Finite Group Databasebeta

← #28556498

Subgroups — #28556498

Order 15104.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437935C211yesmaximal, characteristic
2#21433125C211yesmaximal, characteristic, complemented
3#21422650C212yesmaximal, complemented
4#21422632C212yesmaximal, complemented
5#21422628C211yesmaximal, characteristic, complemented
6#21422624C211yesmaximal, characteristic, complemented
7#21422619C212yesmaximal, complemented
8#21422615C211yesmaximal, characteristic, complemented
9#21422611C211yesmaximal, characteristic, complemented
10#21422585C212yesmaximal, complemented
11C59×#2357C211yesmaximal, 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#16837474—22no—
27#16837474—22no—
28#16837459C2212yescomplemented
29#16837366C2211yescharacteristic
30#16837366C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2212yescomplemented
32#16837138C2212yescomplemented
33#16837138C2212yescomplemented
34#16837138C2212yescomplemented
35#16837138C2212yescomplemented
36#16837135C2212yes—
37#16837135C2212yes—
38#16837135C2212yes—
39#16837135C2212yes—
40C2×C4×D4×C59C2211yescharacteristic
41C59×#390—21no—
42C59×#388—21no—
43C59×#388—21no—
44C59×#382C2211yescharacteristic, complemented
45C59×#381C2211yescharacteristic
46C59×#381C2211yescharacteristic, complemented
47C4×C4⋊C4×C59—21no—
48C4×C4⋊C4×C59C2211yescharacteristic, complemented
49C4×C22⋊C4×C59C2211yescharacteristic
50C4×C22⋊C4×C59C2211yescharacteristic, complemented
51C2×#11745685C2×C412yes—
52C22×Dic118C2×C411yescharacteristic
53C22×D236C2×C411yescharacteristic
54C22×C22⋊D59—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C22×C59⋊C8—22no—
58#14273722—22no—
59#14273721—22no—
60#14273720—22no—
61#14273719C2×C412yes—
62#14273717—22no—
63#14273715C2×C412yes—
64#14273713—22no—
65#14273712—22no—
66#14273710—22no—
67#14273710—22no—
68#14273709—22no—
69#14273708—22no—
70#14273706—21no—
71#14273706—21no—
72#14273706C2×C412yes—
73#14273706C2×C412yes—
74#14273706C2312yes—
75#14273705—22no—
76#14273701—21no—
77#14273701—21no—
78#14273701C2×C412yes—
79#14273701C2×C412yes—
80#14273701C2312yes—
81#14273700—22no—
82#14273699—22no—
83#14273699—22no—
84#14273696—22no—
85#14273695—22no—
86C4×Dic118C2×C412yes—
87C4×D236C2×C412yes—
88C4×C22⋊D59—22no—
89C4×C22⋊D59—22no—
90C4×C22⋊D59—22no—
91(C22⋊D59)⋊C4—22no—
92(C22⋊D59)⋊C4—22no—
93#14273686C2×C412yes—
94#14273684—22no—
95#14273683—22no—
96C2×C4⋊Dic59C2×C411yescharacteristic
97C2×C4⋊Dic59C2×C411yescharacteristic
98C2×Dic59⋊C4—21no—
99C2×C22⋊Dic59—21no—
100C2×(C2×D59)⋊C4—21no—
101C22⋊C4×D59—22no—
102C4⋊(C59⋊C8)—22no—
103C22⋊(C59⋊C8)—22no—
104C22⋊(C59⋊C8)—21no—
105C22⋊(C59⋊C8)—21no—
106C22⋊(C59⋊C8)C2312yes—
107C22⋊(C59⋊C8)C2312yes—
108#14273638—22no—
109C22×D4×C59D411yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C23×C4×C59C2311yescharacteristic
112C4×D4×C59C2×C412yes—
113C4×D4×C59C2×C412yes—
114C4×D4×C59D412yes—
115C4×D4×C59D412yes—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C22⋊C4×C59D411yescharacteristic
125C2×C42×C59—21no—
126C2×C42×C59—21no—
127C2×C42×C59—21no—
128C2×C42×C59C2311yescharacteristic
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59C2311yescharacteristic
133#95×C59C2311yescharacteristic
134#95×C59C2311yescharacteristic
135#95×C59C2311yescharacteristic
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—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—81no—
148C22×C4×C59—81no—
149C22×C4×C59C22×C411yescharacteristic
150C22×C4×C59C22×C411yescharacteristic
151C22×C4×C59C22×C412yes—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D412yes—
159C22×C4×C59C2411yescharacteristic
160C2×C22⋊D59—22no—
161C2×C22⋊D59—22no—
162C2×C22⋊D59—22no—
163C2×C22⋊D59—22no—
164C2×C22⋊D59—42no—
165C2×C22⋊D59—42no—
166C22×Dic59—21no—
167C22×Dic59—21no—
168C22×Dic59—21no—
169#11745685—82no—
170C2×D236—21no—
171C2×D236—81no—
172C2×D236C22×C412yes—
173C2×C4×D59—22no—
174C2×C4×D59—22no—
175C2×Dic118—21no—
176C2×Dic118—81no—
177C2×Dic118C22×C412yes—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59C22⋊C412yes—
181C4⋊C4×C59C22⋊C412yes—
182C22⋊C4×C59—42no—
183C22⋊C4×C59—42no—
184C22⋊C4×C59—81no—
185C22⋊C4×C59—81no—
186C22⋊C4×C59C22⋊C412yes—
187C22⋊C4×C59C22⋊C412yes—
188C22⋊C4×C59C22⋊C412yes—
189C22⋊C4×C59C22⋊C412yes—
190C42×C59—42no—
191C42×C59—81no—
192C42×C59—81no—
193C42×C59C22×C412yes—
194C42×C59C2×D412yes—
195C22⋊Dic59—22no—
196C22⋊Dic59—22no—
197C22⋊Dic59—22no—
198C22⋊Dic59—22no—
199(C2×D59)⋊C4—22no—
200(C2×D59)⋊C4—22no—
201(C2×D59)⋊C4—22no—
202(C2×D59)⋊C4—22no—
203(C2×D59)⋊C4—22no—
204(C2×D59)⋊C4—22no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59C22×C412yes—
208C4⋊Dic59C22×C412yes—
209Dic59⋊C4—22no—
210Dic59⋊C4—22no—
211Dic59⋊C4—22no—
212Dic59⋊C4—22no—
213Dic59⋊C4—22no—
214Dic59⋊C4—22no—
215C4×Dic59—22no—
216C4×Dic59—22no—
217#11745664—81no—
218#11745664—81no—
219C2×C59⋊C8—22no—
220C2×C59⋊C8—22no—
221C2×C59⋊C8—82no—
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—22no—
243C2×C4×C59—42no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—42no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59C2×C22⋊C412yes—
265C2×C4×C59C2×C22⋊C412yes—
266C2×C4×C59C2×C22⋊C411yescharacteristic
267C2×C4×C59C2×C22⋊C412yes—
268C2×C4×C59C2×C22⋊C411yescharacteristic
269C2×C4×C59C23×C411yescharacteristic
270C2×C4×C59C22×D411yescharacteristic
271C22⋊D59—44no—
272C22⋊D59—42no—
273C22⋊D59—42no—
274C22⋊D59—82no—
275C22⋊D59—82no—
276C2×Dic59—22no—
277C2×Dic59—22no—
278C2×Dic59—22no—
279C2×Dic59—42no—
280C2×Dic59—81no—
281C2×Dic59—81no—
282D236—42no—
283D236—81no—
284C4×D59—82no—
285C4×D59—82no—
286Dic118—42no—
287Dic118—81no—
288C59⋊C8—42no—
289C59⋊C8—42no—
290#17625—591nomaximal, Sylow
291C22×C59—21no—
292C22×C59—21no—
293C22×C59—21no—
294C22×C59—22no—
295C22×C59—21no—
296C22×C59—21no—
297C22×C59—21no—
298C22×C59—22no—
299C22×C59—21no—
300C22×C59—22no—
301C22×C59—21no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59C22×C22⋊C411yescharacteristic
307C22×C59#57511yescharacteristic
308C22×C59#57511yescharacteristic
309C2×D59—42no—
310C2×D59—42no—
311C2×D59—42no—
312C2×D59—162no—
313C4×C59—21no—
314C4×C59—21no—
315C4×C59—21no—
316C4×C59—22no—
317C4×C59—21no—
318C4×C59—22no—
319C4×C59—22no—
320C4×C59—42no—
321C4×C59—81no—
322C4×C59—81no—
323C4×C59—81no—
324Dic59—42no—
325Dic59—42no—
326Dic59—42no—
327#3505—591no—
328#2973—591no—
329#2357D5911yescharacteristic, complemented
330#1516—592no—
331#1512—592no—
332#1511—591no—
333#1510—591no—
334#1509—592no—
335#1508—591no—
336#1507—591no—
337#1501—592no—
338C2×C59—21no—
339C2×C59—21no—
340C2×C59—41no—
341C2×C59—41no—
342C2×C59—42no—
343C2×C59#288311yescharacteristic
344C2×C59#289611yescharacteristic
345C2×C59#290111yescharacteristic
346D59—82no—
347#553—592no—
348#553—1182no—
349#552—592no—
350#551—1182no—
351#548—592no—
352#546—592no—
353#539—1181no—
354#538—1181no—
355#534—1181no—
356#529—592no—
357#525—592no—
358C2×C22⋊Q8—591no—
359C2×#121—591no—
360#518—1181no—
361C2×C4×D4C2×D5911yescharacteristic
362#435—1182no—
363#435—1182no—
364#432—592no—
365#407—591no—
366#407—591no—
367C2×C22⋊C8—592no—
368#390—21no—
369#388—21no—
370#388—21no—
371#382C2×D5911yescharacteristic, complemented
372#381C2×D5911yescharacteristic
373#381C2×D5911yescharacteristic, complemented
374C4×C4⋊C4—21no—
375C4×C4⋊C4C2×D5911yescharacteristic, complemented
376C4×C22⋊C4C2×D5911yescharacteristic
377C4×C22⋊C4C2×D5911yescharacteristic, complemented
378#328—592no—
379#328—592no—
380#328—592no—
381#328—592no—
382#327—592no—
383#327—592no—
384#327—592no—
385#327—592no—
386C59#1762511yescharacteristic, complemented, Sylow
387C2×#41—592no—
388C22×Q8—591no—
389C22×D4—591no—
390C22×D4—1181no—
391C22×D4C22⋊D5911yescharacteristic
392C23×C4C22×D5911yescharacteristic
393C23×C4C22×D5911yescharacteristic
394C2×M16—1181no—
395C2×M16—1181no—
396C22×C8—1182no—
397#126—1182no—
398#126—1182no—
399#126—1182no—
400#125—592no—
401#125—1182no—
402#124—592no—
403#124—1182no—
404#123—1182no—
405#123—1182no—
406#123—1182no—
407#123—1182no—
408#123—1182no—
409C22⋊Q8—592no—
410C22⋊Q8—592no—
411C22⋊Q8—592no—
412C22⋊Q8—1181no—
413C22⋊Q8—1181no—
414C22⋊Q8—1182no—
415#121—592no—
416#121—592no—
417#121—592no—
418#121—1182no—
419#121—1182no—
420#121—1181no—
421#121—1181no—
422#121—1182no—
423#120—1182no—
424#120—1182no—
425C4×Q8—592no—
426C4×D4—592no—
427C4×D4—1182no—
428C4×D4—1182no—
429C4×D4—1182no—
430C4×D4—1182no—
431C4×D4—1182no—
432C4×D4C4×D5912yes—
433C4×D4C4×D5912yes—
434C4×D4D23612yes—
435C4×D4D23612yes—
436#117—592no—
437#117—1182no—
438#117—1182no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—591no—
442C2×C4⋊C4—591no—
443C2×C4⋊C4—1181no—
444C2×C4⋊C4C22⋊D5911yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1181no—
450C2×C22⋊C4—1181no—
451C2×C22⋊C4—1182no—
452C2×C22⋊C4C22⋊D5911yescharacteristic
453C2×C22⋊C4C22⋊D5911yescharacteristic
454C2×C42—21no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42C22×D5911yescharacteristic
458C4⋊C8—1182no—
459C22⋊C8—592no—
460C22⋊C8—592no—
461C22⋊C8—1182no—
462C22⋊C8—1181no—
463C22⋊C8—1181no—
464#97—1182no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95C22×D5911yescharacteristic
469#95C22×D5911yescharacteristic
470#95C22×D5911yescharacteristic
471#95C22×D5911yescharacteristic
472C24—1181no—
473C24C2×C22⋊D5911yescharacteristic
474C24C2×C22⋊D5911yescharacteristic
475#41—4722no—
476C2×Q8—592no—
477C2×Q8—1181no—
478C2×Q8—4721no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—592no—
482C2×D4—1182no—
483C2×D4—1182no—
484C2×D4—1182no—
485C2×D4—1182no—
486C2×D4—1181no—
487C2×D4—4721no—
488C2×D4—2362no—
489C2×D4—2362no—
490C2×D4(C2×D59)⋊C412yes—
491C2×D4(C2×D59)⋊C412yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1182no—
502C22×C4—1182no—
503C22×C4C2×C4×D5911yescharacteristic
504C22×C4C2×C4×D5911yescharacteristic
505C22×C4C2×C4×D5912yes—
506C22×C4C2×D23611yescharacteristic
507C22×C4C2×D23611yescharacteristic
508C22×C4C2×D23612yes—
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C23×D5911yescharacteristic
514M16—4721no—
515M16—4721no—
516C2×C8—1182no—
517C2×C8—1182no—
518C2×C8—4722no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—592no—
522C4⋊C4—592no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1182no—
526C4⋊C4—1182no—
527C4⋊C4—1182no—
528C4⋊C4—1182no—
529C4⋊C4—1182no—
530C4⋊C4—1182no—
531C4⋊C4(C2×D59)⋊C412yes—
532C4⋊C4(C2×D59)⋊C412yes—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—81no—
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—1182no—
547C22⋊C4(C2×D59)⋊C412yes—
548C22⋊C4(C2×D59)⋊C412yes—
549C22⋊C4(C2×D59)⋊C412yes—
550C22⋊C4(C2×D59)⋊C412yes—
551C42—42no—
552C42—81no—
553C42—81no—
554C42—1182no—
555C42—1182no—
556C42C2×C4×D5912yes—
557C42C2×D23612yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—1182no—
567C23—4721no—
568C23—4721no—
569C23—2362no—
570C23C2×(C2×D59)⋊C411yescharacteristic
571C23C2×(C2×D59)⋊C412yes—
572C23C2×(C2×D59)⋊C411yescharacteristic
573C23C22×C22⋊D5911yescharacteristic
574Q8—4721no—
575Q8—2362no—
576D4—42no—
577D4—42no—
578D4—4721no—
579D4—4722no—
580D4—4722no—
581D4—2362no—
582D4—2364no—
583D4—2362no—
584D4—2362no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
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—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1182no—
610C2×C4—1182no—
611C2×C4—1182no—
612C2×C4—4722no—
613C2×C4—4722no—
614C2×C4—4721no—
615C2×C4—4721no—
616C2×C4—2362no—
617C2×C4C2×(C2×D59)⋊C412yes—
618C2×C4C2×(C2×D59)⋊C412yes—
619C2×C4C2×(C2×D59)⋊C412yes—
620C2×C4C2×(C2×D59)⋊C411yescharacteristic
621C2×C4C2×(C2×D59)⋊C411yescharacteristic
622C2×C4C22×C4×D5911yescharacteristic
623C2×C4C22×D23611yescharacteristic
624C8—2362no—
625C8—2362no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—9442no—
642C22—2362no—
643C22—2362no—
644C22—2362no—
645C22C22×(C2×D59)⋊C411yescharacteristic
646C22#1683820011yescharacteristic
647C22#1683820011yescharacteristic
648C4—22no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—2362no—
660C4—2362no—
661C4—2362no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—4722no—
668C2#2143255611yescharacteristic
669C2#2143261311yescharacteristic
670C2#2143264411yescharacteristic
subgroupG/H
centerC22C22×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2357C2
socleC22×C59C22×C22⋊C4
prime radical#21437935C2
cosocleC22×C4×C59C24
hypercenter#2357D59
hypocenterC59#17625
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #21433125C2semidirect
3. #21422650C2semidirect
4. #21422632C2semidirect
5. #21422628C2semidirect
6. #21422624C2semidirect
7. #21422619C2semidirect
8. #21422615C2semidirect
9. #21422611C2semidirect
10. #21422585C2semidirect
11. C59×#2357C2semidirect
13. #16838098C4semidirect
14. #16838092C4semidirect
16. #16838061C4semidirect
17. #16838048C4semidirect
28. #16837459C22semidirect
31. C2×C22⋊(C59⋊C8)C22semidirect
32. #16837138C22semidirect
33. #16837138C22semidirect
34. #16837138C22semidirect
35. #16837138C22semidirect
44. C59×#382C22semidirect
46. C59×#381C22semidirect
48. C4×C4⋊C4×C59C22semidirect
50. C4×C22⋊C4×C59C22semidirect
329. #2357D59semidirect
371. #382C2×D59semidirect
373. #381C2×D59semidirect
375. C4×C4⋊C4C2×D59semidirect
377. C4×C22⋊C4C2×D59semidirect
386. C59#17625semidirect

Maximal subgroups

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

subgroupindex
#214379352
#214331252
#214226502
#214226322
#214226282
#214226242
#214226192
#214226152
#214226112
#214225852
C59×#23572
#1762559