Finite Group Databasebeta

← #28657181

Subgroups — #28657181

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427121C211yesmaximal, characteristic, complemented
2#21425413C211yesmaximal, characteristic, complemented
3#21425292C211yesmaximal, characteristic, complemented
4#21425244C211yesmaximal, characteristic, complemented
5#21425099C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic, complemented
7#21424796C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic, complemented
9#21424711C211yesmaximal, characteristic, complemented
10#21424681C211yesmaximal, characteristic, complemented
11#21424454C211yesmaximal, characteristic, complemented
12#21424419C211yesmaximal, characteristic
13#21424188C211yesmaximal, characteristic, complemented
14#21423929C211yesmaximal, characteristic, complemented
15C59×#2490C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
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×#121×C59C2211yescharacteristic
50C59×#397—21no—
51C59×#394—21no—
52C59×#390—21no—
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#388—21no—
56C59×#380C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
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#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#14273658—22no—
85C4⋊(C59⋊C8)—41no—
86#14273650—22no—
87#14273650—22no—
88#14273650—41no—
89#14273648—22no—
90#14273648—22no—
91#14273648—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×D4×C59—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59C2311yescharacteristic
102#125×C59—22no—
103#125×C59—41no—
104#121×C59—22no—
105#121×C59—41no—
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×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C22⋊C4×C59D411yescharacteristic
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—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59—81no—
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×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#4112yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59#4112yes—
170C22⋊C4×C59—41no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—81no—
175C22⋊C4×C59—81no—
176C22⋊C4×C59—81no—
177C22⋊C4×C59—81no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—21no—
181C42×C59—21no—
182C42×C59—81no—
183C22⋊Dic59—24no—
184C22⋊Dic59—42no—
185#11745670—44no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59#4112yes—
191C4⋊Dic59#4112yes—
192Dic59⋊C4—24no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195C4×Dic59—42no—
196C4×Dic59—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—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59C22×D411yescharacteristic
225D4×C59—41no—
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—84no—
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—22no—
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—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—41no—
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C59H(5,𝔽2)11yescharacteristic
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—22no—
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—
276Dic118—22no—
277Dic118—22no—
278Dic118—41no—
279Dic118—41no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#26751—591nomaximal, Sylow
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59—41no—
290C22×C59—41no—
291C22×C59—41no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59#52511yescharacteristic
300C22×C59#54611yescharacteristic
301C22×C59#54611yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57711yescharacteristic
305C22×C59#57711yescharacteristic
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—41no—
312C4×C59—81no—
313C4×C59—81no—
314C4×C59#52911yescharacteristic
315C4×C59#53811yescharacteristic
316C4×C59#55511yescharacteristic
317C4×C59#55911yescharacteristic
318Dic59—42no—
319Dic59—42no—
320Dic59—42no—
321Dic59—81no—
322#2490D5911yescharacteristic
323#2362—591no—
324#2055—591no—
325#2035—591no—
326#2026—591no—
327#2000—591no—
328#1989—591no—
329#1936—591no—
330#1926—591no—
331#1924—591no—
332#1920—591no—
333#1880—591no—
334#1875—591no—
335#1825—591no—
336#1763—591no—
337C2×C59—81no—
338C2×C59—161no—
339C2×C59#245911yescharacteristic
340C2×C59#311111yescharacteristic
341C2×C59#312611yescharacteristic
342C2×C59#330711yescharacteristic
343C2×C59#331111yescharacteristic
344C2×C59#331711yescharacteristic
345C2×C59#332611yescharacteristic
346C22×SD16—591no—
347C2×#125C2×D5911yescharacteristic
348C2×C22⋊Q8—591no—
349C2×#121C2×D5911yescharacteristic
350C2×C4×Q8—591no—
351C2×#117—591no—
352C2×#106—591no—
353C2×C4⋊C8—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×C22⋊C8—591no—
361C2×C22⋊C8—591no—
362C2×#97—591no—
363#397—21no—
364#394—21no—
365#391—591no—
366#390—21no—
367#390C2×D5911yescharacteristic
368#390C2×D5911yescharacteristic
369#389—591no—
370#388—21no—
371#388—1181no—
372#388—1181no—
373#387—1181no—
374#380C2×D5911yescharacteristic
375C4×C4⋊C4—591no—
376C4×C4⋊C4—1181no—
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C22⋊C4C2×D5911yescharacteristic
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#340—591no—
387#336—591no—
388#336—591no—
389C59#2675111yescharacteristic, complemented, Sylow
390C22×Q8—591no—
391C22×D4—21no—
392C22×D4C22×D5911yescharacteristic
393C23×C4C22×D5911yescharacteristic
394C2×SD16—1182no—
395C2×SD16—2361no—
396C2×SD16—2361no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401#125—22no—
402#125—41no—
403C22⋊Q8—1182no—
404C22⋊Q8—1182no—
405#121—22no—
406#121—41no—
407C4×Q8—1182no—
408C4×Q8—2361no—
409#117—1182no—
410#117—2361no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4—1181no—
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C4⋊C4C22×D5911yescharacteristic
422C2×C22⋊C4—21no—
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—1181no—
434C2×C42—1181no—
435C2×C42—1181no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438#106—1182no—
439C4⋊C8—2361no—
440#103—1182no—
441#103—1182no—
442#103—2361no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446C22⋊C8—2361no—
447C22⋊C8—2361no—
448#97—2361no—
449#95—21no—
450#95—21no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—21no—
458C24C2×C22⋊D5911yescharacteristic
459C2×Q8—592no—
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—81no—
474C22×C4—21no—
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—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C23×D5911yescharacteristic
493SD16—2364no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—592no—
514C4⋊C4—592no—
515C4⋊C4—1184no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—2362no—
521C4⋊C4—2362no—
522C4⋊C4#1174568712yes—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C22⋊C4—81no—
531C22⋊C4—1184no—
532C22⋊C4—2362no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—81no—
538C42—2362no—
539C42—2362no—
540C42—2362no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C22×C22⋊D5911yescharacteristic
556Q8—1182no—
557Q8—1182no—
558Q8—2361no—
559Q8—2361no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—84no—
565C2×C4—21no—
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—21no—
580C2×C4—22no—
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—1182no—
594C2×C4—1182no—
595C2×C4—1182no—
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#41×D5911yescharacteristic
606C2×C4#41×D5911yescharacteristic
607C2×C4#41×D5911yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4#1427377211yescharacteristic
610C2×C4#1427377311yescharacteristic
611C8—2361no—
612C8—2361no—
613C8—2361no—
614C8—2361no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#1683790911yescharacteristic
630C22#1683804011yescharacteristic
631C22#1683804611yescharacteristic
632C22#1683820011yescharacteristic
633C22#1683820011yescharacteristic
634C22#1683821311yescharacteristic
635C22#1683821311yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—4721no—
645C4—2362no—
646C4—2362no—
647C4—2362no—
648C4#1683793111yescharacteristic
649C4#1683798611yescharacteristic
650C4#1683811411yescharacteristic
651C4#1683813411yescharacteristic
652C2—81no—
653C2—161no—
654C2#2142802911yescharacteristic
655C2#2143419911yescharacteristic
656C2#2143438211yescharacteristic
657C2#2143606611yescharacteristic
658C2#2143613211yescharacteristic
659C2#2143622211yescharacteristic
660C2#2143634311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2490C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2490D59
hypocenterC59#26751
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2490

Complements

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

NH ≅ G/Ntype
1. #21427121C2semidirect
2. #21425413C2semidirect
3. #21425292C2semidirect
4. #21425244C2semidirect
6. #21425054C2semidirect
7. #21424796C2semidirect
8. #21424720C2semidirect
9. #21424711C2semidirect
10. #21424681C2semidirect
11. #21424454C2semidirect
13. #21424188C2semidirect
14. #21423929C2semidirect
389. C59#26751semidirect

Maximal subgroups

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

subgroupindex
#214271212
#214254132
#214252922
#214252442
#214250992
#214250542
#214247962
#214247202
#214247112
#214246812
#214244542
#214244192
#214241882
#214239292
C59×#24902
#2675159