Finite Group Databasebeta

← #27439866

Subgroups — #27439866

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811176C211yesmaximal, characteristic
2C2×#16811024C211yesmaximal, characteristic, complemented
3C2×#16811023C211yesmaximal, characteristic, complemented
4#20283023C211yesmaximal, characteristic
5#20276047C211yesmaximal, characteristic, complemented
6#20276046C211yesmaximal, characteristic, complemented
7#20276042C211yesmaximal, characteristic, complemented
8#20276041C211yesmaximal, characteristic, complemented
9#20275678C211yesmaximal, characteristic, complemented
10#20275678C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic
12#20274914C211yesmaximal, characteristic
13#20274912C211yesmaximal, characteristic
14C22⋊C8×Dic53C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16#16811176—24no—
17#16811176—24no—
18C2×#13133216C2211yescharacteristic
19C2×#13133216C2211yescharacteristic
20C2×#13133208C2211yescharacteristic
21C2×#13133208C2211yescharacteristic
22C2×#13133208C2211yescharacteristic
23C2×#13133194—21no—
24C2×#13133194—21no—
25C2×C4×Dic106—21no—
26C2×C4×Dic106—21no—
27C2×#13133176C2211yescharacteristic
28C22×C4×Dic53C2211yescharacteristic
29#16811024—24no—
30#16811023—24no—
31C2×#13133151C2211yescharacteristic
32C2×#13133150C2211yescharacteristic
33C2×#13133150C2211yescharacteristic
34C2×#13133149C2211yescharacteristic
35C2×#13133148C2211yescharacteristic
36C2×#13133148C2211yescharacteristic
37C2×C22⋊#88322C2211yescharacteristic
38C2×C8×Dic53C2211yescharacteristic
39C2×C8×Dic53C2211yescharacteristic
40#16810522C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810510—21no—
43#16810506—21no—
44#16810498—21no—
45#16810495—21no—
46#16810488C2211yescharacteristic
47#16810483C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810377C2211yescharacteristic
55#16810377C2211yescharacteristic
56C2×#117×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C53×#340C2211yescharacteristic
63C22×Dic106—21no—
64C22×Dic106—21no—
65C23×Dic53D411yescharacteristic
66C23×Dic53D411yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133216—22no—
70#13133216—22no—
71#13133216—22no—
72#13133216—22no—
73#13133208—22no—
74#13133208—22no—
75#13133208—22no—
76#13133208—22no—
77#13133208D414yes—
78#13133208D414yes—
79#13133194—28no—
80#13133194—28no—
81C4×Dic106—28no—
82C4×Dic106—28no—
83#13133176—22no—
84#13133176—22no—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53—21no—
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—21no—
101C2×C4×Dic53—22no—
102C2×C4×Dic53—22no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53C2311yescharacteristic
107C2×C4×Dic53C2311yescharacteristic
108#13133151—22no—
109#13133150—22no—
110#13133150—22no—
111#13133149—22no—
112#13133148—22no—
113#13133148—22no—
114C22⋊#88322Q814yescomplemented
115C8×Dic53—22no—
116C8×Dic53—22no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121#13133124—21no—
122C23×C4×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125#117×C53—22no—
126#117×C53—22no—
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132C2×C22⋊C4×C53—21no—
133C2×C42×C53—21no—
134#107×C53—22no—
135#106×C53—22no—
136C22⋊C8×C53Q814yescomplemented
137#95×C53—21no—
138C24×C53C2×D411yescharacteristic
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×C53C2×D411yescharacteristic
148C22×C4×C53C2×Q812yes—
149C22×C4×C53C2×Q812yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—41no—
160C22×Dic53—41no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C2×Dic106—24no—
166C2×Dic106—24no—
167C2×Dic106—42no—
168C2×Dic106—42no—
169C2×Dic106—42no—
170C2×Dic106—42no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53C2×Q812yes—
176C2×C8×C53C2×Q812yes—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C2×D412yes—
184C4⋊C4×C53C2×D412yes—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—42no—
187C42×C53—24no—
188C22⋊Dic53—24no—
189C22⋊Dic53—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—24no—
193C4⋊Dic53—24no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—42no—
200C4⋊Dic53C2×D412yes—
201C4⋊Dic53C2×D412yes—
202#11719919—24no—
203#11719919—24no—
204#11719919—24no—
205#11719919—24no—
206#11719919—42no—
207#11719919—42no—
208C4×Dic53—24no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—22no—
212C4×Dic53—22no—
213C4×Dic53C2×D412yes—
214C4×Dic53C2×D412yes—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322C2×Q812yes—
220C2×#88322C2×Q812yes—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53C4⋊Q812yes—
228C23×C53C22×D411yescharacteristic
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
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×C53C4⋊Q812yes—
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C22×Q811yescharacteristic
252C2×C4×C532−1+411yescharacteristic
253C2×C4×C532−1+411yescharacteristic
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—44no—
265C2×Dic53—44no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53#12012yes—
277C2×Dic53#12012yes—
278C2×Dic53#13512yes—
279C2×Dic53#13512yes—
280Dic106—48no—
281Dic106—48no—
282C8×C53—22no—
283C8×C53—22no—
284#88322—22no—
285#88322—22no—
286#25814—531nomaximal, Sylow
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53C2×#12011yescharacteristic
295C22×C53C2×C4⋊Q811yescharacteristic
296C22×C53#53611yescharacteristic
297C22×C53C2×#13511yescharacteristic
298C22×C53C2×#13511yescharacteristic
299C22×C53#57511yescharacteristic
300C22×C53#57511yescharacteristic
301C4×C53—22no—
302C4×C53—42no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53D4×Q811yescharacteristic
307C4×C53D4×Q811yescharacteristic
308C4×C53D4×Q811yescharacteristic
309C4×C53D4×Q811yescharacteristic
310Dic53—22no—
311Dic53—22no—
312Dic53—22no—
313Dic53—22no—
314Dic53—42no—
315Dic53—42no—
316Dic53—81no—
317Dic53—81no—
318C2×#533—531no—
319C2×#499—531no—
320C2×#499—531no—
321#2851—531no—
322#2058—531no—
323#2058—531no—
324#2057—531no—
325#2057—531no—
326#1991—531no—
327#1991—531no—
328#1854—531no—
329#1854—531no—
330#1854—531no—
331#1854D5311yescharacteristic
332C4×C22⋊C8—531no—
333C2×C53—42no—
334C2×C53#243411yescharacteristic
335C2×C53#300211yescharacteristic
336C2×C53#300211yescharacteristic
337C2×C53#316011yescharacteristic
338C2×C53#316611yescharacteristic
339C2×C53#323311yescharacteristic
340C2×C53#323311yescharacteristic
341#533—1064no—
342#533—1064no—
343C2×C4⋊Q8—531no—
344C2×C4⋊Q8—531no—
345C2×#125—531no—
346C2×#125—531no—
347C2×#125—531no—
348C2×C22⋊Q8—1061no—
349C2×C22⋊Q8—1061no—
350C2×C4×Q8—1061no—
351C2×C4×Q8—1061no—
352C2×#117—531no—
353C2×#117C2×D5311yescharacteristic
354C22×C42—531no—
355#499—1064no—
356#499—1064no—
357C2×#107—531no—
358C2×#107—531no—
359C2×#107—531no—
360C2×#107C2×D5311yescharacteristic
361C2×#106—531no—
362C2×#106—531no—
363C2×#106—531no—
364C2×#106C2×D5311yescharacteristic
365C2×C22⋊C8—531no—
366C2×C22⋊C8C2×D5311yescharacteristic
367C2×C4×C8—531no—
368C2×C4×C8—531no—
369#389—531no—
370#389—531no—
371#387—1061no—
372#387—1061no—
373#385—1061no—
374#385—1061no—
375#382—531no—
376#380—531no—
377#380C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340C2×D5311yescharacteristic
385#340C2×D5311yescharacteristic
386#336—531no—
387#336—531no—
388C53#2581411yescharacteristic, complemented, Sylow
389C22×Q8—1061no—
390C22×Q8—1061no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4C22×D5311yescharacteristic
394C22×C8—531no—
395C22×C8—531no—
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C4⋊Q8—1062no—
399C4⋊Q8—1062no—
400C4⋊Q8—1062no—
401C4⋊Q8—1062no—
402#125—534no—
403#125—534no—
404#125—1062no—
405#125—1062no—
406#125—1062no—
407#125—1062no—
408C22⋊Q8—1068no—
409C22⋊Q8—1068no—
410C4×Q8—1068no—
411C4×Q8—1068no—
412#117—22no—
413#117—22no—
414#117—1062no—
415#117—1062no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C42—21no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1062no—
444C2×C42—1062no—
445C2×C42—1061no—
446#107—22no—
447#107—1062no—
448#107—1062no—
449#107—1062no—
450#106—22no—
451#106—1062no—
452#106—1062no—
453#106—1062no—
454C22⋊C8—534no—
455C22⋊C8Dic10614yescomplemented
456C4×C8—1062no—
457C4×C8—1062no—
458#95—21no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464C24D4×D5311yescharacteristic
465C2×Q8—1064no—
466C2×Q8—1064no—
467C2×Q8—2122no—
468C2×Q8—2122no—
469C2×Q8—2122no—
470C2×Q8—2122no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—2121no—
492C22×C4—2121no—
493C22×C4C2×Dic10612yes—
494C22×C4D4×D5311yescharacteristic
495C22×C4Q8×D5312yes—
496C22×C4C23×D5311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—532no—
502C2×C8—532no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8C2×Dic10612yes—
508C2×C8C2×Dic10612yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—532no—
516C4⋊C4—532no—
517C4⋊C4—1064no—
518C4⋊C4—1064no—
519C4⋊C4—1064no—
520C4⋊C4—1064no—
521C4⋊C4—1064no—
522C4⋊C4—1064no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4D4×D5312yes—
534C4⋊C4D4×D5312yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—1064no—
538C22⋊C4—2122no—
539C42—24no—
540C42—532no—
541C42—532no—
542C42—1062no—
543C42—1062no—
544C42—1064no—
545C42—1061no—
546C42—1061no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23#1313321612yes—
554C23C2×D4×D5311yescharacteristic
555Q8—2128no—
556Q8—2128no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
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—532no—
576C2×C4—532no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—2124no—
590C2×C4—2124no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4#1313321612yes—
602C2×C4C2×D4×D5311yescharacteristic
603C2×C4C2×D4×D5311yescharacteristic
604C2×C4C2×Q8×D5311yescharacteristic
605C2×C4C22×Dic10611yescharacteristic
606C2×C4#1313326411yescharacteristic
607C2×C4#1313326411yescharacteristic
608C8—22no—
609C8—22no—
610C8—1062no—
611C8—1062no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22#120×D5311yescharacteristic
620C22C2×#1313321611yescharacteristic
621C22#1681119511yescharacteristic
622C22#135×D5311yescharacteristic
623C22#135×D5311yescharacteristic
624C22#1681141911yescharacteristic
625C22#1681142211yescharacteristic
626C4—22no—
627C4—42no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—1062no—
632C4—1062no—
633C4—1062no—
634C4—1062no—
635C4—4241no—
636C4—4241no—
637C4—2122no—
638C4—2122no—
639C4#1681128611yescharacteristic
640C4#1681128611yescharacteristic
641C4D4×Dic10611yescharacteristic
642C4D4×Dic10611yescharacteristic
643C2—42no—
644C2#2027835711yescharacteristic
645C2#2028393011yescharacteristic
646C2#2028393611yescharacteristic
647C2#2028522711yescharacteristic
648C2#2028526511yescharacteristic
649C2#2028589111yescharacteristic
650C2#2028589711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#25814
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#16811024C2semidirect
3. C2×#16811023C2semidirect
5. #20276047C2semidirect
6. #20276046C2semidirect
7. #20276042C2semidirect
8. #20276041C2semidirect
9. #20275678C2semidirect
10. #20275678C2semidirect
114. C22⋊#88322Q8semidirect
136. C22⋊C8×C53Q8semidirect
388. C53#25814semidirect
455. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168111762
C2×#168110242
C2×#168110232
#202830232
#202760472
#202760462
#202760422
#202760412
#202756782
#202749162
#202749142
#202749122
C22⋊C8×Dic532
C53×#18542
#2581453