Finite Group Databasebeta

← #28560102

Subgroups — #28560102

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434116C211yesmaximal, characteristic
2#21433258C211yesmaximal, characteristic, complemented
3#21433225C211yesmaximal, characteristic, complemented
4#21423000C211yesmaximal, characteristic, complemented
5#21422982C211yesmaximal, characteristic, complemented
6#21422976C211yesmaximal, characteristic, complemented
7#21422976C211yesmaximal, characteristic, complemented
8#21422948C211yesmaximal, characteristic, complemented
9#21422927C211yesmaximal, characteristic, complemented
10#21422910C211yesmaximal, characteristic, complemented
11C8×Dic236C211yesmaximal, characteristic, complemented
12#21422871C211yesmaximal, characteristic, complemented
13#21422562C211yesmaximal, characteristic, complemented
14#21422484C211yesmaximal, characteristic, complemented
15C59×#1454C211yesmaximal, characteristic, complemented
16#16837936C2211yescharacteristic
17C4×#11745685C2211yescharacteristic
18#16837744C411yescharacteristic
19#16837727C411yescharacteristic
20#16837710C411yescharacteristic
21#16837703C411yescharacteristic
22#16837625C411yescharacteristic
23#16837618C411yescharacteristic
24#16837566C411yescharacteristic
25#16837549C411yescharacteristic
26#16837532C2211yescharacteristic
27C8×Dic118C2211yescharacteristic
28C8×Dic118C2211yescharacteristic
29C8×Dic118C2211yescharacteristic
30C4×Dic236C2211yescharacteristic
31C4×#11745660C2211yescharacteristic
32C4×#11745660C2211yescharacteristic
33C4×D472C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837481C2211yescharacteristic
37#16837477—21no—
38#16837477—21no—
39#16837470—21no—
40C8×D236C2211yescharacteristic
41(C22⋊D59)⋊C8—21no—
42#16837462—21no—
43#16837454—21no—
44#16837427—21no—
45#16837426C2211yescharacteristic
46#16837424—21no—
47C2×C4⋊(C59⋊C8)C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837155C2211yescharacteristic
50#16837155C2211yescharacteristic
51#16837155C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837134C2211yescharacteristic
54#16837134C2211yescharacteristic
55#16837134C2211yescharacteristic
56#16837131C2211yescharacteristic
57#16837131C2211yescharacteristic
58#16837131C2211yescharacteristic
59#16837131C2211yescharacteristic
60C59×#431C2211yescharacteristic
61C59×#431C2211yescharacteristic
62C59×#431C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C59×#322C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×#11745664—21no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C22×C59⋊C8—21no—
77C2×C8×D59—21no—
78#14273721—21no—
79#14273719C2×C411yescharacteristic
80#14273715C2×C411yescharacteristic
81#14273710—21no—
82#14273706C2×C411yescharacteristic
83#14273701C2×C411yescharacteristic
84C4×Dic118C2×C411yescharacteristic
85C4×Dic118C2311yescharacteristic
86C4×Dic118C2311yescharacteristic
87C4×D236C2×C411yescharacteristic
88C4×D236C2311yescharacteristic
89C4×D236C2311yescharacteristic
90C4×C22⋊D59—21no—
91C4×C22⋊D59—21no—
92C4×C22⋊D59—21no—
93#14273686C2×C411yescharacteristic
94#14273684—21no—
95#14273684—21no—
96C42×D59—21no—
97#14273661C2×C411yescharacteristic
98#14273661C2×C411yescharacteristic
99#14273659C2×C411yescharacteristic
100#14273659C2×C411yescharacteristic
101C4⋊(C59⋊C8)—21no—
102C4⋊(C59⋊C8)—21no—
103C4⋊(C59⋊C8)C2311yescharacteristic
104C4⋊(C59⋊C8)C2311yescharacteristic
105C4⋊(C59⋊C8)C2311yescharacteristic
106C4⋊(C59⋊C8)C2311yescharacteristic
107Dic59⋊C8—21no—
108Dic59⋊C8—21no—
109Dic59⋊C8—21no—
110Dic59⋊C8—21no—
111#14273652C2×C411yescharacteristic
112#14273652C2×C411yescharacteristic
113#14273652C2×C411yescharacteristic
114#14273652C2×C411yescharacteristic
115#14273649C2×C411yescharacteristic
116#14273649C2×C411yescharacteristic
117#14273649C2×C411yescharacteristic
118#14273649C2×C411yescharacteristic
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8—21no—
121(C2×D59)⋊C8—21no—
122(C2×D59)⋊C8—21no—
123#14273637—21no—
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59—21no—
127C2×C42×C59C2311yescharacteristic
128C4⋊C8×C59—21no—
129C4⋊C8×C59C2×C411yescharacteristic
130C4⋊C8×C59C2×C411yescharacteristic
131C4⋊C8×C59D411yescharacteristic
132C4⋊C8×C59D411yescharacteristic
133C22⋊C8×C59—21no—
134C22⋊C8×C59C2×C411yescharacteristic
135C22⋊C8×C59C2×C411yescharacteristic
136C22⋊C8×C59D411yescharacteristic
137C22⋊C8×C59D411yescharacteristic
138C4×C8×C59C2311yescharacteristic
139C4×C8×C59C2311yescharacteristic
140C4×C8×C59C2311yescharacteristic
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4111yescharacteristic
147C2×C22⋊D59—21no—
148C2×C22⋊D59—21no—
149#11745685—22no—
150#11745685—41no—
151C2×D236C22×C411yescharacteristic
152C2×D236C22×C411yescharacteristic
153C2×C4×D59—21no—
154C2×C4×D59—21no—
155C2×C4×D59—21no—
156C2×C4×D59—21no—
157C2×Dic118C22×C411yescharacteristic
158C2×Dic118C22×C411yescharacteristic
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59C22×C411yescharacteristic
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C22×C411yescharacteristic
165C2×C8×C59C2×D411yescharacteristic
166C2×C8×C59C2×D411yescharacteristic
167C2×C8×C59C2×D411yescharacteristic
168C2×C8×C59C2×D411yescharacteristic
169C42×C59C22×C411yescharacteristic
170C42×C59C2×D411yescharacteristic
171C42×C59#4111yescharacteristic
172C42×C59C2411yescharacteristic
173C22⋊Dic59—21no—
174C22⋊Dic59—21no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—21no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180(C2×D59)⋊C4—21no—
181C4⋊Dic59C22×C411yescharacteristic
182C4⋊Dic59C22×C411yescharacteristic
183C4⋊Dic59C22×C411yescharacteristic
184C4⋊Dic59C22×C411yescharacteristic
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—21no—
195#11745664—81no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—81no—
201Dic236—22no—
202D472—41no—
203#11745660—22no—
204#11745660—41no—
205#11745659—81no—
206#11745659—81no—
207#11745659—81no—
208C8×D59—81no—
209C23×C59C4×D411yescharacteristic
210C22×D59—21no—
211C22×D59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59C4×D411yescharacteristic
218C2×C4×C59C4×D411yescharacteristic
219C2×C4×C59C4×D411yescharacteristic
220C2×C4×C59C23×C411yescharacteristic
221C2×C4×C59C22×D411yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C22⋊D59—42no—
224C22⋊D59—42no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231D236—21no—
232D236—21no—
233D236—21no—
234D236—21no—
235C4×D59—42no—
236C4×D59—42no—
237C4×D59—41no—
238C4×D59—41no—
239C4×D59—41no—
240C4×D59—41no—
241Dic118—21no—
242Dic118—21no—
243Dic118—21no—
244Dic118#13112yes—
245C8×C59—21no—
246C8×C59—21no—
247C8×C59—21no—
248C8×C59—21no—
249C8×C59—21no—
250C8×C59—21no—
251C8×C59—21no—
252C8×C59—21no—
253C8×C59—41no—
254C8×C59—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259#18033—591nomaximal, Sylow
260C22×C59—21no—
261C22×C59—21no—
262C22×C59—21no—
263C22×C59C2×C4×D411yescharacteristic
264C2×D59—41no—
265C2×D59—41no—
266C2×D59—41no—
267C2×D59—41no—
268C2×D59—41no—
269C2×D59—41no—
270C4×C59—21no—
271C4×C59—21no—
272C4×C59—21no—
273C4×C59—21no—
274C4×C59C2×#13111yescharacteristic
275C4×C59#56811yescharacteristic
276C4×C59#57611yescharacteristic
277C4×C59#57811yescharacteristic
278Dic59—22no—
279Dic59—41no—
280Dic59—41no—
281Dic59—41no—
282Dic59—41no—
283Dic59—41no—
284#3102—591no—
285#2994—591no—
286#2991—591no—
287#1592—591no—
288#1589—591no—
289#1588—591no—
290#1588—591no—
291#1584—591no—
292#1581—591no—
293SD16⋊C8—591no—
294C8×Q16—591no—
295#1571—591no—
296#1495—591no—
297#1477—591no—
298#1454D5911yescharacteristic, complemented
299C2×C59—41no—
300C2×C59#293311yescharacteristic
301C2×C59#295111yescharacteristic
302C2×C59#295511yescharacteristic
303D59—81no—
304D59—81no—
305#529—591no—
306C4×#41—591no—
307#485—591no—
308#482—591no—
309#479—591no—
310#478—591no—
311#464—591no—
312#463—591no—
313#452—591no—
314#449—591no—
315#446—591no—
316C8×Q8—591no—
317C8×Q8—591no—
318C8×Q8—591no—
319C4×Q16—591no—
320C4×SD16—591no—
321C4×SD16—591no—
322C4×D8—591no—
323#436—591no—
324#436—591no—
325#436—591no—
326#436—1181no—
327#436—1181no—
328#435—1181no—
329C8×D4—591no—
330C8×D4—1181no—
331#433—1181no—
332#431—1181no—
333#431C2×D5911yescharacteristic
334#431C2×D5911yescharacteristic
335#431C2×D5911yescharacteristic
336#424—1181no—
337#423—591no—
338#423—1181no—
339C2×C4⋊C8—591no—
340#335—591no—
341#335—591no—
342#335—591no—
343#335—591no—
344Q8⋊C8—591no—
345Q8⋊C8—591no—
346Q8⋊C8—591no—
347Q8⋊C8—591no—
348D4⋊C8—591no—
349D4⋊C8—591no—
350D4⋊C8—591no—
351D4⋊C8—591no—
352#322C2×D5911yescharacteristic
353#322C2×D5911yescharacteristic
354#322C2×D5911yescharacteristic
355C82C2×D5911yescharacteristic
356C59#1803311yescharacteristic, complemented, Sylow
357C2×#41—591no—
358C2×Q16—591no—
359C2×SD16—591no—
360C2×SD16—591no—
361C2×D8—591no—
362C2×M16—1181no—
363C2×M16—1181no—
364C2×M16—1181no—
365C2×M16—1181no—
366C22×C8—1181no—
367C22×C8—1181no—
368#126—1181no—
369#125—591no—
370#124—591no—
371#123—1181no—
372C22⋊Q8—591no—
373#121—591no—
374C4×Q8—591no—
375C4×Q8—591no—
376C4×Q8—591no—
377C4×D4—591no—
378C4×D4—591no—
379C4×D4—591no—
380C4×D4—1181no—
381C4×D4—1181no—
382C4×D4—1181no—
383#117—591no—
384#117—1181no—
385#117—1181no—
386C2×C42—1181no—
387C2×C42C22×D5911yescharacteristic
388#107—591no—
389#107—591no—
390#106—591no—
391#106—591no—
392C4⋊C8—21no—
393C4⋊C8—591no—
394C4⋊C8—591no—
395C4⋊C8—591no—
396C4⋊C8—591no—
397C4⋊C8—1181no—
398C4⋊C8—1181no—
399C4⋊C8—1181no—
400C4⋊C8—1181no—
401C4⋊C8—1181no—
402C4⋊C8—1181no—
403C4⋊C8C4×D5911yescharacteristic
404C4⋊C8C4×D5911yescharacteristic
405C4⋊C8D23611yescharacteristic
406C4⋊C8D23611yescharacteristic
407#103—591no—
408#103—591no—
409#103—591no—
410#103—591no—
411#102—591no—
412#102—591no—
413#102—591no—
414#102—591no—
415C22⋊C8—21no—
416C22⋊C8—1181no—
417C22⋊C8—1181no—
418C22⋊C8—1181no—
419C22⋊C8—1181no—
420C22⋊C8C4×D5911yescharacteristic
421C22⋊C8C4×D5911yescharacteristic
422C22⋊C8D23611yescharacteristic
423C22⋊C8D23611yescharacteristic
424#97—1181no—
425C4×C8—1181no—
426C4×C8—1181no—
427C4×C8—1181no—
428C4×C8C22×D5911yescharacteristic
429C4×C8C22×D5911yescharacteristic
430C4×C8C22×D5911yescharacteristic
431C4×C8C22×D5911yescharacteristic
432C4×C8C22×D5911yescharacteristic
433C4×C8C22×D5911yescharacteristic
434#41—1182no—
435#41—2361no—
436C2×Q8—591no—
437C2×Q8—591no—
438C2×D4—591no—
439C2×D4—591no—
440C2×D4—1181no—
441C2×D4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4C2×C4×D5911yescharacteristic
447C22×C4C2×D23611yescharacteristic
448C22×C4#1174568511yescharacteristic
449Q16—1182no—
450SD16—1182no—
451SD16—2361no—
452D8—2361no—
453M16—4721no—
454M16—4721no—
455M16—4721no—
456M16—4721no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—4721no—
464C2×C8—4721no—
465C2×C8C2×C4×D5911yescharacteristic
466C2×C8C2×C4×D5911yescharacteristic
467C2×C8C2×C4×D5911yescharacteristic
468C2×C8C2×C4×D5911yescharacteristic
469C2×C8C2×D23611yescharacteristic
470C2×C8C2×D23611yescharacteristic
471C2×C8C2×D23611yescharacteristic
472C2×C8C2×D23611yescharacteristic
473C4⋊C4—591no—
474C4⋊C4—591no—
475C4⋊C4—591no—
476C4⋊C4—591no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C22⋊C4—1181no—
484C22⋊C4—1181no—
485C22⋊C4—1181no—
486C22⋊C4—1181no—
487C22⋊C4—1181no—
488C22⋊C4—1181no—
489C22⋊C4—1181no—
490C22⋊C4—1181no—
491C42—1181no—
492C42—1181no—
493C42—1181no—
494C42—1181no—
495C42C2×C4×D5911yescharacteristic
496C42C2×D23611yescharacteristic
497C42#1174568511yescharacteristic
498C42C23×D5911yescharacteristic
499C23—1181no—
500C23—1181no—
501C23C4×D23611yescharacteristic
502Q8—592no—
503Q8—1181no—
504Q8—1181no—
505Q8—1181no—
506D4—1181no—
507D4—1181no—
508D4—1181no—
509D4—1181no—
510D4—2362no—
511D4—2362no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—2362no—
524C2×C4—2362no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4C4×D23611yescharacteristic
530C2×C4C4×D23611yescharacteristic
531C2×C4C4×D23611yescharacteristic
532C2×C4C22×C4×D5911yescharacteristic
533C2×C4C22×D23611yescharacteristic
534C2×C4C2×#1174568511yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—2361no—
546C8—2361no—
547C8—2361no—
548C8—2361no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—2361no—
553C22—2361no—
554C22—2361no—
555C22—2361no—
556C22—2361no—
557C22—2361no—
558C22C2×C4×D23611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—1182no—
564C4—2361no—
565C4—2361no—
566C4—2361no—
567C4—2361no—
568C4—2361no—
569C4#131×D5911yescharacteristic
570C4#1683816711yescharacteristic
571C4#1683820711yescharacteristic
572C4#1683821911yescharacteristic
573C2—41no—
574C2—4721no—
575C2—4721no—
576C2#2143282411yescharacteristic
577C2#2143297311yescharacteristic
578C2#2143300011yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1454C2
socleC22×C59C2×C4×D4
prime radical#16837549C4
cosocleC42×C59C24
hypercenter#1454D59
hypocenterC59#18033
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #21433258C2semidirect
3. #21433225C2semidirect
4. #21423000C2semidirect
5. #21422982C2semidirect
6. #21422976C2semidirect
7. #21422976C2semidirect
8. #21422948C2semidirect
9. #21422927C2semidirect
10. #21422910C2semidirect
11. C8×Dic236C2semidirect
12. #21422871C2semidirect
13. #21422562C2semidirect
14. #21422484C2semidirect
15. C59×#1454C2semidirect
298. #1454D59semidirect
356. C59#18033semidirect

Maximal subgroups

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

subgroupindex
#214341162
#214332582
#214332252
#214230002
#214229822
#214229762
#214229482
#214229272
#214229102
C8×Dic2362
#214228712
#214225622
#214224842
C59×#14542
#1803359