837526C2211yescharacteristic
31C4×#11745660C2211yescharacteristic
32C4×#11745671C2211yescharacteristic
33C4×#11745670C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837471—21no—
36#16837470—21no—
37D4×C59⋊C8C2211yescharacteristic
38#16837467C2211yescharacteristic
39#16837464—21no—
40#16837462—21no—
41#16837458C2211yescharacteristic
42#16837458C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837453C2211yescharacteristic
45#16837370—21no—
46#16837367—21no—
47C4×C8×D59C2211yescharacteristic
48#16837154C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837153C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53#16837132C2211yescharacteristic
54#16837131C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C4×SD16×C59C2211yescharacteristic
62C59×#436C2211yescharacteristic
63C59×#334C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745660C2×C411yescharacteristic
69C2×#11745670C2×C411yescharacteristic
70C2×#11745671C2×C411yescharacteristic
71C2×#11745659—21no—
72C22×C59⋊C8—21no—
73C2×C8×D59—21no—
74C2×C8×D59C2×C411yescharacteristic
75C2×C8×D59C2×C411yescharacteristic
76#14273726C2×C411yescharacteristic
77#14273726C2×C411yescharacteristic
78#14273725C2×C411yescharacteristic
79#14273723C2×C411yescharacteristic
80#14273717—21no—
81#14273713—21no—
82#14273707C2×C411yescharacteristic
83#14273705—21no—
84#14273703C2×C411yescharacteristic
85#14273702C2×C411yescharacteristic
86#14273700—21no—
87#14273698C2×C411yescharacteristic
88C4×Dic118—21no—
89C4×Dic118C2311yescharacteristic
90C4×D236C2311yescharacteristic
91C4×C22⋊D59—21no—
92#14273684—21no—
93C42×D59C2311yescharacteristic
94#14273659C2×C411yescharacteristic
95#14273658C2×C411yescharacteristic
96#14273658C2×C411yescharacteristic
97C4⋊(C59⋊C8)—21no—
98C4⋊(C59⋊C8)C2311yescharacteristic
99C4⋊(C59⋊C8)C2311yescharacteristic
100Dic59⋊C8—21no—
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8D411yescharacteristic
104Dic59⋊C8D411yescharacteristic
105#14273652C2×C411yescharacteristic
106#14273651C2×C411yescharacteristic
107#14273650C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273648C2×C411yescharacteristic
110#14273647C2×C411yescharacteristic
111C22⋊(C59⋊C8)—21no—
112(C2×D59)⋊C8—21no—
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8D411yescharacteristic
118(C2×D59)⋊C8D411yescharacteristic
119#14273637—21no—
120#14273637—21no—
121#14273637—21no—
122C4×C59⋊C8—21no—
123C4×C59⋊C8C2311yescharacteristic
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C8×Dic59—21no—
127C8×Dic59C2×C411yescharacteristic
128C8×Dic59C2×C411yescharacteristic
129C2×SD16×C59C2×C411yescharacteristic
130C2×M16×C59—21no—
131C4×Q8×C59C2311yescharacteristic
132C4×D4×C59C2311yescharacteristic
133#106×C59C2×C411yescharacteristic
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140#97×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×D4×C59C22×C411yescharacteristic
146C22×C4×C59—21no—
147C2×C22⋊D59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150#11745685—81no—
151#11745685—81no—
152C2×D236—21no—
153C2×D236C22×C411yescharacteristic
154C2×C4×D59—21no—
155C2×C4×D59C22×C411yescharacteristic
156C2×C4×D59C2×D411yescharacteristic
157C2×C4×D59#4111yescharacteristic
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C22×C411yescharacteristic
161SD16×C59—22no—
162M16×C59—81no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59C22×C411yescharacteristic
167C2×C8×C59C22×C411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C2×C8×C59C2×D411yescharacteristic
170C4⋊C4×C59—21no—
171C4⋊C4×C59C22×C411yescharacteristic
172C4⋊C4×C59C22×C411yescharacteristic
173C22⋊C4×C59—21no—
174C42×C59—21no—
175C42×C59C2411yescharacteristic
176C22⋊Dic59—21no—
177#11745671—22no—
178#11745670—22no—
179(C2×D59)⋊C4—21no—
180(C2×D59)⋊C4—21no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59C22×C411yescharacteristic
185C4⋊Dic59C22×C411yescharacteristic
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192C4×Dic59—21no—
193C4×Dic59C22×C411yescharacteristic
194C4×Dic59C2×D411yescharacteristic
195C4×Dic59#4111yescharacteristic
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—81no—
200C2×C59⋊C8C22×C411yescharacteristic
201C2×C59⋊C8C22×C411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204#11745660—22no—
205#11745659—81no—
206C8×D59—22no—
207C8×D59—22no—
208C8×D59—81no—
209C23×C59—21no—
210C22×D59—21no—
211C22×D59C4×D411yescharacteristic
212Q8×C59—21no—
213Q8×C59—21no—
214D4×C59—21no—
215D4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59C23×C411yescharacteristic
222C2×C4×C59C22×D411yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C22⋊D59—81no—
225C22⋊D59—81no—
226C22⋊D59—81no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59C4×D411yescharacteristic
231C2×Dic59C4×D411yescharacteristic
232C2×Dic59C4×D411yescharacteristic
233D236—21no—
234D236—21no—
235D236—81no—
236C4×D59—21no—
237C4×D59—21no—
238C4×D59—21no—
239C4×D59—21no—
240C4×D59—41no—
241C4×D59—41no—
242C4×D59—41no—
243Dic118—21no—
244Dic118—21no—
245Dic118—81no—
246Dic118—81no—
247C8×C59—21no—
248C8×C59—21no—
249C8×C59—21no—
250C8×C59—41no—
251C8×C59—41no—
252C8×C59—41no—
253C8×C59#13112yes—
254C59⋊C8—21no—
255C59⋊C8—21no—
256C59⋊C8—21no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260C59⋊C8#13112yes—
261#17971—591nomaximal, Sylow
262C22×C59—41no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59C2×C4×D411yescharacteristic
266C2×D59—21no—
267C2×D59—21no—
268C2×D59—21no—
269C2×D59—41no—
270C2×D59—41no—
271C2×D59—41no—
272C4×C59—21no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59C2×#13111yescharacteristic
277C4×C59C2×#13111yescharacteristic
278C4×C59#57711yescharacteristic
279C4×C59#57711yescharacteristic
280Dic59—21no—
281Dic59—21no—
282Dic59—21no—
283Dic59—41no—
284Dic59—41no—
285Dic59—41no—
286#3167—591no—
287#2985—591no—
288#2981—591no—
289#1592—591no—
290#1592—591no—
291#1590—591no—
292#1590D5911yescharacteristic, complemented
293SD16⋊C8—591no—
294SD16⋊C8—591no—
295C8×SD16—591no—
296C8×SD16—591no—
297#1568—591no—
298#1497—591no—
299#1496—591no—
300#1450—591no—
301C2×C59—81no—
302C2×C59#293111yescharacteristic
303C2×C59#294911yescharacteristic
304C2×C59#295711yescharacteristic
305D59—41no—
306D59—81no—
307#533—591no—
308#532—591no—
309#498—591no—
310#492—591no—
311#487—591no—
312#486—591no—
313#465—591no—
314#465—591no—
315#465—591no—
316#465—591no—
317#446—591no—
318#446—1181no—
319#446—1181no—
320#446C2×D5911yescharacteristic
321C8×Q8—591no—
322C8×Q8—591no—
323C4×SD16—591no—
324C4×SD16—591no—
325C4×SD16—591no—
326C4×SD16C2×D5911yescharacteristic
327#436—591no—
328#436C2×D5911yescharacteristic
329#435—1181no—
330#435—1181no—
331C8×D4—591no—
332C8×D4—591no—
333#433—1181no—
334#433—1181no—
335#432—591no—
336#432—591no—
337#431—591no—
338#431—591no—
339#408—1181no—
340#408—1181no—
341C2×C4×C8—591no—
342#335—591no—
343#335—591no—
344#334—591no—
345#334C2×D5911yescharacteristic
346Q8⋊C8—591no—
347Q8⋊C8—591no—
348Q8⋊C8—591no—
349Q8⋊C8C2×D5911yescharacteristic
350D4⋊C8—591no—
351D4⋊C8—591no—
352D4⋊C8—591no—
353D4⋊C8C2×D5911yescharacteristic
354#322—591no—
355#322—591no—
356C82—591no—
357C82C2×D5911yescharacteristic
358C59#1797111yescharacteristic, complemented, Sylow
359C2×#41—1181no—
360C2×SD16—591no—
361C2×SD16—591no—
362C2×SD16—591no—
363C2×SD16C4×D5911yescharacteristic
364C2×M16—21no—
365C2×M16—1181no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—1181no—
369C22×C8—1181no—
370C4⋊Q8—591no—
371C4⋊Q8—591no—
372C4⋊Q8—591no—
373#127—591no—
374#125—1181no—
375#124—1181no—
376C22⋊Q8—591no—
377C22⋊Q8—591no—
378C22⋊Q8—1181no—
379#121—591no—
380#121—591no—
381#121—1181no—
382C4×Q8—591no—
383C4×Q8—1181no—
384C4×Q8C22×D5911yescharacteristic
385C4×D4—591no—
386C4×D4—1181no—
387C4×D4C22×D5911yescharacteristic
388#117—1181no—
389C2×C42—591no—
390#106—591no—
391#106—591no—
392#106—591no—
393#106C4×D5911yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—591no—
396C4⋊C8—591no—
397C4⋊C8—591no—
398C4⋊C8—591no—
399C4⋊C8—1181no—
400C4⋊C8—1181no—
401C4⋊C8—1181no—
402C4⋊C8—1181no—
403C4⋊C8C22×D5911yescharacteristic
404C4⋊C8C22×D5911yescharacteristic
405#103—591no—
406#103—591no—
407#103—591no—
408#103C4×D5911yescharacteristic
409#102—591no—
410#102—591no—
411#102—591no—
412#102C4×D5911yescharacteristic
413C22⋊C8—21no—
414C22⋊C8—591no—
415C22⋊C8—591no—
416C22⋊C8—1181no—
417C22⋊C8—1181no—
418C22⋊C8—1181no—
419C22⋊C8—1181no—
420C22⋊C8—1181no—
421C22⋊C8—1181no—
422#97—21no—
423#97—1181no—
424#97—1181no—
425#97—1181no—
426C4×C8—591no—
427C4×C8—591no—
428C4×C8—591no—
429C4×C8—591no—
430C4×C8—591no—
431C4×C8—1181no—
432C4×C8—1181no—
433C4×C8C22×D5911yescharacteristic
434C4×C8C22×D5911yescharacteristic
435C4×C8C22×D5911yescharacteristic
436#41—4721no—
437#41—4721no—
438C2×Q8—591no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8C2×C4×D5911yescharacteristic
442C2×D4—591no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4C2×C4×D5911yescharacteristic
448C22×C4—21no—
449C22×C4—591no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—1181no—
453SD16—22no—
454SD16—1182no—
455SD16—1182no—
456SD16—1182no—
457M16—81no—
458M16—4721no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—591no—
463C2×C8—591no—
464C2×C8—591no—
465C2×C8—591no—
466C2×C8—1182no—
467C2×C8—1182no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—4721no—
472C2×C8—4721no—
473C2×C8C2×C4×D5911yescharacteristic
474C2×C8C2×C4×D5911yescharacteristic
475C2×C8D4×D5911yescharacteristic
476C2×C8D4×D5911yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—591no—
479C4⋊C4—591no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4C2×C4×D5911yescharacteristic
488C4⋊C4C2×C4×D5911yescharacteristic
489C22⋊C4—21no—
490C22⋊C4—1181no—
491C22⋊C4—1181no—
492C22⋊C4—1181no—
493C22⋊C4—1181no—
494C22⋊C4—1181no—
495C42—21no—
496C42—591no—
497C42—591no—
498C42—591no—
499C42—1181no—
500C42C23×D5911yescharacteristic
501C23—21no—
502C23—591no—
503C23—1181no—
504Q8—21no—
505Q8—21no—
506Q8—1181no—
507Q8—1181no—
508Q8—4721no—
509Q8—4721no—
510D4—21no—
511D4—21no—
512D4—1181no—
513D4—1181no—
514D4—4721no—
515D4—4721no—
516D4—4721no—
517D4—4721no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—591no—
524C2×C4—591no—
525C2×C4—591no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4C22×C4×D5911yescharacteristic
537C2×C4C2×D4×D5911yescharacteristic
538C2×C4#41×D5911yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—592no—
546C8—1181no—
547C8—1181no—
548C8—1181no—
549C8—2361no—
550C8—2361no—
551C8—2361no—
552C8#1427373512yes—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—1181no—
557C22—1181no—
558C22—1181no—
559C22—2361no—
560C22—2361no—
561C22—2361no—
562C22C4×D4×D5911yescharacteristic
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—1181no—
568C4—1181no—
569C4—1181no—
570C4—2361no—
571C4—2361no—
572C4—2361no—
573C4#131×D5911yescharacteristic
574C4C2×#1427373511yescharacteristic
575C4#1683820811yescharacteristic
576C4#1683820911yescharacteristic
577C2—81no—
578C2—4721no—
579C2—2361no—
580C2#2143280511yescharacteristic
581C2#2143295011yescharacteristic
582C2#2143300811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1590C2
socleC22×C59C2×C4×D4
prime radical#16837949C22
cosocleC42×C59C24
hypercenter#1590D59
hypocenterC59#17971
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #21433192C2semidirect
4. #21422998C2semidirect
5. #21422996C2semidirect
6. #21422988C2semidirect
7. #21422911C2semidirect
8. #21422905C2semidirect
9. SD16×C59⋊C8C2semidirect
10. #21422895C2semidirect
11. #21422856C2semidirect
12. #21422567C2semidirect
14. #21422378C2semidirect
15. C59×#1590C2semidirect
292. #1590D59semidirect
358. C59#17971semidirect

Maximal subgroups

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

subgroupindex
#214346522
#214331922
#214331732
#214229982
#214229962
#214229882
#214229112
#214229052
SD16×C59⋊C82
#214228952
#214228562
#214225672
#214225632
#214223782
C59×#15902
#1797159