Finite Group Databasebeta

← #27452196

Subgroups — #27452196

Order 13568.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279056C211yesmaximal, characteristic
2#20277718C211yesmaximal, characteristic
3#20276038C211yesmaximal, characteristic, complemented
4#20275912C211yesmaximal, characteristic
5#20275867C211yesmaximal, characteristic, complemented
6#20275725C211yesmaximal, characteristic
7#20275670C211yesmaximal, characteristic, complemented
8#20275391C211yesmaximal, characteristic, complemented
9#20275350C211yesmaximal, characteristic, complemented
10#20275301C211yesmaximal, characteristic, complemented
11#20275055C211yesmaximal, characteristic, complemented
12#20275020C211yesmaximal, characteristic
13#20274802C211yesmaximal, characteristic
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1923C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133196C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29#16810564—21no—
30#16810551—21no—
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810529C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810520—21no—
36#16810511—21no—
37#16810510—21no—
38#16810495—21no—
39#16810487—21no—
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×Dic106C2311yescharacteristic
61C23×Dic53—21no—
62C2×Dic212—22no—
63C2×Dic212—41no—
64C2×Dic212—41no—
65#13133209—22no—
66#13133209—41no—
67#13133196—22no—
68#13133196—22no—
69#13133196—22no—
70#13133196—41no—
71C4×Dic106—22no—
72C4×Dic106—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133149—22no—
95#13133142—22no—
96#13133142—22no—
97#13133142—22no—
98#13133142—22no—
99#13133142—41no—
100#13133142—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C23×C4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C4⋊C8×C53—41no—
121C22⋊C8×C53—41no—
122C22⋊C8×C53—41no—
123C4×C8×C53—41no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127C22×C4×C53—21no—
128C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—81no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106#4112yes—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C22⋊Dic53—24no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53#4112yes—
194C4⋊Dic53#4112yes—
195C4⋊Dic53#4112yes—
196#11719919—24no—
197#11719919—42no—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201#11719919—81no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53—81no—
205Dic212—44no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#13512yes—
213C23×C53C22×D411yescharacteristic
214C2×C4×C53—21no—
215C2×C4×C53—21no—
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—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53#13712yes—
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
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—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—84no—
264Dic106—22no—
265Dic106—22no—
266Dic106—41no—
267Dic106—41no—
268Dic106—41no—
269Dic106—41no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—41no—
273C8×C53—41no—
274C8×C53—41no—
275C8×C53—41no—
276#26725—531nomaximal, Sylow
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53#52511yescharacteristic
285C22×C53#54611yescharacteristic
286C22×C53#54611yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13711yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53#52911yescharacteristic
297C4×C53#53811yescharacteristic
298C4×C53#55511yescharacteristic
299C4×C53#55911yescharacteristic
300Dic53—42no—
301Dic53—42no—
302Dic53—42no—
303Dic53—42no—
304Dic53—81no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308#2489—531no—
309#2360—531no—
310#2056—531no—
311#2035—531no—
312#2026—531no—
313#2001—531no—
314#1989—531no—
315#1933—531no—
316#1927—531no—
317#1923D5311yescharacteristic
318#1920—531no—
319#1878—531no—
320#1872—531no—
321#1823—531no—
322C4×#103—531no—
323C2×C53—42no—
324C2×C53#245911yescharacteristic
325C2×C53#310811yescharacteristic
326C2×C53#312011yescharacteristic
327C2×C53#329611yescharacteristic
328C2×C53#330011yescharacteristic
329C2×C53#331211yescharacteristic
330C2×C53#333111yescharacteristic
331C22×Q16—531no—
332C2×#125—531no—
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8—531no—
335C2×C4×Q8—531no—
336C22×C4⋊C4—531no—
337C2×#106—531no—
338C2×C4⋊C8C2×D5311yescharacteristic
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×C22⋊C8C2×D5311yescharacteristic
346C2×C22⋊C8C2×D5311yescharacteristic
347C2×C4×C8C2×D5311yescharacteristic
348#400—1061no—
349#397—1061no—
350#391—531no—
351#391—531no—
352#391—531no—
353#389—531no—
354#389—1061no—
355#388—1061no—
356#387—1061no—
357#385—1061no—
358#382—1061no—
359#380—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C22⋊C4—1061no—
363C4×C22⋊C4C2×D5311yescharacteristic
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#336C2×D5311yescharacteristic
373#336C2×D5311yescharacteristic
374C53#2672511yescharacteristic, complemented, Sylow
375C22×Q8—531no—
376C22×Q8—531no—
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×Q16—1062no—
380C2×Q16—2121no—
381C2×Q16—2121no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384C22×C8C22×D5311yescharacteristic
385C22×C8C22×D5311yescharacteristic
386#125—1062no—
387#125—2121no—
388C22⋊Q8—1062no—
389C22⋊Q8—1062no—
390C22⋊Q8—1062no—
391C22⋊Q8—2121no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
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⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—2121no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4D21211yescharacteristic
413C2×C22⋊C4D21211yescharacteristic
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418C2×C42C22×D5311yescharacteristic
419#106—1062no—
420C4⋊C8—41no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425#103—2121no—
426#103—2121no—
427C22⋊C8—41no—
428C22⋊C8—41no—
429C4×C8—41no—
430#95—1061no—
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—
440#95D21211yescharacteristic
441#95D21211yescharacteristic
442C24C2×D21211yescharacteristic
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×Q8—1061no—
450C2×Q8—1061no—
451C2×Q8—1061no—
452C2×Q8—1061no—
453C2×Q8—1061no—
454C2×Q8—1061no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4C2×D21211yescharacteristic
473C22×C4C2×D21211yescharacteristic
474C22×C4C2×D21211yescharacteristic
475C22×C4C2×D21211yescharacteristic
476C22×C4C2×D21211yescharacteristic
477C22×C4C23×D5311yescharacteristic
478Q16—2124no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C4⋊C4—532no—
492C4⋊C4—532no—
493C4⋊C4—532no—
494C4⋊C4—1064no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—4241no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4—2122no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—1064no—
513C22⋊C4—2122no—
514C22⋊C4—2122no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—4241no—
520C42—2122no—
521C42—2122no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313323912yes—
529C23C22×D21211yescharacteristic
530Q8—1062no—
531Q8—1062no—
532Q8—2121no—
533Q8—2121no—
534Q8—2121no—
535Q8—2121no—
536Q8—2121no—
537Q8—2121no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—4244no—
561C2×C4—2121no—
562C2×C4—2121no—
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×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4#1313325012yes—
582C2×C4#41×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326211yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1681113511yescharacteristic
600C22#1681126811yescharacteristic
601C22#1681126811yescharacteristic
602C22C2×#1313323911yescharacteristic
603C22C2×#1313325011yescharacteristic
604C22#1681143511yescharacteristic
605C22#1681144011yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—4241no—
612C4—4241no—
613C4—4241no—
614C4—4241no—
615C4—2122no—
616C4—2122no—
617C4—2122no—
618C4—2122no—
619C4#1681115611yescharacteristic
620C4#1681120511yescharacteristic
621C4#1681133511yescharacteristic
622C4#1681135711yescharacteristic
623C2—42no—
624C2#2027865011yescharacteristic
625C2#2028479711yescharacteristic
626C2#2028493111yescharacteristic
627C2#2028655211yescharacteristic
628C2#2028660411yescharacteristic
629C2#2028676811yescharacteristic
630C2#2028703211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1923D53
hypocenterC53#26725
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #20276038C2semidirect
5. #20275867C2semidirect
7. #20275670C2semidirect
8. #20275391C2semidirect
9. #20275350C2semidirect
10. #20275301C2semidirect
11. #20275055C2semidirect
14. C4×#13133142C2semidirect
374. C53#26725semidirect

Maximal subgroups

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

subgroupindex
#202790562
#202777182
#202760382
#202759122
#202758672
#202757252
#202756702
#202753912
#202753502
#202753012
#202750552
#202750202
#202748022
C4×#131331422
C53×#19232
#2672553