Finite Group Databasebeta

← #28560654

Subgroups — #28560654

Order 15104.

572 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
1#21434703C211yesmaximal, characteristic
2#21433256C211yesmaximal, characteristic
3#21433197C211yesmaximal, characteristic, complemented
4#21423002C211yesmaximal, characteristic, complemented
5#21422989C211yesmaximal, characteristic, complemented
6#21422977C211yesmaximal, characteristic, complemented
7#21422964C211yesmaximal, characteristic, complemented
8#21422949C211yesmaximal, characteristic, complemented
9#21422929C211yesmaximal, characteristic, complemented
10#21422905C211yesmaximal, characteristic, complemented
11#21422900C211yesmaximal, characteristic, complemented
12#21422849C211yesmaximal, characteristic, complemented
13#21422500C211yesmaximal, characteristic, complemented
14#21422391C211yesmaximal, characteristic, complemented
15C59×#1482C211yesmaximal, characteristic
16#16837959C2211yescharacteristic
17#16837812C411yescharacteristic
18#16837797C411yescharacteristic
19#16837772C411yescharacteristic
20#16837761C411yescharacteristic
21#16837658C411yescharacteristic
22#16837658C411yescharacteristic
23#16837646C411yescharacteristic
24#16837646C411yescharacteristic
25#16837533C2211yescharacteristic
26#16837532—21no—
27#16837529C2211yescharacteristic
28#16837529C2211yescharacteristic
29Q8×C59⋊C8C2211yescharacteristic
30C8×Dic118—21no—
31#16837526C2211yescharacteristic
32#16837526C2211yescharacteristic
33C4×#11745672C2211yescharacteristic
34C4×#11745672C2211yescharacteristic
35C4×#11745670C2211yescharacteristic
36C4×#11745670C2211yescharacteristic
37#16837480C2211yescharacteristic
38#16837474—21no—
39D4×C59⋊C8—21no—
40D4×C59⋊C8C2211yescharacteristic
41#16837463—21no—
42#16837460—21no—
43#16837460—21no—
44#16837455C2211yescharacteristic
45#16837429—21no—
46C2×C4⋊(C59⋊C8)C2211yescharacteristic
47C4×#11745664C2211yescharacteristic
48#16837154C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837152C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837134C2211yescharacteristic
54#16837134C2211yescharacteristic
55#16837134C2211yescharacteristic
56#16837124C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837123C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C59×#529C2211yescharacteristic
61C59×#431C2211yescharacteristic
62C59×#431C2211yescharacteristic
63C59×Q8⋊C8C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C59×D4⋊C8C2211yescharacteristic
67C2×#11745672C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745670C2×C411yescharacteristic
70C2×#11745670C2×C411yescharacteristic
71C2×#11745664—21no—
72C2×#11745664C2×C411yescharacteristic
73C2×#11745664C2×C411yescharacteristic
74C22×C59⋊C8—21no—
75C22×C59⋊C8—21no—
76#14273727C2×C411yescharacteristic
77#14273727C2×C411yescharacteristic
78#14273719—21no—
79#14273706—21no—
80#14273706C2×C411yescharacteristic
81#14273706C2×C411yescharacteristic
82C4×Dic118—21no—
83C4×Dic118C2311yescharacteristic
84C4×Dic118C2311yescharacteristic
85#14273686—21no—
86#14273660C2×C411yescharacteristic
87#14273660C2×C411yescharacteristic
88#14273658C2×C411yescharacteristic
89#14273658C2×C411yescharacteristic
90C4⋊(C59⋊C8)—21no—
91C4⋊(C59⋊C8)—21no—
92C4⋊(C59⋊C8)—21no—
93C4⋊(C59⋊C8)D411yescharacteristic
94C4⋊(C59⋊C8)D411yescharacteristic
95C4⋊(C59⋊C8)C2311yescharacteristic
96C4⋊(C59⋊C8)C2311yescharacteristic
97Dic59⋊C8—21no—
98Dic59⋊C8—21no—
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101#14273651C2×C411yescharacteristic
102#14273651C2×C411yescharacteristic
103#14273650C2×C411yescharacteristic
104#14273650C2×C411yescharacteristic
105#14273650C2×C411yescharacteristic
106#14273650C2×C411yescharacteristic
107#14273648C2×C411yescharacteristic
108#14273648C2×C411yescharacteristic
109C22⋊(C59⋊C8)—21no—
110C22⋊(C59⋊C8)—21no—
111C22⋊(C59⋊C8)D411yescharacteristic
112C22⋊(C59⋊C8)D411yescharacteristic
113#14273638—21no—
114#14273638C2×C411yescharacteristic
115#14273638C2×C411yescharacteristic
116#14273637—21no—
117#14273637—21no—
118C4×C59⋊C8—21no—
119C4×C59⋊C8C2311yescharacteristic
120C4×C59⋊C8C2311yescharacteristic
121C4×C59⋊C8C2311yescharacteristic
122C4×C59⋊C8C2311yescharacteristic
123C8×Dic59—21no—
124C8×Dic59—21no—
125#126×C59—21no—
126#125×C59C2×C411yescharacteristic
127#124×C59C2×C411yescharacteristic
128#123×C59—21no—
129C22⋊Q8×C59C2×C411yescharacteristic
130#121×C59C2×C411yescharacteristic
131C4×Q8×C59C2311yescharacteristic
132C4×D4×C59—21no—
133C4×D4×C59C2311yescharacteristic
134#117×C59—21no—
135C2×C42×C59C2311yescharacteristic
136C4⋊C8×C59—21no—
137C4⋊C8×C59C2311yescharacteristic
138C4⋊C8×C59C2311yescharacteristic
139C22⋊C8×C59—21no—
140C22⋊C8×C59—21no—
141C4×C8×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×D4×C59—21no—
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C22×C4×C59C22×C411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59#4111yescharacteristic
151C2×Dic118—21no—
152C2×Dic118C22×C411yescharacteristic
153C2×Dic118C22×C411yescharacteristic
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59C22×C411yescharacteristic
163C4⋊C4×C59C22×C411yescharacteristic
164C22⋊C4×C59—21no—
165C22⋊C4×C59—21no—
166C22⋊C4×C59—21no—
167C22⋊C4×C59—21no—
168C22⋊C4×C59—21no—
169C42×C59—21no—
170C42×C59C22×C411yescharacteristic
171C42×C59C2×D411yescharacteristic
172C42×C59#4111yescharacteristic
173C42×C59C2411yescharacteristic
174C22⋊Dic59—21no—
175C22⋊Dic59—21no—
176#11745672—22no—
177#11745672—41no—
178#11745670—22no—
179#11745670—41no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59C22×C411yescharacteristic
184C4⋊Dic59C22×C411yescharacteristic
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191#11745664—22no—
192#11745664—41no—
193#11745664—81no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—81no—
197C2×C59⋊C8—81no—
198C2×C59⋊C8C22×C411yescharacteristic
199C2×C59⋊C8C22×C411yescharacteristic
200C2×C59⋊C8C22×C411yescharacteristic
201C2×C59⋊C8C22×C411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206C23×C59—21no—
207C23×C59C4×D411yescharacteristic
208Q8×C59—21no—
209Q8×C59—21no—
210D4×C59—21no—
211D4×C59—21no—
212D4×C59—42no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C4×D411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C23×C411yescharacteristic
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233Dic118—21no—
234Dic118—21no—
235Dic118—21no—
236Dic118—21no—
237Dic118—42no—
238C8×C59—41no—
239C8×C59—41no—
240C8×C59—41no—
241C8×C59—41no—
242C59⋊C8—21no—
243C59⋊C8—21no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8#13112yes—
252#18083—591nomaximal, Sylow
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59C2×C4×D411yescharacteristic
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—41no—
265C4×C59—41no—
266C4×C59—41no—
267C4×C59C2×#13111yescharacteristic
268C4×C59#56811yescharacteristic
269C4×C59#57711yescharacteristic
270C4×C59#57811yescharacteristic
271Dic59—41no—
272Dic59—41no—
273Dic59—41no—
274Dic59—41no—
275#3172—591no—
276#2994—591no—
277#2986—591no—
278#1593—591no—
279#1590—591no—
280#1588—591no—
281#1587—591no—
282#1584—591no—
283#1582—591no—
284SD16⋊C8—591no—
285C8×Q16—591no—
286#1565—591no—
287#1482D5911yescharacteristic
288#1481—591no—
289#1453—591no—
290C2×C59—41no—
291C2×C59—81no—
292C2×C59#293511yescharacteristic
293C2×C59#294711yescharacteristic
294C2×C59#295511yescharacteristic
295#533—591no—
296#529C2×D5911yescharacteristic
297#501—591no—
298#497—591no—
299#491—591no—
300#489—591no—
301#470—591no—
302#470—591no—
303#468—591no—
304#468—591no—
305#446—591no—
306#446—591no—
307#446—591no—
308#446—1181no—
309C8×Q8—591no—
310C8×Q8—591no—
311C8×Q8—591no—
312C8×Q8—1181no—
313C4×Q16—591no—
314C4×Q16—591no—
315C4×SD16—591no—
316C4×SD16—591no—
317#436—591no—
318#435—1181no—
319C8×D4—591no—
320C8×D4—1181no—
321#433—1181no—
322#433—1181no—
323#433—1181no—
324#431—591no—
325#431C2×D5911yescharacteristic
326#431C2×D5911yescharacteristic
327#424—1181no—
328C2×C4⋊C8—591no—
329C4×M16—591no—
330#335—591no—
331#335—591no—
332#334—591no—
333#334—591no—
334Q8⋊C8—591no—
335Q8⋊C8—591no—
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8C2×D5911yescharacteristic
339Q8⋊C8C2×D5911yescharacteristic
340D4⋊C8C2×D5911yescharacteristic
341D4⋊C8C2×D5911yescharacteristic
342#322—591no—
343#322—591no—
344#322—591no—
345C82—591no—
346C59#1808311yescharacteristic, complemented, Sylow
347C2×Q16—591no—
348C2×Q16—591no—
349C2×SD16—591no—
350C2×SD16—591no—
351C2×M16—591no—
352C2×M16—591no—
353C2×M16—1181no—
354C22×C8—1181no—
355C22×C8—1181no—
356C4⋊Q8—591no—
357C4⋊Q8—591no—
358#126—21no—
359#125—1181no—
360#125C4×D5911yescharacteristic
361#124C4×D5911yescharacteristic
362#123—21no—
363C22⋊Q8—591no—
364C22⋊Q8—591no—
365C22⋊Q8—1181no—
366C22⋊Q8C4×D5911yescharacteristic
367#121C4×D5911yescharacteristic
368C4×Q8—591no—
369C4×Q8—591no—
370C4×Q8—1181no—
371C4×Q8C22×D5911yescharacteristic
372C4×D4—21no—
373C4×D4C22×D5911yescharacteristic
374#117—21no—
375#117—1181no—
376C2×C42C22×D5911yescharacteristic
377#107—591no—
378#107—591no—
379#106—591no—
380#106—591no—
381C4⋊C8—21no—
382C4⋊C8—591no—
383C4⋊C8—591no—
384C4⋊C8—591no—
385C4⋊C8—591no—
386C4⋊C8—1181no—
387C4⋊C8—1181no—
388C4⋊C8—1181no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8C22×D5911yescharacteristic
394C4⋊C8C22×D5911yescharacteristic
395#103—591no—
396#103—591no—
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#102—591no—
402#102—591no—
403C22⋊C8—21no—
404C22⋊C8—21no—
405C22⋊C8—591no—
406C22⋊C8—591no—
407C22⋊C8—1181no—
408C22⋊C8—1181no—
409#97—591no—
410#97—591no—
411#97—1181no—
412#97—1181no—
413#97—1181no—
414C4×C8—21no—
415C4×C8—591no—
416C4×C8—591no—
417C4×C8—591no—
418C4×C8—591no—
419C4×C8—1181no—
420C4×C8—1181no—
421C4×C8—1181no—
422C4×C8C22×D5911yescharacteristic
423C4×C8C22×D5911yescharacteristic
424C2×Q8—591no—
425C2×Q8—591no—
426C2×Q8—1181no—
427C2×Q8C2×C4×D5911yescharacteristic
428C2×D4—21no—
429C2×D4C2×C4×D5911yescharacteristic
430C22×C4—21no—
431C22×C4C2×C4×D5911yescharacteristic
432C22×C4D4×D5911yescharacteristic
433C22×C4#1174568711yescharacteristic
434Q16—1182no—
435Q16—2361no—
436SD16—1182no—
437SD16—2361no—
438M16—1182no—
439M16—4721no—
440M16—2361no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—591no—
446C2×C8—591no—
447C2×C8—591no—
448C2×C8—591no—
449C2×C8—591no—
450C2×C8—591no—
451C2×C8—591no—
452C2×C8—591no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—4721no—
456C2×C8—4721no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—591no—
462C4⋊C4—591no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4C2×C4×D5911yescharacteristic
471C4⋊C4C2×C4×D5911yescharacteristic
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C42—21no—
480C42—1181no—
481C42—1181no—
482C42C2×C4×D5911yescharacteristic
483C42D4×D5911yescharacteristic
484C42#1174568711yescharacteristic
485C42C23×D5911yescharacteristic
486C23—21no—
487C23(C22⋊D59)⋊C411yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—1181no—
491Q8—1181no—
492Q8—1181no—
493Q8—1181no—
494Q8—2362no—
495D4—21no—
496D4—21no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4(C22⋊D59)⋊C411yescharacteristic
513C2×C4(C22⋊D59)⋊C411yescharacteristic
514C2×C4(C22⋊D59)⋊C411yescharacteristic
515C2×C4C22×C4×D5911yescharacteristic
516C2×C4C2×D4×D5911yescharacteristic
517C2×C4C2×#1174568711yescharacteristic
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—592no—
523C8—1181no—
524C8—1181no—
525C8—1181no—
526C8—1181no—
527C8—1181no—
528C8—1181no—
529C8—1181no—
530C8—2361no—
531C8—2361no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22C2×(C22⋊D59)⋊C411yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—2361no—
547C4—2361no—
548C4—2361no—
549C4—2361no—
550C4#131×D5911yescharacteristic
551C4#1683816711yescharacteristic
552C4#1683820911yescharacteristic
553C4#1683821611yescharacteristic
554C2—41no—
555C2—81no—
556C2#2143283611yescharacteristic
557C2#2143294211yescharacteristic
558C2#2143299511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1482C2
socleC22×C59C2×C4×D4
prime radical#121×C59C2×C4
cosocleC42×C59C24
hypercenter#1482D59
hypocenterC59#18083
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1482

Complements

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

NH ≅ G/Ntype
3. #21433197C2semidirect
4. #21423002C2semidirect
5. #21422989C2semidirect
6. #21422977C2semidirect
7. #21422964C2semidirect
8. #21422949C2semidirect
9. #21422929C2semidirect
10. #21422905C2semidirect
11. #21422900C2semidirect
12. #21422849C2semidirect
13. #21422500C2semidirect
14. #21422391C2semidirect
346. C59#18083semidirect

Maximal subgroups

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

subgroupindex
#214347032
#214332562
#214331972
#214230022
#214229892
#214229772
#214229642
#214229492
#214229292
#214229052
#214229002
#214228492
#214225002
#214223912
C59×#14822
#1808359