Finite Group Databasebeta

← #28555894

Subgroups — #28555894

Order 15104.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21438145C211yesmaximal, characteristic
2#21433131C211yesmaximal, characteristic, complemented
3#21422619C211yesmaximal, characteristic, complemented
4#21422615C212yesmaximal, complemented
5#21422607C211yesmaximal, characteristic, complemented
6#21422598C212yesmaximal, complemented
7#21422594C211yesmaximal, characteristic, complemented
8#21422585C212yesmaximal, complemented
9#21422580C211yesmaximal, characteristic, complemented
10C2×#16837138C212yesmaximal, complemented
11C59×#2275C211yesmaximal, characteristic, complemented
12#16838122C411yescharacteristic, complemented
13#16838107—21no—
14#16838092C412yescomplemented
15#16838078—22no—
16D4×Dic118C411yescharacteristic, complemented
17#16838061C412yescomplemented
18#16838057C411yescharacteristic, complemented
19#16838056C411yescharacteristic, complemented
20#16838046—21no—
21#16838017—21no—
22#16837994—22no—
23#16837959C2211yescharacteristic
24#16837936C2211yescharacteristic
25#16837913C2212yes—
26C2×#14273706C2212yes—
27#16837881—21no—
28#16837480—21no—
29#16837474—22no—
30D4×C59⋊C8—21no—
31#16837459C2211yescharacteristic
32#16837455C2211yescharacteristic
33#16837366C2212yes—
34C2×C22⋊(C59⋊C8)C2212yes—
35#16837138C2212yes—
36#16837138C2212yes—
37#16837138C2212yes—
38#16837138C2212yes—
39#16837138C2211yescharacteristic
40#16837138C2212yes—
41#16837138C2211yescharacteristic
42#16837135C2211yescharacteristic
43#16837135C2212yes—
44#16837135C2211yescharacteristic
45C2×C4×D4×C59—21no—
46C2×C4×D4×C59C2211yescharacteristic
47C4×C4⋊C4×C59—21no—
48C4×C4⋊C4×C59C2211yescharacteristic
49C4×C22⋊C4×C59—22no—
50C4×C22⋊C4×C59C2212yes—
51C59×#376C2211yescharacteristic
52C2×#95×C59C2212yes—
53C2×#11745685C2×C412yescomplemented
54C2×#11745687—21no—
55C22×Dic118C2×C412yescomplemented
56C2×#11745664—22no—
57C22×C59⋊C8—22no—
58#14273727C2×C411yescharacteristic, complemented
59#14273727C2×C411yescharacteristic, complemented
60#14273726—21no—
61#14273721—21no—
62#14273720—22no—
63#14273720—22no—
64#14273719—21no—
65#14273719C2×C411yescharacteristic, complemented
66#14273717—22no—
67#14273715C2×C411yescharacteristic, complemented
68#14273714—21no—
69#14273713—22no—
70#14273711—22no—
71#14273710—22no—
72#14273710—21no—
73#14273709—22no—
74#14273708—22no—
75#14273707—21no—
76#14273706—22no—
77#14273706—22no—
78#14273706—21no—
79#14273706C2×C412yescomplemented
80#14273706C2×C412yescomplemented
81#14273706C2×C412yescomplemented
82#14273706C2311yescharacteristic
83#14273706C2312yes—
84#14273705—22no—
85#14273704—22no—
86#14273701C2×C412yescomplemented
87#14273701C2311yescharacteristic
88#14273699—22no—
89#14273698—21no—
90#14273696—22no—
91C4×Dic118—21no—
92C4×Dic118C2×C411yescharacteristic, complemented
93C4×Dic118C2×C412yescomplemented
94#14273692—21no—
95C4×D236C2×C411yescharacteristic, complemented
96D4×Dic59—21no—
97C4×C22⋊D59—22no—
98C4×C22⋊D59—22no—
99C4×C22⋊D59—21no—
100(C22⋊D59)⋊C4—22no—
101#14273686—21no—
102#14273686C2×C412yescomplemented
103#14273684—21no—
104#14273683—22no—
105C2×C4⋊Dic59C2×C412yescomplemented
106C2×Dic59⋊C4—22no—
107C4⋊C4×D59—21no—
108C2×C22⋊Dic59—22no—
109C4⋊(C59⋊C8)—21no—
110C4⋊(C59⋊C8)—21no—
111C22⋊(C59⋊C8)—22no—
112C22⋊(C59⋊C8)—22no—
113C22⋊(C59⋊C8)C2311yescharacteristic
114C22⋊(C59⋊C8)C2311yescharacteristic
115C22⋊(C59⋊C8)C2312yes—
116#14273638—21no—
117C4×C59⋊C8—21no—
118C22×D4×C59D411yescharacteristic
119C23×C4×C59—22no—
120C23×C4×C59C2312yes—
121C4×D4×C59—28no—
122C4×D4×C59C2×C411yescharacteristic
123C4×D4×C59C2×C411yescharacteristic
124C4×D4×C59C2×C412yes—
125C4×D4×C59D411yescharacteristic
126C4×D4×C59D412yes—
127C4×D4×C59D411yescharacteristic
128C2×C4⋊C4×C59—21no—
129C2×C4⋊C4×C59D411yescharacteristic
130C2×C22⋊C4×C59—22no—
131C2×C22⋊C4×C59D412yes—
132C2×C42×C59—21no—
133C2×C42×C59—21no—
134C2×C42×C59—22no—
135C2×C42×C59C2311yescharacteristic
136#95×C59—22no—
137#95×C59—21no—
138#95×C59C2311yescharacteristic
139#95×C59C2312yes—
140#95×C59C2311yescharacteristic
141C24×C59C2×D412yes—
142C2×D4×C59—24no—
143C2×D4×C59C22⋊C411yescharacteristic
144C2×D4×C59C22⋊C412yes—
145C2×D4×C59C22⋊C411yescharacteristic
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—22no—
149C22×C4×C59—28no—
150C22×C4×C59—82no—
151C22×C4×C59—82no—
152C22×C4×C59—82no—
153C22×C4×C59C22×C412yes—
154C22×C4×C59C22×C412yes—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D412yes—
158C22×C4×C59C2×D412yes—
159C22×C4×C59C2×D412yes—
160C22×C4×C59C2411yescharacteristic
161C2×C22⋊D59—22no—
162C2×C22⋊D59—21no—
163C2×C22⋊D59—22no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166#11745687—44no—
167#11745685—82no—
168C2×D236C22×C411yescharacteristic
169C2×C4×D59—21no—
170C2×C4×D59—22no—
171C2×Dic118—21no—
172C2×Dic118—22no—
173C2×Dic118—82no—
174C2×Dic118C22×C411yescharacteristic
175C2×Dic118C22×C412yes—
176C4⋊C4×C59—24no—
177C4⋊C4×C59C22⋊C411yescharacteristic
178C4⋊C4×C59C22⋊C412yes—
179C4⋊C4×C59C22⋊C411yescharacteristic
180C22⋊C4×C59—28no—
181C22⋊C4×C59C22⋊C412yes—
182C22⋊C4×C59C22⋊C412yes—
183C22⋊C4×C59C22⋊C412yes—
184C22⋊C4×C59C22⋊C412yes—
185C42×C59—24no—
186C42×C59—81no—
187C42×C59—82no—
188C42×C59C22×C411yescharacteristic
189C42×C59C22×C411yescharacteristic
190C42×C59C2×D411yescharacteristic
191C42×C59C2×D411yescharacteristic
192C22⋊Dic59—22no—
193C22⋊Dic59—22no—
194C22⋊Dic59—22no—
195C22⋊Dic59—22no—
196C22⋊Dic59—21no—
197C22⋊Dic59—22no—
198(C2×D59)⋊C4—21no—
199(C2×D59)⋊C4—21no—
200(C2×D59)⋊C4—22no—
201(C2×D59)⋊C4—21no—
202(C2×D59)⋊C4—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—22no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59C22×C411yescharacteristic
208C4⋊Dic59C22×C412yes—
209C4⋊Dic59C22×C411yescharacteristic
210Dic59⋊C4—22no—
211Dic59⋊C4—22no—
212Dic59⋊C4—21no—
213Dic59⋊C4—22no—
214Dic59⋊C4—21no—
215Dic59⋊C4—22no—
216Dic59⋊C4—21no—
217Dic59⋊C4—21no—
218Dic59⋊C4—22no—
219C4×Dic59—22no—
220C4×Dic59—22no—
221C4×Dic59—21no—
222#11745664—82no—
223C2×C59⋊C8—21no—
224C2×C59⋊C8—21no—
225C2×C59⋊C8—22no—
226C2×C59⋊C8—82no—
227C23×C59—22no—
228C23×C59—22no—
229C23×C59—24no—
230C23×C59—42no—
231C23×C59C2×C22⋊C412yes—
232C23×C59C2×C22⋊C412yes—
233C23×C59C22×D411yescharacteristic
234C22×D59—21no—
235D4×C59—24no—
236D4×C59—22no—
237D4×C59#10414yes—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—24no—
244C2×C4×C59—42no—
245C2×C4×C59—42no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—42no—
252C2×C4×C59—41no—
253C2×C4×C59—42no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—88no—
257C2×C4×C59C2×C22⋊C411yescharacteristic
258C2×C4×C59C2×C22⋊C412yes—
259C2×C4×C59C2×C22⋊C411yescharacteristic
260C2×C4×C59C2×C22⋊C411yescharacteristic
261C2×C4×C59C2×C22⋊C412yes—
262C2×C4×C59C2×C22⋊C411yescharacteristic
263C2×C4×C59C23×C411yescharacteristic
264C2×C4×C59C22×D411yescharacteristic
265C22⋊D59—41no—
266C22⋊D59—41no—
267C22⋊D59—44no—
268C22⋊D59—82no—
269C2×Dic59—21no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—21no—
273C2×Dic59—21no—
274C2×Dic59—44no—
275C2×Dic59—82no—
276D236—41no—
277C4×D59—42no—
278C4×D59—82no—
279Dic118—42no—
280Dic118—41no—
281Dic118—42no—
282Dic118—82no—
283C59⋊C8—41no—
284C59⋊C8—42no—
285C59⋊C8—41no—
286#17548—591nomaximal, Sylow
287C22×C59—22no—
288C22×C59—21no—
289C22×C59—22no—
290C22×C59—21no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—44no—
295C22×C59—44no—
296C22×C59C2×#10414yes—
297C22×C59C22×C22⋊C411yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57811yescharacteristic
300C2×D59—41no—
301C2×D59—41no—
302C2×D59—41no—
303C4×C59—21no—
304C4×C59—21no—
305C4×C59—22no—
306C4×C59—22no—
307C4×C59—21no—
308C4×C59—21no—
309C4×C59—21no—
310C4×C59—42no—
311C4×C59—44no—
312C4×C59—42no—
313C4×C59C2×#10412yes—
314Dic59—41no—
315Dic59—42no—
316Dic59—41no—
317Dic59—42no—
318Dic59—41no—
319#3520—591no—
320#2974—591no—
321#2275D5911yescharacteristic, complemented
322#1509—591no—
323#1508—592no—
324#1506—591no—
325#1504—592no—
326#1503—591no—
327#1501—592no—
328#1500—591no—
329C2×#328—592no—
330C2×C59—21no—
331C2×C59—21no—
332C2×C59—24no—
333C2×C59—42no—
334C2×C59#288311yescharacteristic
335C2×C59#289811yescharacteristic
336C2×C59C22×#10411yescharacteristic
337D59—81no—
338#556—591no—
339#554—1181no—
340#552—592no—
341#551—1182no—
342D4×Q8—591no—
343#548—592no—
344#547—591no—
345#547—591no—
346#546—1181no—
347#542—1181no—
348#539—1182no—
349#533—591no—
350#529—591no—
351#525—592no—
352C2×C22⋊Q8—592no—
353#520—1181no—
354C2×C4×D4—21no—
355C2×C4×D4C2×D5911yescharacteristic
356#436—1181no—
357#435—1182no—
358C8×D4—1181no—
359#432—591no—
360#431—591no—
361#407—592no—
362C2×C22⋊C8—592no—
363C4×C4⋊C4—21no—
364C4×C4⋊C4C2×D5911yescharacteristic
365C4×C22⋊C4—22no—
366C4×C22⋊C4C2×D5912yes—
367#376C2×D5911yescharacteristic
368C2×#95C2×D5912yes—
369#328—591no—
370#328—592no—
371#328—592no—
372#328—592no—
373#328—591no—
374#328—592no—
375#328—592no—
376#327—591no—
377#327—591no—
378#327—592no—
379C59#1754811yescharacteristic, complemented, Sylow
380C2×#41—592no—
381C2×#41—1181no—
382C22×Q8—592no—
383C22×D4C22⋊D5911yescharacteristic
384C23×C4—22no—
385C23×C4C22×D5912yes—
386C2×M16—1182no—
387C22×C8—1182no—
388C4⋊Q8—591no—
389C4⋊Q8—591no—
390C4⋊Q8—1181no—
391#126—1182no—
392#126—1182no—
393#126—1181no—
394#125—591no—
395#125—1182no—
396#125—1181no—
397#124—591no—
398#124—1181no—
399#124—1182no—
400#123—1182no—
401#123—1182no—
402#123—1181no—
403#123—1182no—
404#123—1182no—
405C22⋊Q8—592no—
406C22⋊Q8—592no—
407C22⋊Q8—592no—
408C22⋊Q8—591no—
409C22⋊Q8—592no—
410C22⋊Q8—1182no—
411C22⋊Q8—1182no—
412C22⋊Q8—1181no—
413C22⋊Q8—1182no—
414C22⋊Q8—1181no—
415C22⋊Q8—1182no—
416#121—592no—
417#121—591no—
418#121—1182no—
419#121—1181no—
420#120—1182no—
421C4×Q8—591no—
422C4×Q8—592no—
423C4×Q8—1181no—
424C4×Q8—1181no—
425C4×D4—28no—
426C4×D4—591no—
427C4×D4—1181no—
428C4×D4—1182no—
429C4×D4—1182no—
430C4×D4—1181no—
431C4×D4—1182no—
432C4×D4C4×D5911yescharacteristic
433C4×D4C4×D5911yescharacteristic
434C4×D4C4×D5912yes—
435C4×D4D23611yescharacteristic
436C4×D4D23612yes—
437C4×D4D23611yescharacteristic
438#117—592no—
439#117—1181no—
440#117—1182no—
441#117—1181no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—592no—
444C2×C4⋊C4—1182no—
445C2×C4⋊C4—1181no—
446C2×C4⋊C4C22⋊D5911yescharacteristic
447C2×C22⋊C4—22no—
448C2×C22⋊C4—1182no—
449C2×C22⋊C4C22⋊D5912yes—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—22no—
453C2×C42C22×D5911yescharacteristic
454C4⋊C8—1181no—
455C4⋊C8—1181no—
456C22⋊C8—591no—
457C22⋊C8—592no—
458C22⋊C8—591no—
459C22⋊C8—1182no—
460C22⋊C8—1182no—
461#97—1181no—
462C4×C8—1181no—
463#95—22no—
464#95—21no—
465#95C22×D5911yescharacteristic
466#95C22×D5912yes—
467#95C22×D5911yescharacteristic
468C24C2×C22⋊D5912yes—
469#41—4722no—
470#41—2364no—
471C2×Q8—591no—
472C2×Q8—592no—
473C2×Q8—1182no—
474C2×Q8—1181no—
475C2×Q8—4722no—
476C2×D4—24no—
477C2×D4—591no—
478C2×D4—1182no—
479C2×D4—1182no—
480C2×D4—1181no—
481C2×D4(C2×D59)⋊C412yes—
482C2×D4(C2×D59)⋊C411yescharacteristic
483C2×D4(C2×D59)⋊C411yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—28no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—1182no—
492C22×C4—1182no—
493C22×C4—1182no—
494C22×C4—1181no—
495C22×C4C2×C4×D5912yes—
496C22×C4C2×C4×D5912yes—
497C22×C4C2×D23612yes—
498C22×C4C2×D23612yes—
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5912yes—
502C22×C4C23×D5911yescharacteristic
503M16—4722no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1182no—
507C2×C8—4722no—
508C4⋊C4—24no—
509C4⋊C4—591no—
510C4⋊C4—592no—
511C4⋊C4—591no—
512C4⋊C4—1182no—
513C4⋊C4—1182no—
514C4⋊C4—1181no—
515C4⋊C4—1182no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1182no—
520C4⋊C4—1181no—
521C4⋊C4—1182no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1182no—
525C4⋊C4(C2×D59)⋊C411yescharacteristic
526C4⋊C4(C2×D59)⋊C412yes—
527C4⋊C4(C2×D59)⋊C411yescharacteristic
528C22⋊C4—28no—
529C22⋊C4—1182no—
530C22⋊C4—1182no—
531C22⋊C4—1182no—
532C22⋊C4—1182no—
533C22⋊C4—1181no—
534C22⋊C4—1182no—
535C22⋊C4—1181no—
536C22⋊C4—1181no—
537C22⋊C4—1182no—
538C22⋊C4—1181no—
539C22⋊C4—1181no—
540C22⋊C4(C2×D59)⋊C412yes—
541C22⋊C4(C2×D59)⋊C412yes—
542C22⋊C4(C2×D59)⋊C412yes—
543C22⋊C4(C2×D59)⋊C412yes—
544C42—24no—
545C42—81no—
546C42—82no—
547C42—1182no—
548C42—1182no—
549C42—1181no—
550C42C2×C4×D5911yescharacteristic
551C42C2×C4×D5911yescharacteristic
552C42C2×D23611yescharacteristic
553C42C2×D23611yescharacteristic
554C23—24no—
555C23—22no—
556C23—22no—
557C23—42no—
558C23—1181no—
559C23C2×(C2×D59)⋊C412yes—
560C23C2×(C2×D59)⋊C412yes—
561C23C22×C22⋊D5911yescharacteristic
562Q8—4722no—
563Q8—2361no—
564Q8—2362no—
565Q8—2362no—
566D4—24no—
567D4—22no—
568D4—4722no—
569D4—2361no—
570D4—2364no—
571D4—2361no—
572D4—2361no—
573D4#1427365514yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—88no—
593C2×C4—1181no—
594C2×C4—1182no—
595C2×C4—1181no—
596C2×C4—1182no—
597C2×C4—1181no—
598C2×C4—4722no—
599C2×C4—4722no—
600C2×C4—2364no—
601C2×C4—2362no—
602C2×C4C2×(C2×D59)⋊C411yescharacteristic
603C2×C4C2×(C2×D59)⋊C412yes—
604C2×C4C2×(C2×D59)⋊C411yescharacteristic
605C2×C4C2×(C2×D59)⋊C411yescharacteristic
606C2×C4C2×(C2×D59)⋊C412yes—
607C2×C4C2×(C2×D59)⋊C411yescharacteristic
608C2×C4C22×C4×D5911yescharacteristic
609C2×C4C22×D23611yescharacteristic
610C8—2361no—
611C8—2362no—
612C8—2361no—
613C22—22no—
614C22—21no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—2361no—
623C22—2361no—
624C22—2361no—
625C22C2×#1427365514yes—
626C22C22×(C2×D59)⋊C411yescharacteristic
627C22#1683821311yescharacteristic
628C22#1683821711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—2361no—
640C4—2361no—
641C4—2362no—
642C4—2362no—
643C4—2361no—
644C4C2×#1427365512yes—
645C2—21no—
646C2—21no—
647C2—24no—
648C2—42no—
649C2—4721no—
650C2#2143255611yescharacteristic
651C2#2143262611yescharacteristic
652C2C22×#1427365511yescharacteristic
subgroupG/H
centerC22C22×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2275C2
socleC22×C59C22×C22⋊C4
prime radical#21438145C2
cosocleC22×C4×C59C24
hypercenter#2275D59
hypocenterC59#17548
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #21433131C2semidirect
3. #21422619C2semidirect
4. #21422615C2semidirect
5. #21422607C2semidirect
6. #21422598C2semidirect
7. #21422594C2semidirect
8. #21422585C2semidirect
9. #21422580C2semidirect
10. C2×#16837138C2semidirect
11. C59×#2275C2semidirect
12. #16838122C4semidirect
14. #16838092C4semidirect
16. D4×Dic118C4semidirect
17. #16838061C4semidirect
18. #16838057C4semidirect
19. #16838056C4semidirect
53. C2×#11745685C2×C4semidirect
55. C22×Dic118C2×C4semidirect
58. #14273727C2×C4semidirect
59. #14273727C2×C4semidirect
65. #14273719C2×C4semidirect
67. #14273715C2×C4semidirect
79. #14273706C2×C4semidirect
80. #14273706C2×C4semidirect
81. #14273706C2×C4semidirect
86. #14273701C2×C4semidirect
92. C4×Dic118C2×C4semidirect
93. C4×Dic118C2×C4semidirect
95. C4×D236C2×C4semidirect
102. #14273686C2×C4semidirect
105. C2×C4⋊Dic59C2×C4semidirect
321. #2275D59semidirect
379. C59#17548semidirect

Maximal subgroups

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

subgroupindex
#214381452
#214331312
#214226192
#214226152
#214226072
#214225982
#214225942
#214225852
#214225802
C2×#168371382
C59×#22752
#1754859