Finite Group Databasebeta

← #28661504

Subgroups — #28661504

Order 15104.

742 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428531C211yesmaximal, characteristic, complemented
2#21427138C211yesmaximal, characteristic
3#21425502C211yesmaximal, characteristic, complemented
4#21425480C211yesmaximal, characteristic, complemented
5#21425275C211yesmaximal, characteristic
6#21424851C211yesmaximal, characteristic, complemented
7#21424828C211yesmaximal, characteristic, complemented
8#21424699C211yesmaximal, characteristic, complemented
9#21424690C211yesmaximal, characteristic, complemented
10#21424677C211yesmaximal, characteristic, complemented
11#21424355C211yesmaximal, characteristic, complemented
12#21424293C211yesmaximal, characteristic, complemented
13#21424192C211yesmaximal, characteristic, complemented
14#102×Dic59C211yesmaximal, characteristic, complemented
15C59×#2033C211yesmaximal, characteristic
16C2×D4×Dic59C2211yescharacteristic
17C2×#14273686C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837343—21no—
27#16837325—21no—
28#16837305—21no—
29#16837305C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837296—21no—
34#16837295—21no—
35#16837287—21no—
36#16837282—21no—
37#16837271C2211yescharacteristic
38#16837262C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C22⋊C4×Dic59—21no—
42C4×C22⋊Dic59C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#121×C59C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62D4×Dic59—22no—
63D4×Dic59—41no—
64#14273686—22no—
65#14273686—41no—
66C2×C4⋊Dic59—21no—
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×Dic59⋊C4—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59D411yescharacteristic
81C2×C22⋊Dic59D411yescharacteristic
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—22no—
88#14273660—41no—
89#14273658—22no—
90Dic59⋊C8—22no—
91#14273648—22no—
92#14273648—22no—
93C22⋊(C59⋊C8)—22no—
94C8×Dic59—41no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C22×D4×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108#121×C59—22no—
109#121×C59—22no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115#107×C59—22no—
116#102×C59—22no—
117#102×C59—41no—
118C22⋊C8×C59—41no—
119#95×C59—21no—
120C24×C59—21no—
121C24×C59C2×D411yescharacteristic
122C2×D4×C59—24no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—81no—
129C2×D4×C59—81no—
130C2×D4×C59#4112yes—
131C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59#4112yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—81no—
167C22⋊Dic59—22no—
168C22⋊Dic59—22no—
169C22⋊Dic59—42no—
170C22⋊Dic59—42no—
171C22⋊Dic59—42no—
172C22⋊Dic59—81no—
173C22⋊Dic59—81no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—42no—
182C4⋊Dic59—81no—
183C4⋊Dic59#4112yes—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187C4×Dic59—21no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8#4112yes—
200C23×C59—22no—
201C23×C59—21no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59C22×D411yescharacteristic
215D4×C59—22no—
216D4×C59—22no—
217D4×C59—41no—
218D4×C59—41no—
219D4×C59—84no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
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—41no—
235C2×C4×C59—41no—
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—22no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—41no—
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—
265C2×Dic59#13512yes—
266C2×Dic59#13712yes—
267C8×C59—41no—
268C8×C59—41no—
269C59⋊C8—22no—
270C59⋊C8—41no—
271#27041—591nomaximal, Sylow
272C22×C59—41no—
273C22×C59—41no—
274C22×C59—41no—
275C22×C59—42no—
276C22×C59—41no—
277C22×C59—42no—
278C22×C59—41no—
279C22×C59—42no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—81no—
283C22×C59—81no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59#52511yescharacteristic
287C22×C59#54611yescharacteristic
288C22×C59#54711yescharacteristic
289C22×C59C2×#13511yescharacteristic
290C22×C59C2×#13711yescharacteristic
291C22×C59#57611yescharacteristic
292C22×C59#57611yescharacteristic
293C4×C59—42no—
294C4×C59—41no—
295C4×C59—81no—
296C4×C59—81no—
297C4×C59#52911yescharacteristic
298C4×C59#53911yescharacteristic
299C4×C59#55311yescharacteristic
300C4×C59#55611yescharacteristic
301Dic59—22no—
302Dic59—22no—
303Dic59—41no—
304Dic59—41no—
305Dic59—42no—
306Dic59—81no—
307Dic59—81no—
308Dic59—81no—
309#2497—591no—
310#2363—591no—
311#2071—591no—
312#2067—591no—
313#2033—591no—
314#2033D5911yescharacteristic
315#1947—591no—
316#1942—591no—
317#1923—591no—
318#1921—591no—
319#1919—591no—
320#1864—591no—
321#1854—591no—
322#1825—591no—
323C4×#102—591no—
324C2×C59—81no—
325C2×C59—82no—
326C2×C59#246611yescharacteristic
327C2×C59#310711yescharacteristic
328C2×C59#312211yescharacteristic
329C2×C59#330511yescharacteristic
330C2×C59#331411yescharacteristic
331C2×C59#333911yescharacteristic
332C2×C59#335711yescharacteristic
333C2×#121C2×D5911yescharacteristic
334C2×C4×D4—591no—
335C2×#117—591no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#107C2×D5911yescharacteristic
339C2×#106—591no—
340C2×C4⋊C8—591no—
341C2×#102—591no—
342C2×#102—591no—
343C2×#102C2×D5911yescharacteristic
344C2×#102C2×D5911yescharacteristic
345C2×C22⋊C8—591no—
346C2×C22⋊C8C2×D5911yescharacteristic
347C2×C4×C8—591no—
348#400—1181no—
349#396—1181no—
350#390—591no—
351#390—1181no—
352#389—591no—
353#389—591no—
354#389—591no—
355#388—1181no—
356#388—1181no—
357#387—1181no—
358#386—1181no—
359#384—591no—
360#380—591no—
361#380C2×D5911yescharacteristic
362C4×C4⋊C4—591no—
363C4×C4⋊C4—591no—
364C4×C22⋊C4—591no—
365C4×C22⋊C4—1181no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#340C2×D5911yescharacteristic
374#336—591no—
375#336—591no—
376C59#2704111yescharacteristic, complemented, Sylow
377C22×D4—21no—
378C22×D4C22×D5911yescharacteristic
379C23×C4—1181no—
380C23×C4C22×D5911yescharacteristic
381C22×C8—591no—
382C22×C8—591no—
383C22×C8C22×D5911yescharacteristic
384C22×C8C22×D5911yescharacteristic
385#121—22no—
386#121—22no—
387C4×D4—1182no—
388C4×D4—2361no—
389#117—1182no—
390#117—2361no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C4⋊C4C22×D5911yescharacteristic
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—591no—
406C2×C22⋊C4—591no—
407C2×C22⋊C4—1181no—
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42—1181no—
417#107—22no—
418#107—1182no—
419#107—2361no—
420#106—1182no—
421C4⋊C8—1182no—
422#102—22no—
423#102—41no—
424#102—1182no—
425#102—1182no—
426C22⋊C8—41no—
427C22⋊C8—1182no—
428C4×C8—2361no—
429#95—21no—
430#95—591no—
431#95—591no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438C24—21no—
439C24D4×D5911yescharacteristic
440C2×D4—24no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4#1174568712yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—591no—
459C22×C4—591no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—4721no—
467C22×C4—2362no—
468C22×C4—2362no—
469C22×C4D4×D5911yescharacteristic
470C22×C4C23×D5911yescharacteristic
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—592no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—592no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—4721no—
497C4⋊C4—4721no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C4⋊C4—2362no—
501C4⋊C4#1174568712yes—
502C22⋊C4—24no—
503C22⋊C4—81no—
504C22⋊C4—1182no—
505C22⋊C4—1182no—
506C22⋊C4—4721no—
507C22⋊C4—4721no—
508C22⋊C4—2362no—
509C22⋊C4—2362no—
510C22⋊C4—2362no—
511C42—1181no—
512C42—1181no—
513C42—1181no—
514C42—1181no—
515C42—4721no—
516C42—4721no—
517C42—2362no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D4×D5911yescharacteristic
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537D4—84no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
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—592no—
555C2×C4—592no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1181no—
561C2×C4—1181no—
562C2×C4—1181no—
563C2×C4—1181no—
564C2×C4—1181no—
565C2×C4—1182no—
566C2×C4—1181no—
567C2×C4—4724no—
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—2361no—
578C2×C4—2361no—
579C2×C4#41×D5911yescharacteristic
580C2×C4#41×D5911yescharacteristic
581C2×C4C2×#1174568711yescharacteristic
582C2×C4C2×#1174568711yescharacteristic
583C2×C4#1427377211yescharacteristic
584C2×C4#1427377411yescharacteristic
585C8—41no—
586C8—41no—
587C8—1182no—
588C8—2361no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1683790811yescharacteristic
604C22#1683804111yescharacteristic
605C22#1683805011yescharacteristic
606C22#135×D5911yescharacteristic
607C22#137×D5911yescharacteristic
608C22#1683820311yescharacteristic
609C22#1683820611yescharacteristic
610C4—41no—
611C4—42no—
612C4—81no—
613C4—81no—
614C4—1182no—
615C4—1182no—
616C4—4721no—
617C4—4721no—
618C4—4721no—
619C4—2362no—
620C4—2361no—
621C4—2361no—
622C4#1683792911yescharacteristic
623C4#1683799711yescharacteristic
624C4#1683809911yescharacteristic
625C4#1683812111yescharacteristic
626C2—81no—
627C2—82no—
628C2#2142812511yescharacteristic
629C2#2143416011yescharacteristic
630C2#2143433311yescharacteristic
631C2#2143604111yescharacteristic
632C2#2143617211yescharacteristic
633C2#2143649111yescharacteristic
634C2#2143671511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2033C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2033D59
hypocenterC59#27041
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
1. #21428531C2semidirect
3. #21425502C2semidirect
4. #21425480C2semidirect
6. #21424851C2semidirect
7. #21424828C2semidirect
8. #21424699C2semidirect
9. #21424690C2semidirect
10. #21424677C2semidirect
11. #21424355C2semidirect
12. #21424293C2semidirect
13. #21424192C2semidirect
14. #102×Dic59C2semidirect
376. C59#27041semidirect

Maximal subgroups

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

subgroupindex
#214285312
#214271382
#214255022
#214254802
#214252752
#214248512
#214248282
#214246992
#214246902
#214246772
#214243552
#214242932
#214241922
#102×Dic592
C59×#20332
#2704159