Finite Group Databasebeta

← #28589693

Subgroups — #28589693

Order 15104.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837442C211yesmaximal, characteristic, complemented
2C2×#16837403C211yesmaximal, characteristic
3C2×#16837398C211yesmaximal, characteristic
4#21426712C211yesmaximal, characteristic, complemented
5#21424364C212yesmaximal, complemented
6#21424357C212yesmaximal, complemented
7#21424350C212yesmaximal, complemented
8#21424343C212yesmaximal, complemented
9C2×#16837164C211yesmaximal, characteristic, complemented
10#21423793C211yesmaximal, characteristic, complemented
11C2×C59×#520C211yesmaximal, characteristic
12C22×#11745664C2211yescharacteristic
13C23×C59⋊C8C2211yescharacteristic
14C2×#14273686C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16#16837442—24no—
17C2×#14273660C2212yes—
18C2×#14273658C2212yes—
19#16837403C418yescomplemented
20#16837398—24no—
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2212yes—
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2212yes—
27#16837300C2212yes—
28#16837300C2212yes—
29#16837298—21no—
30#16837298—21no—
31#16837260—22no—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837157C2212yes—
35#16837157C2212yes—
36C22×#41×C59C2211yescharacteristic
37C59×#520—24no—
38C59×#520—24no—
39C2×C4×Q8×C59C2212yes—
40C2×C4×D4×C59—22no—
41C2×C4×D4×C59C2212yes—
42C2×#117×C59—21no—
43C2×#117×C59C2211yescharacteristic
44C22×C4⋊C4×C59—21no—
45C22×C4⋊C4×C59C2211yescharacteristic
46C23×Dic59—21no—
47C2×#11745664—22no—
48C2×#11745664—22no—
49C2×#11745664C2×C414yescomplemented
50C22×C59⋊C8—22no—
51C22×C59⋊C8—22no—
52C22×C59⋊C8—22no—
53C22×C59⋊C8C2311yescharacteristic
54C22×C59⋊C8C2311yescharacteristic
55C22×C59⋊C8C2312yes—
56#14273686—22no—
57#14273686—22no—
58C2×C4⋊Dic59—21no—
59C2×C4⋊Dic59—22no—
60C2×C4⋊Dic59C2×C414yescomplemented
61C2×C4⋊Dic59C2311yescharacteristic
62C2×C4⋊Dic59C2311yescharacteristic
63C2×C4⋊Dic59C2312yes—
64C2×Dic59⋊C4—22no—
65C2×Dic59⋊C4—22no—
66C2×C22⋊Dic59—21no—
67C2×C4×Dic59—21no—
68#14273660—24no—
69#14273658—24no—
70#14273651—22no—
71#14273651—22no—
72#14273651C2×C418yescomplemented
73#14273648—22no—
74#14273648—22no—
75#14273648C2×C418yescomplemented
76C59⋊#95—22no—
77C59⋊#95—22no—
78C2×#41×C59—22no—
79C2×#41×C59—22no—
80C2×#41×C59—22no—
81C2×#41×C59C2×C414yes—
82C2×#41×C59D414yes—
83C2×#41×C59Q814yes—
84C22×Q8×C59C2311yescharacteristic
85C22×D4×C59—21no—
86C22×D4×C59C2311yescharacteristic
87C23×C4×C59—21no—
88C23×C4×C59—21no—
89C23×C4×C59—22no—
90C23×C4×C59C2311yescharacteristic
91C4×Q8×C59—24no—
92C4×Q8×C59—24no—
93C4×D4×C59—216no—
94C4×D4×C59—24no—
95C4×D4×C59—24no—
96#117×C59—28no—
97#117×C59—22no—
98#117×C59—22no—
99C2×C4⋊C4×C59—28no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59—22no—
103C2×C4⋊C4×C59D411yescharacteristic
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59D414yes—
106C2×C4⋊C4×C59C2312yes—
107C2×C22⋊C4×C59—22no—
108C2×C22⋊C4×C59D412yes—
109C2×C42×C59—22no—
110C2×C42×C59C2312yes—
111C24×C59—21no—
112C24×C59C2×D411yescharacteristic
113#41×C59—28no—
114#41×C59—28no—
115#41×C59—28no—
116#41×C59C4⋊C4116yescomplemented
117C2×Q8×C59—21no—
118C2×Q8×C59—21no—
119C2×Q8×C59—21no—
120C2×Q8×C59C22×C412yes—
121C2×Q8×C59C2×D412yes—
122C2×Q8×C59C2×Q812yes—
123C2×D4×C59—24no—
124C2×D4×C59—24no—
125C2×D4×C59—24no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59C22×C412yes—
130C2×D4×C59C2×D412yes—
131C2×D4×C59C2×Q812yes—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—24no—
135C22×C4×C59—24no—
136C22×C4×C59—22no—
137C22×C4×C59—28no—
138C22×C4×C59—24no—
139C22×C4×C59—82no—
140C22×C4×C59—82no—
141C22×C4×C59—81no—
142C22×C4×C59—81no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×D412yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2×Q812yes—
150C22×C4×C59#4112yes—
151C22×C4×C59#4112yes—
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—22no—
154C22×Dic59—24no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—81no—
158C22×Dic59—81no—
159C4⋊C4×C59—28no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59C2×D414yes—
165C22⋊C4×C59—28no—
166C22⋊C4×C59—24no—
167C42×C59—28no—
168C42×C59—22no—
169C42×C59—22no—
170C22⋊Dic59—42no—
171C4⋊Dic59—24no—
172C4⋊Dic59—22no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59C22×C414yes—
178Dic59⋊C4—82no—
179Dic59⋊C4—82no—
180C4×Dic59—42no—
181#11745664—28no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—44no—
191C2×C59⋊C8C22×C414yes—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59C4×D412yes—
204C23×C59#12012yes—
205C23×C59C22⋊Q812yes—
206C23×C59C22×D411yescharacteristic
207Q8×C59—22no—
208Q8×C59—22no—
209Q8×C59—22no—
210Q8×C59C2×C4⋊C414yes—
211D4×C59—216no—
212D4×C59—22no—
213D4×C59—22no—
214D4×C59—22no—
215D4×C59C2×C4⋊C414yes—
216C2×C4×C59—22no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—216no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—24no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—22no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—84no—
230C2×C4×C59—88no—
231C2×C4×C59C2×C4⋊C414yes—
232C2×C4×C59C4×D414yes—
233C2×C4×C59C4×D412yes—
234C2×C4×C59#12012yes—
235C2×C4×C59C22⋊Q814yes—
236C2×C4×C59C22⋊Q812yes—
237C2×C4×C59#12314yes—
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×Q811yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×Dic59—24no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—84no—
254C59⋊C8—24no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257#20878—591nomaximal, Sylow
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—24no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59#38514yes—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×#12011yescharacteristic
274C22×C59C2×C22⋊Q811yescharacteristic
275C22×C59#57611yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57811yescharacteristic
279C4×C59—24no—
280C4×C59—42no—
281C4×C59—44no—
282C4×C59—42no—
283C4×C59#38514yes—
284C4×C59C22×C4⋊C411yescharacteristic
285C4×C59C2×C4×D411yescharacteristic
286C4×C59C2×C22⋊Q811yescharacteristic
287C4×C59C2×#12311yescharacteristic
288Dic59—44no—
289Dic59—81no—
290Dic59—81no—
291C2×#520D5911yescharacteristic
292C2×#428—591no—
293C2×#417—591no—
294C2×#416—591no—
295#2316—591no—
296#1865—592no—
297#1864—592no—
298#1863—592no—
299#1862—592no—
300C2×#340—591no—
301#1729—591no—
302C2×C59—24no—
303C2×C59—44no—
304C2×C59C2×#38511yescharacteristic
305C2×C59#294611yescharacteristic
306C2×C59#294711yescharacteristic
307C2×C59#301211yescharacteristic
308C2×C59#301311yescharacteristic
309C2×C59#308311yescharacteristic
310C2×C59#308411yescharacteristic
311C22×#41C2×D5911yescharacteristic
312C22×M16—591no—
313C23×C8—591no—
314#520—24no—
315#520—24no—
316C2×C4×Q8C2×D5912yes—
317C2×C4×D4—22no—
318C2×C4×D4C2×D5912yes—
319C2×#117—21no—
320C2×#117—591no—
321C2×#117C2×D5911yescharacteristic
322C22×C4⋊C4—21no—
323C22×C4⋊C4—591no—
324C22×C4⋊C4C2×D5911yescharacteristic
325#428—1184no—
326C2×#107—592no—
327C2×#106—592no—
328#417—598no—
329#416—1184no—
330C2×#103—592no—
331C2×#103—591no—
332C2×#103—591no—
333C2×#102—592no—
334C2×#102—591no—
335C2×#102—591no—
336#389—592no—
337#389—592no—
338#389—1181no—
339#389—1181no—
340#380—1182no—
341#340—592no—
342#340—592no—
343#336—592no—
344#336—592no—
345C59#2087811yescharacteristic, complemented, Sylow
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41Dic11814yes—
350C2×#41C4×D5914yes—
351C2×#41C22⋊D5914yes—
352C22×Q8C22×D5911yescharacteristic
353C22×D4—21no—
354C22×D4C22×D5911yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—22no—
358C23×C4—1181no—
359C23×C4C22×D5911yescharacteristic
360C2×M16—594no—
361C2×M16—1182no—
362C2×M16—1182no—
363C22×C8—592no—
364C22×C8—591no—
365C22×C8—591no—
366C22×C8—1182no—
367C22×C8—1182no—
368C22×C8—1182no—
369C4×Q8—24no—
370C4×Q8—24no—
371C4×D4—216no—
372C4×D4—24no—
373C4×D4—24no—
374#117—28no—
375#117—22no—
376#117—22no—
377#117—1182no—
378#117—1182no—
379C2×C4⋊C4—28no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—594no—
384C2×C4⋊C4—592no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1182no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1182no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4C22⋊D5911yescharacteristic
392C2×C4⋊C4C22⋊D5911yescharacteristic
393C2×C4⋊C4C22⋊D5914yes—
394C2×C4⋊C4C22×D5912yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4C22⋊D5912yes—
398C2×C42—22no—
399C2×C42—1181no—
400C2×C42C22×D5912yes—
401#107—1184no—
402#106—1184no—
403#103—598no—
404#103—1182no—
405#103—1182no—
406#102—598no—
407#102—1182no—
408#102—1182no—
409#95—1182no—
410#95—1182no—
411C24—21no—
412C24C2×C22⋊D5911yescharacteristic
413#41—28no—
414#41—28no—
415#41—28no—
416#41Dic59⋊C4116yescomplemented
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×Dic11812yes—
421C2×Q8C2×C4×D5912yes—
422C2×Q8C2×C22⋊D5912yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×Dic11812yes—
430C2×D4C2×C4×D5912yes—
431C2×D4C2×C22⋊D5912yes—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1184no—
444C22×C4—1182no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—4721no—
448C22×C4—4721no—
449C22×C4C2×Dic11812yes—
450C22×C4C2×C4×D5912yes—
451C22×C4#1174568512yes—
452C22×C4#1174568512yes—
453C22×C4C2×C22⋊D5912yes—
454C22×C4C2×C22⋊D5912yes—
455C22×C4C2×C22⋊D5912yes—
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5912yes—
458C22×C4C23×D5911yescharacteristic
459M16—1188no—
460C2×C8—594no—
461C2×C8—1182no—
462C2×C8—1182no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—2364no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—594no—
476C4⋊C4—1184no—
477C4⋊C4—1182no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—4722no—
483C4⋊C4—4722no—
484C4⋊C4C2×C22⋊D5914yes—
485C22⋊C4—28no—
486C22⋊C4—24no—
487C22⋊C4—2362no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—2362no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C4×C22⋊D5912yes—
504C23#1427369612yes—
505C23#1427370612yes—
506C23C22×C22⋊D5911yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8C2×Dic59⋊C414yes—
511D4—216no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C2×Dic59⋊C414yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—84no—
530C2×C4—88no—
531C2×C4—1184no—
532C2×C4—4724no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2362no—
540C2×C4—2362no—
541C2×C4C2×Dic59⋊C414yes—
542C2×C4C4×C22⋊D5912yes—
543C2×C4C4×C22⋊D5914yes—
544C2×C4#1427369612yes—
545C2×C4#1427370614yes—
546C2×C4#1427370612yes—
547C2×C4#1427371014yes—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C22×C22⋊D5911yescharacteristic
550C2×C4C22×C22⋊D5911yescharacteristic
551C2×C4C22×Dic11811yescharacteristic
552C2×C4C2×#1174568511yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C8—1184no—
555C8—2361no—
556C8—2361no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#1683727514yes—
571C22C2×C4×C22⋊D5911yescharacteristic
572C22C2×#1427369611yescharacteristic
573C22C2×#1427370611yescharacteristic
574C22#1683820511yescharacteristic
575C22#1683821011yescharacteristic
576C22#1683821311yescharacteristic
577C22#1683821711yescharacteristic
578C4—24no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—4721no—
583C4—4721no—
584C4—2364no—
585C4#1683727514yes—
586C4C22×Dic59⋊C411yescharacteristic
587C4C2×C4×C22⋊D5911yescharacteristic
588C4C2×#1427370611yescharacteristic
589C4C2×#1427371011yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×#1683727511yescharacteristic
593C2#2143292411yescharacteristic
594C2#2143293511yescharacteristic
595C2#2143340011yescharacteristic
596C2#2143341111yescharacteristic
597C2#2143394711yescharacteristic
598C2#2143395811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#520C2
socleC23×C59C22×D4
prime radicalC22×#41×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#520D59
hypocenterC59#20878
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16837442C2semidirect
4. #21426712C2semidirect
5. #21424364C2semidirect
6. #21424357C2semidirect
7. #21424350C2semidirect
8. #21424343C2semidirect
9. C2×#16837164C2semidirect
10. #21423793C2semidirect
19. #16837403C4semidirect
49. C2×#11745664C2×C4semidirect
60. C2×C4⋊Dic59C2×C4semidirect
72. #14273651C2×C4semidirect
75. #14273648C2×C4semidirect
116. #41×C59C4⋊C4semidirect
345. C59#20878semidirect
416. #41Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168374422
C2×#168374032
C2×#168373982
#214267122
#214243642
#214243572
#214243502
#214243432
C2×#168371642
#214237932
C2×C59×#5202
#2087859