Finite Group Databasebeta

← #28596459

Subgroups — #28596459

Order 15104.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837692C211yesmaximal, characteristic
2#21426844C211yesmaximal, characteristic, complemented
3#21425048C211yesmaximal, characteristic, complemented
4#21424864C211yesmaximal, characteristic, complemented
5#21424681C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424357C211yesmaximal, characteristic
8#21424343C211yesmaximal, characteristic
9#21424106C211yesmaximal, characteristic
10#21423994C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12#21423953C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2275C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17#16837692C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837306—21no—
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837285—21no—
35#16837285—21no—
36#16837268C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C4×C4⋊C4×C59C2211yescharacteristic
53C4×C4⋊C4×C59C2211yescharacteristic
54C4×C4⋊C4×C59C2211yescharacteristic
55C4×C22⋊C4×C59—21no—
56C4×C22⋊C4×C59—21no—
57C4×C22⋊C4×C59—21no—
58C59×#376C2211yescharacteristic
59C2×#95×C59—21no—
60C22×Dic118—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273727—22no—
66#14273727C2×C414yescomplemented
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76#14273660—22no—
77#14273658—22no—
78#14273658C2×C414yescomplemented
79#14273658C2×C414yescomplemented
80C4⋊(C59⋊C8)—22no—
81C4⋊(C59⋊C8)C2×C414yescomplemented
82#14273648—22no—
83#14273648—22no—
84#14273648C2×C414yes—
85#14273648C2×C414yes—
86#14273638—22no—
87C4×C59⋊C8—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×D4×C59C2311yescharacteristic
95C23×C4×C59—21no—
96C23×C4×C59—21no—
97C23×C4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59C2×C414yes—
103C4×D4×C59Q814yescomplemented
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121#95×C59D411yescharacteristic
122#95×C59D411yescharacteristic
123C24×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59C22×C412yes—
128C2×D4×C59C2×Q812yes—
129C2×D4×C59#4112yes—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—24no—
134C22×C4×C59—24no—
135C22×C4×C59—24no—
136C22×C4×C59—81no—
137C22×C4×C59—81no—
138C22×C4×C59—81no—
139C22×C4×C59—81no—
140C22×C4×C59—81no—
141C22×C4×C59—81no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×Dic118—24no—
154C2×Dic118—81no—
155C2×Dic118—81no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59C22×C412yes—
160C4⋊C4×C59C2×Q812yes—
161C4⋊C4×C59#4112yes—
162C22⋊C4×C59—24no—
163C22⋊C4×C59—24no—
164C22⋊C4×C59—24no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—81no—
168C42×C59—81no—
169C42×C59—81no—
170C42×C59C22×C412yes—
171C42×C59C2×Q812yes—
172C42×C59#4112yes—
173C42×C59#4112yes—
174C4⋊Dic59—24no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183Dic59⋊C4—81no—
184Dic59⋊C4—81no—
185C4×Dic59—81no—
186C4×Dic59—81no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8C22×C412yes—
198C2×C59⋊C8C22×C412yes—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59—41no—
206C23×C59C22×D411yescharacteristic
207D4×C59—22no—
208D4×C59—22no—
209D4×C59—22no—
210D4×C59C4×Q814yes—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—84no—
235C2×C4×C59—84no—
236C2×C4×C59—84no—
237C2×C4×C59C4×D412yes—
238C2×C4×C59C4×D412yes—
239C2×C4×C59C22⋊Q812yes—
240C2×C4×C59C22⋊Q812yes—
241C2×C4×C59C23×C411yescharacteristic
242C2×C4×C59C22×Q811yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259Dic118—84no—
260C59⋊C8—22no—
261C59⋊C8—22no—
262C59⋊C8—41no—
263C59⋊C8—41no—
264#21623—591nomaximal, Sylow
265C22×C59—24no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×C22⋊Q811yescharacteristic
274C22×C59#52511yescharacteristic
275C22×C59#57611yescharacteristic
276C22×C59#57711yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57711yescharacteristic
279C4×C59—22no—
280C4×C59—22no—
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—42no—
284C4×C59—41no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59C2×C4×Q811yescharacteristic
289C4×C59C4×#4111yescharacteristic
290C4×C59#52911yescharacteristic
291C4×C59#53311yescharacteristic
292Dic59—42no—
293Dic59—42no—
294Dic59—81no—
295Dic59—81no—
296C2×#476—591no—
297#2331—591no—
298#2275D5911yescharacteristic
299#1988—591no—
300#1949—591no—
301#1920—591no—
302#1919—591no—
303#1864—591no—
304#1862—591no—
305#1804—591no—
306#1776—591no—
307C4×#106—591no—
308#1768—591no—
309#1762—591no—
310C4×#102—591no—
311C2×C59—44no—
312C2×C59C4×C22⋊Q811yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#294911yescharacteristic
315C2×C59#308111yescharacteristic
316C2×C59#308411yescharacteristic
317C2×C59#311811yescharacteristic
318C2×C59#312511yescharacteristic
319C2×C4⋊Q8—591no—
320C2×C4×D4C2×D5911yescharacteristic
321C2×C4×D4C2×D5911yescharacteristic
322C2×C4×D4C2×D5911yescharacteristic
323#476—598no—
324C2×#107—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×#106—591no—
328C2×C4⋊C8—591no—
329C2×C4⋊C8—591no—
330C2×#102—591no—
331C2×#102—591no—
332C2×#102—591no—
333C2×#102—591no—
334C2×#97—591no—
335C2×C4×C8—591no—
336#391—1181no—
337#391—1181no—
338#389—591no—
339#389—591no—
340#387—1181no—
341#387—1181no—
342#382—591no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4C2×D5911yescharacteristic
346C4×C4⋊C4C2×D5911yescharacteristic
347C4×C4⋊C4C2×D5911yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4—21no—
350C4×C22⋊C4—21no—
351#376C2×D5911yescharacteristic
352C2×#95—21no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#336—591no—
362#336—591no—
363C59#2162311yescharacteristic, complemented, Sylow
364C22×Q8—1181no—
365C22×D4C22×D5911yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C22×C8—591no—
370C22×C8—591no—
371C22×C8—591no—
372C22×C8—591no—
373C4⋊Q8—594no—
374C4⋊Q8—1182no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4Dic11814yescomplemented
380C4×D4C4×D5914yes—
381C2×C4⋊C4—591no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—1181no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C4⋊C4C22×D5911yescharacteristic
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1181no—
398C2×C42—1181no—
399C2×C42C22×D5911yescharacteristic
400C2×C42C22×D5911yescharacteristic
401C2×C42C22×D5911yescharacteristic
402#107—1182no—
403#106—594no—
404#106—594no—
405#106—1182no—
406C4⋊C8—594no—
407C4⋊C8—1182no—
408#102—594no—
409#102—594no—
410#102—1182no—
411#102—1182no—
412#97—1182no—
413C4×C8—1182no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95C22⋊D5911yescharacteristic
424#95C22⋊D5911yescharacteristic
425#95C22⋊D5911yescharacteristic
426#95C22⋊D5911yescharacteristic
427C24—21no—
428C2×Q8—1184no—
429C2×Q8—4721no—
430C2×Q8—4721no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic11812yes—
435C2×D4C2×C4×D5912yes—
436C2×D4#1174568512yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—592no—
461C2×C8—592no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—592no—
476C4⋊C4—592no—
477C4⋊C4—1184no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—4721no—
485C4⋊C4—4721no—
486C4⋊C4C2×Dic11812yes—
487C4⋊C4C2×C4×D5912yes—
488C4⋊C4#1174568512yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—81no—
497C42—4721no—
498C42—4721no—
499C42C2×Dic11812yes—
500C42C2×C4×D5912yes—
501C42#1174568512yes—
502C42#1174568512yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D5911yescharacteristic
511Q8—4724no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C4×Dic11814yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—1182no—
543C2×C4—1182no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4C4×C22⋊D5912yes—
555C2×C4C4×C22⋊D5912yes—
556C2×C4#1427370612yes—
557C2×C4#1427370612yes—
558C2×C4C22×C4×D5911yescharacteristic
559C2×C4C22×Dic11811yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568511yescharacteristic
562C2×C4C2×#1174568511yescharacteristic
563C2×C4C2×#1174568511yescharacteristic
564C8—1182no—
565C8—1182no—
566C8—2361no—
567C8—2361no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C2×C4×C22⋊D5911yescharacteristic
576C22C2×#1427370611yescharacteristic
577C22#1683791311yescharacteristic
578C22#1683820511yescharacteristic
579C22#1683821011yescharacteristic
580C22#1683821311yescharacteristic
581C22#1683821311yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—4721no—
592C4—4721no—
593C4—2362no—
594C4—2362no—
595C4C2×C4×Dic11811yescharacteristic
596C4C4×#1174568511yescharacteristic
597C4#1683793611yescharacteristic
598C4#1683795911yescharacteristic
599C2—44no—
600C2C4×#1427370611yescharacteristic
601C2#2143292411yescharacteristic
602C2#2143295311yescharacteristic
603C2#2143392711yescharacteristic
604C2#2143395811yescharacteristic
605C2#2143428011yescharacteristic
606C2#2143437311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2275C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2275D59
hypocenterC59#21623
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #21426844C2semidirect
3. #21425048C2semidirect
4. #21424864C2semidirect
5. #21424681C2semidirect
6. #21424674C2semidirect
10. #21423994C2semidirect
11. C4×#14273658C2semidirect
12. #21423953C2semidirect
17. #16837692C4semidirect
66. #14273727C2×C4semidirect
78. #14273658C2×C4semidirect
79. #14273658C2×C4semidirect
81. C4⋊(C59⋊C8)C2×C4semidirect
103. C4×D4×C59Q8semidirect
363. C59#21623semidirect
379. C4×D4Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168376922
#214268442
#214250482
#214248642
#214246812
#214246742
#214243572
#214243432
#214241062
#214239942
C4×#142736582
#214239532
#214239212
C4×#142736482
C59×#22752
#2162359