Finite Group Databasebeta

← #28656836

Subgroups — #28656836

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427132C211yesmaximal, characteristic
2#21425411C211yesmaximal, characteristic, complemented
3#21425285C211yesmaximal, characteristic
4#21425239C211yesmaximal, characteristic, complemented
5#21425097C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21424775C211yesmaximal, characteristic, complemented
8#21424727C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic, complemented
10#21424681C211yesmaximal, characteristic, complemented
11#21424433C211yesmaximal, characteristic, complemented
12#21424409C211yesmaximal, characteristic, complemented
13#21424188C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2489C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×#14273701C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837302—21no—
33#16837300C2211yescharacteristic
34#16837291—21no—
35#16837278—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37#16837249—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×D236C2311yescharacteristic
60C22×C22⋊D59—21no—
61C22×C4×D59—21no—
62C2×#11745671—22no—
63C2×#11745671—41no—
64C2×#11745671—41no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273701—22no—
70#14273701—22no—
71C4×D236—22no—
72C4×D236—41no—
73#14273686—22no—
74#14273686—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×C22⋊Dic59—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273658—22no—
87C4⋊(C59⋊C8)—41no—
88#14273651—22no—
89#14273651—22no—
90#14273651—41no—
91#14273647—22no—
92#14273647—22no—
93#14273647—41no—
94C22⋊(C59⋊C8)—41no—
95C22⋊(C59⋊C8)—41no—
96C4×C59⋊C8—41no—
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
127C23×D59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×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×C59—21no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C2×C22⋊D59—24no—
151C2×C22⋊D59—81no—
152C2×C22⋊D59—81no—
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236#4112yes—
162C2×C4×D59—42no—
163C2×C4×D59—42no—
164C2×C4×D59—81no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—42no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59#4112yes—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—41no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—42no—
183C22⋊Dic59—42no—
184#11745671—44no—
185(C2×D59)⋊C4—24no—
186(C2×D59)⋊C4—42no—
187(C2×D59)⋊C4—42no—
188(C2×D59)⋊C4—81no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59#4112yes—
194C4⋊Dic59#4112yes—
195Dic59⋊C4—81no—
196C4×Dic59—42no—
197C4×Dic59—81no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59C22×D411yescharacteristic
218C22×D59—22no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225Q8×C59—41no—
226Q8×C59—41no—
227Q8×C59—41no—
228Q8×C59—41no—
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—22no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59#13512yes—
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59H(5,𝔽2)11yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C22⋊D59—84no—
267C2×Dic59—22no—
268C2×Dic59—22no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—22no—
277D236—22no—
278D236—41no—
279D236—41no—
280C4×D59—84no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#26728—591nomaximal, Sylow
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—41no—
293C22×C59#52511yescharacteristic
294C22×C59#54611yescharacteristic
295C22×C59#54611yescharacteristic
296C22×C59C2×#13511yescharacteristic
297C22×C59#57511yescharacteristic
298C22×C59#57711yescharacteristic
299C22×C59#57711yescharacteristic
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—22no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—81no—
314C4×C59—81no—
315C4×C59—81no—
316C4×C59#52911yescharacteristic
317C4×C59#53811yescharacteristic
318C4×C59#55511yescharacteristic
319C4×C59#55911yescharacteristic
320Dic59—42no—
321Dic59—42no—
322Dic59—81no—
323#2489D5911yescharacteristic, complemented
324#2363—591no—
325#2055—591no—
326#2034—591no—
327#2025—591no—
328#2000—591no—
329#1989—591no—
330#1933—591no—
331#1927—591no—
332#1923—591no—
333#1920—591no—
334#1877—591no—
335#1873—591no—
336#1825—591no—
337C4×#102—591no—
338C2×C59—81no—
339C2×C59#245911yescharacteristic
340C2×C59#310211yescharacteristic
341C2×C59#312311yescharacteristic
342C2×C59#329611yescharacteristic
343C2×C59#330311yescharacteristic
344C2×C59#331711yescharacteristic
345C2×C59#332611yescharacteristic
346D59—82no—
347C22×SD16—591no—
348C2×#125C2×D5911yescharacteristic
349C2×C22⋊Q8C2×D5911yescharacteristic
350C2×#121—591no—
351C2×C4×D4—591no—
352C2×#117—591no—
353C2×#106—591no—
354C2×C4⋊C8—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×C22⋊C8—591no—
362C2×C22⋊C8—591no—
363C2×C4×C8—591no—
364#400—21no—
365#397—21no—
366#391C2×D5911yescharacteristic
367#391C2×D5911yescharacteristic
368#390—591no—
369#390—1181no—
370#389—591no—
371#388—21no—
372#388—1181no—
373#386—1181no—
374#382—21no—
375#380C2×D5911yescharacteristic
376C4×C4⋊C4—591no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4—1181no—
379C4×C22⋊C4C2×D5911yescharacteristic
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#340—591no—
388#336—591no—
389#336—591no—
390C59#2672811yescharacteristic, complemented, Sylow
391C22×Q8C22×D5911yescharacteristic
392C22×D4—591no—
393C22×D4—1181no—
394C23×C4—1181no—
395C23×C4C22×D5911yescharacteristic
396C2×SD16—1182no—
397C2×SD16—2361no—
398C2×SD16—2361no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—591no—
402C22×C8—591no—
403#125—22no—
404#125—41no—
405C22⋊Q8—22no—
406C22⋊Q8—41no—
407#121—1182no—
408#121—1182no—
409C4×D4—1182no—
410C4×D4—2361no—
411#117—1182no—
412#117—2361no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—591no—
418C2×C4⋊C4—591no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C4⋊C4C22×D5911yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4C22⋊D5911yescharacteristic
431C2×C22⋊C4C22⋊D5911yescharacteristic
432C2×C42—21no—
433C2×C42—1181no—
434C2×C42—1181no—
435C2×C42C22×D5911yescharacteristic
436C2×C42C22×D5911yescharacteristic
437#106—1182no—
438C4⋊C8—2361no—
439#103—1182no—
440#103—1182no—
441#103—2361no—
442#102—1182no—
443#102—1182no—
444#102—2361no—
445C22⋊C8—2361no—
446C22⋊C8—2361no—
447C4×C8—2361no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1181no—
455#95—1181no—
456#95C22⋊D5911yescharacteristic
457#95C22⋊D5911yescharacteristic
458C24—1181no—
459C24C2×C22⋊D5911yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×D4—592no—
467C2×D4—1184no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C2×D4—4721no—
474C2×D4—4721no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—4721no—
489C22×C4—2362no—
490C22×C4—2362no—
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C23×D5911yescharacteristic
497SD16—2364no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—592no—
519C4⋊C4—592no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—4721no—
525C4⋊C4#1174568712yes—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—1184no—
531C22⋊C4—4721no—
532C22⋊C4—2362no—
533C22⋊C4—2362no—
534C22⋊C4—2362no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—4721no—
541C42—2362no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—1182no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23—2361no—
556C23C22×C22⋊D5911yescharacteristic
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561D4—1182no—
562D4—1182no—
563D4—4724no—
564D4—2361no—
565D4—2361no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—4724no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4#1427374712yes—
607C2×C4#41×D5911yescharacteristic
608C2×C4#41×D5911yescharacteristic
609C2×C4#41×D5911yescharacteristic
610C2×C4C2×#1174568711yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377311yescharacteristic
613C8—2361no—
614C8—2361no—
615C8—2361no—
616C8—2361no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—4721no—
628C22—2362no—
629C22—2362no—
630C22—2362no—
631C22#1683790911yescharacteristic
632C22#1683804011yescharacteristic
633C22#1683804611yescharacteristic
634C22C2×#1427374711yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683821011yescharacteristic
637C22#1683821011yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—4721no—
648C4—2362no—
649C4—2362no—
650C4#1683793111yescharacteristic
651C4#1683798611yescharacteristic
652C4#1683811411yescharacteristic
653C4#1683813411yescharacteristic
654C2—81no—
655C2—4722no—
656C2#2142802911yescharacteristic
657C2#2143410811yescharacteristic
658C2#2143434311yescharacteristic
659C2#2143592511yescharacteristic
660C2#2143602411yescharacteristic
661C2#2143621611yescharacteristic
662C2#2143634911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2489C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2489D59
hypocenterC59#26728
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. #21425411C2semidirect
4. #21425239C2semidirect
6. #21425054C2semidirect
7. #21424775C2semidirect
8. #21424727C2semidirect
9. #21424704C2semidirect
10. #21424681C2semidirect
11. #21424433C2semidirect
12. #21424409C2semidirect
13. #21424188C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2489C2semidirect
323. #2489D59semidirect
390. C59#26728semidirect

Maximal subgroups

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

subgroupindex
#214271322
#214254112
#214252852
#214252392
#214250972
#214250542
#214247752
#214247272
#214247042
#214246812
#214244332
#214244092
#214241882
C4×#142736472
C59×#24892
#2672859