Finite Group Databasebeta

← #28556482

Subgroups — #28556482

Order 15104.

1014 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437935C211yesmaximal, characteristic
2#21427036C211yesmaximal, characteristic, complemented
3#21422674C212yesmaximal, complemented
4#21422640C212yesmaximal, complemented
5#21422630C211yesmaximal, characteristic, complemented
6#21422626C211yesmaximal, characteristic, complemented
7#21422617C211yesmaximal, characteristic, complemented
8#21422613C211yesmaximal, characteristic, complemented
9#21422601C212yesmaximal, complemented
10#21422583C212yesmaximal, complemented
11C59×#2975C211yesmaximal, characteristic, complemented
12#16838101—22no—
13#16838098C412yescomplemented
14#16838092C412yescomplemented
15#16838081—22no—
16#16838061C412yescomplemented
17#16838048C412yescomplemented
18#16837999—21no—
19#16837990—21no—
20#16837961—21no—
21#16837936C2212yes—
22#16837913C2212yes—
23C2×#14273706C2211yescharacteristic
24C2×#14273701C2211yescharacteristic
25#16837875—21no—
26#16837295—21no—
27#16837288—21no—
28#16837282—21no—
29#16837270C2211yescharacteristic, complemented
30#16837265C2211yescharacteristic
31#16837262C2211yescharacteristic, complemented
32C22⋊C4×Dic59—21no—
33C4×C22⋊Dic59C2211yescharacteristic
34#16837248C2211yescharacteristic, complemented
35C2×C59⋊#95C2211yescharacteristic, complemented
36#16837140C2212yescomplemented
37#16837140C2212yescomplemented
38#16837140C2212yescomplemented
39#16837140C2212yescomplemented
40#16837137C2212yes—
41#16837137C2212yes—
42#16837137C2212yes—
43#16837137C2212yes—
44C2×C4×D4×C59C2211yescharacteristic
45C59×#436—22no—
46C59×#436—22no—
47C59×#431C2212yescomplemented
48C59×#407C2211yescharacteristic
49C59×#407C2211yescharacteristic
50C59×#407C2212yescomplemented
51C2×#11745685C2×C412yes—
52C22×Dic118C2×C411yescharacteristic, complemented
53C22×D236C2×C411yescharacteristic, complemented
54C22×C22⋊D59—21no—
55C23×Dic59—21no—
56#14273722—22no—
57#14273721—22no—
58#14273720—22no—
59#14273719C2×C412yescomplemented
60#14273717—22no—
61#14273715C2×C412yescomplemented
62#14273713—22no—
63#14273712—22no—
64#14273710—22no—
65#14273710—22no—
66#14273709—22no—
67#14273708—22no—
68#14273706—21no—
69#14273706—21no—
70#14273706C2×C412yes—
71#14273706C2×C412yescomplemented
72#14273706C2312yes—
73#14273705—22no—
74#14273701—21no—
75#14273701—21no—
76#14273701C2×C412yes—
77#14273701C2×C412yescomplemented
78#14273701C2312yes—
79#14273700—22no—
80#14273699—22no—
81#14273699—22no—
82#14273696—22no—
83#14273695—22no—
84C4×Dic118C2×C412yescomplemented
85C4×D236C2×C412yescomplemented
86C4×C22⋊D59—22no—
87C4×C22⋊D59—22no—
88C4×C22⋊D59—22no—
89(C22⋊D59)⋊C4—22no—
90(C22⋊D59)⋊C4—22no—
91#14273686C2×C412yes—
92#14273684—22no—
93#14273683—22no—
94C2×C4⋊Dic59—21no—
95C2×C4⋊Dic59C2×C411yescharacteristic, complemented
96C2×C4⋊Dic59C2×C411yescharacteristic, complemented
97C2×Dic59⋊C4—21no—
98C2×C22⋊Dic59—21no—
99C2×C22⋊Dic59—21no—
100C2×C22⋊Dic59—21no—
101C2×C22⋊Dic59—21no—
102C2×(C2×D59)⋊C4—21no—
103C22⋊C4×D59—22no—
104C2×C4×Dic59—21no—
105C2×C4×Dic59—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95C2311yescharacteristic
112C59⋊#95C2311yescharacteristic
113C59⋊#95C2311yescharacteristic
114C59⋊#95C2311yescharacteristic
115C22×D4×C59D411yescharacteristic
116C23×C4×C59C2311yescharacteristic
117C23×C4×C59C2311yescharacteristic
118C2×M16×C59—21no—
119C2×M16×C59—21no—
120C2×M16×C59—22no—
121C4×D4×C59C2×C412yes—
122C4×D4×C59C2×C412yes—
123C4×D4×C59D412yes—
124C4×D4×C59D412yes—
125C2×C4⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C22⋊C4×C59D411yescharacteristic
128C2×C42×C59C2311yescharacteristic
129C4⋊C8×C59—22no—
130C22⋊C8×C59—21no—
131C22⋊C8×C59—21no—
132C22⋊C8×C59—22no—
133C22⋊C8×C59C2312yes—
134C22⋊C8×C59C2312yes—
135C4×C8×C59—22no—
136C24×C59C2×D411yescharacteristic
137C24×C59C2×D411yescharacteristic
138C23×D59—21no—
139C2×D4×C59—41no—
140C2×D4×C59—41no—
141C2×D4×C59C22⋊C412yes—
142C2×D4×C59C22⋊C412yes—
143C22×C4×C59—81no—
144C22×C4×C59—81no—
145C22×C4×C59C22×C411yescharacteristic
146C22×C4×C59C22×C411yescharacteristic
147C22×C4×C59C22×C412yes—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D412yes—
155C22×C4×C59C2411yescharacteristic
156C2×C22⋊D59—22no—
157C2×C22⋊D59—22no—
158C2×C22⋊D59—22no—
159C2×C22⋊D59—22no—
160C2×C22⋊D59—42no—
161C2×C22⋊D59—42no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C22×Dic59—21no—
167C22×Dic59—21no—
168C22×Dic59—21no—
169C22×Dic59—42no—
170C22×Dic59—42no—
171C22×Dic59—81no—
172#11745685—82no—
173C2×D236—21no—
174C2×D236—81no—
175C2×D236C22×C412yes—
176C2×C4×D59—22no—
177C2×C4×D59—22no—
178C2×Dic118—21no—
179C2×Dic118—81no—
180C2×Dic118C22×C412yes—
181M16×C59—81no—
182M16×C59—81no—
183M16×C59—82no—
184C2×C8×C59—22no—
185C2×C8×C59—22no—
186C4⋊C4×C59C22⋊C412yes—
187C4⋊C4×C59C22⋊C412yes—
188C22⋊C4×C59C22⋊C412yes—
189C22⋊C4×C59C22⋊C412yes—
190C22⋊C4×C59C22⋊C412yes—
191C22⋊C4×C59C22⋊C412yes—
192C42×C59C22×C412yes—
193C42×C59C2×D412yes—
194C22⋊Dic59—22no—
195C22⋊Dic59—22no—
196C22⋊Dic59—22no—
197C22⋊Dic59—22no—
198C22⋊Dic59—42no—
199C22⋊Dic59—42no—
200C22⋊Dic59—81no—
201(C2×D59)⋊C4—22no—
202(C2×D59)⋊C4—22no—
203(C2×D59)⋊C4—22no—
204(C2×D59)⋊C4—22no—
205(C2×D59)⋊C4—22no—
206(C2×D59)⋊C4—22no—
207C4⋊Dic59—21no—
208C4⋊Dic59—21no—
209C4⋊Dic59—81no—
210C4⋊Dic59C22×C412yes—
211C4⋊Dic59C22×C412yes—
212Dic59⋊C4—22no—
213Dic59⋊C4—22no—
214Dic59⋊C4—22no—
215Dic59⋊C4—22no—
216Dic59⋊C4—22no—
217Dic59⋊C4—22no—
218C4×Dic59—22no—
219C4×Dic59—22no—
220C4×Dic59—42no—
221C4×Dic59—81no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59C2×C22⋊C411yescharacteristic
231C23×C59C2×C22⋊C412yes—
232C23×C59C2×C22⋊C411yescharacteristic
233C23×C59C22×D411yescharacteristic
234C22×D59—22no—
235C22×D59—42no—
236C22×D59—81no—
237C22×D59—81no—
238D4×C59—42no—
239D4×C59—42no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—42no—
251C2×C4×C59—41no—
252C2×C4×C59C2×C22⋊C412yes—
253C2×C4×C59C2×C22⋊C412yes—
254C2×C4×C59C2×C22⋊C411yescharacteristic
255C2×C4×C59C2×C22⋊C412yes—
256C2×C4×C59C2×C22⋊C411yescharacteristic
257C2×C4×C59C23×C411yescharacteristic
258C2×C4×C59C22×D411yescharacteristic
259C22⋊D59—44no—
260C22⋊D59—42no—
261C22⋊D59—42no—
262C22⋊D59—82no—
263C22⋊D59—82no—
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—22no—
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—42no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—81no—
281C2×Dic59—81no—
282C2×Dic59—84no—
283D236—42no—
284D236—81no—
285C4×D59—82no—
286C4×D59—82no—
287Dic118—42no—
288Dic118—81no—
289C8×C59—42no—
290C8×C59—42no—
291#17623—591nomaximal, Sylow
292C22×C59—21no—
293C22×C59—21no—
294C22×C59—21no—
295C22×C59—22no—
296C22×C59—21no—
297C22×C59—21no—
298C22×C59—21no—
299C22×C59—22no—
300C22×C59—21no—
301C22×C59—22no—
302C22×C59—21no—
303C22×C59—42no—
304C22×C59—42no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59C22×C22⋊C411yescharacteristic
308C22×C59#57511yescharacteristic
309C22×C59#57511yescharacteristic
310C2×D59—42no—
311C2×D59—42no—
312C2×D59—42no—
313C2×D59—162no—
314C4×C59—21no—
315C4×C59—21no—
316C4×C59—21no—
317C4×C59—22no—
318C4×C59—21no—
319C4×C59—22no—
320C4×C59—22no—
321Dic59—42no—
322Dic59—42no—
323Dic59—42no—
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328#3505—591no—
329#2975D5911yescharacteristic, complemented
330#2353—591no—
331#1521—592no—
332#1513—592no—
333#1511—591no—
334#1510—591no—
335#1508—591no—
336#1507—591no—
337#1504—592no—
338#1500—592no—
339C2×C59—21no—
340C2×C59—21no—
341C2×C59—41no—
342C2×C59—42no—
343C2×C59—41no—
344C2×C59#288311yescharacteristic
345C2×C59#289611yescharacteristic
346C2×C59#290111yescharacteristic
347D59—82no—
348#553—592no—
349#553—1182no—
350#552—592no—
351#551—1182no—
352#548—592no—
353#546—592no—
354#539—1181no—
355#538—1181no—
356#534—1181no—
357#529—592no—
358#525—592no—
359C2×C22⋊Q8—591no—
360C2×#121—591no—
361#518—1181no—
362C2×C4×D4C2×D5911yescharacteristic
363#436—22no—
364#436—22no—
365#431C2×D5912yescomplemented
366#407C2×D5911yescharacteristic
367#407C2×D5911yescharacteristic
368#407C2×D5912yescomplemented
369#388—1181no—
370#387—1181no—
371#386—1181no—
372#383—591no—
373#381—591no—
374#380—591no—
375C4×C22⋊C4—591no—
376C4×C22⋊C4—1181no—
377#376—591no—
378C2×#95—591no—
379#328—592no—
380#328—592no—
381#328—592no—
382#328—592no—
383#327—592no—
384#327—592no—
385#327—592no—
386#327—592no—
387C59#1762311yescharacteristic, complemented, Sylow
388C2×#41—592no—
389C22×Q8—591no—
390C22×D4—591no—
391C22×D4—1181no—
392C22×D4D23611yescharacteristic
393C23×C4—1181no—
394C23×C4C22×D5911yescharacteristic
395C23×C4C22×D5911yescharacteristic
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—22no—
399#126—1182no—
400#126—1182no—
401#126—1182no—
402#125—592no—
403#125—1182no—
404#124—592no—
405#124—1182no—
406#123—1182no—
407#123—1182no—
408#123—1182no—
409#123—1182no—
410#123—1182no—
411C22⋊Q8—592no—
412C22⋊Q8—592no—
413C22⋊Q8—592no—
414C22⋊Q8—1181no—
415C22⋊Q8—1181no—
416C22⋊Q8—1182no—
417#121—592no—
418#121—592no—
419#121—592no—
420#121—1182no—
421#121—1182no—
422#121—1181no—
423#121—1181no—
424#121—1182no—
425#120—1182no—
426#120—1182no—
427C4×Q8—592no—
428C4×D4—592no—
429C4×D4—1182no—
430C4×D4—1182no—
431C4×D4—1182no—
432C4×D4—1182no—
433C4×D4—1182no—
434C4×D4C4×D5912yes—
435C4×D4C4×D5912yes—
436C4×D4C22⋊D5912yes—
437C4×D4C22⋊D5912yes—
438#117—592no—
439#117—1182no—
440#117—1182no—
441C2×C4⋊C4—591no—
442C2×C4⋊C4—591no—
443C2×C4⋊C4—1181no—
444C2×C4⋊C4—1181no—
445C2×C4⋊C4D23611yescharacteristic
446C2×C22⋊C4—1181no—
447C2×C22⋊C4—1181no—
448C2×C22⋊C4—1182no—
449C2×C22⋊C4—1181no—
450C2×C22⋊C4—1181no—
451C2×C22⋊C4—1181no—
452C2×C22⋊C4D23611yescharacteristic
453C2×C22⋊C4D23611yescharacteristic
454C2×C42—1181no—
455C2×C42—1181no—
456C2×C42C22×D5911yescharacteristic
457C4⋊C8—22no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—22no—
461C22⋊C8C22×D5912yes—
462C22⋊C8C22×D5912yes—
463C4×C8—22no—
464#95—591no—
465#95—591no—
466#95—591no—
467#95—591no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473C24—1181no—
474C24C2×D23611yescharacteristic
475C24C2×D23611yescharacteristic
476#41—4722no—
477C2×Q8—592no—
478C2×Q8—1181no—
479C2×Q8—4721no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—592no—
483C2×D4—1182no—
484C2×D4—1182no—
485C2×D4—1182no—
486C2×D4—1182no—
487C2×D4—1181no—
488C2×D4—4721no—
489C2×D4—2362no—
490C2×D4—2362no—
491C2×D4(C2×D59)⋊C412yes—
492C2×D4(C2×D59)⋊C412yes—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1182no—
500C22×C4—1182no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—4721no—
505C22×C4—2362no—
506C22×C4—2362no—
507C22×C4C2×C4×D5911yescharacteristic
508C22×C4C2×C4×D5911yescharacteristic
509C22×C4C2×C4×D5912yes—
510C22×C4C2×D23611yescharacteristic
511C22×C4C2×D23611yescharacteristic
512C22×C4C2×D23611yescharacteristic
513C22×C4C2×D23611yescharacteristic
514C22×C4C2×C22⋊D5911yescharacteristic
515C22×C4C2×C22⋊D5911yescharacteristic
516C22×C4C2×C22⋊D5912yes—
517C22×C4C23×D5911yescharacteristic
518M16—81no—
519M16—81no—
520M16—82no—
521C2×C8—22no—
522C2×C8—22no—
523C4⋊C4—592no—
524C4⋊C4—592no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1182no—
528C4⋊C4—1182no—
529C4⋊C4—1182no—
530C4⋊C4—1182no—
531C4⋊C4—1182no—
532C4⋊C4—1182no—
533C4⋊C4—4721no—
534C4⋊C4(C2×D59)⋊C412yes—
535C4⋊C4(C2×D59)⋊C412yes—
536C22⋊C4—1182no—
537C22⋊C4—1182no—
538C22⋊C4—1182no—
539C22⋊C4—1182no—
540C22⋊C4—1182no—
541C22⋊C4—1182no—
542C22⋊C4—1182no—
543C22⋊C4—1182no—
544C22⋊C4—1182no—
545C22⋊C4—1182no—
546C22⋊C4—4721no—
547C22⋊C4—2362no—
548C22⋊C4—2362no—
549C22⋊C4(C2×D59)⋊C412yes—
550C22⋊C4(C2×D59)⋊C412yes—
551C22⋊C4(C2×D59)⋊C412yes—
552C22⋊C4(C2×D59)⋊C412yes—
553C42—1182no—
554C42—1182no—
555C42—4721no—
556C42—2362no—
557C42C2×C4×D5912yes—
558C42C2×C22⋊D5912yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—1182no—
568C23—4721no—
569C23—4721no—
570C23—2362no—
571C23C2×(C2×D59)⋊C411yescharacteristic
572C23C2×(C2×D59)⋊C412yes—
573C23C2×(C2×D59)⋊C411yescharacteristic
574C23C22×D23611yescharacteristic
575Q8—4721no—
576Q8—2362no—
577D4—42no—
578D4—42no—
579D4—4721no—
580D4—4722no—
581D4—4722no—
582D4—2362no—
583D4—2364no—
584D4—2362no—
585D4—2362no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—1182no—
599C2×C4—1182no—
600C2×C4—1182no—
601C2×C4—1182no—
602C2×C4—4722no—
603C2×C4—4722no—
604C2×C4—4724no—
605C2×C4—4721no—
606C2×C4—4721no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2362no—
618C2×C4—2361no—
619C2×C4C2×(C2×D59)⋊C412yes—
620C2×C4C2×(C2×D59)⋊C412yes—
621C2×C4C2×(C2×D59)⋊C412yes—
622C2×C4C2×(C2×D59)⋊C411yescharacteristic
623C2×C4C2×(C2×D59)⋊C411yescharacteristic
624C2×C4C22×C4×D5911yescharacteristic
625C2×C4C22×C22⋊D5911yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—22no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—9442no—
644C22—2362no—
645C22—2362no—
646C22—2362no—
647C22C22×(C2×D59)⋊C411yescharacteristic
648C22#1683820211yescharacteristic
649C22#1683820211yescharacteristic
650C4—22no—
651C4—22no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—4721no—
658C4—4721no—
659C4—4721no—
660C4—2362no—
661C4—2362no—
662C4—2362no—
663C4—2362no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—41no—
668C2—41no—
669C2—4722no—
670C2#2143255611yescharacteristic
671C2#2143261611yescharacteristic
672C2#2143264111yescharacteristic
subgroupG/H
centerC22C22×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2975C2
socleC22×C59C22×C22⋊C4
prime radical#21437935C2
cosocleC22×C4×C59C24
hypercenter#2975D59
hypocenterC59#17623
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21427036C2semidirect
3. #21422674C2semidirect
4. #21422640C2semidirect
5. #21422630C2semidirect
6. #21422626C2semidirect
7. #21422617C2semidirect
8. #21422613C2semidirect
9. #21422601C2semidirect
10. #21422583C2semidirect
11. C59×#2975C2semidirect
13. #16838098C4semidirect
14. #16838092C4semidirect
16. #16838061C4semidirect
17. #16838048C4semidirect
29. #16837270C22semidirect
31. #16837262C22semidirect
34. #16837248C22semidirect
35. C2×C59⋊#95C22semidirect
36. #16837140C22semidirect
37. #16837140C22semidirect
38. #16837140C22semidirect
39. #16837140C22semidirect
47. C59×#431C22semidirect
50. C59×#407C22semidirect
52. C22×Dic118C2×C4semidirect
53. C22×D236C2×C4semidirect
59. #14273719C2×C4semidirect
61. #14273715C2×C4semidirect
71. #14273706C2×C4semidirect
77. #14273701C2×C4semidirect
84. C4×Dic118C2×C4semidirect
85. C4×D236C2×C4semidirect
95. C2×C4⋊Dic59C2×C4semidirect
96. C2×C4⋊Dic59C2×C4semidirect
329. #2975D59semidirect
365. #431C2×D59semidirect
368. #407C2×D59semidirect
387. C59#17623semidirect

Maximal subgroups

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

subgroupindex
#214379352
#214270362
#214226742
#214226402
#214226302
#214226262
#214226172
#214226132
#214226012
#214225832
C59×#29752
#1762359