Finite Group Databasebeta

← #27452194

Subgroups — #27452194

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#20277718C211yesmaximal, characteristic
2#20276037C211yesmaximal, characteristic, complemented
3#20275911C211yesmaximal, characteristic
4#20275863C211yesmaximal, characteristic, complemented
5#20275723C211yesmaximal, characteristic
6#20275673C211yesmaximal, characteristic, complemented
7#20275394C211yesmaximal, characteristic, complemented
8#20275346C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic
10#20275300C211yesmaximal, characteristic, complemented
11#20275059C211yesmaximal, characteristic, complemented
12#20275023C211yesmaximal, characteristic
13#20274799C211yesmaximal, characteristic
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2489C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×C22⋊#88322C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810520—21no—
34#16810510—21no—
35#16810495—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37C22⋊C4×Dic53—21no—
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C2×C22⋊Q8×C53C2211yescharacteristic
50C53×#400—21no—
51C53×#397—21no—
52C53×#391C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#388—21no—
55C53×#382—21no—
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C2×#11719925—22no—
62C2×#11719925—41no—
63C2×#11719925—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133196—22no—
69#13133196—22no—
70C4×Dic106—22no—
71C4×Dic106—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84#13133148—22no—
85C4⋊#88322—41no—
86#13133141—22no—
87#13133141—22no—
88#13133141—41no—
89#13133140—22no—
90#13133140—22no—
91#13133140—41no—
92C22⋊#88322—41no—
93C22⋊#88322—41no—
94C4×#88322—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×Q8×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101#125×C53—22no—
102#125×C53—41no—
103C22⋊Q8×C53—22no—
104C22⋊Q8×C53—41no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
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×C53D411yescharacteristic
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124#95×C53D411yescharacteristic
125#95×C53D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127C2×Q8×C53—21no—
128C2×Q8×C53—21no—
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×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×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106#4112yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53#4112yes—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—42no—
180C22⋊Dic53—24no—
181#11719925—44no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53#4112yes—
188C4⋊Dic53#4112yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—42no—
193C4×Dic53—81no—
194C2×#88322—21no—
195C2×#88322—21no—
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—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53#13512yes—
213C23×C53C22×D411yescharacteristic
214Q8×C53—41no—
215Q8×C53—41no—
216Q8×C53—41no—
217Q8×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
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#13712yes—
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53H(5,𝔽2)11yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—84no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272#88322—41no—
273#88322—41no—
274#88322—41no—
275#88322—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—42no—
294C4×C53—41no—
295C4×C53—41no—
296C4×C53—41no—
297C4×C53—81no—
298C4×C53—81no—
299C4×C53—81no—
300C4×C53#52911yescharacteristic
301C4×C53#53811yescharacteristic
302C4×C53#55511yescharacteristic
303C4×C53#55911yescharacteristic
304Dic53—42no—
305Dic53—42no—
306Dic53—42no—
307Dic53—81no—
308#2489D5311yescharacteristic
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#1923—531no—
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×#125C2×D5311yescharacteristic
333C2×C22⋊Q8—531no—
334C2×C22⋊Q8C2×D5311yescharacteristic
335C2×C4×Q8—531no—
336C22×C4⋊C4—531no—
337C2×#106—531no—
338C2×C4⋊C8—531no—
339C2×#103—531no—
340C2×#103—531no—
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×C22⋊C8—531no—
346C2×C22⋊C8—531no—
347C2×C4×C8—531no—
348#400—21no—
349#397—21no—
350#391—531no—
351#391C2×D5311yescharacteristic
352#391C2×D5311yescharacteristic
353#389—531no—
354#389—1061no—
355#388—21no—
356#387—1061no—
357#385—1061no—
358#382—21no—
359#380C2×D5311yescharacteristic
360C4×C4⋊C4—531no—
361C4×C4⋊C4C2×D5311yescharacteristic
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#336—531no—
373#336—531no—
374C53#2672511yescharacteristic, complemented, Sylow
375C22×Q8—531no—
376C22×Q8C22×D5311yescharacteristic
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×Q16—1062no—
380C2×Q16—2121no—
381C2×Q16—2121no—
382C22×C8—531no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—531no—
386#125—22no—
387#125—41no—
388C22⋊Q8—22no—
389C22⋊Q8—41no—
390C22⋊Q8—1062no—
391C22⋊Q8—1062no—
392C4×Q8—1062no—
393C4×Q8—2121no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
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—2121no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4C22⋊D5311yescharacteristic
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C42—21no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418C2×C42C22×D5311yescharacteristic
419#106—1062no—
420C4⋊C8—2121no—
421#103—1062no—
422#103—1062no—
423#103—1062no—
424#103—1062no—
425#103—2121no—
426#103—2121no—
427C22⋊C8—2121no—
428C22⋊C8—2121no—
429C4×C8—2121no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95C22⋊D5311yescharacteristic
441#95C22⋊D5311yescharacteristic
442C24C2×C22⋊D5311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—532no—
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—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—4241no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478Q16—2124no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—1061no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—532no—
500C4⋊C4—532no—
501C4⋊C4—1064no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—4241no—
507C4⋊C4—2122no—
508C4⋊C4—2122no—
509C4⋊C4#1171994712yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—1064no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—4241no—
521C42—2122no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1313323712yes—
529C23C22×C22⋊D5311yescharacteristic
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—1062no—
535Q8—1062no—
536Q8—2121no—
537Q8—2121no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
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—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—4244no—
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#1313324712yes—
582C2×C4#41×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994711yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326311yescharacteristic
588C8—2121no—
589C8—2121no—
590C8—2121no—
591C8—2121no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1681113011yescharacteristic
600C22#1681126111yescharacteristic
601C22#1681126711yescharacteristic
602C22C2×#1313323711yescharacteristic
603C22C2×#1313324711yescharacteristic
604C22#1681143111yescharacteristic
605C22#1681143811yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—4241no—
616C4—2122no—
617C4—2122no—
618C4—2122no—
619C4#1681115211yescharacteristic
620C4#1681120711yescharacteristic
621C4#1681133511yescharacteristic
622C4#1681135511yescharacteristic
623C2—42no—
624C2#2027864811yescharacteristic
625C2#2028479111yescharacteristic
626C2#2028492311yescharacteristic
627C2#2028654411yescharacteristic
628C2#2028660211yescharacteristic
629C2#2028676011yescharacteristic
630C2#2028703111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2489C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2489D53
hypocenterC53#26725
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #20276037C2semidirect
4. #20275863C2semidirect
6. #20275673C2semidirect
7. #20275394C2semidirect
8. #20275346C2semidirect
10. #20275300C2semidirect
11. #20275059C2semidirect
14. C4×#13133140C2semidirect
374. C53#26725semidirect

Maximal subgroups

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

subgroupindex
#202777182
#202760372
#202759112
#202758632
#202757232
#202756732
#202753942
#202753462
#202753232
#202753002
#202750592
#202750232
#202747992
C4×#131331402
C53×#24892
#2672553