Finite Group Databasebeta

← #28600522

Subgroups — #28600522

Order 15104.

780 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837740C211yesmaximal, characteristic
2#21426716C211yesmaximal, characteristic
3#21424836C211yesmaximal, characteristic, complemented
4#21424826C211yesmaximal, characteristic, complemented
5#21424789C211yesmaximal, characteristic, complemented
6#21424775C211yesmaximal, characteristic, complemented
7#21424674C211yesmaximal, characteristic, complemented
8#21424667C211yesmaximal, characteristic, complemented
9#21424404C211yesmaximal, characteristic
10#21424350C211yesmaximal, characteristic, complemented
11#21424350C211yesmaximal, characteristic, complemented
12#21424283C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14#21423862C211yesmaximal, characteristic
15C59×#2323C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837740C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837274—21no—
36#16837274—21no—
37#16837271—21no—
38#16837267—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
51C2×#117×C59C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#387—21no—
58C59×#380C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273686—22no—
66#14273686C2×C414yes—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—41no—
69C2×C4⋊Dic59—41no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80#14273660—41no—
81#14273660—41no—
82#14273660C2×C414yescomplemented
83#14273658C2×C414yescomplemented
84#14273651—41no—
85#14273651—41no—
86#14273651C2×C414yescomplemented
87#14273651C2×C414yescomplemented
88C22⋊(C59⋊C8)—22no—
89C22⋊(C59⋊C8)C2×C414yes—
90#14273638—41no—
91C4×C59⋊C8—41no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C22×Q8×C59C2311yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C22⋊Q8×C59—22no—
99C22⋊Q8×C59C2×C414yes—
100C4×Q8×C59—41no—
101C4×Q8×C59—41no—
102#117×C59—41no—
103#117×C59—41no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C22⋊C4×C59D411yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
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×C59C22×C412yes—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4112yes—
147C22×C4×C59C2411yescharacteristic
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—42no—
153C22×Dic59—81no—
154C22×Dic59—81no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—41no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C22×C412yes—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—41no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—42no—
175C42×C59—81no—
176C22⋊Dic59—24no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C22×C412yes—
185C4⋊Dic59C22×C412yes—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C4×Dic59—24no—
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—21no—
201C2×C59⋊C8C22×C412yes—
202C2×C59⋊C8C22×C412yes—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59C4×D412yes—
210C23×C59C22×D411yescharacteristic
211Q8×C59—22no—
212Q8×C59—22no—
213Q8×C59—41no—
214Q8×C59—41no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
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—41no—
236C2×C4×C59—41no—
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×C59C4×D412yes—
246C2×C4×C59C23×C411yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59H(5,𝔽2)11yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C2×Dic59—22no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
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—84no—
265C59⋊C8—22no—
266C59⋊C8—22no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269#22094—591nomaximal, Sylow
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59C2×C4×D411yescharacteristic
278C22×C59#53411yescharacteristic
279C22×C59#53611yescharacteristic
280C22×C59#57611yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C22×C59#57811yescharacteristic
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—41no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59#52011yescharacteristic
294C4×C59#52011yescharacteristic
295C4×C59#53911yescharacteristic
296C4×C59#53911yescharacteristic
297Dic59—42no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301C2×#485—591no—
302#2323D5911yescharacteristic
303#2317—591no—
304#1944—591no—
305#1942—591no—
306#1935—591no—
307#1933—591no—
308#1919—591no—
309#1918—591no—
310#1872—591no—
311#1863—591no—
312#1863—591no—
313#1852—591no—
314#1825—591no—
315#1750—591no—
316C2×C59—42no—
317C2×C59#232211yescharacteristic
318C2×C59#294711yescharacteristic
319C2×C59#295111yescharacteristic
320C2×C59#319811yescharacteristic
321C2×C59#320211yescharacteristic
322C2×C59#324211yescharacteristic
323C2×C59#324311yescharacteristic
324C2×C22⋊Q8C2×D5911yescharacteristic
325C2×C4×Q8C2×D5911yescharacteristic
326C2×#117—591no—
327C2×#117C2×D5911yescharacteristic
328C22×C4⋊C4—591no—
329#485—598no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#106—591no—
334C2×#103—591no—
335C2×#103—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×C22⋊C8—591no—
339C2×C22⋊C8—591no—
340C2×#97—591no—
341C2×C4×C8—591no—
342#391C2×D5911yescharacteristic
343#389—21no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389C2×D5911yescharacteristic
349#388—21no—
350#388—21no—
351#387—21no—
352#385—1181no—
353#385—1181no—
354#384—1181no—
355#382—1181no—
356#380C2×D5911yescharacteristic
357C4×C4⋊C4C2×D5911yescharacteristic
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#336—591no—
367#336—591no—
368C59#2209411yescharacteristic, complemented, Sylow
369C22×Q8C22×D5911yescharacteristic
370C23×C4—1181no—
371C23×C4C22×D5911yescharacteristic
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—591no—
376C22⋊Q8—22no—
377C22⋊Q8C4×D5914yes—
378C4×Q8—41no—
379C4×Q8—41no—
380#117—41no—
381#117—41no—
382#117—594no—
383#117—1182no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—2361no—
397C2×C4⋊C4—2361no—
398C2×C4⋊C4C22⋊D5911yescharacteristic
399C2×C4⋊C4C22⋊D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—21no—
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×C42C22×D5911yescharacteristic
412C2×C42C22×D5911yescharacteristic
413#107—594no—
414#107—2361no—
415#107—2361no—
416#106—594no—
417#103—594no—
418#103—594no—
419#103—2361no—
420#103—2361no—
421C22⋊C8—594no—
422C22⋊C8—1182no—
423#97—2361no—
424C4×C8—2361no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434C24C2×C22⋊D5911yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D5912yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—4721no—
454C22×C4—4721no—
455C22×C4—2362no—
456C22×C4C2×C4×D5912yes—
457C22×C4#1174568512yes—
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C23×D5911yescharacteristic
464C2×C8—592no—
465C2×C8—592no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—592no—
487C4⋊C4—592no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—4721no—
498C4⋊C4—2362no—
499C4⋊C4C2×C4×D5912yes—
500C22⋊C4—24no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—81no—
504C22⋊C4—1184no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—81no—
511C42—1184no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C4×C22⋊D5912yes—
519C23C22×C22⋊D5911yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
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—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—4724no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4C4×C22⋊D5912yes—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4#1427377211yescharacteristic
571C2×C4#1427377211yescharacteristic
572C2×C4#1427377411yescharacteristic
573C2×C4#1427377411yescharacteristic
574C8—1182no—
575C8—1182no—
576C8—2361no—
577C8—2361no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×C22⋊D5911yescharacteristic
586C22#1683796111yescharacteristic
587C22#1683797311yescharacteristic
588C22#1683820511yescharacteristic
589C22#1683821011yescharacteristic
590C22#1683821711yescharacteristic
591C22#1683821711yescharacteristic
592C4—22no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—4721no—
602C4—4721no—
603C4—2362no—
604C4—2362no—
605C4#1683788111yescharacteristic
606C4#1683788111yescharacteristic
607C4#1683799411yescharacteristic
608C4#1683799411yescharacteristic
609C2—42no—
610C2#2142676111yescharacteristic
611C2#2143293911yescharacteristic
612C2#2143296911yescharacteristic
613C2#2143494211yescharacteristic
614C2#2143498711yescharacteristic
615C2#2143535811yescharacteristic
616C2#2143537311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2323C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#2323D59
hypocenterC59#22094
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. #21424836C2semidirect
4. #21424826C2semidirect
5. #21424789C2semidirect
6. #21424775C2semidirect
7. #21424674C2semidirect
8. #21424667C2semidirect
10. #21424350C2semidirect
11. #21424350C2semidirect
18. #16837740C4semidirect
82. #14273660C2×C4semidirect
83. #14273658C2×C4semidirect
86. #14273651C2×C4semidirect
87. #14273651C2×C4semidirect
368. C59#22094semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
#214267162
#214248362
#214248262
#214247892
#214247752
#214246742
#214246672
#214244042
#214243502
#214242832
#214241882
#214238622
C59×#23232
#2209459