Finite Group Databasebeta

← #27352078

Subgroups — #27352078

Order 13568.

1028 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288676C211yesmaximal, characteristic
2#20283143C211yesmaximal, characteristic, complemented
3#20273336C212yesmaximal, complemented
4#20273327C211yesmaximal, characteristic, complemented
5#20273323C211yesmaximal, characteristic, complemented
6#20273314C212yesmaximal, complemented
7#20273306C211yesmaximal, characteristic, complemented
8#20273304C211yesmaximal, characteristic, complemented
9#20273301C212yesmaximal, complemented
10C2×#16810362C212yesmaximal, complemented
11C53×#2970C211yesmaximal, characteristic, complemented
12C2×#13133224—21no—
13#16811358C411yescharacteristic
14#16811346C411yescharacteristic
15#16811336—21no—
16#16811297—22no—
17D4×Dic106C411yescharacteristic
18#16811282C412yes—
19#16811269C412yes—
20#16811261—21no—
21D4×D212C411yescharacteristic
22#16811245—21no—
23#16811206—22no—
24#16811173C2211yescharacteristic
25#16811157C2212yes—
26#16811157C2211yescharacteristic
27C2×#13133205C2212yes—
28#16811095—21no—
29#16810579—22no—
30#16810579—21no—
31C4×#11719917C2212yescomplemented
32C4×#11719917C2211yescharacteristic, complemented
33C2×#13133128C2212yescomplemented
34C2×#13133128C2211yescharacteristic, complemented
35#16810364C2211yescharacteristic, complemented
36#16810364C2212yescomplemented
37#16810364C2211yescharacteristic, complemented
38#16810362C2211yescharacteristic
39#16810362C2212yes—
40#16810362C2211yescharacteristic
41#16810362C2212yescomplemented
42#16810362C2212yescomplemented
43#16810362C2212yescomplemented
44#16810362C2212yescomplemented
45C4×#41×C53C2211yescharacteristic
46C53×#446—21no—
47C53×#435—22no—
48C8×D4×C53—21no—
49C53×#432C2212yescomplemented
50C53×#432C2211yescharacteristic, complemented
51C4×M16×C53C2211yescharacteristic, complemented
52C2×#97×C53C2212yescomplemented
53C2×#11719945C2×C412yes—
54C22×Dic106C2×C412yes—
55C22×D212C2×C412yes—
56C2×D4×D53—21no—
57#13133224—28no—
58C2×#11719917—22no—
59C2×#11719917—21no—
60C22×#88322—22no—
61C22×#88322—21no—
62#13133217C2×C411yescharacteristic
63#13133216—21no—
64#13133214C2×C411yescharacteristic
65#13133212—22no—
66#13133211—21no—
67#13133211—22no—
68#13133209C2311yescharacteristic
69#13133206—21no—
70#13133205—22no—
71#13133205—22no—
72#13133205C2×C412yes—
73#13133205C2×C412yes—
74#13133205C2311yescharacteristic
75#13133205C2312yes—
76#13133203—22no—
77#13133202—22no—
78#13133202—22no—
79#13133200—22no—
80#13133200—21no—
81#13133196C2×C412yes—
82#13133196C2×C411yescharacteristic
83#13133195—22no—
84#13133194—22no—
85#13133193—21no—
86#13133192—21no—
87#13133191—21no—
88#13133191—21no—
89#13133191C2×C411yescharacteristic
90#13133191C2×C412yes—
91#13133190—22no—
92#13133189—22no—
93#13133187—22no—
94#13133185—22no—
95C4×Dic106C2×C412yes—
96C4×Dic106C2×C411yescharacteristic
97C4×D212—21no—
98C4×D212—21no—
99C4×D212C2×C412yes—
100C4×D212C2×C411yescharacteristic
101D4×Dic53—21no—
102#13133179—21no—
103C4×C22⋊D53—21no—
104C4×C22⋊D53—22no—
105#13133177—22no—
106#13133175—21no—
107#13133174—22no—
108#13133174—21no—
109C2×#11719921—22no—
110C2×#11719921—22no—
111C22⋊C4×D53—22no—
112#13133128—22no—
113#13133128—21no—
114#13133128C2311yescharacteristic
115#13133128C2312yes—
116#13133128C2311yescharacteristic
117C4×#88322—22no—
118C4×#88322—21no—
119C2×#41×C53D411yescharacteristic
120C2×M16×C53—21no—
121C22×C8×C53—22no—
122C4×Q8×C53C2×C411yescharacteristic
123C4×Q8×C53D411yescharacteristic
124C4×D4×C53C2×C412yes—
125C4×D4×C53C2×C411yescharacteristic
126C4×D4×C53D411yescharacteristic
127C4×D4×C53D412yes—
128#117×C53D412yes—
129#117×C53D411yescharacteristic
130C2×C42×C53C2312yes—
131C2×C42×C53C2311yescharacteristic
132C4⋊C8×C53—21no—
133C4⋊C8×C53—22no—
134C22⋊C8×C53—22no—
135C22⋊C8×C53—21no—
136#97×C53—22no—
137#97×C53C2311yescharacteristic
138#97×C53C2312yes—
139#97×C53C2311yescharacteristic
140C4×C8×C53—21no—
141C23×D53—22no—
142#41×C53—24no—
143C2×Q8×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C412yes—
145C2×D4×C53C22⋊C411yescharacteristic
146C22×C4×C53C22×C412yes—
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C2×C22⋊D53—22no—
153C2×C22⋊D53—21no—
154C2×C22⋊D53—22no—
155C22×Dic53—22no—
156D4×D53—44no—
157#11719945—82no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—22no—
161C2×D212—82no—
162C2×D212C22×C411yescharacteristic
163C2×D212C22×C412yes—
164C2×C4×D53—21no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×Dic106—22no—
168C2×Dic106—82no—
169C2×Dic106C22×C411yescharacteristic
170C2×Dic106C22×C412yes—
171M16×C53—42no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—22no—
175C2×C8×C53—44no—
176C4⋊C4×C53C22⋊C411yescharacteristic
177C4⋊C4×C53C22⋊C412yes—
178C4⋊C4×C53C22⋊C411yescharacteristic
179C4⋊C4×C53C22⋊C412yes—
180C22⋊C4×C53C22⋊C412yes—
181C22⋊C4×C53C22⋊C412yes—
182C22⋊C4×C53C22⋊C412yes—
183C42×C53C22×C412yes—
184C42×C53C22×C411yescharacteristic
185C42×C53C2×D412yes—
186C42×C53C2×D411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D412yes—
189C42×C53C2411yescharacteristic
190C22⋊Dic53—22no—
191C22⋊Dic53—21no—
192#11719921—21no—
193#11719921—22no—
194#11719921—22no—
195#11719921—22no—
196#11719921—22no—
197#11719921—22no—
198#11719921—22no—
199#11719921—21no—
200#11719921—21no—
201#11719921—21no—
202#11719921—22no—
203#11719921—22no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C411yescharacteristic
207C4⋊Dic53C22×C411yescharacteristic
208#11719919—21no—
209#11719919—22no—
210#11719919—22no—
211#11719919—21no—
212#11719919—21no—
213#11719919—21no—
214C4×Dic53—22no—
215C4×Dic53—21no—
216#11719917—28no—
217#11719917—24no—
218C2×#88322—28no—
219C2×#88322—24no—
220C2×#88322—21no—
221C2×#88322—22no—
222C2×#88322—21no—
223C23×C53C2×C22⋊C411yescharacteristic
224C23×C53C2×C22⋊C412yes—
225C22×D53—22no—
226C22×D53—21no—
227C22×D53—44no—
228C22×D53—82no—
229C22×D53—82no—
230Q8×C53—22no—
231D4×C53—24no—
232D4×C53#10414yes—
233C2×C4×C53—24no—
234C2×C4×C53—22no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C412yes—
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C22⋊C411yescharacteristic
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C22⋊D53—44no—
249C22⋊D53—82no—
250C22⋊D53—81no—
251C2×Dic53—21no—
252C2×Dic53—22no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—82no—
256D212—41no—
257D212—42no—
258D212—42no—
259D212—81no—
260D212—82no—
261C4×D53—42no—
262C4×D53—82no—
263C4×D53—82no—
264Dic106—41no—
265Dic106—42no—
266Dic106—82no—
267C8×C53—22no—
268C8×C53—42no—
269C8×C53—41no—
270#88322—22no—
271#88322—24no—
272#88322—22no—
273#17652—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53C2×#10414yes—
280C22×C53C22×C22⋊C411yescharacteristic
281C2×D53—42no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—42no—
285C2×D53—42no—
286C2×D53—41no—
287C2×D53—164no—
288C4×C53—21no—
289C4×C53—22no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—22no—
293C4×C53—21no—
294C4×C53—22no—
295C4×C53—21no—
296C4×C53C2×#10412yes—
297Dic53—41no—
298Dic53—42no—
299Dic53—41no—
300Dic53—41no—
301#3514—531no—
302#2970D5311yescharacteristic, complemented
303#2875—531no—
304#1532—532no—
305#1530—531no—
306#1529—531no—
307#1527—532no—
308#1525—531no—
309#1524—531no—
310#1524—532no—
311C2×#329—532no—
312C2×C53—24no—
313C2×C53—41no—
314C2×C53#288811yescharacteristic
315C2×C53C22×#10411yescharacteristic
316C2×C53#290111yescharacteristic
317D53—82no—
318D53—81no—
319C2×#131—1061no—
320#559—531no—
321#556—531no—
322#555—1061no—
323#551—1062no—
324D4×Q8—531no—
325#548—532no—
326#546—532no—
327#546—1061no—
328D42—531no—
329#542—1061no—
330#538—1062no—
331#532—531no—
332#529—531no—
333#529—532no—
334C2×#124—532no—
335#518—1061no—
336C4×#41C2×D5311yescharacteristic
337#446—21no—
338#435—22no—
339C8×D4—21no—
340#432C2×D5312yescomplemented
341#432C2×D5311yescharacteristic, complemented
342#405—1062no—
343#405—1061no—
344C4×M16—532no—
345C4×M16—531no—
346C4×M16C2×D5311yescharacteristic, complemented
347C2×#97—532no—
348C2×#97—531no—
349C2×#97C2×D5312yescomplemented
350#330—531no—
351#330—532no—
352#330—531no—
353#329—531no—
354#329—532no—
355#329—531no—
356#329—532no—
357#329—532no—
358#329—532no—
359#329—532no—
360C53#1765211yescharacteristic, complemented, Sylow
361C2×#41—532no—
362C2×#41D21211yescharacteristic
363C22×Q8—532no—
364C22×D4—532no—
365C22×D4—1061no—
366#131—1068no—
367C2×M16—21no—
368C2×M16—1062no—
369C2×M16—1061no—
370C22×C8—22no—
371C22×C8—1062no—
372C22×C8—1061no—
373C4⋊Q8—531no—
374C4⋊Q8—1061no—
375#127—531no—
376#126—1062no—
377#126—1061no—
378#126—1062no—
379#125—531no—
380#124—532no—
381#124—532no—
382#124—531no—
383#124—532no—
384#124—1061no—
385#124—1062no—
386#124—1062no—
387#124—1062no—
388#123—1062no—
389#123—1061no—
390#123—1062no—
391#123—1062no—
392C22⋊Q8—532no—
393C22⋊Q8—531no—
394C22⋊Q8—1061no—
395C22⋊Q8—1062no—
396C22⋊Q8—1062no—
397#121—532no—
398#121—531no—
399#121—1062no—
400#121—1061no—
401#121—1061no—
402#121—1061no—
403#121—1062no—
404#120—1062no—
405#120—1062no—
406C4×Q8—532no—
407C4×Q8—531no—
408C4×Q8C4×D5311yescharacteristic
409C4×Q8C22⋊D5311yescharacteristic
410C4×D4—531no—
411C4×D4—532no—
412C4×D4—1061no—
413C4×D4—1061no—
414C4×D4—1062no—
415C4×D4—1061no—
416C4×D4—1062no—
417C4×D4—1061no—
418C4×D4—1061no—
419C4×D4C4×D5311yescharacteristic
420C4×D4C4×D5312yes—
421C4×D4C22⋊D5311yescharacteristic
422C4×D4C22⋊D5312yes—
423#117—1061no—
424#117—1062no—
425#117—1061no—
426#117D21212yes—
427#117D21211yescharacteristic
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—1062no—
430C2×C22⋊C4—1062no—
431C2×C42C22×D5312yes—
432C2×C42C22×D5311yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8—21no—
437#97—22no—
438#97—531no—
439#97—532no—
440#97—531no—
441#97—1062no—
442#97—1061no—
443#97C22×D5311yescharacteristic
444#97C22×D5312yes—
445#97C22×D5311yescharacteristic
446C4×C8—21no—
447C4×C8—1062no—
448C4×C8—1061no—
449C24—1062no—
450#41—24no—
451#41—4242no—
452C2×Q8—531no—
453C2×Q8—532no—
454C2×Q8—1062no—
455C2×Q8—4242no—
456C2×Q8#1171992111yescharacteristic
457C2×D4—531no—
458C2×D4—532no—
459C2×D4—1061no—
460C2×D4—1062no—
461C2×D4—1062no—
462C2×D4—1061no—
463C2×D4—1062no—
464C2×D4—1061no—
465C2×D4—4242no—
466C2×D4—2124no—
467C2×D4#1171992112yes—
468C2×D4#1171992111yescharacteristic
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1061no—
473C22×C4C2×C4×D5312yes—
474C22×C4C2×C4×D5311yescharacteristic
475C22×C4C2×D21212yes—
476C22×C4C2×D21211yescharacteristic
477C22×C4C2×C22⋊D5312yes—
478C22×C4C2×C22⋊D5311yescharacteristic
479M16—42no—
480M16—1068no—
481M16—1064no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—1068no—
487C2×C8—1064no—
488C2×C8—1061no—
489C2×C8—1062no—
490C2×C8—1061no—
491C4⋊C4—531no—
492C4⋊C4—531no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1062no—
496C4⋊C4—1061no—
497C4⋊C4—1062no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4#1171992111yescharacteristic
502C4⋊C4#1171992111yescharacteristic
503C4⋊C4#1171992112yes—
504C4⋊C4#1171992112yes—
505C22⋊C4—1062no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1062no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—1062no—
512C22⋊C4—1062no—
513C22⋊C4—1061no—
514C22⋊C4—1062no—
515C22⋊C4—1061no—
516C22⋊C4—1062no—
517C22⋊C4—1061no—
518C22⋊C4—1062no—
519C22⋊C4#1171992112yes—
520C22⋊C4#1171992112yes—
521C22⋊C4#1171992112yes—
522C42—1062no—
523C42—1061no—
524C42C2×C4×D5312yes—
525C42C2×C4×D5311yescharacteristic
526C42C2×D21212yes—
527C42C2×D21211yescharacteristic
528C42C2×C22⋊D5311yescharacteristic
529C42C2×C22⋊D5312yes—
530C42C23×D5311yescharacteristic
531C23—1061no—
532C23—1062no—
533C23—4242no—
534C23—4242no—
535C23—2124no—
536C23C2×#1171992111yescharacteristic
537C23C2×#1171992112yes—
538Q8—22no—
539Q8—4242no—
540Q8—2121no—
541Q8—2122no—
542D4—24no—
543D4—4241no—
544D4—4242no—
545D4—4241no—
546D4—4242no—
547D4—2121no—
548D4—2122no—
549D4—2124no—
550D4—2122no—
551D4#1313314314yes—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—4242no—
563C2×C4—4242no—
564C2×C4—4242no—
565C2×C4—2122no—
566C2×C4C2×#1171992112yes—
567C2×C4C2×#1171992111yescharacteristic
568C2×C4C2×#1171992112yes—
569C2×C4C2×#1171992111yescharacteristic
570C2×C4C2×#1171992112yes—
571C2×C4C2×#1171992111yescharacteristic
572C2×C4C22×C4×D5311yescharacteristic
573C2×C4C22×C22⋊D5311yescharacteristic
574C2×C4C22×D21211yescharacteristic
575C8—22no—
576C8—42no—
577C8—41no—
578C8—1062no—
579C8—1064no—
580C8—1062no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—8484no—
587C22—2122no—
588C22—2121no—
589C22—2121no—
590C22—2122no—
591C22—2122no—
592C22—2121no—
593C22C2×#1313314314yes—
594C22C22×#1171992111yescharacteristic
595C4—21no—
596C4—21no—
597C4—22no—
598C4—21no—
599C4—22no—
600C4—21no—
601C4—22no—
602C4—21no—
603C4—2122no—
604C4—2121no—
605C4—2121no—
606C4—2121no—
607C4C2×#1313314312yes—
608C2—24no—
609C2—41no—
610C2—4241no—
611C2—4242no—
612C2#2028319311yescharacteristic
613C2C22×#1313314311yescharacteristic
614C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#2970C2
socleC22×C53C22×C22⋊C4
prime radical#20288676C2
cosocleC42×C53C24
hypercenter#2970D53
hypocenterC53#17652
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #20283143C2semidirect
3. #20273336C2semidirect
4. #20273327C2semidirect
5. #20273323C2semidirect
6. #20273314C2semidirect
7. #20273306C2semidirect
8. #20273304C2semidirect
9. #20273301C2semidirect
10. C2×#16810362C2semidirect
11. C53×#2970C2semidirect
31. C4×#11719917C22semidirect
32. C4×#11719917C22semidirect
33. C2×#13133128C22semidirect
34. C2×#13133128C22semidirect
35. #16810364C22semidirect
36. #16810364C22semidirect
37. #16810364C22semidirect
41. #16810362C22semidirect
42. #16810362C22semidirect
43. #16810362C22semidirect
44. #16810362C22semidirect
49. C53×#432C22semidirect
50. C53×#432C22semidirect
51. C4×M16×C53C22semidirect
52. C2×#97×C53C22semidirect
302. #2970D53semidirect
340. #432C2×D53semidirect
341. #432C2×D53semidirect
346. C4×M16C2×D53semidirect
349. C2×#97C2×D53semidirect
360. C53#17652semidirect

Maximal subgroups

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

subgroupindex
#202886762
#202831432
#202733362
#202733272
#202733232
#202733142
#202733062
#202733042
#202733012
C2×#168103622
C53×#29702
#1765253