Finite Group Databasebeta

← #27351296

Subgroups — #27351296

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288766C211yesmaximal, characteristic
2#20283749C211yesmaximal, characteristic, complemented
3#20277077C211yesmaximal, characteristic, complemented
4#20273240C211yesmaximal, characteristic, complemented
5#20273233C212yesmaximal, complemented
6#20273226C211yesmaximal, characteristic, complemented
7#20273216C212yesmaximal, complemented
8#20273215C211yesmaximal, characteristic, complemented
9#20273203C212yesmaximal, complemented
10#328×D53C212yesmaximal, complemented
11C53×#1500C211yesmaximal, characteristic, complemented
12#16811346C411yescharacteristic, complemented
13#16811330—21no—
14#16811311C412yescomplemented
15#16811296—22no—
16Q8×D212C411yescharacteristic, complemented
17#16811281C412yescomplemented
18#16811278C411yescharacteristic, complemented
19#16811277C411yescharacteristic, complemented
20#16811269—21no—
21#16811244—21no—
22#16811214—22no—
23#16811180C2211yescharacteristic
24#16811133C2212yes—
25C22⋊Q8×D53C2212yes—
26#16811103—21no—
27C2×C4×D212—21no—
28C2×C4×D212C2211yescharacteristic
29#16810702—21no—
30#16810693—22no—
31C8×D212—21no—
32#16810680C2211yescharacteristic
33#16810585C2212yes—
34C22⋊C8×D53C2212yes—
35C4×C4⋊Dic53—21no—
36C4×C4⋊Dic53C2211yescharacteristic
37#16810470—22no—
38#16810470C2212yes—
39#95×D53C2212yes—
40#16810361C2212yes—
41#16810361C2212yes—
42#16810360C2212yes—
43#16810359C2212yes—
44#16810359C2212yes—
45#16810357C2212yes—
46C53×#529C2211yescharacteristic
47C53×#431C2211yescharacteristic
48C53×#376C2211yescharacteristic
49C53×#328C2211yescharacteristic
50C53×#328C2211yescharacteristic
51C53×#327C2211yescharacteristic
52C53×#327C2211yescharacteristic
53C2×#11719945—21no—
54C2×#11719947C2×C412yescomplemented
55C2×Q8×D53C2×C412yescomplemented
56C22×D212D411yescharacteristic
57C22×C4×D53—22no—
58C22×C4×D53C2312yes—
59C2×#11719912—22no—
60C2×C8×D53—22no—
61#13133217—21no—
62#13133217C2×C411yescharacteristic, complemented
63#13133217C2×C411yescharacteristic, complemented
64#13133212—22no—
65#13133212—22no—
66#13133209—21no—
67#13133208—22no—
68#13133205—21no—
69#13133203—22no—
70#13133202—22no—
71#13133201—22no—
72#13133199—22no—
73#13133199—22no—
74#13133197—22no—
75#13133197C2×C412yescomplemented
76#13133197C2×C412yescomplemented
77#13133196—21no—
78#13133196—21no—
79#13133196C2312yes—
80#13133195—22no—
81#13133194—22no—
82#13133194—22no—
83#13133193C2×C412yescomplemented
84#13133191—21no—
85#13133190—22no—
86#13133188C2×C412yescomplemented
87#13133185—22no—
88C4×Dic106—21no—
89C4×Dic106—21no—
90C4×Dic106C2×C412yescomplemented
91C4×D212—28no—
92C4×D212—21no—
93C4×D212C2×C411yescharacteristic
94C4×D212C2×C411yescharacteristic
95C4×D212C2×C412yes—
96C4×D212D411yescharacteristic
97C4×D212D412yes—
98C4×D212D411yescharacteristic
99D4×Dic53—22no—
100#13133177—22no—
101#13133177—22no—
102#13133176—21no—
103#13133175—22no—
104#13133175C2×C412yescomplemented
105C2×C4⋊Dic53—21no—
106C2×C4⋊Dic53—21no—
107C2×C4⋊Dic53D411yescharacteristic
108C4⋊C4×D53—22no—
109C4⋊C4×D53C2×C412yescomplemented
110C2×#11719921—22no—
111C2×#11719921D412yes—
112C22⋊C4×D53—22no—
113C2×C4×Dic53—21no—
114C2×C4×Dic53—22no—
115C4⋊#88322—21no—
116C22⋊#88322C2312yes—
117#13133129—22no—
118#13133129—22no—
119#13133128—21no—
120#13133124—22no—
121#13133124C2312yes—
122#126×C53—21no—
123#125×C53C2×C411yescharacteristic, complemented
124#124×C53C2×C411yescharacteristic, complemented
125#123×C53—21no—
126C22⋊Q8×C53C2311yescharacteristic
127#121×C53C2311yescharacteristic
128C4×Q8×C53C2×C411yescharacteristic, complemented
129C4×D4×C53—21no—
130C4×D4×C53C2×C411yescharacteristic, complemented
131#117×C53—21no—
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53—21no—
135C22⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53C2311yescharacteristic
137C4×C8×C53—21no—
138#95×C53—21no—
139#95×C53C2311yescharacteristic
140#95×C53C2311yescharacteristic
141C23×D53C2×D412yes—
142C2×Q8×C53C22×C411yescharacteristic
143C2×D4×C53—21no—
144C2×D4×C53C22×C411yescharacteristic
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2411yescharacteristic
151C2×C22⋊D53—22no—
152C2×C22⋊D53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53C2×D412yes—
156Q8×D53—82no—
157#11719947—82no—
158#11719945—44no—
159C2×D212—24no—
160C2×D212C22⋊C412yes—
161C2×D212C22⋊C411yescharacteristic
162C2×D212C22⋊C411yescharacteristic
163C2×C4×D53—22no—
164C2×C4×D53—28no—
165C2×C4×D53—22no—
166C2×C4×D53—82no—
167C2×C4×D53—82no—
168C2×C4×D53—82no—
169C2×C4×D53C22×C412yes—
170C2×C4×D53C22×C412yes—
171C2×C4×D53C2×D412yes—
172C2×C4×D53C2×D412yes—
173C2×Dic106—21no—
174C2×Dic106—22no—
175C2×Dic106C22×C412yes—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53C22×C411yescharacteristic
183C4⋊C4×C53C22×C411yescharacteristic
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C22⋊C4×C53—21no—
187C22⋊C4×C53—21no—
188C22⋊C4×C53—21no—
189C42×C53—24no—
190C42×C53—21no—
191C42×C53C22×C411yescharacteristic
192C42×C53C22×C411yescharacteristic
193C42×C53C2×D411yescharacteristic
194C42×C53C2×D411yescharacteristic
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197#11719921—28no—
198#11719921—22no—
199#11719921—22no—
200#11719921—22no—
201#11719921—22no—
202#11719921C22⋊C412yes—
203#11719921C22⋊C412yes—
204#11719921C22⋊C412yes—
205#11719921C22⋊C412yes—
206C4⋊Dic53—24no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—22no—
211C4⋊Dic53—22no—
212C4⋊Dic53C22⋊C411yescharacteristic
213C4⋊Dic53C22⋊C412yes—
214C4⋊Dic53C22⋊C411yescharacteristic
215C4⋊Dic53C22×C412yes—
216#11719919—22no—
217#11719919—22no—
218#11719919—22no—
219#11719919—22no—
220C4×Dic53—22no—
221C4×Dic53—22no—
222C4×Dic53—81no—
223C4×Dic53—82no—
224C2×#88322—22no—
225#11719912—82no—
226C8×D53—82no—
227C23×C53—21no—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—24no—
232C22×D53—42no—
233C22×D53C2×C22⋊C412yes—
234C22×D53C2×C22⋊C412yes—
235Q8×C53—41no—
236D4×C53—41no—
237D4×C53—41no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—42no—
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C22×D411yescharacteristic
259C22⋊D53—44no—
260C22⋊D53—82no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—82no—
268C2×Dic53C2×C22⋊C412yes—
269C2×Dic53C2×C22⋊C412yes—
270D212—24no—
271D212—22no—
272D212#10414yes—
273C4×D53—42no—
274C4×D53—42no—
275C4×D53—42no—
276C4×D53—42no—
277C4×D53—44no—
278C4×D53—82no—
279C4×D53—88no—
280Dic106—42no—
281Dic106—42no—
282Dic106—82no—
283C8×C53—41no—
284C8×C53—41no—
285#88322—42no—
286#17548—531nomaximal, Sylow
287C22×C53—21no—
288C22×C53—21no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C22×C22⋊C411yescharacteristic
293C22×C53#57711yescharacteristic
294C22×C53#57811yescharacteristic
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—44no—
301C2×D53—44no—
302C2×D53C2×#10414yes—
303C4×C53—21no—
304C4×C53—21no—
305C4×C53—21no—
306C4×C53—21no—
307C4×C53—21no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—41no—
313C4×C53C2×#10412yes—
314Dic53—22no—
315Dic53—22no—
316Dic53—44no—
317Dic53—42no—
318Dic53—42no—
319#3520—531no—
320#2974—531no—
321#2275—531no—
322#1509—531no—
323#1508—532no—
324#1506—531no—
325#1504—532no—
326#1503—531no—
327#1501—532no—
328#1500D5311yescharacteristic, complemented
329C2×#328—532no—
330C2×C53—21no—
331C2×C53—21no—
332C2×C53—81no—
333C2×C53#288311yescharacteristic
334C2×C53#289811yescharacteristic
335C2×C53C22×#10411yescharacteristic
336D53—24no—
337D53—42no—
338#556—531no—
339#554—1061no—
340#552—532no—
341#551—1062no—
342D4×Q8—531no—
343#548—532no—
344#547—531no—
345#547—531no—
346#546—1061no—
347#542—1061no—
348#539—1062no—
349#533—531no—
350#529C2×D5311yescharacteristic
351#525—532no—
352C2×C22⋊Q8—532no—
353#520—1061no—
354C2×C4×D4—531no—
355C2×C4×D4—1061no—
356#436—1061no—
357#435—1062no—
358C8×D4—1061no—
359#432—531no—
360#431C2×D5311yescharacteristic
361#407—532no—
362C2×C22⋊C8—532no—
363C4×C4⋊C4—531no—
364C4×C4⋊C4—1061no—
365C4×C22⋊C4—532no—
366C4×C22⋊C4—1062no—
367#376C2×D5311yescharacteristic
368C2×#95—532no—
369#328—532no—
370#328—532no—
371#328—532no—
372#328—532no—
373#328—532no—
374#328C2×D5311yescharacteristic
375#328C2×D5311yescharacteristic
376#327—532no—
377#327C2×D5311yescharacteristic
378#327C2×D5311yescharacteristic
379C53#1754811yescharacteristic, complemented, Sylow
380C2×#41—532no—
381C2×#41—1061no—
382C22×Q8—532no—
383C22×D4—531no—
384C23×C4—532no—
385C23×C4—1062no—
386C2×M16—1062no—
387C22×C8—1062no—
388C4⋊Q8—531no—
389C4⋊Q8—531no—
390C4⋊Q8—1061no—
391#126—21no—
392#126—1062no—
393#126—1062no—
394#125—1062no—
395#125—1061no—
396#125C4×D5311yescharacteristic, complemented
397#124—1061no—
398#124—1062no—
399#124C4×D5311yescharacteristic, complemented
400#123—21no—
401#123—1062no—
402#123—1062no—
403#123—1062no—
404#123—1062no—
405C22⋊Q8—532no—
406C22⋊Q8—532no—
407C22⋊Q8—532no—
408C22⋊Q8—532no—
409C22⋊Q8—1062no—
410C22⋊Q8—1062no—
411C22⋊Q8—1062no—
412C22⋊Q8—1062no—
413C22⋊Q8—1061no—
414C22⋊Q8—1061no—
415C22⋊Q8C22×D5311yescharacteristic
416#121—532no—
417#121—1062no—
418#121—1061no—
419#121C22×D5311yescharacteristic
420#120—1062no—
421C4×Q8—532no—
422C4×Q8—1061no—
423C4×Q8—1061no—
424C4×Q8C4×D5311yescharacteristic, complemented
425C4×D4—21no—
426C4×D4—531no—
427C4×D4—531no—
428C4×D4—532no—
429C4×D4—531no—
430C4×D4—532no—
431C4×D4—531no—
432C4×D4—1061no—
433C4×D4—1068no—
434C4×D4—1062no—
435C4×D4—1062no—
436C4×D4—1062no—
437C4×D4C4×D5311yescharacteristic, complemented
438#117—21no—
439#117—532no—
440#117—1062no—
441#117—1061no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—532no—
444C2×C4⋊C4—1062no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C22⋊C4—532no—
448C2×C22⋊C4—1062no—
449C2×C22⋊C4—1062no—
450C2×C42—21no—
451C2×C42—1062no—
452C2×C42—1061no—
453C2×C42C22×D5311yescharacteristic
454C4⋊C8—21no—
455C4⋊C8—1061no—
456C22⋊C8—532no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C22⋊C8C22×D5311yescharacteristic
460C22⋊C8C22×D5311yescharacteristic
461#97—1061no—
462C4×C8—21no—
463#95—21no—
464#95—532no—
465#95—1062no—
466#95C22×D5311yescharacteristic
467#95C22×D5311yescharacteristic
468C24—532no—
469#41—4242no—
470#41—2124no—
471C2×Q8—532no—
472C2×Q8—1062no—
473C2×Q8—1061no—
474C2×Q8—4242no—
475C2×Q8C2×C4×D5311yescharacteristic
476C2×D4—21no—
477C2×D4—531no—
478C2×D4—532no—
479C2×D4—531no—
480C2×D4—1064no—
481C2×D4—1062no—
482C2×D4—1062no—
483C2×D4C2×C4×D5311yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—532no—
492C22×C4—1062no—
493C22×C4—1068no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—4242no—
498C22×C4—4242no—
499C22×C4—4242no—
500C22×C4D4×D5311yescharacteristic
501C22×C4D4×D5311yescharacteristic
502C22×C4C23×D5311yescharacteristic
503M16—4242no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1062no—
507C2×C8—4242no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—531no—
513C4⋊C4—532no—
514C4⋊C4—531no—
515C4⋊C4—532no—
516C4⋊C4—1062no—
517C4⋊C4—1062no—
518C4⋊C4—1062no—
519C4⋊C4—1062no—
520C4⋊C4—1064no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C4⋊C4C2×C4×D5311yescharacteristic
527C4⋊C4C2×C4×D5311yescharacteristic
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—532no—
534C22⋊C4—532no—
535C22⋊C4—532no—
536C22⋊C4—532no—
537C22⋊C4—1062no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1068no—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C42—24no—
545C42—21no—
546C42—1062no—
547C42—1062no—
548C42—4241no—
549C42—4242no—
550C42C2×C4×D5311yescharacteristic
551C42C2×C4×D5311yescharacteristic
552C42D4×D5311yescharacteristic
553C42D4×D5311yescharacteristic
554C23—21no—
555C23—532no—
556C23—532no—
557C23—1064no—
558C23—1062no—
559C23—1062no—
560C23—2122no—
561C23C2×D4×D5311yescharacteristic
562Q8—41no—
563Q8—4242no—
564Q8—2122no—
565Q8—2122no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—534no—
570D4—1064no—
571D4—1062no—
572D4—4242no—
573D4—2124no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—1064no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—4242no—
594C2×C4—4248no—
595C2×C4—4242no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4—2124no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4C22⋊C4×D5311yescharacteristic
605C2×C4C22⋊C4×D5311yescharacteristic
606C2×C4C22⋊C4×D5311yescharacteristic
607C2×C4C22⋊C4×D5311yescharacteristic
608C2×C4C22×C4×D5311yescharacteristic
609C2×C4C2×D4×D5311yescharacteristic
610C8—41no—
611C8—41no—
612C8—2122no—
613C22—21no—
614C22—21no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—534no—
619C22—1062no—
620C22—1062no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—2124no—
625C22—2124no—
626C22C2×C22⋊C4×D5311yescharacteristic
627C22#1681143011yescharacteristic
628C22#1681143711yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—1062no—
640C4—1062no—
641C4—2124no—
642C4—2122no—
643C4—2122no—
644C4#104×D5312yes—
645C2—21no—
646C2—21no—
647C2—81no—
648C2—1064no—
649C2—2122no—
650C2#2028317411yescharacteristic
651C2#2028324411yescharacteristic
652C2C2×#104×D5311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1500C2
socleC22×C53C22×C22⋊C4
prime radical#20288766C2
cosocleC22×C4×C53C24
hypercenter#1500D53
hypocenterC53#17548
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
2. #20283749C2semidirect
3. #20277077C2semidirect
4. #20273240C2semidirect
5. #20273233C2semidirect
6. #20273226C2semidirect
7. #20273216C2semidirect
8. #20273215C2semidirect
9. #20273203C2semidirect
10. #328×D53C2semidirect
11. C53×#1500C2semidirect
12. #16811346C4semidirect
14. #16811311C4semidirect
16. Q8×D212C4semidirect
17. #16811281C4semidirect
18. #16811278C4semidirect
19. #16811277C4semidirect
54. C2×#11719947C2×C4semidirect
55. C2×Q8×D53C2×C4semidirect
62. #13133217C2×C4semidirect
63. #13133217C2×C4semidirect
75. #13133197C2×C4semidirect
76. #13133197C2×C4semidirect
83. #13133193C2×C4semidirect
86. #13133188C2×C4semidirect
90. C4×Dic106C2×C4semidirect
104. #13133175C2×C4semidirect
109. C4⋊C4×D53C2×C4semidirect
123. #125×C53C2×C4semidirect
124. #124×C53C2×C4semidirect
128. C4×Q8×C53C2×C4semidirect
130. C4×D4×C53C2×C4semidirect
328. #1500D53semidirect
379. C53#17548semidirect
396. #125C4×D53semidirect
399. #124C4×D53semidirect
424. C4×Q8C4×D53semidirect
437. C4×D4C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202887662
#202837492
#202770772
#202732402
#202732332
#202732262
#202732162
#202732152
#202732032
#328×D532
C53×#15002
#1754853