Finite Group Databasebeta

← #27452209

Subgroups — #27452209

Order 13568.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277718C211yesmaximal, characteristic
2#20276037C211yesmaximal, characteristic, complemented
3#20275918C211yesmaximal, characteristic
4#20275853C211yesmaximal, characteristic, complemented
5#20275723C211yesmaximal, characteristic
6#20275664C211yesmaximal, characteristic, complemented
7#20275394C211yesmaximal, characteristic, complemented
8#20275353C211yesmaximal, characteristic, complemented
9#20275323C211yesmaximal, characteristic
10#20275307C211yesmaximal, characteristic, complemented
11#20275059C211yesmaximal, characteristic, complemented
12#20275023C211yesmaximal, characteristic
13#20274803C211yesmaximal, characteristic
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2492C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
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×C4⋊Q8×C53C2211yescharacteristic
49C2×C22⋊Q8×C53C2211yescharacteristic
50C53×#400—21no—
51C53×#393—21no—
52C53×#391—21no—
53C53×#391C2211yescharacteristic
54C53×#391C2211yescharacteristic
55C53×#388—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#13133150—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×C53—21no—
100C22×Q8×C53C2311yescharacteristic
101C23×C4×C53C2311yescharacteristic
102C4⋊Q8×C53—22no—
103C4⋊Q8×C53—41no—
104C22⋊Q8×C53—22no—
105C22⋊Q8×C53—41no—
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—
133C2×Q8×C53—42no—
134C2×Q8×C53—42no—
135C2×Q8×C53—81no—
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×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106#4112yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53#4112yes—
173C22⋊C4×C53—41no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—81no—
176C22⋊C4×C53Q1614yescomplemented
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—81no—
182C22⋊Dic53—24no—
183#11719925—44no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—42no—
189C4⋊Dic53#4112yes—
190C4⋊Dic53#4112yes—
191#11719919—24no—
192#11719919—42no—
193#11719919—81no—
194C4×Dic53—42no—
195C4×Dic53—81no—
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×C53C2×Q1612yes—
215C23×C53C22×D411yescharacteristic
216Q8×C53—41no—
217Q8×C53—41no—
218Q8×C53—41no—
219Q8×C53—41no—
220Q8×C53—84no—
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—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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—41no—
249C2×C4×C53—41no—
250C2×C4×C53C2×Q1612yes—
251C2×C4×C53C2×Q1612yes—
252C2×C4×C53C2×#4111yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—84no—
271Dic106—22no—
272Dic106—22no—
273Dic106—41no—
274Dic106—41no—
275#88322—41no—
276#88322—41no—
277#88322—41no—
278#88322—41no—
279#26726—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53#52511yescharacteristic
288C22×C53#54611yescharacteristic
289C22×C53#54611yescharacteristic
290C22×C53C22×Q1611yescharacteristic
291C22×C53#57511yescharacteristic
292C22×C53#57711yescharacteristic
293C22×C53#57711yescharacteristic
294C4×C53—22no—
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—42no—
300C4×C53—81no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53#52911yescharacteristic
304C4×C53#53811yescharacteristic
305C4×C53#55511yescharacteristic
306C4×C53#55911yescharacteristic
307Dic53—42no—
308Dic53—42no—
309Dic53—42no—
310Dic53—81no—
311#2492D5311yescharacteristic
312#2360—531no—
313#2056—531no—
314#2036—531no—
315#2024—531no—
316#2001—531no—
317#1987—531no—
318#1933—531no—
319#1928—531no—
320#1923—531no—
321#1921—531no—
322#1878—531no—
323#1872—531no—
324#1824—531no—
325C4×#103—531no—
326C2×C53—42no—
327C2×C53#245911yescharacteristic
328C2×C53#310411yescharacteristic
329C2×C53#312511yescharacteristic
330C2×C53#329911yescharacteristic
331C2×C53#329911yescharacteristic
332C2×C53#331111yescharacteristic
333C2×C53#332311yescharacteristic
334C22×Q16—531no—
335C2×C4⋊Q8C2×D5311yescharacteristic
336C2×C22⋊Q8—531no—
337C2×C22⋊Q8C2×D5311yescharacteristic
338C2×C4×Q8—531no—
339C22×C4⋊C4—531no—
340C2×#107—531no—
341C2×C4⋊C8—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#103—531no—
345C2×#103—531no—
346C2×#103—531no—
347C2×#103—531no—
348C2×C22⋊C8—531no—
349C2×C22⋊C8—531no—
350C2×C4×C8—531no—
351#400—21no—
352#393—21no—
353#391—21no—
354#391—531no—
355#391C2×D5311yescharacteristic
356#391C2×D5311yescharacteristic
357#389—531no—
358#389—1061no—
359#388—21no—
360#387—1061no—
361#385—1061no—
362#380C2×D5311yescharacteristic
363C4×C4⋊C4—531no—
364C4×C4⋊C4C2×D5311yescharacteristic
365C4×C22⋊C4—1061no—
366C4×C22⋊C4C2×D5311yescharacteristic
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#336—531no—
376#336—531no—
377C53#2672611yescharacteristic, complemented, Sylow
378C22×Q8—21no—
379C22×Q8—531no—
380C22×Q8C22×D5311yescharacteristic
381C23×C4—1061no—
382C23×C4C22×D5311yescharacteristic
383C2×Q16—1062no—
384C2×Q16—2121no—
385C2×Q16—2121no—
386C22×C8—531no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C4×Q8—1062no—
397C4×Q8—2121no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—2121no—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C4⋊C4C22×D5311yescharacteristic
411C2×C4⋊C4C22×D5311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1061no—
415C2×C22⋊C4C22⋊D5311yescharacteristic
416C2×C22⋊C4C22⋊D5311yescharacteristic
417C2×C42—21no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42C22×D5311yescharacteristic
421C2×C42C22×D5311yescharacteristic
422#107—1062no—
423C4⋊C8—2121no—
424#103—1062no—
425#103—1062no—
426#103—1062no—
427#103—1062no—
428#103—2121no—
429#103—2121no—
430C22⋊C8—2121no—
431C22⋊C8—2121no—
432C4×C8—2121no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95C22⋊D5311yescharacteristic
444#95C22⋊D5311yescharacteristic
445C24C2×C22⋊D5311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—532no—
456C2×Q8—1061no—
457C2×Q8—1061no—
458C2×Q8—1061no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—4241no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C23×D5311yescharacteristic
484Q16—2124no—
485C2×C8—1061no—
486C2×C8—1061no—
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—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—532no—
505C4⋊C4—532no—
506C4⋊C4—1064no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—4241no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4#1171994712yes—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—1064no—
519C22⋊C4#1171992514yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—4241no—
526C42—2122no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1171992512yes—
534C23C22×C22⋊D5311yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—84no—
540Q8—1062no—
541Q8—1062no—
542Q8—2121no—
543Q8—2121no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
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—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—4244no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4C2×#1171992512yes—
587C2×C4C2×#1171992512yes—
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C2×C4#41×D5311yescharacteristic
591C2×C4C2×#1171994711yescharacteristic
592C2×C4#1313326211yescharacteristic
593C2×C4#1313326311yescharacteristic
594C8—2121no—
595C8—2121no—
596C8—2121no—
597C8—2121no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1681113011yescharacteristic
606C22#1681126111yescharacteristic
607C22#1681126711yescharacteristic
608C22C22×#1171992511yescharacteristic
609C22#1681142111yescharacteristic
610C22#1681143111yescharacteristic
611C22#1681143411yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—4241no—
622C4—2122no—
623C4—2122no—
624C4—2122no—
625C4#1681115211yescharacteristic
626C4#1681120711yescharacteristic
627C4#1681133511yescharacteristic
628C4#1681135511yescharacteristic
629C2—42no—
630C2#2027864811yescharacteristic
631C2#2028475011yescharacteristic
632C2#2028498611yescharacteristic
633C2#2028658511yescharacteristic
634C2#2028659111yescharacteristic
635C2#2028674511yescharacteristic
636C2#2028693111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2492C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2492D53
hypocenterC53#26726
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #20276037C2semidirect
4. #20275853C2semidirect
6. #20275664C2semidirect
7. #20275394C2semidirect
8. #20275353C2semidirect
10. #20275307C2semidirect
11. #20275059C2semidirect
14. C4×#13133140C2semidirect
176. C22⋊C4×C53Q16semidirect
377. C53#26726semidirect
519. C22⋊C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202777182
#202760372
#202759182
#202758532
#202757232
#202756642
#202753942
#202753532
#202753232
#202753072
#202750592
#202750232
#202748032
C4×#131331402
C53×#24922
#2672653