Finite Group Databasebeta

← #27366518

Subgroups — #27366518

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285275C211yesmaximal, characteristic, complemented
2#20284170C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic, complemented
4#20274143C211yesmaximal, characteristic, complemented
5#20274136C211yesmaximal, characteristic, complemented
6#20274133C211yesmaximal, characteristic, complemented
7#20274126C211yesmaximal, characteristic, complemented
8#20274099C211yesmaximal, characteristic, complemented
9#20274092C211yesmaximal, characteristic, complemented
10#20274065C211yesmaximal, characteristic, complemented
11#20274058C211yesmaximal, characteristic, complemented
12C2×#16810373C211yesmaximal, characteristic, complemented
13#20273372C211yesmaximal, characteristic, complemented
14#20273365C211yesmaximal, characteristic, complemented
15C53×#3076C211yesmaximal, characteristic, complemented
16C2×#13133237—21no—
17#16811173C2211yescharacteristic
18#16811019C2211yescharacteristic
19#16810997C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810971C2211yescharacteristic
22#16810854—21no—
23#16810854—21no—
24#16810845—21no—
25#16810838—21no—
26#16810833C2211yescharacteristic
27#16810828C2211yescharacteristic
28#16810823C2211yescharacteristic
29#16810818C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31Q8×#88322C2211yescharacteristic
32C4×#11719924—21no—
33C4×#11719924—21no—
34D4×#88322—21no—
35D4×#88322—21no—
36D4×#88322C2211yescharacteristic
37D4×#88322C2211yescharacteristic
38#16810676—21no—
39#16810676C2211yescharacteristic
40C2×C4⋊#88322C2211yescharacteristic
41#16810617—21no—
42C2×C4×#88322—21no—
43C2×C4×#88322C2211yescharacteristic
44#16810373C2211yescharacteristic
45#16810373C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C4×#41×C53C2211yescharacteristic
58C53×#479—21no—
59C53×#478—21no—
60C53×#477C2211yescharacteristic
61C53×#476C2211yescharacteristic
62C53×#475C2211yescharacteristic
63C53×#474C2211yescharacteristic
64C53×#427—21no—
65C2×C4⋊C8×C53C2211yescharacteristic
66C53×#416—21no—
67C2×#11719945—21no—
68#13133237—44no—
69C2×#11719925—21no—
70C2×#11719923—21no—
71C2×#11719924—21no—
72C2×#11719924—21no—
73C2×#11719924—21no—
74C2×#11719922—21no—
75C22×#88322—21no—
76C22×#88322—21no—
77C22×#88322—21no—
78C22×#88322D411yescharacteristic
79C22×#88322D411yescharacteristic
80#13133217C2311yescharacteristic
81#13133214C2311yescharacteristic
82#13133205—21no—
83#13133191—21no—
84#13133191—21no—
85C4×D212—21no—
86C4×D212—21no—
87#13133148—21no—
88#13133148—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322—21no—
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95C4⋊#88322C2311yescharacteristic
96#13133141—21no—
97#13133141—21no—
98#13133140—21no—
99#13133140—21no—
100#13133137—21no—
101#13133137—21no—
102#13133137—21no—
103#13133137—21no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106C22⋊#88322—21no—
107C4×#88322—21no—
108C4×#88322—21no—
109C4×#88322—21no—
110C4×#88322D411yescharacteristic
111C4×#88322D411yescharacteristic
112C4×#88322C2311yescharacteristic
113C4×#88322C2311yescharacteristic
114C2×#41×C53D411yescharacteristic
115C22×C8×C53—21no—
116C4⋊Q8×C53C2311yescharacteristic
117C4⋊Q8×C53C2311yescharacteristic
118#125×C53—21no—
119C22⋊Q8×C53—21no—
120C22⋊Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53D411yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C4×D4×C53—21no—
126C4×D4×C53—21no—
127C4×D4×C53D411yescharacteristic
128C4×D4×C53C2311yescharacteristic
129#117×C53—21no—
130#117×C53—21no—
131#117×C53D411yescharacteristic
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—21no—
135#107×C53—21no—
136#106×C53—21no—
137#106×C53—21no—
138C4⋊C8×C53—21no—
139C4⋊C8×C53C2311yescharacteristic
140C4⋊C8×C53C2311yescharacteristic
141#103×C53—21no—
142#103×C53—21no—
143#102×C53—21no—
144#102×C53—21no—
145#41×C53—22no—
146#41×C53SD1614yescomplemented
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53C2×D411yescharacteristic
150C2×D4×C53—21no—
151C2×D4×C53C2×D411yescharacteristic
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53#4111yescharacteristic
158C2×C22⋊D53—41no—
159#11719945—42no—
160#11719945—42no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212—21no—
164C2×C4×D53—41no—
165C2×Dic106—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—42no—
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—41no—
176C4⋊C4×C53—41no—
177C4⋊C4×C53C2×D411yescharacteristic
178C4⋊C4×C53C2×D411yescharacteristic
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—41no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—41no—
187C42×C53C2×D411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53#4111yescharacteristic
190C42×C53C2411yescharacteristic
191#11719925—42no—
192#11719924—42no—
193#11719924—42no—
194#11719924—42no—
195#11719923—42no—
196#11719922—42no—
197#11719921—41no—
198#11719921—41no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C2×#88322—22no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—41no—
207C2×#88322—42no—
208C2×#88322—42no—
209C2×#88322—42no—
210C2×#88322C2×D411yescharacteristic
211C2×#88322C2×D411yescharacteristic
212C2×#88322C2×D411yescharacteristic
213C2×#88322C2×D411yescharacteristic
214C23×C53—21no—
215C23×C53#12111yescharacteristic
216C22×D53—41no—
217Q8×C53—21no—
218Q8×C53—41no—
219Q8×C53—41no—
220Q8×C53—81no—
221Q8×C53C2×SD1612yes—
222D4×C53—24no—
223D4×C53—21no—
224D4×C53C2×SD1612yes—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—24no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53C2×SD1612yes—
241C2×C4×C53#13512yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C22⋊D53—82no—
246C2×Dic53—41no—
247D212—41no—
248D212—41no—
249D212—41no—
250D212—41no—
251D212—81no—
252C4×D53—82no—
253Dic106—41no—
254Dic106—41no—
255C8×C53—41no—
256C8×C53—41no—
257#88322—21no—
258#88322—21no—
259#88322—21no—
260#88322—41no—
261#88322—41no—
262#88322—41no—
263#88322—41no—
264#88322#13512yescomplemented
265#19124—531nomaximal, Sylow
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—22no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53#46512yes—
272C22×C53C2×#12111yescharacteristic
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C4×C53—21no—
277C4×C53—22no—
278C4×C53—21no—
279C4×C53—21no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53#46512yes—
285C4×C53C22×SD1611yescharacteristic
286C4×C53C2×#13511yescharacteristic
287C4×C53C2×#13511yescharacteristic
288C4×C53#57511yescharacteristic
289Dic53—81no—
290#3167—531no—
291#3076D5311yescharacteristic, complemented
292#3035—531no—
293C8×#41—531no—
294#1680—531no—
295#1679—531no—
296#1678—531no—
297#1677—531no—
298#1672—531no—
299#1671—531no—
300#1666—531no—
301#1665—531no—
302C2×#334—531no—
303#1542—531no—
304#1540—531no—
305C2×C53—22no—
306C2×C53—42no—
307C2×C53#303511yescharacteristic
308C2×C53C2×#46511yescharacteristic
309C2×C53#306711yescharacteristic
310D53—161no—
311C2×#135—1061no—
312#533C2×D5311yescharacteristic
313#532—531no—
314C4×#41C2×D5311yescharacteristic
315#498—531no—
316#492—531no—
317#487—531no—
318#486—531no—
319#479—21no—
320#478—21no—
321#477C2×D5311yescharacteristic
322#476C2×D5311yescharacteristic
323#475C2×D5311yescharacteristic
324#474C2×D5311yescharacteristic
325#465—1061no—
326#465—1061no—
327#464—1061no—
328#463—1061no—
329#462—531no—
330#461—531no—
331#460—531no—
332#459—531no—
333C8×Q8—531no—
334C8×Q8—531no—
335C4×SD16—1061no—
336C4×SD16—1061no—
337C8×D4—531no—
338C8×D4—531no—
339C8×D4—1061no—
340C8×D4—1061no—
341#431—531no—
342#431—1061no—
343#427—21no—
344C2×C4⋊C8—531no—
345C2×C4⋊C8C2×D5311yescharacteristic
346#416—21no—
347#416—1061no—
348C2×C4×C8—531no—
349C2×C4×C8—1061no—
350#334—531no—
351#334—531no—
352#334—531no—
353#334—531no—
354#333—531no—
355#333—531no—
356#332—531no—
357#332—531no—
358#332—531no—
359#332—531no—
360#331—531no—
361#331—531no—
362C53#1912411yescharacteristic, complemented, Sylow
363C2×#41—1061no—
364C2×#41C22⋊D5311yescharacteristic
365#135—2124no—
366C2×Q16—1061no—
367C2×SD16—1061no—
368C2×SD16—1061no—
369C2×SD16—1061no—
370C2×SD16—1061no—
371C2×D8—1061no—
372C22×C8—21no—
373C22×C8—531no—
374C22×C8—531no—
375C22×C8—1061no—
376C22×C8—1061no—
377C22×C8—1061no—
378C4⋊Q8—531no—
379C4⋊Q8C22×D5311yescharacteristic
380C4⋊Q8C22×D5311yescharacteristic
381#127—531no—
382#125—21no—
383#124—1061no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386#121—1061no—
387#121—1061no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8C22⋊D5311yescharacteristic
391C4×Q8C22×D5311yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—1061no—
395C4×D4—1061no—
396C4×D4C22⋊D5311yescharacteristic
397C4×D4C22×D5311yescharacteristic
398#117—21no—
399#117—21no—
400#117C22⋊D5311yescharacteristic
401C2×C42—21no—
402C2×C42C22×D5311yescharacteristic
403#107—21no—
404#107—21no—
405#106—21no—
406#106—21no—
407#106—1061no—
408#106—1061no—
409C4⋊C8—21no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—1061no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8C22×D5311yescharacteristic
418C4⋊C8C22×D5311yescharacteristic
419#103—21no—
420#103—21no—
421#103—1061no—
422#103—1061no—
423#103—1061no—
424#103—1061no—
425#102—21no—
426#102—21no—
427#102—1061no—
428#102—1061no—
429#102—1061no—
430#102—1061no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C4×C8—531no—
435C4×C8—531no—
436C4×C8—531no—
437C4×C8—531no—
438C4×C8—1061no—
439C4×C8—1061no—
440C4×C8—1061no—
441#41—22no—
442#41—2122no—
443#41—2122no—
444#41#1171992414yescomplemented
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—1061no—
448C2×Q8C2×C22⋊D5311yescharacteristic
449C2×D4—21no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—2121no—
454C2×D4C2×C22⋊D5311yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—2121no—
459C22×C4D4×D5311yescharacteristic
460C22×C4#1171994711yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462Q16—2122no—
463SD16—2122no—
464SD16—2122no—
465SD16—2122no—
466SD16—2122no—
467D8—2122no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—42no—
471C2×C8—531no—
472C2×C8—531no—
473C2×C8—531no—
474C2×C8—531no—
475C2×C8—1062no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—2122no—
481C2×C8—2121no—
482C2×C8—2122no—
483C2×C8—2122no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4C2×C22⋊D5311yescharacteristic
495C4⋊C4C2×C22⋊D5311yescharacteristic
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—2121no—
501C22⋊C4—2121no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42D4×D5311yescharacteristic
507C42#1171994711yescharacteristic
508C42C2×C22⋊D5311yescharacteristic
509C42C23×D5311yescharacteristic
510C23—21no—
511C23—2121no—
512C23#1313318911yescharacteristic
513Q8—21no—
514Q8—41no—
515Q8—41no—
516Q8—81no—
517Q8—2121no—
518Q8—2121no—
519Q8C2×#1171992412yes—
520D4—24no—
521D4—21no—
522D4—4241no—
523D4—4242no—
524D4—2121no—
525D4—2121no—
526D4—2121no—
527D4—2121no—
528D4C2×#1171992412yes—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—4242no—
542C2×C4—2121no—
543C2×C4#1313318911yescharacteristic
544C2×C4#1313318911yescharacteristic
545C2×C4#1313318911yescharacteristic
546C2×C4C2×#1171992412yes—
547C2×C4#1313323712yes—
548C2×C4C2×D4×D5311yescharacteristic
549C2×C4C22×C22⋊D5311yescharacteristic
550C2×C4C2×#1171994711yescharacteristic
551C8—41no—
552C8—41no—
553C8—532no—
554C8—1061no—
555C8—1061no—
556C8—1061no—
557C8—2121no—
558C8—2121no—
559C8—2121no—
560C8—2121no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—4241no—
567C22—4241no—
568C22—4241no—
569C22#1681085412yes—
570C22C2×#1313318911yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—4241no—
580C4#1681085412yes—
581C4C22×#1171992411yescharacteristic
582C4#135×D5311yescharacteristic
583C4C2×#1313323711yescharacteristic
584C4#1681142011yescharacteristic
585C2—22no—
586C2—42no—
587C2—8481no—
588C2#2028416611yescharacteristic
589C2C2×#1681085411yescharacteristic
590C2#2028443211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3076C2
socleC22×C53C2×#121
prime radical#20284170C2
hypercenter#3076D53
hypocenterC53#19124
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #20285275C2semidirect
3. #41×#88322C2semidirect
4. #20274143C2semidirect
5. #20274136C2semidirect
6. #20274133C2semidirect
7. #20274126C2semidirect
8. #20274099C2semidirect
9. #20274092C2semidirect
10. #20274065C2semidirect
11. #20274058C2semidirect
12. C2×#16810373C2semidirect
13. #20273372C2semidirect
14. #20273365C2semidirect
15. C53×#3076C2semidirect
146. #41×C53SD16semidirect
264. #88322#135semidirect
291. #3076D53semidirect
362. C53#19124semidirect
444. #41#11719924semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202841702
#41×#883222
#202741432
#202741362
#202741332
#202741262
#202740992
#202740922
#202740652
#202740582
C2×#168103732
#202733722
#202733652
C53×#30762
#1912453