Finite Group Databasebeta

← #27413093

Subgroups — #27413093

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810664C211yesmaximal, characteristic
2C2×#16810662C211yesmaximal, characteristic
3#20277196C211yesmaximal, characteristic
4#382×D53C211yesmaximal, characteristic
5#20276202C211yesmaximal, characteristic, complemented
6#20276201C211yesmaximal, characteristic, complemented
7#20276200C211yesmaximal, characteristic, complemented
8#20276199C211yesmaximal, characteristic, complemented
9#20276198C211yesmaximal, characteristic, complemented
10#20276197C211yesmaximal, characteristic, complemented
11#20276196C211yesmaximal, characteristic, complemented
12#20274433C211yesmaximal, characteristic
13#20274433C211yesmaximal, characteristic
14#20274407C211yesmaximal, characteristic
15C53×#2085C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C42×D53C2211yescharacteristic
23#16810664—24no—
24#16810662—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810497—21no—
35#16810494—21no—
36#16810489C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810488C2211yescharacteristic
39#16810487—21no—
40#16810487—21no—
41C4⋊C4×Dic53—21no—
42#16810474—21no—
43#95×D53—21no—
44#95×D53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53D411yescharacteristic
64C22×C4×D53D411yescharacteristic
65C22×C4×D53C2311yescharacteristic
66C2×#11719912—22no—
67C2×#11719912—22no—
68C2×#11719912—22no—
69C2×#11719912—22no—
70C2×#11719912Q814yes—
71C2×#11719912Q814yes—
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133175—22no—
75#13133175—22no—
76#13133175—22no—
77#13133175D414yes—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C4⋊C4×D53—22no—
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53D411yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98C42×D53—22no—
99C42×D53—22no—
100#13133151—22no—
101#13133150—22no—
102#13133150—22no—
103#13133149—22no—
104#13133148—22no—
105#13133148—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123#107×C53—22no—
124#106×C53—22no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128C23×D53C2×D411yescharacteristic
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53C2×D412yes—
156C2×C4×D53C2×Q812yes—
157C2×C4×D53C2×Q812yes—
158C2×C4×D53#4112yes—
159C2×C4×D53#4112yes—
160C2×C4×D53#4112yes—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53C2×Q812yes—
166C2×C8×C53C2×Q812yes—
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—21no—
174C4⋊C4×C53—42no—
175C4⋊C4×C53C2×D412yes—
176C42×C53—21no—
177C42×C53—21no—
178#11719921—24no—
179#11719921—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53C2×D412yes—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—24no—
196C4×Dic53—22no—
197C4×Dic53—22no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C2×Q812yes—
204C2×#88322C2×Q812yes—
205#11719912—28no—
206#11719912—28no—
207C23×C53C22×D411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53#12112yes—
213C22×D53#12312yes—
214C22×D53C4⋊Q812yes—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53#12112yes—
258C2×Dic53#12312yes—
259C2×Dic53C4⋊Q812yes—
260C4×D53—44no—
261C4×D53—44no—
262C4×D53—84no—
263C4×D53—84no—
264C4×D53C22⋊Q814yes—
265C4×D53C22⋊Q814yes—
266C4×D53#12514yes—
267C4×D53#12514yes—
268C8×C53—22no—
269C8×C53—22no—
270#88322—22no—
271#88322—22no—
272#23659—531nomaximal, Sylow
273C22×C53C2×#12111yescharacteristic
274C22×C53C2×#12311yescharacteristic
275C22×C53C2×C4⋊Q811yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53#39814yes—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53C2×C22⋊Q811yescharacteristic
294C4×C53C2×C22⋊Q811yescharacteristic
295C4×C53C2×#12511yescharacteristic
296C4×C53C2×#12511yescharacteristic
297Dic53—41no—
298Dic53—41no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302Dic53—81no—
303Dic53#39814yes—
304C2×#428—531no—
305C2×#428—531no—
306#2296—531no—
307C2×#382—531no—
308#2085—531no—
309#2085—531no—
310#2085—531no—
311#2085—531no—
312#2085—531no—
313#2085—531no—
314#2085—531no—
315#2085D5311yescharacteristic, complemented
316#1736—531no—
317#1736—531no—
318#1728—531no—
319C2×C53C2×#39811yescharacteristic
320C2×C53#304011yescharacteristic
321C2×C53#304011yescharacteristic
322C2×C53#309311yescharacteristic
323C2×C53#309311yescharacteristic
324C2×C53#316611yescharacteristic
325C2×C53#316611yescharacteristic
326D53—24no—
327C22×M16—531no—
328C22×M16—531no—
329C2×#117—531no—
330C2×#117—531no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4—531no—
333C22×C42—531no—
334#428—1064no—
335#428—1064no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#107—531no—
339C2×#107C2×D5311yescharacteristic
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106C2×D5311yescharacteristic
344#389—531no—
345#389—531no—
346#389—531no—
347#389C2×D5311yescharacteristic
348#385—1061no—
349#385—1061no—
350#382—531no—
351#382—531no—
352#382—531no—
353#382—1061no—
354#382—1061no—
355#382C2×D5311yescharacteristic
356C4×C4⋊C4—1061no—
357C4×C4⋊C4—1061no—
358C2×#95—1061no—
359C2×#95—1061no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340C2×D5311yescharacteristic
367#340C2×D5311yescharacteristic
368#336—531no—
369#336—531no—
370#336—531no—
371#336C2×D5311yescharacteristic
372C53#2365911yescharacteristic, complemented, Sylow
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—1061no—
377C23×C4—1061no—
378C2×M16—534no—
379C2×M16—534no—
380C2×M16—1062no—
381C2×M16—1062no—
382C2×M16—1062no—
383C2×M16—1062no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388#117—534no—
389#117—1062no—
390#117—1062no—
391#117—1062no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—1062no—
416C2×C42—1062no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42C22×D5311yescharacteristic
420#107—22no—
421#107—1062no—
422#107—1062no—
423#107—1062no—
424#106—22no—
425#106—1062no—
426#106—1062no—
427#106—1062no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440C24—531no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—531no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—532no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—2121no—
461C22×C4—2121no—
462C22×C4—2121no—
463C22×C4—2121no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4C23×D5311yescharacteristic
473M16—1068no—
474M16—1068no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8Q8×D5312yes—
486C2×C8Q8×D5312yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—532no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4D4×D5312yes—
509C22⋊C4—1064no—
510C22⋊C4—2122no—
511C42—21no—
512C42—21no—
513C42—1062no—
514C42—1062no—
515C42—1064no—
516C42—1061no—
517C42—1061no—
518C42—2122no—
519C23—532no—
520C23—532no—
521C23—532no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1061no—
526C23C2×D4×D5311yescharacteristic
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—534no—
549C2×C4—534no—
550C2×C4—534no—
551C2×C4—534no—
552C2×C4—1062no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—4244no—
560C2×C4—4244no—
561C2×C4—2124no—
562C2×C4—2124no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4C2×D4×D5311yescharacteristic
575C2×C4C2×Q8×D5311yescharacteristic
576C2×C4C2×Q8×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C8—22no—
581C8—22no—
582C8—1062no—
583C8—1062no—
584C22—534no—
585C22—1062no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22#121×D5311yescharacteristic
592C22#123×D5311yescharacteristic
593C22C4⋊Q8×D5311yescharacteristic
594C22#1681141911yescharacteristic
595C22#1681142011yescharacteristic
596C22#1681142011yescharacteristic
597C22#1681142211yescharacteristic
598C4—42no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—534no—
605C4—4241no—
606C4—4241no—
607C4—4241no—
608C4—2122no—
609C4—2121no—
610C4—2121no—
611C4C22⋊Q8×D5311yescharacteristic
612C4C22⋊Q8×D5311yescharacteristic
613C4#125×D5311yescharacteristic
614C4#125×D5311yescharacteristic
615C2—1064no—
616C2#398×D5311yescharacteristic
617C2#2028420411yescharacteristic
618C2#2028420611yescharacteristic
619C2#2028464311yescharacteristic
620C2#2028464711yescharacteristic
621C2#2028526311yescharacteristic
622C2#2028526611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2085C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2085D53
hypocenterC53#23659
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
5. #20276202C2semidirect
6. #20276201C2semidirect
7. #20276200C2semidirect
8. #20276199C2semidirect
9. #20276198C2semidirect
10. #20276197C2semidirect
11. #20276196C2semidirect
15. C53×#2085C2semidirect
315. #2085D53semidirect
372. C53#23659semidirect

Maximal subgroups

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

subgroupindex
C2×#168106642
C2×#168106622
#202771962
#382×D532
#202762022
#202762012
#202762002
#202761992
#202761982
#202761972
#202761962
#202744332
#202744072
C53×#20852
#2365953