Finite Group Databasebeta

← #28656834

Subgroups — #28656834

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#21428437C211yesmaximal, characteristic, complemented
2#21425413C211yesmaximal, characteristic, complemented
3#21425282C211yesmaximal, characteristic
4#21425242C211yesmaximal, characteristic, complemented
5#21425099C211yesmaximal, characteristic
6#21425053C211yesmaximal, characteristic, complemented
7#21424771C211yesmaximal, characteristic, complemented
8#21424730C211yesmaximal, characteristic, complemented
9#21424704C211yesmaximal, characteristic, complemented
10#21424678C211yesmaximal, characteristic, complemented
11#21424430C211yesmaximal, characteristic, complemented
12#21424408C211yesmaximal, characteristic, complemented
13#21424191C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2363C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273650C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837343—21no—
31#16837330—21no—
32#16837308C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837290—21no—
35#16837266—21no—
36#16837262C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#121×C59C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×#117×C59C2211yescharacteristic
51C59×#390—21no—
52C59×#390C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#388—21no—
55C59×#386—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C22⋊C4×C59—21no—
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C2×#11745670—22no—
61C2×#11745670—41no—
62C2×#11745670—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273719—22no—
68#14273719—41no—
69#14273706—22no—
70#14273706—41no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C4×Dic59—21no—
80#14273658—22no—
81C4⋊(C59⋊C8)—41no—
82#14273650—22no—
83#14273650—22no—
84#14273650—41no—
85#14273648—22no—
86#14273648—22no—
87#14273648—41no—
88C22⋊(C59⋊C8)—41no—
89C22⋊(C59⋊C8)—41no—
90C4×C59⋊C8—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×D4×C59—21no—
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100C23×C4×C59C2311yescharacteristic
101#121×C59—22no—
102#121×C59—22no—
103C4×D4×C59—22no—
104C4×D4×C59—41no—
105#117×C59—22no—
106#117×C59—41no—
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×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121C2×C42×C59C2311yescharacteristic
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C24×C59C2×D411yescharacteristic
128C2×D4×C59—24no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—81no—
135C2×D4×C59—81no—
136C2×D4×C59#4112yes—
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×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—81no—
170C4⋊C4×C59#4112yes—
171C4⋊C4×C59#4112yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—41no—
175C22⋊C4×C59—42no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—81no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—42no—
184C42×C59—81no—
185C22⋊Dic59—42no—
186C22⋊Dic59—42no—
187#11745670—44no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59#4112yes—
194Dic59⋊C4—42no—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197C4×Dic59—42no—
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—22no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59C22×D411yescharacteristic
225D4×C59—22no—
226D4×C59—22no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—84no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
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—84no—
258C2×C4×C59#13512yes—
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59H(5,𝔽2)11yescharacteristic
264C2×C4×C59H(5,𝔽2)11yescharacteristic
265C2×Dic59—22no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277Dic118—41no—
278Dic118—41no—
279Dic118—41no—
280Dic118—41no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285#26728—591nomaximal, Sylow
286C22×C59—41no—
287C22×C59—42no—
288C22×C59—41no—
289C22×C59—42no—
290C22×C59—41no—
291C22×C59—42no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59#52511yescharacteristic
301C22×C59#54611yescharacteristic
302C22×C59#54611yescharacteristic
303C22×C59C2×#13511yescharacteristic
304C22×C59#57511yescharacteristic
305C22×C59#57711yescharacteristic
306C22×C59#57711yescharacteristic
307C4×C59—22no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—81no—
315C4×C59#52911yescharacteristic
316C4×C59#53811yescharacteristic
317C4×C59#55511yescharacteristic
318C4×C59#55911yescharacteristic
319Dic59—42no—
320Dic59—81no—
321Dic59—81no—
322Dic59—81no—
323#2489—591no—
324#2363D5911yescharacteristic
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—82no—
340C2×C59#245911yescharacteristic
341C2×C59#310211yescharacteristic
342C2×C59#312311yescharacteristic
343C2×C59#329611yescharacteristic
344C2×C59#330311yescharacteristic
345C2×C59#331711yescharacteristic
346C2×C59#332611yescharacteristic
347C22×SD16—591no—
348C2×#125—591no—
349C2×C22⋊Q8—591no—
350C2×#121C2×D5911yescharacteristic
351C2×C4×D4C2×D5911yescharacteristic
352C2×#117C2×D5911yescharacteristic
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—1181no—
365#397—1181no—
366#391—591no—
367#391—591no—
368#390—21no—
369#390C2×D5911yescharacteristic
370#389C2×D5911yescharacteristic
371#388—21no—
372#388—1181no—
373#386—21no—
374#382—1181no—
375#380—591no—
376C4×C4⋊C4—591no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4—21no—
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×Q8—591no—
392C22×D4—21no—
393C22×D4C22×D5911yescharacteristic
394C23×C4—21no—
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—1182no—
404#125—2361no—
405C22⋊Q8—1182no—
406C22⋊Q8—2361no—
407#121—22no—
408#121—22no—
409C4×D4—22no—
410C4×D4—41no—
411#117—22no—
412#117—41no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—591no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
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—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C22⋊C4C22⋊D5911yescharacteristic
431C2×C22⋊C4C22⋊D5911yescharacteristic
432C2×C42—21no—
433C2×C42—21no—
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—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95C22⋊D5911yescharacteristic
457#95C22⋊D5911yescharacteristic
458C24—21no—
459C24C2×C22⋊D5911yescharacteristic
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×D4—24no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—81no—
473C2×D4—81no—
474C2×D4#1174568512yes—
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—42no—
485C22×C4—42no—
486C22×C4—81no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—1181no—
490C22×C4—1181no—
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—81no—
515C4⋊C4—592no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—2362no—
522C4⋊C4—2362no—
523C4⋊C4—2362no—
524C4⋊C4#1174568512yes—
525C4⋊C4#1174568712yes—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—41no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C22⋊C4—2362no—
534C22⋊C4—2362no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—81no—
541C42—2362no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C22×C22⋊D5911yescharacteristic
557Q8—2361no—
558Q8—2361no—
559Q8—2361no—
560Q8—2361no—
561D4—22no—
562D4—22no—
563D4—41no—
564D4—41no—
565D4—84no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
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—21no—
580C2×C4—21no—
581C2×C4—22no—
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—84no—
594C2×C4—1182no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
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×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568511yescharacteristic
610C2×C4C2×#1174568511yescharacteristic
611C2×C4#1427377211yescharacteristic
612C2×C4#1427377311yescharacteristic
613C8—2361no—
614C8—2361no—
615C8—2361no—
616C8—2361no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#1683791311yescharacteristic
632C22#1683804011yescharacteristic
633C22#1683804611yescharacteristic
634C22C2×#1427374711yescharacteristic
635C22#1683820011yescharacteristic
636C22#1683821311yescharacteristic
637C22#1683821311yescharacteristic
638C4—22no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—42no—
645C4—81no—
646C4—4721no—
647C4—4721no—
648C4—4721no—
649C4—2362no—
650C4#1683793211yescharacteristic
651C4#1683798511yescharacteristic
652C4#1683811711yescharacteristic
653C4#1683813511yescharacteristic
654C2—81no—
655C2—82no—
656C2#2142802711yescharacteristic
657C2#2143411411yescharacteristic
658C2#2143434811yescharacteristic
659C2#2143593111yescharacteristic
660C2#2143601811yescharacteristic
661C2#2143622211yescharacteristic
662C2#2143634311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2363C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2363D59
hypocenterC59#26728
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
1. #21428437C2semidirect
2. #21425413C2semidirect
4. #21425242C2semidirect
6. #21425053C2semidirect
7. #21424771C2semidirect
8. #21424730C2semidirect
9. #21424704C2semidirect
10. #21424678C2semidirect
11. #21424430C2semidirect
12. #21424408C2semidirect
13. #21424191C2semidirect
14. C4×#14273648C2semidirect
390. C59#26728semidirect

Maximal subgroups

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

subgroupindex
#214284372
#214254132
#214252822
#214252422
#214250992
#214250532
#214247712
#214247302
#214247042
#214246782
#214244302
#214244082
#214241912
C4×#142736482
C59×#23632
#2672859