Finite Group Databasebeta

← #28559910

Subgroups — #28559910

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434163C211yesmaximal, characteristic
2#21433224C211yesmaximal, characteristic
3#21433194C211yesmaximal, characteristic, complemented
4#21422989C211yesmaximal, characteristic, complemented
5#21422975C211yesmaximal, characteristic, complemented
6#21422972C211yesmaximal, characteristic, complemented
7#21422966C211yesmaximal, characteristic, complemented
8#21422965C211yesmaximal, characteristic, complemented
9C8×Dic236C211yesmaximal, characteristic, complemented
10#21422900C211yesmaximal, characteristic, complemented
11#21422883C211yesmaximal, characteristic, complemented
12#21422549C211yesmaximal, characteristic, complemented
13#21422475C211yesmaximal, characteristic
14#21422378C211yesmaximal, characteristic, complemented
15C59×#1590C211yesmaximal, characteristic, complemented
16#16837956C2211yescharacteristic
17C4×#11745687C2211yescharacteristic
18#16837746C411yescharacteristic
19#16837742C411yescharacteristic
20#16837686C411yescharacteristic
21#16837680C411yescharacteristic
22#16837611C411yescharacteristic
23#16837605C411yescharacteristic
24#16837565C411yescharacteristic
25#16837561C411yescharacteristic
26#16837530—21no—
27#16837529C2211yescharacteristic
28Q8×C59⋊C8C2211yescharacteristic
29C8×Dic118C2211yescharacteristic
30C8×Dic118C2211yescharacteristic
31#16837526—21no—
32#16837526C2211yescharacteristic
33C4×Dic236C2211yescharacteristic
34C4×#11745672C2211yescharacteristic
35C4×#11745670C2211yescharacteristic
36#16837480C2211yescharacteristic
37#16837478—21no—
38#16837477—21no—
39#16837464—21no—
40#16837462—21no—
41#16837454C2211yescharacteristic
42#16837453C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837452—21no—
45#16837425—21no—
46#16837424C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52#16837134C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54#16837132C2211yescharacteristic
55#16837132C2211yescharacteristic
56D4⋊(C59⋊C8)C2211yescharacteristic
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×#11745687C2×C411yescharacteristic
68C2×Dic236C2×C411yescharacteristic
69C2×#11745672C2×C411yescharacteristic
70C2×#11745670C2×C411yescharacteristic
71C2×#11745664—21no—
72C2×#11745659—21no—
73C2×C8×D59—21no—
74#14273726C2×C411yescharacteristic
75#14273725C2×C411yescharacteristic
76#14273717—21no—
77#14273707C2×C411yescharacteristic
78#14273705—21no—
79#14273703C2×C411yescharacteristic
80C4×Dic118—21no—
81C4×Dic118C2311yescharacteristic
82C4×Dic118C2311yescharacteristic
83#14273692C2×C411yescharacteristic
84D4×Dic59C2×C411yescharacteristic
85C4×C22⋊D59—21no—
86(C22⋊D59)⋊C4—21no—
87#14273685C2×C411yescharacteristic
88#14273684—21no—
89#14273683—21no—
90C2×C4×Dic59—21no—
91C42×D59C2311yescharacteristic
92#14273661C2×C411yescharacteristic
93#14273660C2×C411yescharacteristic
94#14273658C2×C411yescharacteristic
95C4⋊(C59⋊C8)—21no—
96C4⋊(C59⋊C8)C2311yescharacteristic
97C4⋊(C59⋊C8)C2311yescharacteristic
98Dic59⋊C8—21no—
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8C2×C411yescharacteristic
104Dic59⋊C8C2×C411yescharacteristic
105Dic59⋊C8D411yescharacteristic
106Dic59⋊C8D411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273652C2×C411yescharacteristic
109#14273651C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111#14273650C2×C411yescharacteristic
112#14273648C2×C411yescharacteristic
113C22⋊(C59⋊C8)—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8C2×C411yescharacteristic
116(C2×D59)⋊C8C2×C411yescharacteristic
117(C2×D59)⋊C8D411yescharacteristic
118(C2×D59)⋊C8D411yescharacteristic
119#14273637—21no—
120C4×C59⋊C8—21no—
121C4×C59⋊C8C2311yescharacteristic
122C4×C59⋊C8C2311yescharacteristic
123C4×C59⋊C8C2311yescharacteristic
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59—21no—
127C8×Dic59—21no—
128C8×Dic59—21no—
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—
148C22×Dic59—21no—
149C22×Dic59—21no—
150#11745687—22no—
151#11745687—41no—
152C2×C4×D59C22×C411yescharacteristic
153C2×C4×D59C2×D411yescharacteristic
154C2×C4×D59#4111yescharacteristic
155C2×Dic118—21no—
156C2×Dic118C22×C411yescharacteristic
157C2×Dic118C22×C411yescharacteristic
158SD16×C59—41no—
159M16×C59—81no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C22×C411yescharacteristic
165C2×C8×C59C2×D411yescharacteristic
166C2×C8×C59C2×D411yescharacteristic
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C411yescharacteristic
169C4⋊C4×C59C22×C411yescharacteristic
170C22⋊C4×C59—21no—
171C42×C59—21no—
172C42×C59C2411yescharacteristic
173C22⋊Dic59—21no—
174#11745672—22no—
175#11745670—41no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—21no—
178(C2×D59)⋊C4—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59C22×C411yescharacteristic
181C4⋊Dic59C22×C411yescharacteristic
182Dic59⋊C4—21no—
183Dic59⋊C4—21no—
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59C22×C411yescharacteristic
194C4×Dic59C2×D411yescharacteristic
195C4×Dic59#4111yescharacteristic
196#11745664—81no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C411yescharacteristic
201C2×C59⋊C8C22×C411yescharacteristic
202C2×C59⋊C8C2×D411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204Dic236—22no—
205#11745659—81no—
206C8×D59—24no—
207C23×C59—21no—
208C22×D59C4×D411yescharacteristic
209Q8×C59—21no—
210Q8×C59—21no—
211D4×C59—21no—
212D4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C22×D411yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C22⋊D59—42no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—42no—
227C2×Dic59—42no—
228C2×Dic59C4×D411yescharacteristic
229C2×Dic59C4×D411yescharacteristic
230C2×Dic59C4×D411yescharacteristic
231C4×D59—21no—
232C4×D59—21no—
233C4×D59—21no—
234C4×D59—21no—
235C4×D59—22no—
236Dic118—21no—
237Dic118—21no—
238Dic118—21no—
239Dic118—42no—
240Dic118#13112yes—
241C8×C59—22no—
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—21no—
245C8×C59—21no—
246C8×C59—41no—
247C8×C59—41no—
248C59⋊C8—22no—
249C59⋊C8—21no—
250C59⋊C8—21no—
251C59⋊C8—21no—
252C59⋊C8—21no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255#18019—591nomaximal, Sylow
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59C2×C4×D411yescharacteristic
260C2×D59—21no—
261C2×D59—21no—
262C2×D59—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—22no—
272Dic59—21no—
273Dic59—21no—
274Dic59—41no—
275Dic59—41no—
276Dic59—41no—
277Dic59#44312yes—
278#3107—591no—
279#2991—591no—
280#2986—591no—
281#1590—591no—
282#1590D5911yescharacteristic, complemented
283#1588—591no—
284#1588—591no—
285#1587—591no—
286#1587—591no—
287C8×Q16—591no—
288C8×Q16—591no—
289#1574—591no—
290#1493—591no—
291#1476—591no—
292#1450—591no—
293C2×C59—81no—
294C2×C59#293311yescharacteristic
295C2×C59#294711yescharacteristic
296C2×C59C2×#44311yescharacteristic
297D59—41no—
298#533—591no—
299C4×#41—591no—
300#485—591no—
301#485—591no—
302#475—591no—
303#474—591no—
304#462—591no—
305#461—591no—
306#452—591no—
307#452—591no—
308#446—591no—
309#446—1181no—
310#446C2×D5911yescharacteristic
311C8×Q8—591no—
312C8×Q8—591no—
313C8×Q8—591no—
314C8×Q8—591no—
315C8×Q8—1181no—
316C4×Q16—591no—
317C4×Q16—591no—
318C4×SD16—591no—
319C4×SD16C2×D5911yescharacteristic
320#436—591no—
321#436—1181no—
322#436—1181no—
323#436C2×D5911yescharacteristic
324#433—1181no—
325#433—1181no—
326#431—591no—
327#431—591no—
328#431—591no—
329#431—1181no—
330#423—591no—
331#423—1181no—
332C2×C4×C8—591no—
333#335—591no—
334#335—591no—
335#334—591no—
336#334C2×D5911yescharacteristic
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8—591no—
341Q8⋊C8—591no—
342Q8⋊C8C2×D5911yescharacteristic
343D4⋊C8—591no—
344D4⋊C8C2×D5911yescharacteristic
345#322—591no—
346#322—591no—
347C82—591no—
348C82C2×D5911yescharacteristic
349C59#1801911yescharacteristic, complemented, Sylow
350C2×#41—591no—
351C2×Q16—591no—
352C2×Q16—591no—
353C2×SD16—591no—
354C2×SD16C4×D5911yescharacteristic
355C2×M16—21no—
356C2×M16—1181no—
357C2×M16—1181no—
358C22×C8—1181no—
359C4⋊Q8—591no—
360C4⋊Q8—591no—
361#125—1181no—
362C22⋊Q8—591no—
363C22⋊Q8—591no—
364C22⋊Q8—1181no—
365C4×Q8—591no—
366C4×Q8—591no—
367C4×Q8—591no—
368C4×Q8—1181no—
369C4×Q8C22×D5911yescharacteristic
370C4×D4—591no—
371C4×D4—1181no—
372C4×D4—1181no—
373C4×D4C22×D5911yescharacteristic
374#117—591no—
375#117—1181no—
376#117—1181no—
377C2×C42—591no—
378C2×C42—1181no—
379#107—591no—
380#107—591no—
381#106—591no—
382#106C4×D5911yescharacteristic
383C4⋊C8—21no—
384C4⋊C8—591no—
385C4⋊C8—591no—
386C4⋊C8—591no—
387C4⋊C8—591no—
388C4⋊C8—591no—
389C4⋊C8—591no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8—1181no—
395C4⋊C8—1181no—
396C4⋊C8C22×D5911yescharacteristic
397C4⋊C8C22×D5911yescharacteristic
398#103—591no—
399#103—591no—
400#103—591no—
401#103—591no—
402#103—591no—
403#103C4×D5911yescharacteristic
404#102—591no—
405#102C4×D5911yescharacteristic
406C22⋊C8—21no—
407C22⋊C8—591no—
408C22⋊C8—591no—
409C22⋊C8—591no—
410C22⋊C8—591no—
411C22⋊C8—1181no—
412C22⋊C8—1181no—
413#97—21no—
414#97—1181no—
415C4×C8—591no—
416C4×C8—591no—
417C4×C8—591no—
418C4×C8—1181no—
419C4×C8—1181no—
420C4×C8—1181no—
421C4×C8—1181no—
422C4×C8—1181no—
423C4×C8—1181no—
424C4×C8C22×D5911yescharacteristic
425C4×C8C22×D5911yescharacteristic
426C4×C8C22×D5911yescharacteristic
427#41—1182no—
428#41—2361no—
429C2×Q8—591no—
430C2×Q8—591no—
431C2×Q8—1181no—
432C2×Q8C2×C4×D5911yescharacteristic
433C2×D4—1181no—
434C2×D4C2×C4×D5911yescharacteristic
435C22×C4—21no—
436C22×C4—591no—
437C22×C4—591no—
438C22×C4—591no—
439C22×C4—1181no—
440C22×C4—1181no—
441Q16—1182no—
442Q16—1182no—
443SD16—41no—
444SD16—2361no—
445M16—81no—
446M16—4721no—
447M16—4721no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—591no—
452C2×C8—591no—
453C2×C8—591no—
454C2×C8—591no—
455C2×C8—1184no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8C2×C4×D5911yescharacteristic
460C2×C8C2×C4×D5911yescharacteristic
461C2×C8D4×D5911yescharacteristic
462C2×C8D4×D5911yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—591no—
465C4⋊C4—591no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4C2×C4×D5911yescharacteristic
475C4⋊C4C2×C4×D5911yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C22⋊C4—1181no—
480C22⋊C4—1181no—
481C42—21no—
482C42—591no—
483C42—591no—
484C42—591no—
485C42—1181no—
486C42—1181no—
487C42—1181no—
488C42—1181no—
489C42C23×D5911yescharacteristic
490C23—21no—
491C23—591no—
492Q8—21no—
493Q8—21no—
494Q8—592no—
495Q8—1181no—
496Q8—1181no—
497Q8—1181no—
498Q8—2362no—
499D4—21no—
500D4—21no—
501D4—2362no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—591no—
508C2×C4—591no—
509C2×C4—591no—
510C2×C4—1182no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—2362no—
520C2×C4—2362no—
521C2×C4C22×C4×D5911yescharacteristic
522C2×C4C2×D4×D5911yescharacteristic
523C2×C4#41×D5911yescharacteristic
524C8—22no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—1182no—
532C8—1181no—
533C8—1181no—
534C8—1181no—
535C8—1181no—
536C8—2361no—
537C8—2361no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—1181no—
542C22—1181no—
543C22—1181no—
544C22C4×D4×D5911yescharacteristic
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—592no—
550C4—1181no—
551C4—1181no—
552C4—1182no—
553C4—2361no—
554C4—2361no—
555C4—2361no—
556C4#131×D5911yescharacteristic
557C4#1683816811yescharacteristic
558C4#1683820811yescharacteristic
559C4#1683821611yescharacteristic
560C2—81no—
561C2—2361no—
562C2#2143282511yescharacteristic
563C2#2143293211yescharacteristic
564C2#443×D5911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1590C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745687C2×C4
cosocleC42×C59C24
hypercenter#1590D59
hypocenterC59#18019
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
3. #21433194C2semidirect
4. #21422989C2semidirect
5. #21422975C2semidirect
6. #21422972C2semidirect
7. #21422966C2semidirect
8. #21422965C2semidirect
9. C8×Dic236C2semidirect
10. #21422900C2semidirect
11. #21422883C2semidirect
12. #21422549C2semidirect
14. #21422378C2semidirect
15. C59×#1590C2semidirect
282. #1590D59semidirect
349. C59#18019semidirect

Maximal subgroups

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

subgroupindex
#214341632
#214332242
#214331942
#214229892
#214229752
#214229722
#214229662
#214229652
C8×Dic2362
#214229002
#214228832
#214225492
#214224752
#214223782
C59×#15902
#1801959