Finite Group Databasebeta

← #28662000

Subgroups — #28662000

Order 15104.

740 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428704C211yesmaximal, characteristic, complemented
2#21425644C211yesmaximal, characteristic, complemented
3#21425418C211yesmaximal, characteristic, complemented
4#21425332C211yesmaximal, characteristic
5#21425190C211yesmaximal, characteristic
6#21424949C211yesmaximal, characteristic
7#21424737C211yesmaximal, characteristic, complemented
8#21424713C211yesmaximal, characteristic
9#21424681C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424437C211yesmaximal, characteristic, complemented
12#21424430C211yesmaximal, characteristic
13#21424106C211yesmaximal, characteristic
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2367C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30C59⋊#401—21no—
31#16837340—21no—
32#16837308C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837308C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837283—21no—
37#16837269—21no—
38#16837268C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#124×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#385—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C22×Dic118C2311yescharacteristic
60C2×#11745672—22no—
61C2×#11745672—41no—
62C2×#11745672—41no—
63C2×#11745670—22no—
64C2×#11745670—22no—
65C2×#11745670—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75#14273660—41no—
76#14273658—22no—
77C4⋊(C59⋊C8)—41no—
78#14273651—22no—
79#14273651—22no—
80#14273651—41no—
81#14273650—22no—
82#14273650—41no—
83#14273648—22no—
84#14273648—41no—
85#14273648—41no—
86C4×C59⋊C8—41no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100#124×C59—22no—
101#124×C59—22no—
102C4×Q8×C59—22no—
103C4×Q8×C59—41no—
104C4×D4×C59—22no—
105C4×D4×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59#4112yes—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59#4112yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—81no—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2411yescharacteristic
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59#4112yes—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—81no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—42no—
181C42×C59—81no—
182#11745672—44no—
183#11745670—44no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190Dic59⋊C4—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#4112yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216D4×C59—22no—
217D4×C59—22no—
218D4×C59—41no—
219D4×C59—41no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—22no—
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—21no—
238C2×C4×C59—21no—
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×C59#13512yes—
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C59H(5,𝔽2)11yescharacteristic
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—41no—
265C2×Dic59—41no—
266Dic118—41no—
267Dic118—41no—
268Dic118—41no—
269Dic118—41no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274#27074—591nomaximal, Sylow
275C22×C59—42no—
276C22×C59—42no—
277C22×C59—42no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59#52511yescharacteristic
283C22×C59#54611yescharacteristic
284C22×C59#55011yescharacteristic
285C22×C59C2×#13511yescharacteristic
286C22×C59#57511yescharacteristic
287C22×C59#57711yescharacteristic
288C22×C59#57811yescharacteristic
289C4×C59—22no—
290C4×C59—41no—
291C4×C59—41no—
292C4×C59—42no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—42no—
297C4×C59—81no—
298C4×C59#52911yescharacteristic
299C4×C59#53811yescharacteristic
300C4×C59#55111yescharacteristic
301C4×C59#55311yescharacteristic
302Dic59—81no—
303Dic59—81no—
304Dic59—81no—
305Dic59—81no—
306#2514—591no—
307#2367D5911yescharacteristic
308#2097—591no—
309#2056—591no—
310#2042—591no—
311#2015—591no—
312#1968—591no—
313#1928—591no—
314#1925—591no—
315#1920—591no—
316#1916—591no—
317#1878—591no—
318#1877—591no—
319#1804—591no—
320C4×#103—591no—
321C2×C59—82no—
322C2×C59#246711yescharacteristic
323C2×C59#310211yescharacteristic
324C2×C59#312011yescharacteristic
325C2×C59#330011yescharacteristic
326C2×C59#331111yescharacteristic
327C2×C59#339411yescharacteristic
328C2×C59#340211yescharacteristic
329C22×Q16—591no—
330C22×SD16—591no—
331C2×#124C2×D5911yescharacteristic
332C2×C4×Q8C2×D5911yescharacteristic
333C2×C4×D4C2×D5911yescharacteristic
334C2×#107—591no—
335C2×#106—591no—
336C2×C4⋊C8—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#103—591no—
340C2×#103—591no—
341C2×#103—591no—
342C2×#102—591no—
343C2×#102—591no—
344C2×#102—591no—
345C2×C4×C8—591no—
346#401—1181no—
347#400—1181no—
348#391—591no—
349#391—591no—
350#391—591no—
351#391C2×D5911yescharacteristic
352#390C2×D5911yescharacteristic
353#389—591no—
354#388—21no—
355#388—21no—
356#387—1181no—
357#385—21no—
358#383—1181no—
359#382—591no—
360C4×C4⋊C4—591no—
361C4×C4⋊C4C2×D5911yescharacteristic
362C4×C4⋊C4C2×D5911yescharacteristic
363C4×C22⋊C4—21no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336—591no—
369#336—591no—
370#336—591no—
371#336—591no—
372C59#2707411yescharacteristic, complemented, Sylow
373C22×Q8—591no—
374C22×Q8C22×D5911yescharacteristic
375C22×D4C22×D5911yescharacteristic
376C23×C4—21no—
377C2×Q16—1182no—
378C2×Q16—2361no—
379C2×Q16—2361no—
380C2×SD16—1182no—
381C2×SD16—1182no—
382C2×SD16—2361no—
383C22×C8—591no—
384C22×C8—591no—
385C22×C8—591no—
386C22×C8—591no—
387#124—22no—
388#124—22no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×D4—22no—
392C4×D4—41no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4C22⋊D5911yescharacteristic
399C2×C4⋊C4C22⋊D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411C2×C42C22×D5911yescharacteristic
412C2×C42C22×D5911yescharacteristic
413#107—2361no—
414#106—1182no—
415C4⋊C8—2361no—
416#103—1182no—
417#103—1182no—
418#103—1182no—
419#103—2361no—
420#103—2361no—
421#102—1182no—
422#102—2361no—
423#102—2361no—
424C4×C8—2361no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95C22⋊D5911yescharacteristic
439#95C22⋊D5911yescharacteristic
440C24—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8#1174568512yes—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4#1174568512yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4C2×C22⋊D5911yescharacteristic
470C22×C4C2×C22⋊D5911yescharacteristic
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C23×D5911yescharacteristic
476Q16—2364no—
477SD16—2364no—
478C2×C8—592no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—2362no—
503C4⋊C4#1174568712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—81no—
517C42—2362no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C22⋊D5911yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—2361no—
531Q8—2361no—
532Q8—2361no—
533Q8—2361no—
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—2361no—
566C2×C4—2361no—
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—2361no—
577C2×C4#1427374712yes—
578C2×C4#41×D5911yescharacteristic
579C2×C4C2×#1174568711yescharacteristic
580C2×C4C2×#1174568511yescharacteristic
581C2×C4C2×#1174568511yescharacteristic
582C2×C4#1427377211yescharacteristic
583C2×C4#1427377311yescharacteristic
584C8—1182no—
585C8—2361no—
586C8—2361no—
587C8—2361no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22#1683791311yescharacteristic
596C22#1683804611yescharacteristic
597C22#1683807111yescharacteristic
598C22C2×#1427374711yescharacteristic
599C22#1683820011yescharacteristic
600C22#1683821311yescharacteristic
601C22#1683821711yescharacteristic
602C4—22no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—81no—
611C4—4721no—
612C4—4721no—
613C4—4721no—
614C4—4721no—
615C4#1683793211yescharacteristic
616C4#1683798511yescharacteristic
617C4#1683808111yescharacteristic
618C4#1683809811yescharacteristic
619C2—82no—
620C2#2142814111yescharacteristic
621C2#2143411411yescharacteristic
622C2#2143431011yescharacteristic
623C2#2143598311yescharacteristic
624C2#2143613211yescharacteristic
625C2#2143708811yescharacteristic
626C2#2143717111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2367C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2367D59
hypocenterC59#27074
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #21428704C2semidirect
2. #21425644C2semidirect
3. #21425418C2semidirect
7. #21424737C2semidirect
9. #21424681C2semidirect
10. #21424655C2semidirect
11. #21424437C2semidirect
14. C4×#14273651C2semidirect
372. C59#27074semidirect

Maximal subgroups

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

subgroupindex
#214287042
#214256442
#214254182
#214253322
#214251902
#214249492
#214247372
#214247132
#214246812
#214246552
#214244372
#214244302
#214241062
C4×#142736512
C59×#23672
#2707459