Finite Group Databasebeta

← #28556532

Subgroups — #28556532

Order 15104.

1010 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#21437948C211yesmaximal, characteristic
2#21433125C211yesmaximal, characteristic, complemented
3#21422650C212yesmaximal, complemented
4#21422646C211yesmaximal, characteristic, complemented
5#21422642C211yesmaximal, characteristic, complemented
6#21422637C212yesmaximal, complemented
7#21422607C211yesmaximal, characteristic, complemented
8#21422603C211yesmaximal, characteristic, complemented
9#21422598C212yesmaximal, complemented
10#21422585C212yesmaximal, complemented
11C59×#2353C211yesmaximal, characteristic, complemented
12#16838135—22no—
13#16838122C412yescomplemented
14#16838117C412yescomplemented
15#16838102—22no—
16#16838073C412yescomplemented
17#16838057C412yescomplemented
18#16837990—21no—
19#16837990—21no—
20#16837969—21no—
21#16837959C2211yescharacteristic
22#16837952C2211yescharacteristic
23#16837913C2212yes—
24#16837913C2212yes—
25#16837883—21no—
26#16837474—22no—
27#16837474—22no—
28#16837459C2211yescharacteristic
29#16837459C2211yescharacteristic
30#16837366C2212yes—
31C2×C22⋊(C59⋊C8)C2212yes—
32#16837138C2212yes—
33#16837138C2212yes—
34#16837138C2212yes—
35#16837138C2212yes—
36#16837135C2212yes—
37#16837135C2212yes—
38#16837135C2212yes—
39#16837135C2212yes—
40C2×C4×D4×C59C2211yescharacteristic
41C59×#388—21no—
42C59×#387—21no—
43C59×#386—21no—
44C59×#383C2211yescharacteristic
45C59×#381C2211yescharacteristic
46C59×#380C2211yescharacteristic
47C4×C22⋊C4×C59—21no—
48C4×C22⋊C4×C59C2211yescharacteristic
49C59×#376C2211yescharacteristic
50C2×#95×C59C2211yescharacteristic
51C2×#11745685—21no—
52C2×#11745685C2×C412yes—
53C2×#11745685C2×C412yescomplemented
54C2×#11745664—22no—
55C22×C59⋊C8—22no—
56#14273727C2×C411yescharacteristic, complemented
57#14273727C2×C411yescharacteristic, complemented
58#14273727C2×C411yescharacteristic, complemented
59#14273724C2×C411yescharacteristic, complemented
60#14273723—22no—
61#14273720—22no—
62#14273720—22no—
63#14273719—21no—
64#14273715—21no—
65#14273714—22no—
66#14273713—22no—
67#14273713—22no—
68#14273710—22no—
69#14273710—22no—
70#14273709—22no—
71#14273709—22no—
72#14273706—21no—
73#14273706C2×C412yes—
74#14273706C2×C412yescomplemented
75#14273706C2312yes—
76#14273701—21no—
77#14273701C2×C412yes—
78#14273701C2×C412yescomplemented
79#14273701C2312yes—
80#14273700—22no—
81#14273700—22no—
82#14273698—22no—
83#14273698—22no—
84#14273696—22no—
85#14273696—22no—
86C4×Dic118—21no—
87C4×Dic118C2×C412yescomplemented
88#14273692—22no—
89C4×D236—21no—
90C4×D236C2×C412yescomplemented
91#14273689—22no—
92C4×C22⋊D59—22no—
93C4×C22⋊D59—22no—
94C4×C22⋊D59—22no—
95C4×C22⋊D59—22no—
96#14273686—21no—
97#14273686C2×C412yes—
98#14273686C2×C412yescomplemented
99#14273685—22no—
100C4⋊C4×D59—22no—
101C4⋊(C59⋊C8)—21no—
102C4⋊(C59⋊C8)—21no—
103C22⋊(C59⋊C8)—22no—
104C22⋊(C59⋊C8)—22no—
105C22⋊(C59⋊C8)C2312yes—
106C22⋊(C59⋊C8)C2312yes—
107#14273638—21no—
108#14273638—21no—
109C22×D4×C59D411yescharacteristic
110C23×C4×C59—21no—
111C23×C4×C59C2311yescharacteristic
112C23×C4×C59C2311yescharacteristic
113C4×D4×C59C2×C412yes—
114C4×D4×C59C2×C412yes—
115C4×D4×C59D412yes—
116C4×D4×C59D412yes—
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×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59—21no—
125C2×C42×C59—21no—
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
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
138C2×D4×C59—41no—
139C2×D4×C59—41no—
140C2×D4×C59C22⋊C412yes—
141C2×D4×C59C22⋊C412yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59—81no—
151C22×C4×C59C22×C412yes—
152C22×C4×C59C22×C412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D412yes—
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—22no—
165#11745685—42no—
166#11745685—42no—
167#11745685—82no—
168#11745685—82no—
169C2×D236—21no—
170C2×D236—21no—
171C2×D236C22×C412yes—
172C2×C4×D59—22no—
173C2×C4×D59—22no—
174C2×C4×D59—22no—
175C2×Dic118—21no—
176C2×Dic118—21no—
177C2×Dic118C22×C412yes—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C22⋊C412yes—
180C4⋊C4×C59C22⋊C412yes—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—42no—
183C22⋊C4×C59—81no—
184C22⋊C4×C59C22⋊C412yes—
185C22⋊C4×C59C22⋊C412yes—
186C22⋊C4×C59C22⋊C412yes—
187C22⋊C4×C59C22⋊C412yes—
188C42×C59—42no—
189C42×C59—81no—
190C42×C59C22×C411yescharacteristic
191C42×C59C22×C411yescharacteristic
192C42×C59C2×D411yescharacteristic
193C42×C59C2×D411yescharacteristic
194C22⋊Dic59—22no—
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—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59C22×C412yes—
208C4⋊Dic59C22×C412yes—
209Dic59⋊C4—22no—
210Dic59⋊C4—22no—
211Dic59⋊C4—22no—
212Dic59⋊C4—22no—
213C4×Dic59—22no—
214C4×Dic59—22no—
215C4×Dic59—22no—
216#11745664—82no—
217C2×C59⋊C8—22no—
218C2×C59⋊C8—22no—
219C2×C59⋊C8—82no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59C2×C22⋊C412yes—
229C23×C59C2×C22⋊C412yes—
230C23×C59C22×D411yescharacteristic
231C22×D59—22no—
232D4×C59—42no—
233D4×C59—42no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—42no—
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—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—84no—
257C2×C4×C59C2×C22⋊C411yescharacteristic
258C2×C4×C59C2×C22⋊C412yes—
259C2×C4×C59C2×C22⋊C412yes—
260C2×C4×C59C2×C22⋊C411yescharacteristic
261C2×C4×C59C2×C22⋊C411yescharacteristic
262C2×C4×C59C2×C22⋊C411yescharacteristic
263C2×C4×C59C23×C411yescharacteristic
264C2×C4×C59C22×D411yescharacteristic
265C22⋊D59—42no—
266C22⋊D59—42no—
267C22⋊D59—82no—
268C22⋊D59—82no—
269C22⋊D59—82no—
270C22⋊D59—82no—
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—22no—
274D236—42no—
275D236—42no—
276D236—81no—
277C4×D59—44no—
278C4×D59—82no—
279C4×D59—82no—
280Dic118—42no—
281Dic118—42no—
282Dic118—81no—
283C59⋊C8—42no—
284C59⋊C8—42no—
285#17629—591nomaximal, Sylow
286C22×C59—21no—
287C22×C59—21no—
288C22×C59—22no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59C22×C22⋊C411yescharacteristic
299C22×C59#57511yescharacteristic
300C22×C59#57511yescharacteristic
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—42no—
304C4×C59—21no—
305C4×C59—21no—
306C4×C59—22no—
307C4×C59—22no—
308C4×C59—21no—
309C4×C59—21no—
310C4×C59—21no—
311C4×C59—21no—
312C4×C59—42no—
313C4×C59—81no—
314C4×C59—81no—
315C4×C59—81no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319#3506—591no—
320#2973—591no—
321#2353D5911yescharacteristic, complemented
322#1516—592no—
323#1515—591no—
324#1514—591no—
325#1513—592no—
326#1506—591no—
327#1505—591no—
328#1504—592no—
329#1501—592no—
330C2×C59—21no—
331C2×C59—21no—
332C2×C59—42no—
333C2×C59—42no—
334C2×C59#288311yescharacteristic
335C2×C59#289611yescharacteristic
336C2×C59#290111yescharacteristic
337D59—82no—
338#559—1182no—
339#556—592no—
340#555—592no—
341#553—1182no—
342#550—592no—
343#547—592no—
344#538—1181no—
345#538—1181no—
346#535—1181no—
347#533—591no—
348#532—591no—
349#525—592no—
350#525—592no—
351#520—1181no—
352C2×C4×D4C2×D5911yescharacteristic
353#435—1182no—
354#435—1182no—
355#432—591no—
356#432—591no—
357#407—592no—
358C2×C22⋊C8—592no—
359#388—21no—
360#387—21no—
361#386—21no—
362#383C2×D5911yescharacteristic
363#381C2×D5911yescharacteristic
364#380C2×D5911yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D5911yescharacteristic
367#376C2×D5911yescharacteristic
368C2×#95C2×D5911yescharacteristic
369#328—592no—
370#328—592no—
371#328—592no—
372#328—592no—
373#327—592no—
374#327—592no—
375#327—592no—
376#327—592no—
377C59#1762911yescharacteristic, complemented, Sylow
378C2×#41—592no—
379C2×#41—592no—
380C2×#41—1181no—
381C22×D4C22⋊D5911yescharacteristic
382C23×C4—21no—
383C23×C4C22×D5911yescharacteristic
384C23×C4C22×D5911yescharacteristic
385C2×M16—1182no—
386C22×C8—1182no—
387C4⋊Q8—591no—
388C4⋊Q8—591no—
389C4⋊Q8—591no—
390#127—591no—
391#127—1182no—
392#126—1182no—
393#126—1182no—
394#125—1181no—
395#124—1182no—
396#124—1182no—
397#124—1182no—
398#124—1181no—
399#123—1182no—
400#123—1182no—
401#123—1182no—
402#123—1182no—
403C22⋊Q8—592no—
404C22⋊Q8—592no—
405C22⋊Q8—592no—
406C22⋊Q8—1181no—
407#121—592no—
408#121—592no—
409#121—592no—
410#121—1182no—
411#121—1182no—
412#121—1181no—
413#121—1182no—
414#121—1182no—
415#120—1182no—
416#120—1182no—
417C4×Q8—592no—
418C4×Q8—1182no—
419C4×Q8—1181no—
420C4×D4—592no—
421C4×D4—1182no—
422C4×D4—1182no—
423C4×D4—1182no—
424C4×D4—1182no—
425C4×D4—1182no—
426C4×D4—1181no—
427C4×D4C4×D5912yes—
428C4×D4C4×D5912yes—
429C4×D4D23612yes—
430C4×D4D23612yes—
431#117—592no—
432#117—592no—
433#117—1181no—
434#117—1182no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—1182no—
437C2×C4⋊C4C22⋊D5911yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4C22⋊D5911yescharacteristic
442C2×C22⋊C4C22⋊D5911yescharacteristic
443C2×C42—21no—
444C2×C42—21no—
445C2×C42C22×D5911yescharacteristic
446C4⋊C8—1181no—
447C4⋊C8—1181no—
448C22⋊C8—592no—
449C22⋊C8—592no—
450C22⋊C8—1182no—
451C22⋊C8—1182no—
452#97—1181no—
453#97—1181no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95C22×D5911yescharacteristic
460#95C22×D5911yescharacteristic
461#95C22×D5911yescharacteristic
462#95C22×D5911yescharacteristic
463C24C2×C22⋊D5911yescharacteristic
464C24C2×C22⋊D5911yescharacteristic
465#41—4722no—
466#41—4722no—
467#41—2362no—
468#41—2362no—
469C2×Q8—592no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—592no—
475C2×D4—1182no—
476C2×D4—1182no—
477C2×D4—1182no—
478C2×D4—1182no—
479C2×D4—1182no—
480C2×D4—1181no—
481C2×D4—1181no—
482C2×D4(C2×D59)⋊C412yes—
483C2×D4(C2×D59)⋊C412yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1182no—
494C22×C4—1182no—
495C22×C4—1182no—
496C22×C4C2×C4×D5912yes—
497C22×C4C2×C4×D5912yes—
498C22×C4C2×D23612yes—
499C22×C4C2×D23612yes—
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C23×D5911yescharacteristic
505M16—4722no—
506C2×C8—1182no—
507C2×C8—1182no—
508C2×C8—4722no—
509C4⋊C4—81no—
510C4⋊C4—592no—
511C4⋊C4—592no—
512C4⋊C4—1181no—
513C4⋊C4—1182no—
514C4⋊C4—1182no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1182no—
518C4⋊C4—1181no—
519C4⋊C4—1182no—
520C4⋊C4(C2×D59)⋊C412yes—
521C4⋊C4(C2×D59)⋊C412yes—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—1182no—
526C22⋊C4—1182no—
527C22⋊C4—1182no—
528C22⋊C4—1182no—
529C22⋊C4—1182no—
530C22⋊C4—1182no—
531C22⋊C4—1182no—
532C22⋊C4—1182no—
533C22⋊C4—1182no—
534C22⋊C4(C2×D59)⋊C412yes—
535C22⋊C4(C2×D59)⋊C412yes—
536C22⋊C4(C2×D59)⋊C412yes—
537C22⋊C4(C2×D59)⋊C412yes—
538C42—42no—
539C42—81no—
540C42—1182no—
541C42—1182no—
542C42—1182no—
543C42C2×C4×D5911yescharacteristic
544C42C2×C4×D5911yescharacteristic
545C42C2×D23611yescharacteristic
546C42C2×D23611yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—1182no—
556C23C2×(C2×D59)⋊C412yes—
557C23C2×(C2×D59)⋊C412yes—
558C23C22×C22⋊D5911yescharacteristic
559Q8—4721no—
560Q8—2362no—
561Q8—2362no—
562D4—42no—
563D4—42no—
564D4—4721no—
565D4—4722no—
566D4—4722no—
567D4—4722no—
568D4—4722no—
569D4—2362no—
570D4—2362no—
571D4—2362no—
572D4—2362no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—4722no—
600C2×C4—4722no—
601C2×C4—2364no—
602C2×C4C2×(C2×D59)⋊C411yescharacteristic
603C2×C4C2×(C2×D59)⋊C412yes—
604C2×C4C2×(C2×D59)⋊C412yes—
605C2×C4C2×(C2×D59)⋊C411yescharacteristic
606C2×C4C2×(C2×D59)⋊C411yescharacteristic
607C2×C4C2×(C2×D59)⋊C411yescharacteristic
608C2×C4C22×C4×D5911yescharacteristic
609C2×C4C22×D23611yescharacteristic
610C8—2362no—
611C8—2362no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—2362no—
625C22—2362no—
626C22—2362no—
627C22C22×(C2×D59)⋊C411yescharacteristic
628C22#1683820011yescharacteristic
629C22#1683820011yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—2362no—
643C4—2362no—
644C4—2362no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—4722no—
650C2#2143255611yescharacteristic
651C2#2143261311yescharacteristic
652C2#2143264411yescharacteristic
subgroupG/H
centerC22C22×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2353C2
socleC22×C59C22×C22⋊C4
prime radical#21437948C2
cosocleC22×C4×C59C24
hypercenter#2353D59
hypocenterC59#17629
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #21433125C2semidirect
3. #21422650C2semidirect
4. #21422646C2semidirect
5. #21422642C2semidirect
6. #21422637C2semidirect
7. #21422607C2semidirect
8. #21422603C2semidirect
9. #21422598C2semidirect
10. #21422585C2semidirect
11. C59×#2353C2semidirect
13. #16838122C4semidirect
14. #16838117C4semidirect
16. #16838073C4semidirect
17. #16838057C4semidirect
53. C2×#11745685C2×C4semidirect
56. #14273727C2×C4semidirect
57. #14273727C2×C4semidirect
58. #14273727C2×C4semidirect
59. #14273724C2×C4semidirect
74. #14273706C2×C4semidirect
78. #14273701C2×C4semidirect
87. C4×Dic118C2×C4semidirect
90. C4×D236C2×C4semidirect
98. #14273686C2×C4semidirect
321. #2353D59semidirect
377. C59#17629semidirect

Maximal subgroups

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

subgroupindex
#214379482
#214331252
#214226502
#214226462
#214226422
#214226372
#214226072
#214226032
#214225982
#214225852
C59×#23532
#1762959