Finite Group Databasebeta

← #28559112

Subgroups — #28559112

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434167C211yesmaximal, characteristic
2#21433213C211yesmaximal, characteristic
3#21433193C211yesmaximal, characteristic, complemented
4#21423002C211yesmaximal, characteristic, complemented
5#21422995C211yesmaximal, characteristic, complemented
6#21422990C211yesmaximal, characteristic, complemented
7#21422989C211yesmaximal, characteristic, complemented
8#21422977C211yesmaximal, characteristic, complemented
9#21422970C211yesmaximal, characteristic, complemented
10#21422949C211yesmaximal, characteristic, complemented
11#21422942C211yesmaximal, characteristic, complemented
12#21422882C211yesmaximal, characteristic, complemented
13#21422500C211yesmaximal, characteristic, complemented
14#21422396C211yesmaximal, characteristic, complemented
15C59×#1488C211yesmaximal, characteristic
16#16837959C2211yescharacteristic
17#16837743C411yescharacteristic
18#16837743C411yescharacteristic
19#16837687C411yescharacteristic
20#16837682C411yescharacteristic
21#16837612C411yescharacteristic
22#16837607C411yescharacteristic
23#16837564C411yescharacteristic
24#16837564C411yescharacteristic
25#16837533C2211yescharacteristic
26#16837533C2211yescharacteristic
27#16837530—21no—
28#16837530—21no—
29#16837529C2211yescharacteristic
30#16837529C2211yescharacteristic
31#16837526C2211yescharacteristic
32#16837526C2211yescharacteristic
33C4×#11745672C2211yescharacteristic
34C4×#11745672C2211yescharacteristic
35C4×#11745670C2211yescharacteristic
36C4×#11745670C2211yescharacteristic
37#16837480C2211yescharacteristic
38#16837480C2211yescharacteristic
39#16837474—21no—
40#16837474—21no—
41#16837460—21no—
42#16837460—21no—
43#16837459—21no—
44#16837455C2211yescharacteristic
45#16837455C2211yescharacteristic
46#16837426—21no—
47C2×C4⋊(C59⋊C8)C2211yescharacteristic
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
60C4×#41×C59C2211yescharacteristic
61C59×#432C2211yescharacteristic
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×#11745664—21no—
73C22×C59⋊C8—21no—
74#14273727C2×C411yescharacteristic
75#14273727C2×C411yescharacteristic
76#14273719—21no—
77#14273706—21no—
78#14273706C2×C411yescharacteristic
79#14273706C2×C411yescharacteristic
80C4×Dic118—21no—
81C4×Dic118C2311yescharacteristic
82C4×Dic118C2311yescharacteristic
83#14273686—21no—
84#14273660C2×C411yescharacteristic
85#14273660C2×C411yescharacteristic
86#14273658C2×C411yescharacteristic
87#14273658C2×C411yescharacteristic
88C4⋊(C59⋊C8)—21no—
89C4⋊(C59⋊C8)—21no—
90C4⋊(C59⋊C8)—21no—
91C4⋊(C59⋊C8)C2×C411yescharacteristic
92C4⋊(C59⋊C8)C2×C411yescharacteristic
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)C2×C411yescharacteristic
112C22⋊(C59⋊C8)C2×C411yescharacteristic
113C22⋊(C59⋊C8)D411yescharacteristic
114C22⋊(C59⋊C8)D411yescharacteristic
115#14273638—21no—
116#14273638—21no—
117#14273637—21no—
118#14273637—21no—
119C4×C59⋊C8C2311yescharacteristic
120C4×C59⋊C8C2311yescharacteristic
121C4×C59⋊C8C2311yescharacteristic
122C4×C59⋊C8C2311yescharacteristic
123C8×Dic59—21no—
124C8×Dic59—21no—
125C2×#41×C59C2×C411yescharacteristic
126C4×Q8×C59C2×C411yescharacteristic
127C4×Q8×C59C2311yescharacteristic
128C4×D4×C59—21no—
129C4×D4×C59—21no—
130C4×D4×C59C2×C411yescharacteristic
131C4×D4×C59C2311yescharacteristic
132#117×C59—21no—
133#117×C59C2×C411yescharacteristic
134C2×C42×C59—21no—
135C2×C42×C59C2311yescharacteristic
136C4⋊C8×C59—21no—
137C4⋊C8×C59C2311yescharacteristic
138C4⋊C8×C59C2311yescharacteristic
139C22⋊C8×C59—21no—
140C22⋊C8×C59—21no—
141#97×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144#41×C59—41no—
145#41×C59—41no—
146C2×Q8×C59C22×C411yescharacteristic
147C2×D4×C59—21no—
148C2×D4×C59C22×C411yescharacteristic
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59C22×C411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59#4111yescharacteristic
155C2×Dic118—21no—
156C2×Dic118C22×C411yescharacteristic
157C2×Dic118C22×C411yescharacteristic
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59C22×C411yescharacteristic
165C4⋊C4×C59C22×C411yescharacteristic
166C22⋊C4×C59—21no—
167C22⋊C4×C59—21no—
168C22⋊C4×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59C22×C411yescharacteristic
173C42×C59C2×D411yescharacteristic
174C42×C59#4111yescharacteristic
175C42×C59C2411yescharacteristic
176C22⋊Dic59—21no—
177C22⋊Dic59—21no—
178#11745672—41no—
179#11745672—41no—
180#11745670—41no—
181#11745670—41no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C411yescharacteristic
186C4⋊Dic59C22×C411yescharacteristic
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193#11745664—81no—
194#11745664—81no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
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—42no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—81no—
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C4×D411yescharacteristic
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235Dic118—21no—
236Dic118—21no—
237Dic118—21no—
238Dic118—21no—
239Dic118—81no—
240C8×C59—41no—
241C8×C59—41no—
242C8×C59—41no—
243C8×C59—41no—
244C59⋊C8—21no—
245C59⋊C8—21no—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—21no—
250C59⋊C8—21no—
251C59⋊C8—21no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#17949—591nomaximal, Sylow
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59C2×C4×D411yescharacteristic
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—21no—
265C4×C59—21no—
266C4×C59—41no—
267C4×C59—41no—
268C4×C59—41no—
269C4×C59#56811yescharacteristic
270C4×C59#56811yescharacteristic
271C4×C59#57711yescharacteristic
272C4×C59#57811yescharacteristic
273Dic59—41no—
274Dic59—41no—
275Dic59—41no—
276Dic59—41no—
277#3107—591no—
278#2989—591no—
279#2985—591no—
280#1593—591no—
281#1592—591no—
282#1591—591no—
283#1590—591no—
284#1588—591no—
285#1587—591no—
286#1584—591no—
287#1583—591no—
288#1573—591no—
289#1488D5911yescharacteristic
290#1481—591no—
291#1454—591no—
292C2×C59—41no—
293C2×C59—81no—
294C2×C59#293011yescharacteristic
295C2×C59#294711yescharacteristic
296C2×C59#295711yescharacteristic
297#533—591no—
298C4×#41C2×D5911yescharacteristic
299#485—591no—
300#485—591no—
301#475—591no—
302#474—591no—
303#462—591no—
304#461—591no—
305#452—591no—
306#452—591no—
307#446—591no—
308#446—591no—
309#446—591no—
310#446—591no—
311#446—1181no—
312#446—1181no—
313C8×Q8—591no—
314C8×Q8—591no—
315C4×Q16—591no—
316C4×Q16—591no—
317C4×SD16—591no—
318C4×SD16—591no—
319#436—591no—
320#436—591no—
321#435—1181no—
322#435—1181no—
323#433—1181no—
324#433—1181no—
325#432—1181no—
326#432C2×D5911yescharacteristic
327#431—591no—
328#431—591no—
329#431C2×D5911yescharacteristic
330#423—1181no—
331C2×C4⋊C8—591no—
332#335—591no—
333#335—591no—
334#334—591no—
335#334—591no—
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8C2×D5911yescharacteristic
341Q8⋊C8C2×D5911yescharacteristic
342D4⋊C8C2×D5911yescharacteristic
343D4⋊C8C2×D5911yescharacteristic
344#322—591no—
345#322—591no—
346#322—591no—
347C82—591no—
348C59#1794911yescharacteristic, complemented, Sylow
349C2×#41C4×D5911yescharacteristic
350C2×Q16—591no—
351C2×Q16—591no—
352C2×SD16—591no—
353C2×SD16—591no—
354C2×M16—1181no—
355C2×M16—1181no—
356C22×C8—1181no—
357C4⋊Q8—591no—
358C4⋊Q8—591no—
359#125—1181no—
360C22⋊Q8—591no—
361C22⋊Q8—591no—
362C22⋊Q8—1181no—
363C4×Q8—591no—
364C4×Q8—591no—
365C4×Q8—1181no—
366C4×Q8C4×D5911yescharacteristic
367C4×Q8C22×D5911yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D5911yescharacteristic
371C4×D4C22×D5911yescharacteristic
372#117—21no—
373#117—1181no—
374#117C4×D5911yescharacteristic
375C2×C42—21no—
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—591no—
387C4⋊C8—591no—
388C4⋊C8—1181no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8—1181no—
395C4⋊C8C22×D5911yescharacteristic
396C4⋊C8C22×D5911yescharacteristic
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#103—591no—
402#103—591no—
403#102—591no—
404#102—591no—
405C22⋊C8—21no—
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—
415#97—1181no—
416#97—1181no—
417#97—1181no—
418C4×C8—591no—
419C4×C8—591no—
420C4×C8—591no—
421C4×C8—591no—
422C4×C8—1181no—
423C4×C8—1181no—
424C4×C8C22×D5911yescharacteristic
425C4×C8C22×D5911yescharacteristic
426#41—41no—
427#41—41no—
428C2×Q8—591no—
429C2×Q8—591no—
430C2×Q8—1181no—
431C2×Q8C2×C4×D5911yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D5911yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4#1174568711yescharacteristic
440Q16—2361no—
441Q16—2361no—
442SD16—2361no—
443SD16—2361no—
444M16—4721no—
445M16—4721no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—591no—
451C2×C8—591no—
452C2×C8—591no—
453C2×C8—591no—
454C2×C8—591no—
455C2×C8—591no—
456C2×C8—591no—
457C2×C8—591no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—4721no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—591no—
464C4⋊C4—591no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4C2×C4×D5911yescharacteristic
473C4⋊C4C2×C4×D5911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1181no—
483C42—1181no—
484C42C2×C4×D5911yescharacteristic
485C42D4×D5911yescharacteristic
486C42#1174568711yescharacteristic
487C42C23×D5911yescharacteristic
488C23—21no—
489C23(C22⋊D59)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—1181no—
493Q8—1181no—
494Q8—1181no—
495Q8—1181no—
496Q8—4721no—
497D4—21no—
498D4—21no—
499D4—42no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4(C22⋊D59)⋊C411yescharacteristic
517C2×C4(C22⋊D59)⋊C411yescharacteristic
518C2×C4(C22⋊D59)⋊C411yescharacteristic
519C2×C4C22×C4×D5911yescharacteristic
520C2×C4C2×D4×D5911yescharacteristic
521C2×C4C2×#1174568711yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—1181no—
527C8—1181no—
528C8—1181no—
529C8—1181no—
530C8—1181no—
531C8—1181no—
532C8—1181no—
533C8—1181no—
534C8—2361no—
535C8—2361no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×(C22⋊D59)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—2361no—
551C4—2361no—
552C4—2361no—
553C4—2361no—
554C4#1683816811yescharacteristic
555C4#1683816811yescharacteristic
556C4#1683820911yescharacteristic
557C4#1683821611yescharacteristic
558C2—41no—
559C2—81no—
560C2#2143279511yescharacteristic
561C2#2143294211yescharacteristic
562C2#2143300911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1488C2
socleC22×C59C2×C4×D4
prime radicalC2×#41×C59C2×C4
cosocleC42×C59C24
hypercenter#1488D59
hypocenterC59#17949
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
3. #21433193C2semidirect
4. #21423002C2semidirect
5. #21422995C2semidirect
6. #21422990C2semidirect
7. #21422989C2semidirect
8. #21422977C2semidirect
9. #21422970C2semidirect
10. #21422949C2semidirect
11. #21422942C2semidirect
12. #21422882C2semidirect
13. #21422500C2semidirect
14. #21422396C2semidirect
348. C59#17949semidirect

Maximal subgroups

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

subgroupindex
#214341672
#214332132
#214331932
#214230022
#214229952
#214229902
#214229892
#214229772
#214229702
#214229492
#214229422
#214228822
#214225002
#214223962
C59×#14882
#1794959