Finite Group Databasebeta

← #27429891

Subgroups — #27429891

Order 13568.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810879C211yesmaximal, characteristic
2C2×C4×#11719925C211yesmaximal, characteristic, complemented
3#20278240C211yesmaximal, characteristic
4#20276134C211yesmaximal, characteristic, complemented
5#20276123C211yesmaximal, characteristic, complemented
6#20275918C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275585C211yesmaximal, characteristic, complemented
9#20275440C211yesmaximal, characteristic, complemented
10#20274982C211yesmaximal, characteristic, complemented
11#20274969C211yesmaximal, characteristic, complemented
12#20274884C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic
14#20274499C211yesmaximal, characteristic
15C53×#2421C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810879—24no—
21C4×#11719925—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810559—21no—
37#16810557—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810516—21no—
41#16810506—21no—
42#16810493C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#397—21no—
54C53×#391C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#387—21no—
58C53×#382C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719925D414yes—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917—22no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—41no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87#13133150—22no—
88#13133150—41no—
89#13133148—22no—
90#13133148—22no—
91#13133148—41no—
92#13133141—22no—
93#13133141—22no—
94#13133141—22no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98C22⋊#88322—22no—
99#13133128—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—41no—
109C4×Q8×C53—22no—
110C4×Q8×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53C2×D412yes—
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×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106C2×D412yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—41no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—42no—
183C22⋊Dic53—42no—
184C22⋊Dic53—81no—
185#11719925—28no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194#11719919—42no—
195#11719919—42no—
196#11719919—81no—
197C4×Dic53—42no—
198#11719917—44no—
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×#88322C2×D412yes—
208C2×#88322#4112yes—
209C2×#88322#4112yes—
210C2×#88322#4112yes—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53C22×D411yescharacteristic
219Q8×C53—22no—
220Q8×C53—22no—
221Q8×C53—41no—
222Q8×C53—41no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—22no—
228C2×C4×C53—21no—
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—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
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×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53#13712yes—
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C532−1+411yescharacteristic
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—41no—
271C2×Dic53—41no—
272Dic106—22no—
273Dic106—22no—
274Dic106—41no—
275Dic106—41no—
276#88322—22no—
277#88322—41no—
278#88322—41no—
279#88322#12314yes—
280#25039—531nomaximal, Sylow
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53#53211yescharacteristic
289C22×C53D4211yescharacteristic
290C22×C53#54711yescharacteristic
291C22×C53C2×#13711yescharacteristic
292C22×C53#57511yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57811yescharacteristic
295C4×C53—22no—
296C4×C53—41no—
297C4×C53—41no—
298C4×C53—41no—
299C4×C53—42no—
300C4×C53—41no—
301C4×C53—81no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53C2×#12311yescharacteristic
305C4×C53#54211yescharacteristic
306C4×C53#54711yescharacteristic
307C4×C53#55311yescharacteristic
308Dic53—42no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312C2×#470—531no—
313C2×C4×Q16—531no—
314#2423—531no—
315#2421D5311yescharacteristic
316#2074—531no—
317#2072—531no—
318#2036—531no—
319#2035—531no—
320#1971—531no—
321#1941—531no—
322#1865—531no—
323#1863—531no—
324#1847—531no—
325#1809—531no—
326#1753—531no—
327C2×C53—81no—
328C2×C53#242211yescharacteristic
329C2×C53#317211yescharacteristic
330C2×C53#317811yescharacteristic
331C2×C53#327011yescharacteristic
332C2×C53#327111yescharacteristic
333C2×C53#335211yescharacteristic
334C2×C53#335611yescharacteristic
335C22×Q16—531no—
336C22×M16—531no—
337C2×C22⋊Q8—531no—
338C2×C22⋊Q8C2×D5311yescharacteristic
339C2×C4×Q8—531no—
340C2×C4×Q8C2×D5311yescharacteristic
341#470—1064no—
342C4×Q16—1064no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#106—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#103—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×C22⋊C8—531no—
355C2×#97—531no—
356C2×C4×C8—531no—
357#400—21no—
358#399—1061no—
359#398—1061no—
360#397—21no—
361#391—531no—
362#391C2×D5311yescharacteristic
363#389—531no—
364#389C2×D5311yescharacteristic
365#388—21no—
366#388—1061no—
367#387—21no—
368#387—1061no—
369#384—531no—
370#382C2×D5311yescharacteristic
371#380—531no—
372#380C2×D5311yescharacteristic
373C4×C22⋊C4C2×D5311yescharacteristic
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380C53#2503911yescharacteristic, complemented, Sylow
381C22×Q8—531no—
382C22×Q8C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C2×Q16—534no—
385C2×Q16—1062no—
386C2×Q16—1062no—
387C2×M16—1062no—
388C2×M16—1062no—
389C2×M16—1062no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C22⋊Q8—22no—
395C22⋊Q8—41no—
396C22⋊Q8—1062no—
397C22⋊Q8—2121no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—1062no—
401C4×Q8—2121no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4C22⋊D5311yescharacteristic
421C2×C22⋊C4C22⋊D5311yescharacteristic
422C2×C42—21no—
423C2×C42—1061no—
424C2×C42C22×D5311yescharacteristic
425C2×C42C22×D5311yescharacteristic
426#107—1062no—
427#107—2121no—
428#106—1062no—
429#106—1062no—
430#106—2121no—
431#103—1062no—
432#103—1062no—
433#103—1062no—
434#103—1062no—
435#103—1062no—
436#103—1062no—
437C22⋊C8—1062no—
438#97—1062no—
439C4×C8—1062no—
440#95—21no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95C22⋊D5311yescharacteristic
451#95C22⋊D5311yescharacteristic
452C24C2×C22⋊D5311yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—532no—
459C2×Q8—1061no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8D4×D5312yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C23×D5311yescharacteristic
485Q16—1068no—
486M16—2124no—
487C2×C8—532no—
488C2×C8—532no—
489C2×C8—532no—
490C2×C8—532no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—4241no—
515C4⋊C4—4241no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4—2122no—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—4241no—
524C22⋊C4—2122no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—2122no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23C22×C22⋊D5311yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—1062no—
544Q8—1062no—
545Q8—2121no—
546Q8—2121no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
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—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—1062no—
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×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4#1313324712yes—
590C2×C4C2×D4×D5311yescharacteristic
591C2×C4#41×D5311yescharacteristic
592C2×C4#41×D5311yescharacteristic
593C2×C4#41×D5311yescharacteristic
594C2×C4#1313326211yescharacteristic
595C2×C4#1313326411yescharacteristic
596C8—534no—
597C8—1062no—
598C8—2121no—
599C8—2121no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22#1681116911yescharacteristic
608C22D4×C22⋊D5311yescharacteristic
609C22#1681127211yescharacteristic
610C22C2×#1313324711yescharacteristic
611C22#1681142111yescharacteristic
612C22#1681143111yescharacteristic
613C22#1681143811yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—4241no—
624C4—4241no—
625C4—4241no—
626C4—2122no—
627C4#123×D5311yescharacteristic
628C4#1681124111yescharacteristic
629C4#1681127111yescharacteristic
630C4#1681131811yescharacteristic
631C2—81no—
632C2#2027822611yescharacteristic
633C2#2028532211yescharacteristic
634C2#2028538511yescharacteristic
635C2#2028628111yescharacteristic
636C2#2028630111yescharacteristic
637C2#2028728411yescharacteristic
638C2#2028732411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2421C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2421D53
hypocenterC53#25039
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2421

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11719925C2semidirect
4. #20276134C2semidirect
5. #20276123C2semidirect
8. #20275585C2semidirect
9. #20275440C2semidirect
10. #20274982C2semidirect
11. #20274969C2semidirect
12. #20274884C2semidirect
380. C53#25039semidirect

Maximal subgroups

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

subgroupindex
C2×#168108792
C2×C4×#117199252
#202782402
#202761342
#202761232
#202759182
#202759082
#202755852
#202754402
#202749822
#202749692
#202748842
#202747462
#202744992
C53×#24212
#2503953