Finite Group Databasebeta

← #27429159

Subgroups — #27429159

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279637C211yesmaximal, characteristic
2#20276146C211yesmaximal, characteristic, complemented
3#20276141C211yesmaximal, characteristic, complemented
4#20275918C211yesmaximal, characteristic, complemented
5#20275918C211yesmaximal, characteristic, complemented
6#20275514C211yesmaximal, characteristic
7#20275473C211yesmaximal, characteristic, complemented
8#20275027C211yesmaximal, characteristic, complemented
9#20275022C211yesmaximal, characteristic, complemented
10#20274848C211yesmaximal, characteristic, complemented
11#20274810C211yesmaximal, characteristic, complemented
12#20274801C211yesmaximal, characteristic, complemented
13#20274727C211yesmaximal, characteristic, complemented
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2352C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133140C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810546—21no—
32#16810546—21no—
33#16810526—21no—
34#16810519—21no—
35#16810493C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48C2×C4×D4×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#388—21no—
52C53×#387—21no—
53C53×#386—21no—
54C53×#382C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C22×Dic106C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67#13133196—41no—
68C4×Dic106—22no—
69C4×Dic106—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83#13133150—41no—
84#13133150—41no—
85C4⋊#88322—22no—
86#13133140—22no—
87#13133140—22no—
88#13133140—41no—
89#13133140—41no—
90C22⋊#88322—41no—
91C22⋊#88322—41no—
92C22⋊#88322—41no—
93C22⋊#88322—41no—
94#13133128—22no—
95#13133124—21no—
96#13133124—21no—
97C22×D4×C53D411yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103#117×C53—22no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—41no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53C2×D411yescharacteristic
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—22no—
129C2×D4×C53—22no—
130C2×D4×C53—41no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—42no—
148C22×C4×C53—42no—
149C22×C4×C53—81no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106#4112yes—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—81no—
180C42×C53—42no—
181C42×C53—81no—
182C42×C53#4112yes—
183C42×C53#4112yes—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—81no—
187C22⋊Dic53—81no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196#11719919—42no—
197#11719919—42no—
198#11719919—42no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53#13712yes—
226C23×C53C22×D411yescharacteristic
227D4×C53—44no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—84no—
259C2×C4×C53#12312yes—
260C2×C4×C53#12312yes—
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53H(5,𝔽2)11yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×Dic53—22no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279Dic106—22no—
280Dic106—22no—
281Dic106—41no—
282Dic106—41no—
283#88322—41no—
284#88322—41no—
285#88322—41no—
286#88322—41no—
287#24984—531nomaximal, Sylow
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53C2×#12311yescharacteristic
303C22×C53#54611yescharacteristic
304C22×C53#54611yescharacteristic
305C22×C53C2×#13711yescharacteristic
306C22×C53#57511yescharacteristic
307C22×C53#57611yescharacteristic
308C22×C53#57811yescharacteristic
309C4×C53—22no—
310C4×C53—22no—
311C4×C53—41no—
312C4×C53—41no—
313C4×C53—42no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53—81no—
317C4×C53#52911yescharacteristic
318C4×C53#54011yescharacteristic
319C4×C53#55311yescharacteristic
320C4×C53#55511yescharacteristic
321Dic53—42no—
322Dic53—81no—
323Dic53—81no—
324Dic53—81no—
325#2538—531no—
326#2352D5311yescharacteristic
327#2076—531no—
328#2075—531no—
329#2036—531no—
330#2036—531no—
331#1955—531no—
332#1947—531no—
333#1873—531no—
334#1872—531no—
335#1836—531no—
336#1825—531no—
337#1823—531no—
338#1805—531no—
339#1763—531no—
340C2×C53—42no—
341C2×C53—81no—
342C2×C53#242011yescharacteristic
343C2×C53#309511yescharacteristic
344C2×C53#309811yescharacteristic
345C2×C53#330911yescharacteristic
346C2×C53#331211yescharacteristic
347C2×C53#331311yescharacteristic
348C2×C53#331411yescharacteristic
349C2×C22⋊Q8—531no—
350C2×C22⋊Q8—531no—
351C2×C4×Q8—531no—
352C2×C4×D4C2×D5311yescharacteristic
353C2×#117C2×D5311yescharacteristic
354C22×C4⋊C4C2×D5311yescharacteristic
355C2×#107—531no—
356C2×#107—531no—
357C2×C4⋊C8—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×C22⋊C8—531no—
363C2×C22⋊C8—531no—
364C2×C22⋊C8—531no—
365C2×C22⋊C8—531no—
366C2×#97—531no—
367#396—1061no—
368#396—1061no—
369#390—1061no—
370#389—1061no—
371#388—21no—
372#387—21no—
373#386—21no—
374#384—531no—
375#382C2×D5311yescharacteristic
376#380—531no—
377#380—531no—
378#380C2×D5311yescharacteristic
379#380C2×D5311yescharacteristic
380C4×C4⋊C4C2×D5311yescharacteristic
381C4×C22⋊C4—21no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392C53#2498411yescharacteristic, complemented, Sylow
393C22×Q8—531no—
394C22×D4C22⋊D5311yescharacteristic
395C23×C4—21no—
396C23×C4C22×D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C22×C8—531no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C22⋊Q8—1062no—
403C22⋊Q8—1062no—
404C22⋊Q8—2121no—
405C22⋊Q8—2121no—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×D4—22no—
409C4×D4—22no—
410#117—22no—
411#117—41no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—41no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4C22⋊D5311yescharacteristic
424C2×C4⋊C4C22×D5311yescharacteristic
425C2×C4⋊C4C22×D5311yescharacteristic
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4C22⋊D5311yescharacteristic
436C2×C22⋊C4C22⋊D5311yescharacteristic
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—1061no—
440C2×C42C22×D5311yescharacteristic
441#107—2121no—
442#107—2121no—
443C4⋊C8—1062no—
444#103—1062no—
445#103—1062no—
446#103—2121no—
447#103—2121no—
448C22⋊C8—2121no—
449C22⋊C8—2121no—
450C22⋊C8—2121no—
451C22⋊C8—2121no—
452#97—1062no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—1061no—
459#95—1061no—
460C24C2×C22⋊D5311yescharacteristic
461C24C2×C22⋊D5311yescharacteristic
462C2×Q8—532no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—41no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
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—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C23×D5311yescharacteristic
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—4241no—
527C4⋊C4—2122no—
528C4⋊C4—2122no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4#1171994712yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—4241no—
538C22⋊C4—4241no—
539C22⋊C4—2122no—
540C22⋊C4—2122no—
541C42—42no—
542C42—81no—
543C42—2122no—
544C42#1171994512yes—
545C42#1171994512yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23#1313324712yes—
560C23C22×C22⋊D5311yescharacteristic
561Q8—1062no—
562Q8—1062no—
563Q8—2121no—
564Q8—2121no—
565D4—44no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
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—84no—
597C2×C4—1062no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4#1313320012yes—
610C2×C4#1313320012yes—
611C2×C4#41×D5311yescharacteristic
612C2×C4C2×#1171994711yescharacteristic
613C2×C4C2×#1171994511yescharacteristic
614C2×C4C2×#1171994511yescharacteristic
615C2×C4#1313326211yescharacteristic
616C2×C4#1313326311yescharacteristic
617C8—2121no—
618C8—2121no—
619C8—2121no—
620C8—2121no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1313320011yescharacteristic
636C22#1681126111yescharacteristic
637C22#1681126711yescharacteristic
638C22C2×#1313324711yescharacteristic
639C22#1681142111yescharacteristic
640C22#1681142611yescharacteristic
641C22#1681143811yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—4241no—
651C4—4241no—
652C4—4241no—
653C4—2122no—
654C4#1681115511yescharacteristic
655C4#1681122411yescharacteristic
656C4#1681132311yescharacteristic
657C4#1681133811yescharacteristic
658C2—42no—
659C2—81no—
660C2#2027820111yescharacteristic
661C2#2028466311yescharacteristic
662C2#2028469311yescharacteristic
663C2#2028671511yescharacteristic
664C2#2028676011yescharacteristic
665C2#2028678111yescharacteristic
666C2#2028679611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2352C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2352D53
hypocenterC53#24984
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2352

Complements

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

NH ≅ G/Ntype
2. #20276146C2semidirect
3. #20276141C2semidirect
4. #20275918C2semidirect
5. #20275918C2semidirect
7. #20275473C2semidirect
8. #20275027C2semidirect
9. #20275022C2semidirect
10. #20274848C2semidirect
11. #20274810C2semidirect
12. #20274801C2semidirect
13. #20274727C2semidirect
14. #20274548C2semidirect
392. C53#24984semidirect

Maximal subgroups

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

subgroupindex
#202796372
#202761462
#202761412
#202759182
#202755142
#202754732
#202750272
#202750222
#202748482
#202748102
#202748012
#202747272
#202745482
C53×#23522
#2498453