Finite Group Databasebeta

← #27439945

Subgroups — #27439945

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811250C211yesmaximal, characteristic
2C2×#16811022C211yesmaximal, characteristic, complemented
3#20280546C211yesmaximal, characteristic
4#20276055C211yesmaximal, characteristic, complemented
5#20276051C211yesmaximal, characteristic, complemented
6#20276046C211yesmaximal, characteristic, complemented
7#20276041C211yesmaximal, characteristic, complemented
8#20275684C211yesmaximal, characteristic, complemented
9#20275677C211yesmaximal, characteristic, complemented
10#20274913C211yesmaximal, characteristic
11#20274911C211yesmaximal, characteristic
12#20274906C211yesmaximal, characteristic
13#20274901C211yesmaximal, characteristic
14#20274484C211yesmaximal, characteristic
15C2×C53×#501C211yesmaximal, characteristic, complemented
16#16811250—24no—
17#16811250—24no—
18C2×#13133216C2211yescharacteristic
19C2×#13133215—21no—
20C2×#13133209C2211yescharacteristic
21C2×#13133208C2211yescharacteristic
22C2×#13133207—21no—
23C2×#13133195—21no—
24C2×#13133193—21no—
25C2×#13133175C2211yescharacteristic
26C2×#13133174C2211yescharacteristic
27C2×C4⋊C4×D53C2211yescharacteristic
28#16811022—24no—
29C2×#13133150C2211yescharacteristic
30C2×#13133150C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133129C2211yescharacteristic
34C2×#13133129C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810556—21no—
37#16810556—21no—
38#16810545—21no—
39#16810545—21no—
40#16810519C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810493C2211yescharacteristic
43#16810480C2211yescharacteristic
44#16810480C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810377C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×C4⋊Q8×C53C2211yescharacteristic
56C2×#125×C53C2211yescharacteristic
57C53×#501—24no—
58C2×#107×C53C2211yescharacteristic
59C2×#107×C53C2211yescharacteristic
60C2×#106×C53C2211yescharacteristic
61C2×#106×C53C2211yescharacteristic
62C2×#97×C53C2211yescharacteristic
63C22×Dic106—21no—
64C22×C4×D53—21no—
65C22×C4×D53C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133216—22no—
69#13133216—22no—
70#13133215—28no—
71#13133209—22no—
72#13133209D414yes—
73#13133208—22no—
74#13133208—22no—
75#13133207—28no—
76#13133195—28no—
77#13133193—28no—
78#13133175—22no—
79#13133175—22no—
80#13133174—22no—
81#13133174—22no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919D411yescharacteristic
97C2×#11719919D411yescharacteristic
98C4⋊C4×D53—22no—
99C4⋊C4×D53—22no—
100C2×#11719921—21no—
101C2×#11719921—21no—
102C2×#11719921—21no—
103C2×#11719921D411yescharacteristic
104C2×#11719921D411yescharacteristic
105C2×C4×Dic53—21no—
106C2×C4×Dic53C2311yescharacteristic
107#13133150—22no—
108#13133150—22no—
109#13133148—22no—
110#13133148—22no—
111#13133129Q814yescomplemented
112#13133129Q814yescomplemented
113C4×#88322—22no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118C22×Q8×C53—21no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C4⋊Q8×C53—22no—
122C4⋊Q8×C53D414yes—
123#125×C53—22no—
124#125×C53—22no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C42×C53C2311yescharacteristic
133#107×C53—22no—
134#107×C53—22no—
135#106×C53—22no—
136#106×C53—22no—
137#97×C53—22no—
138C23×D53C2×D411yescharacteristic
139C2×Q8×C53—24no—
140C2×Q8×C53—81no—
141C2×Q8×C53—81no—
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×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—24no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×C4×D53C2×Q812yes—
164C2×C4×D53C2×Q812yes—
165C2×Dic106—24no—
166C2×Dic106—81no—
167C2×Dic106—81no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53C2×Q812yes—
173C2×C8×C53C2×Q812yes—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53C2×D412yes—
184C4⋊C4×C53C2×D412yes—
185C42×C53C2×D412yes—
186C42×C53C2×D412yes—
187#11719921—24no—
188#11719921—22no—
189#11719921—22no—
190#11719921—81no—
191#11719921—81no—
192C4⋊Dic53—24no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—81no—
200C4⋊Dic53—81no—
201C4⋊Dic53C2×D412yes—
202C4⋊Dic53C2×D412yes—
203#11719919—24no—
204#11719919—24no—
205#11719919—24no—
206#11719919—24no—
207#11719919—24no—
208#11719919—22no—
209#11719919—22no—
210#11719919—81no—
211#11719919—81no—
212C4×Dic53—24no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322C2×Q812yes—
220C2×#88322C2×Q812yes—
221C23×C53C22×D411yescharacteristic
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—21no—
226C22×D53—21no—
227C22×D53—21no—
228C22×D53C4⋊Q812yes—
229Q8×C53—84no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
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#12012yes—
247C2×C4×C53#12012yes—
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—22no—
258C2×Dic53—22no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
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×Dic53C4⋊Q812yes—
275C4×D53—22no—
276C4×D53—22no—
277C4×D53—22no—
278C4×D53—22no—
279C4×D53—84no—
280Dic106—84no—
281C8×C53—22no—
282C8×C53—22no—
283#88322—22no—
284#88322—22no—
285#25822—531nomaximal, Sylow
286C22×C53C2×#12011yescharacteristic
287C22×C53C2×C4⋊Q811yescharacteristic
288C22×C53#53611yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57711yescharacteristic
291C22×C53#57711yescharacteristic
292C22×C53#57811yescharacteristic
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—41no—
300C4×C53—22no—
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—41no—
312Dic53—42no—
313Dic53—41no—
314Dic53—42no—
315Dic53—81no—
316Dic53—81no—
317C2×#544—531no—
318C2×#501D5311yescharacteristic, complemented
319C2×#499—531no—
320#2629—531no—
321#2059—531no—
322#2059—531no—
323#2058—531no—
324#2057—531no—
325#1992—531no—
326#1990—531no—
327#1854—531no—
328#1854—531no—
329#1853—531no—
330#1852—531no—
331#1750—531no—
332C2×C53#243411yescharacteristic
333C2×C53#301211yescharacteristic
334C2×C53#301311yescharacteristic
335C2×C53#316311yescharacteristic
336C2×C53#316411yescharacteristic
337C2×C53#324111yescharacteristic
338C2×C53#324211yescharacteristic
339D53—42no—
340#544—1064no—
341#544—1064no—
342C2×C4⋊Q8—531no—
343C2×C4⋊Q8—1061no—
344C2×C4⋊Q8C2×D5311yescharacteristic
345C2×#125—531no—
346C2×#125—531no—
347C2×#125—1061no—
348C2×#125C2×D5311yescharacteristic
349C2×C22⋊Q8—1061no—
350C2×C22⋊Q8—1061no—
351C2×#117—531no—
352C2×#117—531no—
353C22×C4⋊C4—531no—
354#501—24no—
355#499—1064no—
356C2×#107—531no—
357C2×#107—531no—
358C2×#107C2×D5311yescharacteristic
359C2×#107C2×D5311yescharacteristic
360C2×#106—531no—
361C2×#106—531no—
362C2×#106C2×D5311yescharacteristic
363C2×#106C2×D5311yescharacteristic
364C2×C22⋊C8—531no—
365C2×C22⋊C8—531no—
366C2×#97C2×D5311yescharacteristic
367C2×C4×C8—531no—
368#398—1061no—
369#398—1061no—
370#396—1061no—
371#396—1061no—
372#389—531no—
373#389—531no—
374#384—531no—
375#380—531no—
376#380—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#336—531no—
386#336—531no—
387C53#2582211yescharacteristic, complemented, Sylow
388C22×Q8—21no—
389C22×Q8—1061no—
390C23×C4—531no—
391C23×C4—1061no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8C22×D5311yescharacteristic
395C22×C8C22×D5311yescharacteristic
396C4⋊Q8—22no—
397C4⋊Q8—1062no—
398C4⋊Q8—1068no—
399C4⋊Q8—1062no—
400C4⋊Q8C22⋊D5314yes—
401#125—22no—
402#125—22no—
403#125—534no—
404#125—1062no—
405#125—1068no—
406#125—1062no—
407#125—1062no—
408C22⋊Q8—1068no—
409C22⋊Q8—1068no—
410#117—1062no—
411#117—1062no—
412#117—1062no—
413#117—1062no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—1062no—
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⋊C4—1062no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C22⋊C4—531no—
440C2×C22⋊C4—531no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C42—531no—
445C2×C42—1061no—
446C2×C42C22×D5311yescharacteristic
447#107—22no—
448#107—22no—
449#107—1062no—
450#107—1062no—
451#106—22no—
452#106—22no—
453#106—1062no—
454#106—1062no—
455C22⋊C8—534no—
456C22⋊C8—534no—
457#97—22no—
458C4×C8—1062no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463C24—531no—
464C2×Q8—24no—
465C2×Q8—81no—
466C2×Q8—81no—
467C2×Q8—1064no—
468C2×Q8—4241no—
469C2×Q8—4241no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—531no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—1064no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4D4×D5311yescharacteristic
491C22×C4D4×D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—532no—
498C2×C8—532no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8Q8×D5312yes—
504C2×C8Q8×D5312yes—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—532no—
515C4⋊C4—532no—
516C4⋊C4—1064no—
517C4⋊C4—1064no—
518C4⋊C4—1064no—
519C4⋊C4—1064no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1064no—
523C4⋊C4—1064no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—4241no—
531C4⋊C4—4241no—
532C4⋊C4—4241no—
533C4⋊C4—4241no—
534C4⋊C4C2×C22⋊D5312yes—
535C4⋊C4C2×C22⋊D5312yes—
536C22⋊C4—1064no—
537C22⋊C4—1062no—
538C22⋊C4—1062no—
539C22⋊C4—4241no—
540C22⋊C4—4241no—
541C42—1064no—
542C42—1061no—
543C42—1061no—
544C42D4×D5312yes—
545C42C2×C22⋊D5312yes—
546C23—532no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—1061no—
552C23—1061no—
553C23C2×D4×D5311yescharacteristic
554Q8—84no—
555Q8—4244no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
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—532no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1062no—
583C2×C4—1061no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—4244no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4#1313318512yes—
599C2×C4#1313318512yes—
600C2×C4C2×D4×D5311yescharacteristic
601C2×C4C22×C22⋊D5311yescharacteristic
602C2×C4C2×Q8×D5311yescharacteristic
603C2×C4C2×Q8×D5311yescharacteristic
604C2×C4#1313326511yescharacteristic
605C2×C4#1313326511yescharacteristic
606C8—22no—
607C8—22no—
608C8—1062no—
609C8—1062no—
610C22—1062no—
611C22—1062no—
612C22—1062no—
613C22—2121no—
614C22—2121no—
615C22—2121no—
616C22—2121no—
617C22C2×#1313318511yescharacteristic
618C22C4⋊Q8×D5311yescharacteristic
619C22#1681119811yescharacteristic
620C22#1681142711yescharacteristic
621C22#1681143211yescharacteristic
622C22#1681143311yescharacteristic
623C22#1681143911yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—1062no—
631C4—4241no—
632C4—4241no—
633C4—2121no—
634C4—2122no—
635C4—2121no—
636C4—2122no—
637C4Q8×C22⋊D5311yescharacteristic
638C4Q8×C22⋊D5311yescharacteristic
639C4#1681128811yescharacteristic
640C4#1681128811yescharacteristic
641C2—2122no—
642C2#2027835611yescharacteristic
643C2#2028401811yescharacteristic
644C2#2028402911yescharacteristic
645C2#2028524211yescharacteristic
646C2#2028525011yescharacteristic
647C2#2028596911yescharacteristic
648C2#2028598411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#501C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#501D53
hypocenterC53#25822
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
2. C2×#16811022C2semidirect
4. #20276055C2semidirect
5. #20276051C2semidirect
6. #20276046C2semidirect
7. #20276041C2semidirect
8. #20275684C2semidirect
9. #20275677C2semidirect
15. C2×C53×#501C2semidirect
111. #13133129Q8semidirect
112. #13133129Q8semidirect
318. C2×#501D53semidirect
387. C53#25822semidirect

Maximal subgroups

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

subgroupindex
C2×#168112502
C2×#168110222
#202805462
#202760552
#202760512
#202760462
#202760412
#202756842
#202756772
#202749132
#202749112
#202749062
#202749012
#202744842
C2×C53×#5012
#2582253