Finite Group Databasebeta

← #27465138

Subgroups — #27465138

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277324C211yesmaximal, characteristic, complemented
2#20276263C211yesmaximal, characteristic, complemented
3#20276236C211yesmaximal, characteristic, complemented
4#20276123C211yesmaximal, characteristic, complemented
5#20276118C211yesmaximal, characteristic, complemented
6#20275353C211yesmaximal, characteristic, complemented
7#20275346C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic, complemented
9#20275332C211yesmaximal, characteristic, complemented
10#20274908C211yesmaximal, characteristic, complemented
11#20274903C211yesmaximal, characteristic, complemented
12#20274786C211yesmaximal, characteristic, complemented
13#20274655C211yesmaximal, characteristic
14#20274655C211yesmaximal, characteristic
15C53×#2311C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133141C2211yescharacteristic
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810516—21no—
31#16810516—21no—
32#16810509—21no—
33#16810509—21no—
34#16810493C2211yescharacteristic
35#16810489C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C22×#41×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#391C2211yescharacteristic
52C53×#391C2211yescharacteristic
53C53×#390C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#386—21no—
56C53×#386—21no—
57C53×#380—21no—
58C53×#380—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73#13133150—41no—
74#13133150—41no—
75#13133148—41no—
76#13133148—41no—
77#13133141—22no—
78#13133141—22no—
79#13133141—41no—
80#13133141—41no—
81#13133138—22no—
82#13133138—22no—
83#13133138—41no—
84#13133138—41no—
85C22⋊#88322—22no—
86C22⋊#88322—22no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93C2×#41×C53—22no—
94C2×#41×C53—22no—
95C2×#41×C53—22no—
96C2×#41×C53—22no—
97C2×#41×C53—41no—
98C2×#41×C53—41no—
99C22×Q8×C53C2311yescharacteristic
100C22×D4×C53—21no—
101C22×D4×C53C2311yescharacteristic
102C23×C4×C53—21no—
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105#117×C53—22no—
106#117×C53—22no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—41no—
109C2×C4⋊C4×C53—41no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122#95×C53—21no—
123#95×C53—21no—
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128#41×C53—44no—
129#41×C53—44no—
130#41×C53—44no—
131#41×C53—44no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53#4112yes—
138C2×D4×C53—24no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—42no—
145C2×D4×C53—42no—
146C2×D4×C53#4112yes—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53—42no—
158C22×C4×C53—81no—
159C22×C4×C53—81no—
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53#4112yes—
166C22×C4×C53#4112yes—
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—22no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—42no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—81no—
182C22⋊C4×C53—81no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53—81no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C22⋊Dic53—81no—
190C22⋊Dic53—81no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—81no—
200C4⋊Dic53—81no—
201C4×Dic53—81no—
202C4×Dic53—81no—
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—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—22no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53#12312yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—48no—
237D4×C53—41no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—48no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—84no—
268C2×C4×C53#12312yes—
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C53H(5,𝔽2)11yescharacteristic
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287#88322—41no—
288#88322—41no—
289#88322—41no—
290#88322—41no—
291#27593—531nomaximal, Sylow
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53C2×#12311yescharacteristic
307C22×C53#52511yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#57611yescharacteristic
310C22×C53#57711yescharacteristic
311C22×C53#57711yescharacteristic
312C22×C53#57811yescharacteristic
313C4×C53—22no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53#55111yescharacteristic
322C4×C53#55111yescharacteristic
323C4×C53#55111yescharacteristic
324C4×C53#55111yescharacteristic
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329#2314—531no—
330#2311D5311yescharacteristic
331#2097—531no—
332#2092—531no—
333#2072—531no—
334#2071—531no—
335#1928—531no—
336#1927—531no—
337#1926—531no—
338#1925—531no—
339#1853—531no—
340#1852—531no—
341#1818—531no—
342#1788—531no—
343#1788—531no—
344C2×C53—42no—
345C2×C53—82no—
346C2×C53#248211yescharacteristic
347C2×C53#309811yescharacteristic
348C2×C53#309911yescharacteristic
349C2×C53#311911yescharacteristic
350C2×C53#312011yescharacteristic
351C2×C53#322411yescharacteristic
352C2×C53#322511yescharacteristic
353C22×#41C2×D5311yescharacteristic
354C2×#117C2×D5311yescharacteristic
355C22×C4⋊C4C2×D5311yescharacteristic
356C2×#107—531no—
357C2×#107—531no—
358C2×#106—531no—
359C2×#106—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×#102—531no—
368C2×C22⋊C8—531no—
369C2×C22⋊C8—531no—
370#391C2×D5311yescharacteristic
371#391C2×D5311yescharacteristic
372#390C2×D5311yescharacteristic
373#390C2×D5311yescharacteristic
374#388—1061no—
375#388—1061no—
376#387—1061no—
377#387—1061no—
378#386—21no—
379#386—21no—
380#384—531no—
381#382—531no—
382#380—21no—
383#380—21no—
384#380—531no—
385#380—531no—
386C4×C4⋊C4—531no—
387C4×C4⋊C4—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#336—531no—
393#336—531no—
394#336—531no—
395#336—531no—
396C53#2759311yescharacteristic, complemented, Sylow
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—22no—
401C2×#41—41no—
402C2×#41—41no—
403C22×Q8C22×D5311yescharacteristic
404C22×D4—21no—
405C22×D4C22×D5311yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4C22×D5311yescharacteristic
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8—531no—
413#117—22no—
414#117—22no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—41no—
417C2×C4⋊C4—41no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
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—21no—
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4C22⋊D5311yescharacteristic
435C2×C22⋊C4C22⋊D5311yescharacteristic
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C22×D5311yescharacteristic
439C2×C42C22×D5311yescharacteristic
440#107—2121no—
441#107—2121no—
442#106—2121no—
443#106—2121no—
444#103—1062no—
445#103—1062no—
446#103—2121no—
447#103—2121no—
448#102—1062no—
449#102—1062no—
450#102—2121no—
451#102—2121no—
452C22⋊C8—1062no—
453C22⋊C8—1062no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464C24—21no—
465C24C2×C22⋊D5311yescharacteristic
466#41—44no—
467#41—44no—
468#41—44no—
469#41—44no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8#1171994512yes—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4#1171994512yes—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4#1171994712yes—
503C22×C4#1171994712yes—
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C23×D5311yescharacteristic
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—4241no—
538C4⋊C4—4241no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—81no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—4241no—
546C22⋊C4—4241no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—4241no—
552C42—4241no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23#1313319812yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—41no—
576D4—41no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4#1313319812yes—
619C2×C4C2×#1171994711yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4C2×#1171994511yescharacteristic
622C2×C4C2×#1171994511yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326211yescharacteristic
625C8—2121no—
626C8—2121no—
627C8—2121no—
628C8—2121no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×#1313319811yescharacteristic
644C22#1681113411yescharacteristic
645C22#1681119011yescharacteristic
646C22#1681142611yescharacteristic
647C22#1681143111yescharacteristic
648C22#1681143411yescharacteristic
649C22#1681143811yescharacteristic
650C4—22no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—4241no—
659C4—4241no—
660C4—4241no—
661C4—4241no—
662C4#1681129911yescharacteristic
663C4#1681129911yescharacteristic
664C4#1681129911yescharacteristic
665C4#1681129911yescharacteristic
666C2—42no—
667C2—82no—
668C2#2027898911yescharacteristic
669C2#2028468911yescharacteristic
670C2#2028470011yescharacteristic
671C2#2028491411yescharacteristic
672C2#2028492911yescharacteristic
673C2#2028580911yescharacteristic
674C2#2028582411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2311C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#2311D53
hypocenterC53#27593
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
1. #20277324C2semidirect
2. #20276263C2semidirect
3. #20276236C2semidirect
4. #20276123C2semidirect
5. #20276118C2semidirect
6. #20275353C2semidirect
7. #20275346C2semidirect
8. #20275339C2semidirect
9. #20275332C2semidirect
10. #20274908C2semidirect
11. #20274903C2semidirect
12. #20274786C2semidirect
396. C53#27593semidirect

Maximal subgroups

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

subgroupindex
#202773242
#202762632
#202762362
#202761232
#202761182
#202753532
#202753462
#202753392
#202753322
#202749082
#202749032
#202747862
#202746552
C53×#23112
#2759353