Finite Group Databasebeta

← #27466217

Subgroups — #27466217

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277338C211yesmaximal, characteristic, complemented
2#20276283C211yesmaximal, characteristic, complemented
3#20276273C211yesmaximal, characteristic, complemented
4#20276248C211yesmaximal, characteristic, complemented
5#20276236C211yesmaximal, characteristic, complemented
6#20275483C211yesmaximal, characteristic, complemented
7#20275478C211yesmaximal, characteristic, complemented
8#20275038C211yesmaximal, characteristic
9#20275033C211yesmaximal, characteristic
10#20274912C211yesmaximal, characteristic, complemented
11#20274907C211yesmaximal, characteristic, complemented
12#20274725C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic, complemented
14#20274699C211yesmaximal, characteristic, complemented
15C53×#2324C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810522—21no—
31#16810522—21no—
32#16810498—21no—
33#16810498—21no—
34#16810493C2211yescharacteristic
35#16810493C2211yescharacteristic
36#16810489C2211yescharacteristic
37#16810489C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C2×#117×C53C2211yescharacteristic
51C53×#390—21no—
52C53×#390C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#388—21no—
55C53×#386—21no—
56C53×#385—21no—
57C53×#380C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176—41no—
65#13133176—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—41no—
80#13133148—41no—
81#13133148—41no—
82#13133148—41no—
83C4⋊#88322—22no—
84C4⋊#88322—22no—
85#13133138—41no—
86#13133138—41no—
87#13133138—41no—
88#13133138—41no—
89C22⋊#88322—22no—
90C22⋊#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×D4×C53—21no—
96C22×D4×C53C2311yescharacteristic
97C23×C4×C53—21no—
98C23×C4×C53C2311yescharacteristic
99#121×C53—22no—
100#121×C53Q814yes—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103#117×C53—22no—
104#117×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
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×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C24×C53—21no—
125C24×C53C2×D411yescharacteristic
126C2×D4×C53—24no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—81no—
133C2×D4×C53—81no—
134C2×D4×C53C2×Q812yes—
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—42no—
143C22×C4×C53—81no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×Q812yes—
151C22×C4×C53#4112yes—
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C22×Dic53—81no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53C2×Q812yes—
169C22⋊C4×C53—24no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—81no—
174C22⋊C4×C53—81no—
175C22⋊C4×C53—81no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—81no—
179C42×C53#4112yes—
180C42×C53#4112yes—
181C22⋊Dic53—42no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191C4⋊Dic53—81no—
192C4⋊Dic53—81no—
193C4⋊Dic53—81no—
194C4⋊Dic53—81no—
195C4×Dic53—42no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—22no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53C22⋊Q812yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—22no—
224D4×C53—22no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—84no—
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—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
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—84no—
254C2×C4×C53C22⋊Q812yes—
255C2×C4×C53#12312yes—
256C2×C4×C53#12312yes—
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—84no—
276#88322—41no—
277#88322—41no—
278#88322—41no—
279#88322—41no—
280#27688—531nomaximal, Sylow
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53C2×C22⋊Q811yescharacteristic
296C22×C53C2×#12311yescharacteristic
297C22×C53#53511yescharacteristic
298C22×C53#57611yescharacteristic
299C22×C53#57711yescharacteristic
300C22×C53#57711yescharacteristic
301C22×C53#57711yescharacteristic
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—42no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53—81no—
309C4×C53—81no—
310C4×C53#55311yescharacteristic
311C4×C53#55311yescharacteristic
312C4×C53#55411yescharacteristic
313C4×C53#55411yescharacteristic
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318#2324D5311yescharacteristic
319#2317—531no—
320#2101—531no—
321#2099—531no—
322#2094—531no—
323#2092—531no—
324#1949—531no—
325#1948—531no—
326#1875—531no—
327#1874—531no—
328#1854—531no—
329#1853—531no—
330#1804—531no—
331#1802—531no—
332#1798—531no—
333C2×C53—42no—
334C2×C53—82no—
335C2×C53#248711yescharacteristic
336C2×C53#308111yescharacteristic
337C2×C53#308411yescharacteristic
338C2×C53#309611yescharacteristic
339C2×C53#309911yescharacteristic
340C2×C53#322111yescharacteristic
341C2×C53#322411yescharacteristic
342C2×#121C2×D5311yescharacteristic
343C2×C4×D4C2×D5311yescharacteristic
344C2×#117—531no—
345C2×#117C2×D5311yescharacteristic
346C22×C4⋊C4—531no—
347C2×#107—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×C4⋊C8—531no—
352C2×C4⋊C8—531no—
353C2×#102—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×C22⋊C8—531no—
358C2×C22⋊C8—531no—
359#390—21no—
360#390C2×D5311yescharacteristic
361#389—1061no—
362#389—1061no—
363#389C2×D5311yescharacteristic
364#388—21no—
365#386—21no—
366#385—21no—
367#385—1061no—
368#385—1061no—
369#384—531no—
370#384—531no—
371#382—531no—
372#382—531no—
373#380C2×D5311yescharacteristic
374C4×C4⋊C4C2×D5311yescharacteristic
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#336—531no—
384#336—531no—
385C53#2768811yescharacteristic, complemented, Sylow
386C22×D4—21no—
387C22×D4C22×D5311yescharacteristic
388C23×C4—21no—
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—531no—
395#121—22no—
396#121Dic10614yes—
397C4×D4—22no—
398C4×D4—22no—
399#117—22no—
400#117—22no—
401#117—2121no—
402#117—2121no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—2121no—
414C2×C4⋊C4—2121no—
415C2×C4⋊C4C22⋊D5311yescharacteristic
416C2×C4⋊C4C22⋊D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C22⋊C4C22⋊D5311yescharacteristic
428C2×C42—21no—
429C2×C42—1061no—
430C2×C42C22×D5311yescharacteristic
431C2×C42C22×D5311yescharacteristic
432#107—2121no—
433#106—2121no—
434#106—2121no—
435#106—2121no—
436C4⋊C8—1062no—
437C4⋊C8—1062no—
438#102—2121no—
439#102—2121no—
440#102—2121no—
441#102—2121no—
442C22⋊C8—1062no—
443C22⋊C8—1062no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451C24—21no—
452C24C2×C22⋊D5311yescharacteristic
453C2×D4—24no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—81no—
460C2×D4—81no—
461C2×D4C2×Dic10612yes—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—4241no—
477C22×C4—4241no—
478C22×C4—2122no—
479C22×C4C2×Dic10612yes—
480C22×C4#1171994512yes—
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C23×D5311yescharacteristic
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—81no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C4⋊C4C2×Dic10612yes—
521C22⋊C4—24no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C22⋊C4—2122no—
529C42—21no—
530C42—21no—
531C42—81no—
532C42—2122no—
533C42#1171994712yes—
534C42#1171994712yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#1313319612yes—
549C23C22×C22⋊D5311yescharacteristic
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—4244no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4#1313319612yes—
595C2×C4#1313319812yes—
596C2×C4#1313319812yes—
597C2×C4C22×Dic10611yescharacteristic
598C2×C4C2×#1171994711yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994511yescharacteristic
601C2×C4#1313326311yescharacteristic
602C2×C4#1313326311yescharacteristic
603C8—2121no—
604C8—2121no—
605C8—2121no—
606C8—2121no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#1313319611yescharacteristic
622C22C2×#1313319811yescharacteristic
623C22#1681118711yescharacteristic
624C22#1681142611yescharacteristic
625C22#1681143111yescharacteristic
626C22#1681143411yescharacteristic
627C22#1681143411yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4#1681131911yescharacteristic
641C4#1681131911yescharacteristic
642C4#1681133011yescharacteristic
643C4#1681133011yescharacteristic
644C2—42no—
645C2—82no—
646C2#2027903211yescharacteristic
647C2#2028454611yescharacteristic
648C2#2028457711yescharacteristic
649C2#2028466711yescharacteristic
650C2#2028470011yescharacteristic
651C2#2028577011yescharacteristic
652C2#2028580311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2324C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2324D53
hypocenterC53#27688
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
1. #20277338C2semidirect
2. #20276283C2semidirect
3. #20276273C2semidirect
4. #20276248C2semidirect
5. #20276236C2semidirect
6. #20275483C2semidirect
7. #20275478C2semidirect
10. #20274912C2semidirect
11. #20274907C2semidirect
12. #20274725C2semidirect
13. #20274715C2semidirect
14. #20274699C2semidirect
385. C53#27688semidirect

Maximal subgroups

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

subgroupindex
#202773382
#202762832
#202762732
#202762482
#202762362
#202754832
#202754782
#202750382
#202750332
#202749122
#202749072
#202747252
#202747152
#202746992
C53×#23242
#2768853