Finite Group Databasebeta

← #28616170

Subgroups — #28616170

Order 15104.

1610 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837913C211yesmaximal, characteristic
2C2×#16837438C211yesmaximal, characteristic, complemented
3C2×#16837396C211yesmaximal, characteristic, complemented
4C2×#16837396C211yesmaximal, characteristic, complemented
5C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
6#21425299C212yesmaximal, complemented
7#21425292C212yesmaximal, complemented
8#21425285C212yesmaximal, complemented
9#21425278C212yesmaximal, complemented
10#21423812C211yesmaximal, characteristic, complemented
11C59×#2280C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C23×C59⋊C8C2211yescharacteristic
14C23×C59⋊C8C2211yescharacteristic
15#16837913—24no—
16#16837913—24no—
17C2×#14273710—21no—
18C2×#14273706C2212yes—
19C2×#14273701C2212yes—
20C2×#14273696—21no—
21C2×C4×C22⋊D59—22no—
22C2×#14273686C2211yescharacteristic
23#16837438—24no—
24C2×#14273660C2212yescomplemented
25C2×#14273658C2212yescomplemented
26#16837396—24no—
27#16837396—24no—
28C2×#14273650C2211yescharacteristic, complemented
29C2×#14273650C2212yescomplemented
30C2×#14273650C2211yescharacteristic, complemented
31C2×#14273647C2211yescharacteristic, complemented
32C2×#14273647C2212yescomplemented
33C2×#14273647C2211yescharacteristic, complemented
34C2×C22⋊(C59⋊C8)—24no—
35C2×C22⋊(C59⋊C8)C2212yes—
36C2×C22⋊(C59⋊C8)C2212yes—
37#16837164C2212yescomplemented
38#16837164C2212yescomplemented
39C24×C4×C59C2211yescharacteristic
40C2×#117×C59C2211yescharacteristic
41C2×#117×C59C2211yescharacteristic
42C59×#381—21no—
43C59×#380C2212yes—
44C59×#380C2212yes—
45C59×#379—21no—
46C4×C22⋊C4×C59—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685—22no—
50C2×#11745685—22no—
51C2×#11745685—22no—
52C2×#11745685D414yes—
53C22×Dic118C2311yescharacteristic
54C22×D236C2311yescharacteristic
55C22×C22⋊D59—22no—
56C22×C22⋊D59—21no—
57C22×C4×D59—21no—
58C22×C59⋊C8—22no—
59C22×C59⋊C8—22no—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8D414yescomplemented
64C22×C59⋊C8C2312yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273710—28no—
68#14273706—24no—
69#14273706—24no—
70#14273701—24no—
71#14273701—24no—
72#14273696—28no—
73C4×C22⋊D59—216no—
74#14273686—22no—
75#14273686D414yes—
76C2×C4⋊Dic59C2312yes—
77C2×Dic59⋊C4—22no—
78C2×C22⋊Dic59—22no—
79C2×C22⋊Dic59—21no—
80C2×(C2×D59)⋊C4—22no—
81C2×C4×Dic59—21no—
82#14273660—24no—
83#14273658—24no—
84#14273650—24no—
85#14273650—22no—
86#14273650—22no—
87#14273647—24no—
88#14273647—22no—
89#14273647—22no—
90C22⋊(C59⋊C8)—24no—
91C22⋊(C59⋊C8)—24no—
92C25×C59D411yescharacteristic
93C23×C4×C59—22no—
94C23×C4×C59—22no—
95C23×C4×C59—22no—
96C23×C4×C59—22no—
97C23×C4×C59—22no—
98C23×C4×C59—22no—
99C23×C4×C59D411yescharacteristic
100C23×C4×C59D412yes—
101C23×C4×C59C2312yes—
102C23×C4×C59C2311yescharacteristic
103#117×C59—22no—
104#117×C59—22no—
105#117×C59—22no—
106#117×C59—22no—
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2312yes—
110C2×C22⋊C4×C59—22no—
111C2×C22⋊C4×C59—22no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116#95×C59—22no—
117#95×C59—22no—
118C24×C59—22no—
119C24×C59—24no—
120C24×C59—22no—
121C24×C59—22no—
122C24×C59—42no—
123C24×C59C2×D412yes—
124C24×C59C2×D411yescharacteristic
125C23×D59—21no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—24no—
130C22×C4×C59—24no—
131C22×C4×C59—24no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—22no—
138C22×C4×C59—22no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—44no—
142C22×C4×C59—44no—
143C22×C4×C59—44no—
144C22×C4×C59—44no—
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D412yes—
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2×D412yes—
150C22×C4×C59#4112yes—
151C22×C4×C59#4112yes—
152C22×C4×C59#4112yes—
153C22×C4×C59C2411yescharacteristic
154C2×C22⋊D59—28no—
155C2×C22⋊D59—24no—
156C2×C22⋊D59—44no—
157C2×C22⋊D59—42no—
158C2×C22⋊D59—42no—
159C2×C22⋊D59—44no—
160C22×Dic59—22no—
161C22×Dic59—21no—
162#11745685—28no—
163#11745685—28no—
164#11745685—44no—
165#11745685—44no—
166C2×D236—21no—
167C2×D236—21no—
168C2×D236—21no—
169C2×D236—21no—
170C2×D236—21no—
171C2×D236C2×D412yes—
172C2×C4×D59—24no—
173C2×C4×D59—42no—
174C2×C4×D59—42no—
175C2×Dic118—21no—
176C2×Dic118—21no—
177C2×Dic118—21no—
178C2×Dic118—21no—
179C2×Dic118—21no—
180C2×Dic118C2×D412yes—
181C4⋊C4×C59—22no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—22no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C22⋊C4×C59—44no—
188C22⋊C4×C59—42no—
189C22⋊C4×C59—42no—
190C22⋊C4×C59—44no—
191C42×C59—42no—
192C42×C59—42no—
193C22⋊Dic59—28no—
194C22⋊Dic59—24no—
195(C2×D59)⋊C4—28no—
196C4⋊Dic59—22no—
197C4⋊Dic59C2×D414yes—
198Dic59⋊C4—28no—
199C4×Dic59—24no—
200C2×C59⋊C8—28no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—22no—
206C2×C59⋊C8—22no—
207C2×C59⋊C8—22no—
208C2×C59⋊C8—44no—
209C2×C59⋊C8C2×D412yes—
210C2×C59⋊C8C2×D412yes—
211C23×C59—22no—
212C23×C59—22no—
213C23×C59—28no—
214C23×C59—22no—
215C23×C59—22no—
216C23×C59—22no—
217C23×C59—22no—
218C23×C59—22no—
219C23×C59—22no—
220C23×C59—48no—
221C23×C59—48no—
222C23×C59—44no—
223C23×C59—44no—
224C23×C59#12012yes—
225C23×C59#12112yes—
226C23×C59#12312yes—
227C23×C59#13514yes—
228C23×C59C22×D411yescharacteristic
229C22×D59—22no—
230C22×D59—41no—
231C22×D59—41no—
232C22×D59—42no—
233C22×D59—42no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—44no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—48no—
243C2×C4×C59—42no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59#12012yes—
251C2×C4×C59#12112yes—
252C2×C4×C59#12114yes—
253C2×C4×C59#12114yes—
254C2×C4×C59#12314yes—
255C2×C4×C59#12312yes—
256C2×C4×C59#12414yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59C22×D411yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C22⋊D59—416no—
264C22⋊D59—48no—
265C2×Dic59—24no—
266C2×Dic59—22no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—42no—
270C2×Dic59—42no—
271D236—22no—
272D236—22no—
273D236—41no—
274D236—41no—
275C4×D59—48no—
276Dic118—22no—
277Dic118—22no—
278Dic118—41no—
279Dic118—41no—
280C59⋊C8—22no—
281C59⋊C8—22no—
282C59⋊C8—42no—
283#23479—591nomaximal, Sylow
284C22×C59—24no—
285C22×C59—24no—
286C22×C59—22no—
287C22×C59—22no—
288C22×C59—24no—
289C22×C59—24no—
290C22×C59—24no—
291C22×C59—416no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59#39414yes—
297C22×C59C2×#12011yescharacteristic
298C22×C59C2×#12111yescharacteristic
299C22×C59C2×#12311yescharacteristic
300C22×C59C2×#13512yes—
301C22×C59#57512yes—
302C2×D59—42no—
303C2×D59—44no—
304C2×D59—82no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—42no—
308C4×C59—82no—
309C4×C59—81no—
310C4×C59—81no—
311C4×C59#39414yes—
312C4×C59C2×#12111yescharacteristic
313C4×C59C2×#12111yescharacteristic
314C4×C59C2×#12311yescharacteristic
315C4×C59C2×#12411yescharacteristic
316Dic59—42no—
317Dic59—44no—
318C2×#525—591no—
319C2×#427—591no—
320C2×#416—591no—
321C2×#416—591no—
322C22×C22⋊C8—591no—
323#2280D5911yescharacteristic, complemented
324#2036—592no—
325#2035—592no—
326#2034—592no—
327#2033—592no—
328#1735—591no—
329C2×C59—24no—
330C2×C59—44no—
331C2×C59C2×#39411yescharacteristic
332C2×C59#300212yes—
333C2×C59#305011yescharacteristic
334C2×C59#305411yescharacteristic
335C2×C59#309112yes—
336D59—82no—
337C22×#41—591no—
338C24×C4C2×D5911yescharacteristic
339C23×C8—591no—
340C23×C8—591no—
341#525—1184no—
342#525—1184no—
343C2×#123—1181no—
344C2×C22⋊Q8—592no—
345C2×#121—592no—
346C2×#120—1181no—
347C2×C4×D4—1182no—
348C2×#117—591no—
349C2×#117C2×D5911yescharacteristic
350C2×#117C2×D5911yescharacteristic
351#427—1184no—
352C2×#107—592no—
353C2×#106—592no—
354#416—1184no—
355#416—1184no—
356C2×#103—591no—
357C2×#103—592no—
358C2×#103—591no—
359C2×#102—591no—
360C2×#102—592no—
361C2×#102—591no—
362C2×C22⋊C8—592no—
363C2×C22⋊C8—592no—
364C2×C22⋊C8—1184no—
365#381—21no—
366#380C2×D5912yes—
367#380C2×D5912yes—
368#379—21no—
369C4×C22⋊C4—22no—
370#340—592no—
371#340—592no—
372C59#2347911yescharacteristic, complemented, Sylow
373C25C22⋊D5911yescharacteristic
374C2×#41—594no—
375C2×#41—1182no—
376C2×#41—1182no—
377C2×#41—1182no—
378C2×#41—1182no—
379C2×#41—1182no—
380C22×Q8—591no—
381C22×D4—591no—
382C22×D4—1182no—
383C22×D4—1181no—
384C23×C4—22no—
385C23×C4—22no—
386C23×C4—22no—
387C23×C4—22no—
388C23×C4—22no—
389C23×C4—22no—
390C23×C4—1181no—
391C23×C4C22⋊D5911yescharacteristic
392C23×C4C22⋊D5912yes—
393C23×C4C22×D5912yes—
394C23×C4C22×D5911yescharacteristic
395C22×C8—594no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—592no—
399C22×C8—1182no—
400C22×C8—1182no—
401C22×C8—1182no—
402C22×C8—1182no—
403C22×C8—1182no—
404#123—1188no—
405C22⋊Q8—1184no—
406C22⋊Q8—1184no—
407#121—1184no—
408#121—1184no—
409#120—1188no—
410C4×D4—11816no—
411#117—22no—
412#117—22no—
413#117—22no—
414#117—22no—
415#117—594no—
416#117—1182no—
417C2×C4⋊C4—592no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5912yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1182no—
428C2×C22⋊C4—1182no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—1181no—
432#107—1184no—
433#106—1184no—
434#103—1182no—
435#103—1182no—
436#103—1184no—
437#102—1182no—
438#102—1182no—
439#102—1184no—
440C22⋊C8—1184no—
441C22⋊C8—1184no—
442#95—22no—
443#95—22no—
444C24—22no—
445C24—24no—
446C24—22no—
447C24—22no—
448C24—42no—
449C24—1181no—
450C24C2×C22⋊D5912yes—
451C24C2×C22⋊D5911yescharacteristic
452#41—1188no—
453#41—1188no—
454#41—2364no—
455#41—2364no—
456C2×Q8—592no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×D4—592no—
463C2×D4—1188no—
464C2×D4—1184no—
465C2×D4—1181no—
466C2×D4—1181no—
467C2×D4—1181no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—2364no—
471C2×D4—2362no—
472C2×D4—2362no—
473C2×D4—2364no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—1182no—
494C22×C4—1181no—
495C22×C4—1184no—
496C22×C4—2362no—
497C22×C4—2362no—
498C22×C4#1174568512yes—
499C22×C4#1174568512yes—
500C22×C4D4×D5912yes—
501C22×C4D4×D5912yes—
502C22×C4D4×D5912yes—
503C22×C4#1174568712yes—
504C22×C4C2×C22⋊D5912yes—
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C23×D5911yescharacteristic
507C2×C8—592no—
508C2×C8—592no—
509C2×C8—1188no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1182no—
515C2×C8—1182no—
516C2×C8—1182no—
517C2×C8—2364no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—594no—
525C4⋊C4—1188no—
526C4⋊C4—1182no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—1188no—
532C22⋊C4—1188no—
533C22⋊C4—1184no—
534C42—42no—
535C42—42no—
536C42—1184no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—48no—
547C23—48no—
548C23—44no—
549C23—44no—
550C23—1182no—
551C23—2361no—
552C23—2361no—
553C23—2362no—
554C23—2362no—
555C23#1427369512yes—
556C23#1427369912yes—
557C23#1427371012yes—
558C23#1427374714yes—
559C23C22×C22⋊D5911yescharacteristic
560Q8—1182no—
561Q8—1182no—
562Q8—2361no—
563Q8—2361no—
564D4—1182no—
565D4—1182no—
566D4—23616no—
567D4—2368no—
568D4—2361no—
569D4—2361no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—1184no—
587C2×C4—1182no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2362no—
591C2×C4—2362no—
592C2×C4—2368no—
593C2×C4#1427369512yes—
594C2×C4#1427369912yes—
595C2×C4#1427370014yes—
596C2×C4#1427370014yes—
597C2×C4#1427370914yes—
598C2×C4#1427371012yes—
599C2×C4#1427371314yes—
600C2×C4C2×D4×D5911yescharacteristic
601C2×C4C2×D4×D5911yescharacteristic
602C2×C4C2×D4×D5911yescharacteristic
603C2×C4C2×#1174568711yescharacteristic
604C2×C4C2×#1174568511yescharacteristic
605C2×C4C2×#1174568511yescharacteristic
606C8—1182no—
607C8—1182no—
608C8—2362no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—4721no—
622C22—4721no—
623C22—4722no—
624C22—2362no—
625C22—2364no—
626C22#1683731714yes—
627C22C2×#1427369511yescharacteristic
628C22C2×#1427369911yescharacteristic
629C22C2×#1427371011yescharacteristic
630C22C2×#1427374712yes—
631C22#1683820012yes—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—2362no—
637C4—2364no—
638C4#1683731714yes—
639C4C2×#1427370011yescharacteristic
640C4C2×#1427370011yescharacteristic
641C4C2×#1427370911yescharacteristic
642C4C2×#1427371311yescharacteristic
643C2—24no—
644C2—44no—
645C2—4722no—
646C2C2×#1683731711yescharacteristic
647C2#2143331212yes—
648C2#2143367411yescharacteristic
649C2#2143370211yescharacteristic
650C2#2143400912yes—
subgroupG/H
centerC22×C4C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2280C2
socleC23×C59C22×D4
prime radicalC2×#16837913C2
hypercenter#2280D59
hypocenterC59#23479
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C2×#16837438C2semidirect
3. C2×#16837396C2semidirect
4. C2×#16837396C2semidirect
5. C22×C22⋊(C59⋊C8)C2semidirect
6. #21425299C2semidirect
7. #21425292C2semidirect
8. #21425285C2semidirect
9. #21425278C2semidirect
10. #21423812C2semidirect
11. C59×#2280C2semidirect
24. C2×#14273660C22semidirect
25. C2×#14273658C22semidirect
28. C2×#14273650C22semidirect
29. C2×#14273650C22semidirect
30. C2×#14273650C22semidirect
31. C2×#14273647C22semidirect
32. C2×#14273647C22semidirect
33. C2×#14273647C22semidirect
37. #16837164C22semidirect
38. #16837164C22semidirect
63. C22×C59⋊C8D4semidirect
323. #2280D59semidirect
372. C59#23479semidirect

Maximal subgroups

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

subgroupindex
C2×#168379132
C2×#168374382
C2×#168373962
C22×C22⋊(C59⋊C8)2
#214252992
#214252922
#214252852
#214252782
#214238122
C59×#22802
#2347959