Finite Group Databasebeta

← #27355154

Subgroups — #27355154

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284758C211yesmaximal, characteristic
2M16×D212C211yesmaximal, characteristic, complemented
3#20273615C211yesmaximal, characteristic, complemented
4#20273598C211yesmaximal, characteristic, complemented
5#20273579C211yesmaximal, characteristic, complemented
6#20273570C211yesmaximal, characteristic, complemented
7#20273558C211yesmaximal, characteristic, complemented
8#20273542C211yesmaximal, characteristic, complemented
9#20273537C211yesmaximal, characteristic, complemented
10C8×#11719924C211yesmaximal, characteristic, complemented
11#20273503C211yesmaximal, characteristic, complemented
12#20273180C211yesmaximal, characteristic, complemented
13#20273105C211yesmaximal, characteristic, complemented
14#20273012C211yesmaximal, characteristic, complemented
15C53×#2969C211yesmaximal, characteristic, complemented
16C2×M16×D53—21no—
17C2×C4×D212C2211yescharacteristic
18#16810956C411yescharacteristic
19#16810939C411yescharacteristic
20#16810920C411yescharacteristic
21#16810920C411yescharacteristic
22#16810835C411yescharacteristic
23#16810835C411yescharacteristic
24#16810773C411yescharacteristic
25#16810756C411yescharacteristic
26C4×#11719924C2211yescharacteristic
27C4×#11719924C2211yescharacteristic
28C4×#11719922C2211yescharacteristic
29C4×#11719922C2211yescharacteristic
30#16810702C2211yescharacteristic
31#16810702C2211yescharacteristic
32#16810696—21no—
33#16810696—21no—
34C8×D212C2211yescharacteristic
35C8×D212C2211yescharacteristic
36#16810676C2211yescharacteristic
37#16810676C2211yescharacteristic
38#16810647C2211yescharacteristic
39#16810586—21no—
40#16810375C2211yescharacteristic, complemented
41#16810375C2211yescharacteristic, complemented
42#16810373C2211yescharacteristic, complemented
43#16810373C2211yescharacteristic, complemented
44Q8⋊#88322C2211yescharacteristic, complemented
45Q8⋊#88322C2211yescharacteristic, complemented
46D4⋊#88322C2211yescharacteristic, complemented
47D4⋊#88322C2211yescharacteristic, complemented
48#16810350C2211yescharacteristic
49#16810350C2211yescharacteristic
50#16810350C2211yescharacteristic
51#16810350C2211yescharacteristic
52#16810345C2211yescharacteristic, complemented
53#16810345C2211yescharacteristic, complemented
54#16810344C2211yescharacteristic, complemented
55C8×#88322C2211yescharacteristic, complemented
56C53×#529C2211yescharacteristic
57C53×#446C2211yescharacteristic, complemented
58C8×Q8×C53C2211yescharacteristic, complemented
59C53×#436—21no—
60C53×#436C2211yescharacteristic, complemented
61C53×#435—21no—
62C8×D4×C53C2211yescharacteristic, complemented
63C53×#433—21no—
64C53×#431C2211yescharacteristic
65C53×#405—21no—
66C4×M16×C53C2211yescharacteristic
67C22×D212C2×C411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719924C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719922C2×C411yescharacteristic
72C2×#11719922C2×C411yescharacteristic
73C2×#11719917—21no—
74C2×#11719912—21no—
75C2×#11719912—21no—
76M16×D53—44no—
77C2×C8×D53—21no—
78C2×C8×D53—21no—
79C4×D212—21no—
80C4×D212—21no—
81C4×D212C2×C411yescharacteristic
82C4×D212C2×C411yescharacteristic
83C4×D212C2311yescharacteristic
84C4×D212C2311yescharacteristic
85C2×C4⋊Dic53C2×C411yescharacteristic
86C2×#11719921—21no—
87#13133150C2×C411yescharacteristic
88#13133150C2×C411yescharacteristic
89#13133148C2×C411yescharacteristic
90#13133148C2×C411yescharacteristic
91C4⋊#88322—21no—
92C4⋊#88322—21no—
93C4⋊#88322C2×C411yescharacteristic
94C4⋊#88322C2×C411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133141C2×C411yescharacteristic
98#13133141C2×C411yescharacteristic
99#13133138C2×C411yescharacteristic
100#13133138C2×C411yescharacteristic
101#13133137C2×C411yescharacteristic
102#13133137C2×C411yescharacteristic
103#13133137C2×C411yescharacteristic
104#13133137C2×C411yescharacteristic
105C22⋊#88322—21no—
106C22⋊#88322C2×C411yescharacteristic
107C22⋊#88322C2×C411yescharacteristic
108#13133129—21no—
109#13133129—21no—
110#13133129—21no—
111#13133129—21no—
112C4×#88322C2311yescharacteristic
113C4×#88322C2311yescharacteristic
114C4×#88322C2311yescharacteristic
115C4×#88322C2311yescharacteristic
116C2×M16×C53—21no—
117C2×M16×C53D411yescharacteristic
118C2×M16×C53D411yescharacteristic
119C22×C8×C53—21no—
120#126×C53—21no—
121#125×C53C2×C411yescharacteristic
122#124×C53C2×C411yescharacteristic
123#123×C53—21no—
124C22⋊Q8×C53C2×C411yescharacteristic
125#121×C53C2×C411yescharacteristic
126C4×Q8×C53C2311yescharacteristic
127C4×D4×C53—21no—
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53C2311yescharacteristic
134C4⋊C8×C53C2311yescharacteristic
135C22⋊C8×C53—21no—
136C22⋊C8×C53—21no—
137C22⋊C8×C53—21no—
138#97×C53—21no—
139#97×C53D411yescharacteristic
140#97×C53D411yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53—21no—
143C4×C8×C53C2311yescharacteristic
144C4×C8×C53C2311yescharacteristic
145C23×D53—21no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×D4×C53—21no—
148C2×D4×C53C22×C411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53#4111yescharacteristic
153C22×Dic53—21no—
154C2×D212—22no—
155C2×D212—21no—
156C2×D212—41no—
157C2×D212C22×C411yescharacteristic
158C2×D212C22×C411yescharacteristic
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—21no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165M16×C53—22no—
166M16×C53—41no—
167M16×C53—42no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—42no—
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C2×C8×C53C2×D411yescharacteristic
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53C22×C411yescharacteristic
180C4⋊C4×C53C22×C411yescharacteristic
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—21no—
184C22⋊C4×C53—21no—
185C22⋊C4×C53—21no—
186C42×C53—21no—
187C42×C53C22×C411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53#4111yescharacteristic
190C42×C53C2411yescharacteristic
191#11719924—22no—
192#11719924—41no—
193#11719922—22no—
194#11719922—41no—
195#11719921—21no—
196#11719921—21no—
197#11719921—21no—
198#11719921—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53C22×C411yescharacteristic
201C4⋊Dic53C22×C411yescharacteristic
202#11719917—42no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C2×#88322C22×C411yescharacteristic
210C2×#88322C22×C411yescharacteristic
211#11719912—42no—
212#11719912—42no—
213C8×D53—42no—
214C8×D53—42no—
215C23×C53—21no—
216C23×C53C4×D411yescharacteristic
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—81no—
222Q8×C53—21no—
223Q8×C53—21no—
224D4×C53—21no—
225D4×C53—21no—
226D4×C53—81no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53C4×D411yescharacteristic
238C2×C4×C53C4×D411yescharacteristic
239C2×C4×C53C4×D411yescharacteristic
240C2×C4×C53C23×C411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—42no—
246D212—21no—
247D212—21no—
248D212—21no—
249D212—42no—
250D212#13112yes—
251C4×D53—42no—
252C4×D53—42no—
253C4×D53—41no—
254C4×D53—41no—
255C4×D53—41no—
256C4×D53—41no—
257C8×C53—21no—
258C8×C53—21no—
259C8×C53—21no—
260C8×C53—41no—
261C8×C53—41no—
262C8×C53AGL(1,ℤ/8)12yes—
263#88322—21no—
264#88322—21no—
265#88322—21no—
266#88322—21no—
267#88322—41no—
268#88322—41no—
269#88322—41no—
270#88322—41no—
271#18008—531nomaximal, Sylow
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×C4×D411yescharacteristic
279C2×D53—22no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—84no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53—41no—
293C4×C53C2×#13111yescharacteristic
294C4×C53#56811yescharacteristic
295C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C53#57511yescharacteristic
297Dic53—41no—
298Dic53—41no—
299#3105—531no—
300#2969D5311yescharacteristic, complemented
301D4×M16—531no—
302#1592—531no—
303#1589—531no—
304#1586—531no—
305#1585—531no—
306#1583—531no—
307#1581—531no—
308#1580—531no—
309C8×SD16—531no—
310#1574—531no—
311#1495—531no—
312#1478—531no—
313#1453—531no—
314C2×C53—41no—
315C2×C53—81no—
316C2×C53#293311yescharacteristic
317C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C53#295711yescharacteristic
319D53—42no—
320D53—81no—
321C22×M16—1061no—
322#529C2×D5311yescharacteristic
323C2×C4×D4—531no—
324#484—531no—
325#481—531no—
326#478—531no—
327#478—531no—
328#463—531no—
329#463—531no—
330#450—531no—
331#447—531no—
332#446C2×D5311yescharacteristic, complemented
333C8×Q8C2×D5311yescharacteristic, complemented
334C4×SD16—531no—
335C4×SD16—531no—
336C4×D8—531no—
337C4×D8—531no—
338#436—21no—
339#436—531no—
340#436—531no—
341#436C2×D5311yescharacteristic, complemented
342#435—21no—
343#435—1061no—
344#435—1061no—
345C8×D4—531no—
346C8×D4—531no—
347C8×D4C2×D5311yescharacteristic, complemented
348#433—21no—
349#431—531no—
350#431—531no—
351#431C2×D5311yescharacteristic
352#423—531no—
353#407—1061no—
354#405—21no—
355C4×M16C2×D5311yescharacteristic
356#335—531no—
357#335—531no—
358#334—531no—
359#334—531no—
360Q8⋊C8—531no—
361Q8⋊C8—531no—
362D4⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8—531no—
365D4⋊C8—531no—
366D4⋊C8—531no—
367D4⋊C8—531no—
368#322—531no—
369#322—531no—
370#322—531no—
371C82—531no—
372C53#1800811yescharacteristic, complemented, Sylow
373C22×D4—531no—
374C23×C4—1061no—
375C2×SD16—531no—
376C2×SD16—531no—
377C2×D8—531no—
378C2×D8—531no—
379C2×M16—21no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—2124no—
384C2×M16C22⋊D5311yescharacteristic
385C2×M16C22⋊D5311yescharacteristic
386C22×C8—21no—
387C22×C8—1061no—
388C22×C8—1061no—
389#126—21no—
390#125C4×D5311yescharacteristic
391#124C4×D5311yescharacteristic
392#123—21no—
393C22⋊Q8C4×D5311yescharacteristic
394#121C4×D5311yescharacteristic
395C4×Q8C22×D5311yescharacteristic
396C4×D4—21no—
397C4×D4—531no—
398C4×D4—531no—
399C4×D4—531no—
400C4×D4—531no—
401C4×D4—1061no—
402C4×D4—1061no—
403C4×D4C22×D5311yescharacteristic
404#117—21no—
405C2×C4⋊C4—531no—
406C2×C22⋊C4—1061no—
407C2×C42C22×D5311yescharacteristic
408#107—531no—
409#107—531no—
410#106—531no—
411#106—531no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—531no—
415C4⋊C8—531no—
416C4⋊C8—531no—
417C4⋊C8—531no—
418C4⋊C8—1061no—
419C4⋊C8—1061no—
420C4⋊C8C22×D5311yescharacteristic
421C4⋊C8C22×D5311yescharacteristic
422#103—531no—
423#103—531no—
424#102—531no—
425#102—531no—
426#102—531no—
427#102—531no—
428#102—531no—
429#102—531no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—531no—
434C22⋊C8—531no—
435C22⋊C8—1061no—
436C22⋊C8—1061no—
437C22⋊C8—1061no—
438C22⋊C8—1061no—
439C22⋊C8—1061no—
440#97—21no—
441#97C22⋊D5311yescharacteristic
442#97C22⋊D5311yescharacteristic
443C4×C8—21no—
444C4×C8—21no—
445C4×C8—531no—
446C4×C8—531no—
447C4×C8—531no—
448C4×C8—531no—
449C4×C8C22×D5311yescharacteristic
450C4×C8C22×D5311yescharacteristic
451C24—1061no—
452C2×Q8C2×C4×D5311yescharacteristic
453C2×D4—21no—
454C2×D4—531no—
455C2×D4—531no—
456C2×D4—1062no—
457C2×D4—1061no—
458C2×D4—2121no—
459C2×D4C2×C4×D5311yescharacteristic
460C22×C4—21no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4C2×C4×D5311yescharacteristic
469C22×C4#1171994511yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471SD16—1062no—
472SD16—2121no—
473D8—1062no—
474D8—2121no—
475M16—22no—
476M16—41no—
477M16—42no—
478M16—2122no—
479M16—2122no—
480M16—2122no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—531no—
485C2×C8—531no—
486C2×C8—531no—
487C2×C8—531no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—2122no—
493C2×C8—2122no—
494C2×C8C2×C22⋊D5311yescharacteristic
495C2×C8C2×C22⋊D5311yescharacteristic
496C2×C8C2×C22⋊D5311yescharacteristic
497C2×C8C2×C22⋊D5311yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—531no—
503C4⋊C4—531no—
504C4⋊C4—1061no—
505C4⋊C4C2×C4×D5311yescharacteristic
506C4⋊C4C2×C4×D5311yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—1061no—
513C22⋊C4—1061no—
514C22⋊C4—1061no—
515C22⋊C4—1061no—
516C42—21no—
517C42C2×C4×D5311yescharacteristic
518C42#1171994511yescharacteristic
519C42C2×C22⋊D5311yescharacteristic
520C42C23×D5311yescharacteristic
521C23—21no—
522C23—1061no—
523C23—1061no—
524C23—4241no—
525C23—2122no—
526C23—2122no—
527C23C4×C22⋊D5311yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—532no—
534D4—1061no—
535D4—1061no—
536D4—1061no—
537D4—2122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4C4×C22⋊D5311yescharacteristic
558C2×C4C4×C22⋊D5311yescharacteristic
559C2×C4C4×C22⋊D5311yescharacteristic
560C2×C4C22×C4×D5311yescharacteristic
561C2×C4C22×C22⋊D5311yescharacteristic
562C2×C4C2×#1171994511yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—1061no—
569C8—1061no—
570C8—1061no—
571C8—1061no—
572C8—2121no—
573C8—2121no—
574C8—2121no—
575C8—2121no—
576C8#1313324212yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1062no—
584C22—4244no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22C2×C4×C22⋊D5311yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—2121no—
599C4—2121no—
600C4#131×D5311yescharacteristic
601C4#1681138811yescharacteristic
602C4C2×#1313324211yescharacteristic
603C4#1681142111yescharacteristic
604C2—41no—
605C2—81no—
606C2—4241no—
607C2—2122no—
608C2#2028343711yescharacteristic
609C2C4×#1313324211yescharacteristic
610C2#2028363011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2969C2
socleC22×C53C2×C4×D4
prime radical#16810756C4
cosocleC42×C53C24
hypercenter#2969D53
hypocenterC53#18008
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D212C2semidirect
3. #20273615C2semidirect
4. #20273598C2semidirect
5. #20273579C2semidirect
6. #20273570C2semidirect
7. #20273558C2semidirect
8. #20273542C2semidirect
9. #20273537C2semidirect
10. C8×#11719924C2semidirect
11. #20273503C2semidirect
12. #20273180C2semidirect
13. #20273105C2semidirect
14. #20273012C2semidirect
15. C53×#2969C2semidirect
40. #16810375C22semidirect
41. #16810375C22semidirect
42. #16810373C22semidirect
43. #16810373C22semidirect
44. Q8⋊#88322C22semidirect
45. Q8⋊#88322C22semidirect
46. D4⋊#88322C22semidirect
47. D4⋊#88322C22semidirect
52. #16810345C22semidirect
53. #16810345C22semidirect
54. #16810344C22semidirect
55. C8×#88322C22semidirect
57. C53×#446C22semidirect
58. C8×Q8×C53C22semidirect
60. C53×#436C22semidirect
62. C8×D4×C53C22semidirect
300. #2969D53semidirect
332. #446C2×D53semidirect
333. C8×Q8C2×D53semidirect
341. #436C2×D53semidirect
347. C8×D4C2×D53semidirect
372. C53#18008semidirect

Maximal subgroups

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

subgroupindex
#202847582
M16×D2122
#202736152
#202735982
#202735792
#202735702
#202735582
#202735422
#202735372
C8×#117199242
#202735032
#202731802
#202731052
#202730122
C53×#29692
#1800853