Finite Group Databasebeta

← #28610050

Subgroups — #28610050

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837735C211yesmaximal, characteristic
2#21426971C211yesmaximal, characteristic
3#21424831C211yesmaximal, characteristic, complemented
4#21424789C211yesmaximal, characteristic, complemented
5#21424727C211yesmaximal, characteristic, complemented
6#21424667C211yesmaximal, characteristic, complemented
7#21424664C211yesmaximal, characteristic, complemented
8#21424520C211yesmaximal, characteristic
9#21424364C211yesmaximal, characteristic, complemented
10#21424289C211yesmaximal, characteristic
11#21423994C211yesmaximal, characteristic, complemented
12#21423926C211yesmaximal, characteristic, complemented
13#21423874C211yesmaximal, characteristic
14C2×#16837164C211yesmaximal, characteristic
15C59×#2360C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837735C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837285—21no—
33#16837285—21no—
34#16837262C2211yescharacteristic
35#16837260—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C22⋊C4×Dic59—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#387—21no—
56C59×#385—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C4×C22⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59C2×C414yes—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C4×Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660C2×C414yescomplemented
81#14273660C2×C414yescomplemented
82#14273658—22no—
83#14273658—41no—
84#14273651—22no—
85#14273651—41no—
86#14273651C2×C414yescomplemented
87#14273651C2×C414yescomplemented
88C22⋊(C59⋊C8)C2×C414yes—
89#14273638—22no—
90#14273638—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59—21no—
98C23×C4×C59C2311yescharacteristic
99C22⋊Q8×C59—22no—
100C22⋊Q8×C59C2×C414yes—
101C4×Q8×C59—22no—
102C4×Q8×C59—41no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—22no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—41no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59C22×C412yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—81no—
143C22×C4×C59C22×C412yes—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59#4112yes—
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—24no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—81no—
157C22×Dic59—81no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—81no—
166C4⋊C4×C59C22×C412yes—
167C4⋊C4×C59#4112yes—
168C22⋊C4×C59—24no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176C42×C59—81no—
177C22⋊Dic59—42no—
178C4⋊Dic59—24no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C412yes—
186C4⋊Dic59C22×C412yes—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—81no—
190C4×Dic59—81no—
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⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—44no—
201C2×C59⋊C8C22×C412yes—
202C2×C59⋊C8C22×C412yes—
203C2×C59⋊C8#4112yes—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4×D412yes—
209C23×C59#13712yes—
210C23×C59#13712yes—
211C23×C59C22×D411yescharacteristic
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—22no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—84no—
247C2×C4×C59C4×D412yes—
248C2×C4×C59C23×C411yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—22no—
255C2×Dic59—22no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—84no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271#22932—591nomaximal, Sylow
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—22no—
276C22×C59—22no—
277C22×C59—22no—
278C22×C59#44114yes—
279C22×C59C2×C4×D411yescharacteristic
280C22×C59#54611yescharacteristic
281C22×C59#54711yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57711yescharacteristic
286C4×C59—22no—
287C4×C59—41no—
288C4×C59—41no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—81no—
295C4×C59C4×#4111yescharacteristic
296C4×C59#52011yescharacteristic
297C4×C59#55111yescharacteristic
298C4×C59#55211yescharacteristic
299Dic59—42no—
300Dic59—42no—
301Dic59—81no—
302Dic59—81no—
303C2×#484—591no—
304#2360D5911yescharacteristic
305#2345—591no—
306#1943—591no—
307#1935—591no—
308#1927—591no—
309#1918—591no—
310#1918—591no—
311#1892—591no—
312#1865—591no—
313#1853—591no—
314#1776—591no—
315#1763—591no—
316#1752—591no—
317C2×#340—591no—
318C2×C59—24no—
319C2×C59#235911yescharacteristic
320C2×C59C2×#44111yescharacteristic
321C2×C59#294611yescharacteristic
322C2×C59#331411yescharacteristic
323C2×C59#332011yescharacteristic
324C2×C59#334411yescharacteristic
325C2×C59#334611yescharacteristic
326C23×C8—591no—
327C2×C22⋊Q8C2×D5911yescharacteristic
328C2×C4×Q8C2×D5911yescharacteristic
329C22×C4⋊C4—591no—
330C22×C4⋊C4C2×D5911yescharacteristic
331#484—598no—
332C2×#107—591no—
333C2×#107—591no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×C22⋊C8—591no—
341C2×#97—591no—
342C2×#97—591no—
343#391C2×D5911yescharacteristic
344#389—21no—
345#389—591no—
346#389—591no—
347#389C2×D5911yescharacteristic
348#387—21no—
349#387—1181no—
350#387—1181no—
351#385—21no—
352#380—591no—
353#380—1181no—
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4C2×D5911yescharacteristic
357C4×C22⋊C4—21no—
358C4×C22⋊C4—1181no—
359C4×C22⋊C4C2×D5911yescharacteristic
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336—591no—
369#336—591no—
370C59#2293211yescharacteristic, complemented, Sylow
371C22×Q8C22×D5911yescharacteristic
372C23×C4—21no—
373C23×C4—1181no—
374C23×C4C22×D5911yescharacteristic
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—1182no—
380C22×C8—1182no—
381C22×C8—2361no—
382C22⋊Q8—22no—
383C22⋊Q8C4×D5914yes—
384C4×Q8—22no—
385C4×Q8—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—41no—
392C2×C4⋊C4—594no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—1182no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4C22⋊D5911yescharacteristic
407C2×C22⋊C4C22⋊D5911yescharacteristic
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1181no—
411C2×C42—1181no—
412C2×C42C22×D5911yescharacteristic
413C2×C42C22×D5911yescharacteristic
414#107—594no—
415#107—594no—
416#106—1182no—
417#106—2361no—
418#103—594no—
419#103—594no—
420#103—1182no—
421#103—2361no—
422C22⋊C8—594no—
423#97—1182no—
424#97—2361no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95C22⋊D5911yescharacteristic
435#95C22⋊D5911yescharacteristic
436C24C2×C22⋊D5911yescharacteristic
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5912yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—81no—
454C22×C4—1184no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—4721no—
460C22×C4—4721no—
461C22×C4C2×C4×D5912yes—
462C22×C4#1174568512yes—
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C23×D5911yescharacteristic
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—592no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—2364no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—592no—
491C4⋊C4—592no—
492C4⋊C4—1184no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—4721no—
500C4⋊C4—4721no—
501C4⋊C4C2×C4×D5912yes—
502C4⋊C4#1174568712yes—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—2362no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—81no—
513C42—4721no—
514C42—4721no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C4×C22⋊D5912yes—
520C23#1427375712yes—
521C23#1427375812yes—
522C23C22×C22⋊D5911yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—1182no—
559C2×C4—1182no—
560C2×C4—4724no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4C4×C22⋊D5912yes—
572C2×C4C22×C4×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4C2×#1174568711yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—1182no—
579C8—1182no—
580C8—1182no—
581C8—2361no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1683750714yes—
589C22C2×C4×C22⋊D5911yescharacteristic
590C22#1683804611yescharacteristic
591C22#1683805111yescharacteristic
592C22C2×#1427375711yescharacteristic
593C22C2×#1427375811yescharacteristic
594C22#1683820511yescharacteristic
595C22#1683821011yescharacteristic
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—81no—
605C4—4721no—
606C4—4721no—
607C4—2362no—
608C4—2362no—
609C4C4×#1174568711yescharacteristic
610C4#1683788111yescharacteristic
611C4#1683808111yescharacteristic
612C4#1683809211yescharacteristic
613C2—24no—
614C2#2142708211yescharacteristic
615C2C2×#1683750711yescharacteristic
616C2#2143292411yescharacteristic
617C2#2143617711yescharacteristic
618C2#2143626711yescharacteristic
619C2#2143655111yescharacteristic
620C2#2143657511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2360C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2360D59
hypocenterC59#22932
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
3. #21424831C2semidirect
4. #21424789C2semidirect
5. #21424727C2semidirect
6. #21424667C2semidirect
7. #21424664C2semidirect
9. #21424364C2semidirect
11. #21423994C2semidirect
12. #21423926C2semidirect
18. #16837735C4semidirect
80. #14273660C2×C4semidirect
81. #14273660C2×C4semidirect
86. #14273651C2×C4semidirect
87. #14273651C2×C4semidirect
370. C59#22932semidirect

Maximal subgroups

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

subgroupindex
C2×#168377352
#214269712
#214248312
#214247892
#214247272
#214246672
#214246642
#214245202
#214243642
#214242892
#214239942
#214239262
#214238742
C2×#168371642
C59×#23602
#2293259