Finite Group Databasebeta

← #27413084

Subgroups — #27413084

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810664C211yesmaximal, characteristic
2C2×#16810662C211yesmaximal, characteristic
3#389×D53C211yesmaximal, characteristic
4#20277121C211yesmaximal, characteristic
5#20276198C211yesmaximal, characteristic, complemented
6#20276197C211yesmaximal, characteristic, complemented
7#20276196C211yesmaximal, characteristic, complemented
8#20276195C211yesmaximal, characteristic, complemented
9#20276194C211yesmaximal, characteristic, complemented
10#20276190C211yesmaximal, characteristic, complemented
11#20276189C211yesmaximal, characteristic, complemented
12#20274433C211yesmaximal, characteristic
13#20274433C211yesmaximal, characteristic
14#20274407C211yesmaximal, characteristic
15C53×#2085C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C42×D53C2211yescharacteristic
24#16810664—24no—
25#16810662—24no—
26C2×#13133151C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810520—21no—
35#16810519C2211yescharacteristic
36#16810518—21no—
37#16810492C2211yescharacteristic
38#16810492C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810482—21no—
41#16810482—21no—
42#16810474—21no—
43#16810474—21no—
44#95×D53—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C22×C4×D53D411yescharacteristic
64C22×C4×D53D411yescharacteristic
65C22×C4×D53C2311yescharacteristic
66C2×#11719912—22no—
67C2×#11719912—22no—
68C2×#11719912—22no—
69C2×#11719912—22no—
70C2×#11719912Q814yes—
71C2×#11719912Q814yes—
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133175—22no—
75#13133175—22no—
76#13133175—22no—
77#13133175—22no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C4⋊C4×D53—22no—
92C4⋊C4×D53—22no—
93C4⋊C4×D53—22no—
94C4⋊C4×D53—44no—
95C4⋊C4×D53D414yes—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53D411yescharacteristic
101C2×C4×Dic53D411yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103C42×D53—22no—
104C42×D53—22no—
105#13133151—22no—
106#13133150—22no—
107#13133150—22no—
108#13133149—22no—
109#13133148—22no—
110#13133148—22no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#107×C53—22no—
125#106×C53—22no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129C23×D53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×C4×D53—24no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×C4×D53C2×D412yes—
157C2×C4×D53C2×Q812yes—
158C2×C4×D53C2×Q812yes—
159C2×C4×D53#4112yes—
160C2×C4×D53#4112yes—
161C2×C4×D53#4112yes—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C2×Q812yes—
167C2×C8×C53C2×Q812yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—42no—
176C4⋊C4×C53C2×D412yes—
177C42×C53—21no—
178C42×C53—21no—
179#11719921—42no—
180#11719921—42no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—42no—
189C4⋊Dic53C2×D412yes—
190#11719919—24no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—42no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322C2×Q812yes—
209C2×#88322C2×Q812yes—
210#11719912—28no—
211#11719912—28no—
212C23×C53C22×D411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53#12112yes—
218C22×D53#12312yes—
219C22×D53C4⋊Q812yes—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
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×C53C22×D411yescharacteristic
239C2×C4×C53C22×Q811yescharacteristic
240C2×C4×C53C22×Q811yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—22no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
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#12112yes—
261C2×Dic53#12312yes—
262C2×Dic53C4⋊Q812yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264C2×Dic53#13712yes—
265C4×D53—44no—
266C4×D53—44no—
267C4×D53—48no—
268C4×D53—84no—
269C4×D53C22⋊Q814yes—
270C4×D53C22⋊Q814yes—
271C4×D53#12514yes—
272C4×D53#12514yes—
273C8×C53—22no—
274C8×C53—22no—
275#88322—22no—
276#88322—22no—
277#23657—531nomaximal, Sylow
278C22×C53C2×#12111yescharacteristic
279C22×C53C2×#12311yescharacteristic
280C22×C53C2×C4⋊Q811yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57511yescharacteristic
284C22×C53#57511yescharacteristic
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53#39814yes—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53—81no—
298C4×C53C2×C22⋊Q811yescharacteristic
299C4×C53C2×C22⋊Q811yescharacteristic
300C4×C53C2×#12511yescharacteristic
301C4×C53C2×#12511yescharacteristic
302Dic53—22no—
303Dic53—22no—
304Dic53—42no—
305Dic53—81no—
306Dic53—81no—
307Dic53—81no—
308Dic53#39814yes—
309C2×#428—531no—
310C2×#428—531no—
311C2×#389—531no—
312#2285—531no—
313#2085—531no—
314#2085—531no—
315#2085—531no—
316#2085D5311yescharacteristic, complemented
317#2084—531no—
318#2084—531no—
319#2083—531no—
320#2083—531no—
321#1736—531no—
322#1736—531no—
323#1728—531no—
324C2×C53C2×#39811yescharacteristic
325C2×C53#303611yescharacteristic
326C2×C53#303711yescharacteristic
327C2×C53#309411yescharacteristic
328C2×C53#309511yescharacteristic
329C2×C53#316511yescharacteristic
330C2×C53#316611yescharacteristic
331D53—24no—
332C22×M16—531no—
333C22×M16—531no—
334C2×#117—531no—
335C2×#117—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—531no—
338C22×C4⋊C4—1061no—
339C22×C42—531no—
340#428—1064no—
341#428—1064no—
342C2×#107—531no—
343C2×#107—531no—
344C2×#107—531no—
345C2×#107C2×D5311yescharacteristic
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106C2×D5311yescharacteristic
350#389—531no—
351#389—531no—
352#389—531no—
353#389—1061no—
354#389—1061no—
355#389C2×D5311yescharacteristic
356#384—531no—
357#384—531no—
358#382—531no—
359#382C2×D5311yescharacteristic
360#380—1061no—
361#380—1061no—
362C4×C4⋊C4—1061no—
363C4×C4⋊C4—1061no—
364C2×#95—1061no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340C2×D5311yescharacteristic
372#340C2×D5311yescharacteristic
373#336—531no—
374#336—531no—
375#336—531no—
376#336C2×D5311yescharacteristic
377C53#2365711yescharacteristic, complemented, Sylow
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—1061no—
382C23×C4—1061no—
383C2×M16—534no—
384C2×M16—534no—
385C2×M16—1062no—
386C2×M16—1062no—
387C2×M16—1062no—
388C2×M16—1062no—
389C22×C8—531no—
390C22×C8—531no—
391C22×C8C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393#117—1062no—
394#117—1062no—
395#117—1062no—
396#117—1062no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—531no—
399C2×C4⋊C4—534no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1062no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1061no—
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—2124no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—1062no—
426C2×C42—1062no—
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430#107—22no—
431#107—1062no—
432#107—1062no—
433#107—1062no—
434#106—22no—
435#106—1062no—
436#106—1062no—
437#106—1062no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446C24—531no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—531no—
452C22×C4—532no—
453C22×C4—532no—
454C22×C4—532no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1064no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—2121no—
470C22×C4—2121no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4—2122no—
476C22×C4D4×D5311yescharacteristic
477C22×C4D4×D5311yescharacteristic
478C22×C4C23×D5311yescharacteristic
479M16—1068no—
480M16—1068no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—532no—
486C2×C8—532no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8Q8×D5312yes—
492C2×C8Q8×D5312yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—532no—
502C4⋊C4—1064no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—2122no—
511C4⋊C4—2122no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4D4×D5312yes—
519C22⋊C4—2122no—
520C22⋊C4—2122no—
521C42—21no—
522C42—21no—
523C42—1062no—
524C42—1062no—
525C42—1061no—
526C42—1061no—
527C42—2122no—
528C42—2122no—
529C23—532no—
530C23—532no—
531C23—532no—
532C23—1061no—
533C23—1061no—
534C23—1061no—
535C23—1061no—
536C23C2×D4×D5311yescharacteristic
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
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—41no—
554C2×C4—41no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—534no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—534no—
562C2×C4—532no—
563C2×C4—532no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1062no—
569C2×C4—4244no—
570C2×C4—2124no—
571C2×C4—2124no—
572C2×C4—2128no—
573C2×C4—2121no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4C2×D4×D5311yescharacteristic
585C2×C4C2×Q8×D5311yescharacteristic
586C2×C4C2×Q8×D5311yescharacteristic
587C2×C4#41×D5311yescharacteristic
588C2×C4#41×D5311yescharacteristic
589C2×C4#41×D5311yescharacteristic
590C8—22no—
591C8—22no—
592C8—1062no—
593C8—1062no—
594C22—534no—
595C22—1062no—
596C22—1062no—
597C22—1062no—
598C22—1062no—
599C22—1062no—
600C22—1062no—
601C22#121×D5311yescharacteristic
602C22#123×D5311yescharacteristic
603C22C4⋊Q8×D5311yescharacteristic
604C22AGL(1,ℤ/8)×D5311yescharacteristic
605C22#137×D5311yescharacteristic
606C22#1681142011yescharacteristic
607C22#1681142011yescharacteristic
608C4—42no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—534no—
615C4—1062no—
616C4—1062no—
617C4—4241no—
618C4—4241no—
619C4—4241no—
620C4—2122no—
621C4C22⋊Q8×D5311yescharacteristic
622C4C22⋊Q8×D5311yescharacteristic
623C4#125×D5311yescharacteristic
624C4#125×D5311yescharacteristic
625C2—1064no—
626C2#398×D5311yescharacteristic
627C2#2028417711yescharacteristic
628C2#2028418511yescharacteristic
629C2#2028465411yescharacteristic
630C2#2028466211yescharacteristic
631C2#2028526011yescharacteristic
632C2#2028526311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2085C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2085D53
hypocenterC53#23657
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
5. #20276198C2semidirect
6. #20276197C2semidirect
7. #20276196C2semidirect
8. #20276195C2semidirect
9. #20276194C2semidirect
10. #20276190C2semidirect
11. #20276189C2semidirect
15. C53×#2085C2semidirect
316. #2085D53semidirect
377. C53#23657semidirect

Maximal subgroups

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

subgroupindex
C2×#168106642
C2×#168106622
#389×D532
#202771212
#202761982
#202761972
#202761962
#202761952
#202761942
#202761902
#202761892
#202744332
#202744072
C53×#20852
#2365753