Finite Group Databasebeta

← #28657151

Subgroups — #28657151

Order 15104.

720 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427121C211yesmaximal, characteristic
2#21425418C211yesmaximal, characteristic, complemented
3#21425299C211yesmaximal, characteristic
4#21425244C211yesmaximal, characteristic, complemented
5#21425104C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21424789C211yesmaximal, characteristic, complemented
8#21424734C211yesmaximal, characteristic, complemented
9#21424711C211yesmaximal, characteristic
10#21424688C211yesmaximal, characteristic, complemented
11#21424440C211yesmaximal, characteristic, complemented
12#21424409C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic
14#21423929C211yesmaximal, characteristic, complemented
15C59×#2489C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31#16837308C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837284—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837253—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#125×C59C2211yescharacteristic
49C2×C22⋊Q8×C59C2211yescharacteristic
50C59×#400—21no—
51C59×#397—21no—
52C59×#391C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#388—21no—
55C59×#382—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C2×#11745672—22no—
61C2×#11745672—41no—
62C2×#11745672—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273706—22no—
68#14273706—22no—
69C4×Dic118—22no—
70C4×Dic118—41no—
71#14273686—22no—
72#14273686—41no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84#14273660—22no—
85C4⋊(C59⋊C8)—41no—
86#14273651—22no—
87#14273651—22no—
88#14273651—41no—
89#14273650—22no—
90#14273650—22no—
91#14273650—41no—
92C22⋊(C59⋊C8)—41no—
93C22⋊(C59⋊C8)—41no—
94#14273638—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C22×Q8×C59C2311yescharacteristic
100C23×C4×C59C2311yescharacteristic
101#125×C59—22no—
102#125×C59—41no—
103C22⋊Q8×C59—22no—
104C22⋊Q8×C59—41no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59D411yescharacteristic
114C2×C22⋊C4×C59D411yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59—21no—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59C2×D411yescharacteristic
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×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—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
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—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118#4112yes—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59#4112yes—
168C22⋊C4×C59—41no—
169C22⋊C4×C59—41no—
170C22⋊C4×C59—42no—
171C22⋊C4×C59—42no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—42no—
177C22⋊Dic59—24no—
178C22⋊Dic59—42no—
179#11745672—44no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59#4112yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189C4×Dic59—42no—
190C4×Dic59—42no—
191C4×Dic59—42no—
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⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59C22×D411yescharacteristic
212Q8×C59—41no—
213Q8×C59—41no—
214Q8×C59—41no—
215Q8×C59—41no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
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—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
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×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59H(5,𝔽2)11yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×Dic59—22no—
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—
265Dic118—22no—
266Dic118—22no—
267Dic118—41no—
268Dic118—41no—
269C59⋊C8—41no—
270C59⋊C8—41no—
271C59⋊C8—41no—
272C59⋊C8—41no—
273#26749—591nomaximal, Sylow
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59#52511yescharacteristic
282C22×C59#54611yescharacteristic
283C22×C59#54611yescharacteristic
284C22×C59#57511yescharacteristic
285C22×C59#57511yescharacteristic
286C22×C59#57811yescharacteristic
287C22×C59#57811yescharacteristic
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—41no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—81no—
295C4×C59—81no—
296C4×C59—81no—
297C4×C59#52911yescharacteristic
298C4×C59#53811yescharacteristic
299C4×C59#55511yescharacteristic
300C4×C59#55911yescharacteristic
301Dic59—42no—
302Dic59—42no—
303Dic59—42no—
304Dic59—81no—
305#2489D5911yescharacteristic
306#2362—591no—
307#2056—591no—
308#2036—591no—
309#2026—591no—
310#2001—591no—
311#1989—591no—
312#1935—591no—
313#1928—591no—
314#1924—591no—
315#1921—591no—
316#1878—591no—
317#1873—591no—
318#1825—591no—
319#1763—591no—
320C2×C59—81no—
321C2×C59#245911yescharacteristic
322C2×C59#311111yescharacteristic
323C2×C59#312711yescharacteristic
324C2×C59#331311yescharacteristic
325C2×C59#331511yescharacteristic
326C2×C59#331511yescharacteristic
327C2×C59#332711yescharacteristic
328C22×Q16—591no—
329C2×#125C2×D5911yescharacteristic
330C2×C22⋊Q8—591no—
331C2×C22⋊Q8C2×D5911yescharacteristic
332C2×C4×Q8—591no—
333C2×#117—591no—
334C2×#107—591no—
335C2×C4⋊C8—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×C22⋊C8—591no—
343C2×C22⋊C8—591no—
344C2×#97—591no—
345#400—21no—
346#397—21no—
347#391—591no—
348#391C2×D5911yescharacteristic
349#391C2×D5911yescharacteristic
350#389—591no—
351#388—21no—
352#388—1181no—
353#388—1181no—
354#387—1181no—
355#382—21no—
356#380C2×D5911yescharacteristic
357C4×C4⋊C4—591no—
358C4×C4⋊C4—1181no—
359C4×C4⋊C4C2×D5911yescharacteristic
360C4×C22⋊C4C2×D5911yescharacteristic
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#336—591no—
370#336—591no—
371C59#2674911yescharacteristic, complemented, Sylow
372C22×Q8—591no—
373C22×Q8C22×D5911yescharacteristic
374C23×C4C22×D5911yescharacteristic
375C2×Q16—1182no—
376C2×Q16—2361no—
377C2×Q16—2361no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8—591no—
382#125—22no—
383#125—41no—
384C22⋊Q8—22no—
385C22⋊Q8—41no—
386C22⋊Q8—1182no—
387C22⋊Q8—1182no—
388C4×Q8—1182no—
389C4×Q8—2361no—
390#117—1182no—
391#117—2361no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
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—21no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4C22⋊D5911yescharacteristic
409C2×C22⋊C4C22⋊D5911yescharacteristic
410C2×C42—21no—
411C2×C42—1181no—
412C2×C42—1181no—
413C2×C42—1181no—
414C2×C42C22×D5911yescharacteristic
415C2×C42C22×D5911yescharacteristic
416#107—1182no—
417C4⋊C8—2361no—
418#103—1182no—
419#103—1182no—
420#103—1182no—
421#103—1182no—
422#103—2361no—
423#103—2361no—
424C22⋊C8—2361no—
425C22⋊C8—2361no—
426#97—2361no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95C22⋊D5911yescharacteristic
438#95C22⋊D5911yescharacteristic
439C24C2×C22⋊D5911yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—592no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C23×D5911yescharacteristic
472Q16—2364no—
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—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—592no—
494C4⋊C4—592no—
495C4⋊C4—1184no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4#1174568712yes—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—1184no—
508C22⋊C4—2362no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—2362no—
515C42—2362no—
516C42—2362no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D5911yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—1182no—
530Q8—1182no—
531Q8—2361no—
532Q8—2361no—
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—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
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—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—1182no—
565C2×C4—1182no—
566C2×C4—1182no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4#41×D5911yescharacteristic
577C2×C4#41×D5911yescharacteristic
578C2×C4#41×D5911yescharacteristic
579C2×C4C2×#1174568711yescharacteristic
580C2×C4#1427377211yescharacteristic
581C2×C4#1427377311yescharacteristic
582C8—2361no—
583C8—2361no—
584C8—2361no—
585C8—2361no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1683790911yescharacteristic
594C22#1683804011yescharacteristic
595C22#1683804611yescharacteristic
596C22#1683820011yescharacteristic
597C22#1683820011yescharacteristic
598C22#1683821711yescharacteristic
599C22#1683821711yescharacteristic
600C4—41no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—4721no—
610C4—2362no—
611C4—2362no—
612C4—2362no—
613C4#1683793111yescharacteristic
614C4#1683798611yescharacteristic
615C4#1683811411yescharacteristic
616C4#1683813411yescharacteristic
617C2—81no—
618C2#2142802911yescharacteristic
619C2#2143419911yescharacteristic
620C2#2143439111yescharacteristic
621C2#2143616211yescharacteristic
622C2#2143618611yescharacteristic
623C2#2143619211yescharacteristic
624C2#2143635811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2489C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2489D59
hypocenterC59#26749
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #21425418C2semidirect
4. #21425244C2semidirect
6. #21425054C2semidirect
7. #21424789C2semidirect
8. #21424734C2semidirect
10. #21424688C2semidirect
11. #21424440C2semidirect
14. #21423929C2semidirect
371. C59#26749semidirect

Maximal subgroups

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

subgroupindex
#214271212
#214254182
#214252992
#214252442
#214251042
#214250542
#214247892
#214247342
#214247112
#214246882
#214244402
#214244092
#214241882
#214239292
C59×#24892
#2674959