Finite Group Databasebeta

← #27387492

Subgroups — #27387492

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810664C211yesmaximal, characteristic
2C2×#16810664C211yesmaximal, characteristic
3C2×#16810577C211yesmaximal, characteristic
4#20277337C211yesmaximal, characteristic
5#20277330C211yesmaximal, characteristic
6#20275456C211yesmaximal, characteristic, complemented
7#20275455C211yesmaximal, characteristic, complemented
8#20275454C211yesmaximal, characteristic, complemented
9#20275450C211yesmaximal, characteristic, complemented
10#20275447C211yesmaximal, characteristic, complemented
11#20275443C211yesmaximal, characteristic, complemented
12#20275442C211yesmaximal, characteristic, complemented
13#20274430C211yesmaximal, characteristic
14#340×D53C211yesmaximal, characteristic
15C53×#1942C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133174C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23#16810664—24no—
24#16810664C418yescomplemented
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31#16810577—24no—
32C2×#13133128C2211yescharacteristic
33C2×#13133127C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810522C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810520—21no—
40#16810519C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810518—21no—
43#16810497—21no—
44#16810494—21no—
45#16810492—21no—
46#16810487—21no—
47#16810482—21no—
48#16810480—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×C4×C8×C53C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53C2311yescharacteristic
65C2×#11719912C2×C414yes—
66C2×#11719912D414yes—
67C2×#11719912Q814yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C2×C8×D53—22no—
71C2×C8×D53—22no—
72C2×C8×D53Q814yes—
73#13133175—22no—
74#13133175—22no—
75#13133175—22no—
76#13133175C2×C414yes—
77#13133174—22no—
78#13133174—22no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919D411yescharacteristic
92C2×#11719919D411yescharacteristic
93C4⋊C4×D53—22no—
94C4⋊C4×D53—22no—
95C4⋊C4×D53—22no—
96C4⋊C4×D53C2×C414yes—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921D411yescharacteristic
100C2×#11719921D411yescharacteristic
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53C2311yescharacteristic
104#13133150—22no—
105#13133150C2×C414yescomplemented
106#13133149—22no—
107#13133149C2×C414yescomplemented
108#13133148—22no—
109#13133148C2×C414yescomplemented
110#13133128—22no—
111#13133127—22no—
112C8×Dic53—22no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53—21no—
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#107×C53—22no—
129#107×C53C2×C414yescomplemented
130C4×C8×C53—22no—
131#95×C53—21no—
132#95×C53—21no—
133C23×D53C2×D411yescharacteristic
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×C4×D53—24no—
149C2×C4×D53—42no—
150C2×C4×D53—81no—
151C2×C4×D53—81no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53C22×C412yes—
155C2×C4×D53C2×D412yes—
156C2×C4×D53C2×Q812yes—
157C2×C4×D53C2×Q812yes—
158C2×C4×D53#4112yes—
159C2×C4×D53#4112yes—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C2×C8×C53C2×D412yes—
164C2×C8×C53C2×Q812yes—
165C2×C8×C53C2×Q812yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—81no—
173C4⋊C4×C53—81no—
174C4⋊C4×C53C22×C412yes—
175C4⋊C4×C53C22×C412yes—
176C42×C53—21no—
177C42×C53—21no—
178#11719921—24no—
179#11719921—22no—
180#11719921—22no—
181#11719921—42no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—81no—
189C4⋊Dic53—81no—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53C22×C412yes—
192#11719919—24no—
193#11719919—22no—
194#11719919—22no—
195#11719919—42no—
196#11719919—81no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—24no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C22×C412yes—
207C2×#88322C2×D412yes—
208C2×#88322C2×Q812yes—
209C2×#88322C2×Q812yes—
210#11719912C4⋊C4116yescomplemented
211C8×D53—28no—
212C23×C53C22×D411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53C4×D412yes—
218C22×D53#12112yes—
219C22×D53C4⋊Q812yes—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
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—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53C23×C411yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×Q811yescharacteristic
241C2×C4×C53C22×Q811yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53C4×D412yes—
263C2×Dic53#12112yes—
264C2×Dic53C4⋊Q812yes—
265C4×D53—84no—
266C4×D53—84no—
267C4×D53C2×C4⋊C414yes—
268C4×D53C4×Q814yes—
269C4×D53C22⋊Q814yes—
270C4×D53#12514yes—
271C8×C53—22no—
272C8×C53C2×C4⋊C414yes—
273#88322—22no—
274#88322C2×C4⋊C414yes—
275#21160—531nomaximal, Sylow
276C22×C53C2×C4×D411yescharacteristic
277C22×C53C2×#12111yescharacteristic
278C22×C53C2×C4⋊Q811yescharacteristic
279C22×C53#57611yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57711yescharacteristic
282C22×C53#57811yescharacteristic
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53#38914yes—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—81no—
295C4×C53—81no—
296C4×C53C22×C4⋊C411yescharacteristic
297C4×C53C2×C4×Q811yescharacteristic
298C4×C53C2×C22⋊Q811yescharacteristic
299C4×C53C2×#12511yescharacteristic
300Dic53—42no—
301Dic53—42no—
302Dic53—41no—
303Dic53—41no—
304Dic53—81no—
305Dic53—81no—
306Dic53#38914yes—
307C2×#428—531no—
308C2×#428—531no—
309C2×#405—531no—
310#2317—531no—
311#2316—531no—
312#1944—531no—
313#1944—531no—
314#1943—531no—
315#1943—531no—
316#1942—531no—
317#1942D5311yescharacteristic, complemented
318#1941—531no—
319#1941—531no—
320#1735—531no—
321C2×#340—531no—
322C2×C53C2×#38911yescharacteristic
323C2×C53#294611yescharacteristic
324C2×C53#294711yescharacteristic
325C2×C53#306211yescharacteristic
326C2×C53#306311yescharacteristic
327C2×C53#316311yescharacteristic
328C2×C53#316411yescharacteristic
329D53—24no—
330C22×M16—531no—
331C23×C8—531no—
332C2×#117—531no—
333C2×#117—531no—
334C2×#117—531no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—531no—
337#428—538no—
338#428—1064no—
339C2×#107—531no—
340C2×#107—531no—
341C2×#107C2×D5311yescharacteristic
342C2×#107C2×D5311yescharacteristic
343C2×#106—531no—
344C2×#106—531no—
345C2×#106—531no—
346C2×#106—531no—
347#405—1064no—
348C2×#97—531no—
349C2×#97—531no—
350C2×C4×C8—531no—
351C2×C4×C8C2×D5311yescharacteristic
352#389—531no—
353#389—531no—
354#389—531no—
355#389—531no—
356#389—531no—
357#389—531no—
358#389—1061no—
359#389—1061no—
360#389C2×D5311yescharacteristic
361#389C2×D5311yescharacteristic
362#385—1061no—
363#385—1061no—
364#384—1061no—
365#382—1061no—
366#380—1061no—
367#380—1061no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—531no—
372#340—531no—
373#340—531no—
374#340C2×D5311yescharacteristic
375#340C2×D5311yescharacteristic
376C53#2116011yescharacteristic, complemented, Sylow
377C23×C4—531no—
378C23×C4—1061no—
379C23×C4—1061no—
380C2×M16—534no—
381C2×M16—534no—
382C2×M16—534no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8—534no—
386C22×C8—1062no—
387C22×C8—1062no—
388C22×C8C22×D5311yescharacteristic
389C22×C8C22×D5311yescharacteristic
390#117—534no—
391#117—1062no—
392#117—1062no—
393#117—1062no—
394#117—1062no—
395#117—1062no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—534no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1062no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C42—531no—
425C2×C42—1061no—
426C2×C42—1061no—
427C2×C42C22×D5311yescharacteristic
428#107—22no—
429#107—534no—
430#107—1062no—
431#107C4×D5314yescomplemented
432#106—534no—
433#106—534no—
434#106—1062no—
435#106—1062no—
436#97—1062no—
437#97—1062no—
438C4×C8—22no—
439C4×C8—1062no—
440#95—21no—
441#95—21no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448C24—531no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—531no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—1064no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—4241no—
470C22×C4—4241no—
471C22×C4—2122no—
472C22×C4D4×D5311yescharacteristic
473C22×C4D4×D5311yescharacteristic
474C22×C4C23×D5311yescharacteristic
475M16—5316no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—532no—
479C2×C8—532no—
480C2×C8—532no—
481C2×C8—532no—
482C2×C8—1068no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8C2×C4×D5312yes—
486C2×C8D4×D5312yes—
487C2×C8Q8×D5312yes—
488C2×C8Q8×D5312yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—4241no—
509C4⋊C4—4241no—
510C4⋊C4—4241no—
511C4⋊C4—4241no—
512C4⋊C4—4241no—
513C4⋊C4—4241no—
514C4⋊C4—2122no—
515C4⋊C4C2×C4×D5312yes—
516C4⋊C4C2×C4×D5312yes—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1064no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—1064no—
524C42—1061no—
525C42—1061no—
526C42—2122no—
527C23—532no—
528C23—532no—
529C23—532no—
530C23—1061no—
531C23—1061no—
532C23—1061no—
533C23—1061no—
534C23C2×D4×D5311yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—532no—
554C2×C4—532no—
555C2×C4—532no—
556C2×C4—534no—
557C2×C4—534no—
558C2×C4—534no—
559C2×C4—534no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—4244no—
569C2×C4—4244no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4C22×C4×D5311yescharacteristic
581C2×C4C2×D4×D5311yescharacteristic
582C2×C4C2×Q8×D5311yescharacteristic
583C2×C4C2×Q8×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4#41×D5311yescharacteristic
586C8—22no—
587C8—534no—
588C8—1062no—
589C8C4⋊C4×D5314yes—
590C22—534no—
591C22—1062no—
592C22—1062no—
593C22—1062no—
594C22—1062no—
595C22—1062no—
596C22—1062no—
597C22C4×D4×D5311yescharacteristic
598C22#121×D5311yescharacteristic
599C22C4⋊Q8×D5311yescharacteristic
600C22#1681142411yescharacteristic
601C22#1681143011yescharacteristic
602C22#1681143011yescharacteristic
603C22#1681143611yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—534no—
611C4—4241no—
612C4—4241no—
613C4—2122no—
614C4—2122no—
615C4—2121no—
616C4—2121no—
617C4C2×C4⋊C4×D5311yescharacteristic
618C4C4×Q8×D5311yescharacteristic
619C4C22⋊Q8×D5311yescharacteristic
620C4#125×D5311yescharacteristic
621C2—1064no—
622C2#389×D5311yescharacteristic
623C2#2028354011yescharacteristic
624C2#2028355311yescharacteristic
625C2#2028439411yescharacteristic
626C2#2028440511yescharacteristic
627C2#2028524411yescharacteristic
628C2#2028524811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1942C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1942D53
hypocenterC53#21160
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
6. #20275456C2semidirect
7. #20275455C2semidirect
8. #20275454C2semidirect
9. #20275450C2semidirect
10. #20275447C2semidirect
11. #20275443C2semidirect
12. #20275442C2semidirect
15. C53×#1942C2semidirect
24. #16810664C4semidirect
105. #13133150C2×C4semidirect
107. #13133149C2×C4semidirect
109. #13133148C2×C4semidirect
129. #107×C53C2×C4semidirect
210. #11719912C4⋊C4semidirect
317. #1942D53semidirect
376. C53#21160semidirect
431. #107C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106642
C2×#168105772
#202773372
#202773302
#202754562
#202754552
#202754542
#202754502
#202754472
#202754432
#202754422
#202744302
#340×D532
C53×#19422
#2116053