Finite Group Databasebeta

← #28330767

Subgroups — #28330767

Order 13568.

1744 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291658C213yesmaximal, complemented
2#20291657C213yesmaximal, complemented
3#20291647C211yesmaximal, characteristic, complemented
4#20291646C213yesmaximal, complemented
5#20291624C213yesmaximal, complemented
6#20291609C211yesmaximal, characteristic, complemented
7#20291608C211yesmaximal, characteristic, complemented
8#20291547C213yesmaximal, complemented
9#20291546C213yesmaximal, complemented
10#20291525C213yesmaximal, complemented
11#41×#88324C211yesmaximal, characteristic, complemented
12Q8×#11719936C211yesmaximal, characteristic, complemented
13Q8×#11719940C211yesmaximal, characteristic, complemented
14#20291503C213yesmaximal, complemented
15#20288331C211yesmaximal, characteristic
16#16811675C2213yes—
17#16811675C2213yes—
18#16811674C2213yes—
19#16811674C2213yes—
20#16811673C2213yes—
21#16811673C2213yes—
22#16811672C2213yes—
23#16811672C2213yes—
24#16811672C2216yes—
25Q8×#88324C2211yescharacteristic
26Q8×#88324C2213yes—
27Q8×#88324C2211yescharacteristic
28Q8×#88324C2213yes—
29Q8×#88324C2211yescharacteristic
30Q8×#88324C2211yescharacteristic
31#16811658C2216yes—
32#16811658C2213yes—
33#16811658C2213yes—
34#16811656C2213yes—
35#16811656C2216yes—
36#16811656C2211yescharacteristic
37#16811654C2213yes—
38#16811654C2213yes—
39#16811652C2213yes—
40#16811652C2213yes—
41D4×#88324C2213yes—
42D4×#88324C2213yes—
43#16811647C2216yes—
44#16811647C2216yes—
45#16811642C2213yes—
46#16811641C2213yes—
47#16811640C2213yes—
48#16811639C2213yes—
49#16811639C2213yes—
50#16811639C2213yes—
51#16811629C2216yes—
52#16811627C2213yes—
53#16811626C2213yes—
54#16811626C2213yes—
55#16811587C2216yes—
56C4×#11719936C2213yes—
57C4×#11719940C2213yes—
58C2×C4×#88324C2213yes—
59#16811349C411yescharacteristic
60#16811342C413yes—
61#16811313C413yes—
62#16811307C413yes—
63Q8×C22⋊D53C411yescharacteristic
64#16811280C411yescharacteristic
65#16811273C413yes—
66#16811175C2213yes—
67#16811154C2213yes—
68#16811152C2213yes—
69#41×Dic53C2211yescharacteristic
70C4×#11719947C2213yes—
71C2×Q8×Dic53C2211yescharacteristic
72C4×Q8×D53C2211yescharacteristic
73C53×#548C411yescharacteristic
74C2×#11719936C2311yescharacteristic
75C2×#11719936C2313yes—
76C2×#11719940C2311yescharacteristic
77C2×#11719940C2313yes—
78C2×#11719940C2316yes—
79C22×#88324C2313yes—
80C22×#88324C2313yes—
81#13133284C2313yes—
82#13133284C2316yes—
83#13133284C2313yes—
84#13133284C2311yescharacteristic
85#13133284C2313yes—
86C4⋊#88324C2313yes—
87C4⋊#88324C2313yes—
88C4⋊#88324C2313yes—
89C4⋊#88324C2316yes—
90C4⋊#88324C2313yes—
91C4⋊#88324C2313yes—
92C4⋊#88324C2313yes—
93#13133274C2311yescharacteristic
94#13133274C2313yes—
95C22⋊#88324C2311yescharacteristic
96C22⋊#88324C2313yes—
97C22⋊#88324C2316yes—
98C22⋊#88324C2313yes—
99C22⋊#88324C2313yes—
100#13133272C2313yes—
101#13133272C2313yes—
102#13133272C2313yes—
103#13133272C2313yes—
104C4×#88324C2311yescharacteristic
105C4×#88324C2313yes—
106C4×#88324C2313yes—
107C4×#88324C2313yes—
108C4×#88324C2313yes—
109C4×#88324C2313yes—
110C4×#88324C2316yes—
111C4×#88324C2316yes—
112C2×#11719947C2×C413yes—
113C2×Q8×D53C2×C411yescharacteristic
114#13133216C2×C413yes—
115#13133215C2×C413yes—
116#13133210C2×C416yes—
117#13133210C2×C416yes—
118#13133209C2×C413yes—
119#13133208C2×C413yes—
120#13133207C2×C416yes—
121#13133204C2×C413yes—
122#13133199C2×C416yes—
123#13133198C2×C416yes—
124#13133196C2×C413yes—
125#13133196C2×C413yes—
126#13133194C2×C416yes—
127#13133193C2×C413yes—
128#13133188C2×C413yes—
129Q8×Dic53C2×C411yescharacteristic
130Q8×Dic53C2×C411yescharacteristic
131Q8×Dic53C2×C413yes—
132Q8×Dic53C2311yescharacteristic
133Q8×Dic53C2313yes—
134C4×Dic106C2×C413yes—
135C4×Dic106C2×C413yes—
136#13133182C2313yes—
137#13133182C2313yes—
138D4×Dic53C2×C413yes—
139D4×Dic53C2313yes—
140D4×Dic53C2313yes—
141C4×C22⋊D53C2×C413yes—
142C4×C22⋊D53C2×C413yes—
143#13133177C2311yescharacteristic
144#13133177C2313yes—
145#13133176C2×C413yes—
146#13133175C2×C413yes—
147#13133173C2×C413yes—
148#13133173C2×C413yes—
149#13133173C2316yes—
150C2×C4⋊Dic53C2×C413yes—
151C4⋊C4×D53C2×C413yes—
152C2×C4×Dic53C2313yes—
153C2×C4×Dic53C2313yes—
154C42×D53C2313yes—
155C2×#41×C53C2×C411yescharacteristic
156C22×Q8×C53C2×C411yescharacteristic
157#124×C53C2×C413yes—
158C22⋊Q8×C53C2×C413yes—
159#121×C53C2×C413yes—
160C4×Q8×C53C2×C411yescharacteristic
161C4×D4×C53C2×C413yes—
162#41×C53—24no—
163C2×Q8×C53—24no—
164C2×Q8×C53C22×C413yes—
165C2×Q8×C53C22×C411yescharacteristic
166C2×D4×C53C22×C413yes—
167C2×D4×C53C22×C413yes—
168C22×C4×C53C22×C413yes—
169C22×C4×C53C22×C413yes—
170C2×C22⋊D53C22×C411yescharacteristic
171C2×C22⋊D53C22×C413yes—
172C22×Dic53C22×C413yes—
173C22×Dic53C22×C413yes—
174C22×Dic53C2411yescharacteristic
175C22×Dic53C2413yes—
176Q8×D53—24no—
177#11719947—46no—
178C2×C4×D53C22×C413yes—
179C2×C4×D53C22×C413yes—
180C2×C4×D53C2411yescharacteristic
181C2×Dic106C22×C413yes—
182C2×Dic106C22×C413yes—
183#11719940—22no—
184#11719940—43no—
185#11719940—46no—
186C2×#88324—26no—
187C2×#88324—43no—
188C2×#88324C2413yes—
189C2×#88324C2411yescharacteristic
190C2×#88324C2411yescharacteristic
191C2×#88324C2411yescharacteristic
192C2×#88324C2413yes—
193C2×#88324C2413yes—
194C2×#88324C2413yes—
195C2×#88324C2411yescharacteristic
196#11719936—22no—
197#11719936—43no—
198C4⋊C4×C53C22×C413yes—
199C4⋊C4×C53C22×C413yes—
200C22⋊C4×C53C22×C411yescharacteristic
201C22⋊C4×C53C22×C413yes—
202C22⋊C4×C53C22×C416yes—
203C42×C53C22×C413yes—
204C22⋊Dic53C22×C413yes—
205C22⋊Dic53C22×C413yes—
206C22⋊Dic53C22×C416yes—
207C22⋊Dic53C22×C416yes—
208#11719921C22×C411yescharacteristic
209#11719921C22×C413yes—
210C4⋊Dic53C22×C413yes—
211C4⋊Dic53C22×C413yes—
212C4⋊Dic53C22×C416yes—
213C4⋊Dic53C22×C413yes—
214C4⋊Dic53C22×C413yes—
215#11719919C22×C416yes—
216#11719919C22×C411yescharacteristic
217#11719919C22×C413yes—
218#11719919C22×C413yes—
219#11719919C22×C413yes—
220#11719919C22×C416yes—
221#11719919C22×C416yes—
222C4×Dic53C22×C413yes—
223C4×Dic53C22×C413yes—
224C4×Dic53C22×C413yes—
225C4×Dic53C22×C413yes—
226C4×Dic53C22×C413yes—
227C4×Dic53C2413yes—
228C4×Dic53C2413yes—
229C4×Dic53C2411yescharacteristic
230C4×Dic53C2413yes—
231C23×C53C23×C411yescharacteristic
232C23×C53C23×C413yes—
233C22×D53C23×C411yescharacteristic
234Q8×C53—26no—
235Q8×C53#13114yes—
236D4×C53—26no—
237D4×C53—43no—
238C2×C4×C53—26no—
239C2×C4×C53—26no—
240C2×C4×C53C23×C413yes—
241C2×C4×C53C23×C411yescharacteristic
242C2×C4×C53C23×C411yescharacteristic
243C2×C4×C53C23×C413yes—
244C2×C4×C53C23×C413yes—
245C22⋊D53—22no—
246C22⋊D53—43no—
247C2×Dic53—26no—
248C2×Dic53—23no—
249C2×Dic53—21no—
250C2×Dic53—23no—
251C2×Dic53—21no—
252C2×Dic53—43no—
253C2×Dic53C23×C413yes—
254C2×Dic53C23×C413yes—
255C2×Dic53C23×C413yes—
256C2×Dic53C23×C411yescharacteristic
257C2×Dic53C23×C413yes—
258C2×Dic53C23×C411yescharacteristic
259C2×Dic53C2511yescharacteristic
260C4×D53—26no—
261C4×D53—21no—
262C4×D53—21no—
263C4×D53—43no—
264Dic106—26no—
265Dic106—26no—
266#88324—21no—
267#88324—23no—
268#88324—21no—
269#88324—23no—
270#88324—23no—
271#88324—21no—
272#88324—23no—
273#883242−1+412yes—
274#60345—531nomaximal, Sylow
275C22×C53—23no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—23no—
279C22×C53—23no—
280C22×C53—21no—
281C22×C53C24×C411yescharacteristic
282C2×D53—21no—
283C2×D53—21no—
284C2×D53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—23no—
288C4×C53—23no—
289C4×C53C2×#13116yes—
290Dic53—23no—
291Dic53—21no—
292Dic53—23no—
293Dic53—23no—
294Dic53—23no—
295Dic53#56812yes—
296Dic53C2×2−1+411yescharacteristic
297Dic53#58511yescharacteristic
298#3482—531no—
299#2993—533no—
300#2993—533no—
301#2991—533no—
302#2991—531no—
303#2986—533no—
304#2982—531no—
305#2982—531no—
306#2969—533no—
307#2969—533no—
308#2965—533no—
309C8×#41—531no—
310Q8×M16—531no—
311Q8×M16—531no—
312#2960—533no—
313C2×C53—41no—
314C2×C53—43no—
315C2×C53C4×2−1+411yescharacteristic
316C2×C53C22×#13111yescharacteristic
317C2×C53C2×#56811yescharacteristic
318D53—41no—
319#557—531no—
320#556—533no—
321#552—533no—
322#552—533no—
323D4×Q8—531no—
324#548—531no—
325#548#615711yescharacteristic
326#547—533no—
327#533—533no—
328#529—533no—
329#529—533no—
330C4×#41—531no—
331C4×#41—533no—
332C2×C4×Q8—531no—
333C2×C4×Q8—531no—
334#446—533no—
335#446—533no—
336#446—533no—
337#446—533no—
338#446—533no—
339#446—533no—
340C8×Q8—533no—
341C8×Q8—533no—
342C8×Q8—536no—
343C8×Q8—531no—
344C8×Q8—531no—
345C8×Q8—531no—
346C8×Q8—531no—
347C8×Q8—533no—
348C8×Q8—533no—
349#436—531no—
350#436—533no—
351#436—536no—
352#436—533no—
353#436—533no—
354#436—536no—
355#435—533no—
356#435—533no—
357#435—533no—
358#435—533no—
359C8×D4—533no—
360C8×D4—533no—
361#433—536no—
362#433—536no—
363#431—533no—
364#431—533no—
365#431—533no—
366#431—533no—
367#431—533no—
368#431—533no—
369#424—536no—
370#423—533no—
371#423—533no—
372#423—533no—
373#405—536no—
374C4×M16—533no—
375C4×M16—533no—
376C2×C4×C8—533no—
377C53#6034511yescharacteristic, complemented, Sylow
378C2×#41—533no—
379C2×#41C2×#615711yescharacteristic
380C22×Q8—531no—
381C22×Q8C2×#615711yescharacteristic
382C2×M16—531no—
383C2×M16—531no—
384C2×M16—533no—
385C2×M16—533no—
386C2×M16—536no—
387C22×C8—533no—
388C22×C8—533no—
389C4⋊Q8—533no—
390C4⋊Q8—533no—
391#126—536no—
392#126—536no—
393#125—533no—
394#125—536no—
395#125—533no—
396#124—533no—
397#124C2×#615713yes—
398#123—536no—
399#123—536no—
400C22⋊Q8—533no—
401C22⋊Q8—533no—
402C22⋊Q8—533no—
403C22⋊Q8—536no—
404C22⋊Q8C2×#615713yes—
405#121—533no—
406#121C2×#615713yes—
407C4×Q8—531no—
408C4×Q8—533no—
409C4×Q8—531no—
410C4×Q8—533no—
411C4×Q8—531no—
412C4×Q8—533no—
413C4×Q8—533no—
414C4×Q8—533no—
415C4×Q8—533no—
416C4×Q8C2×#615711yescharacteristic
417C4×D4—533no—
418C4×D4—533no—
419C4×D4—533no—
420C4×D4—533no—
421C4×D4—533no—
422C4×D4—531no—
423C4×D4—533no—
424C4×D4C2×#615713yes—
425#117—533no—
426#117—533no—
427#117—533no—
428#117—533no—
429#117—536no—
430C2×C4⋊C4—533no—
431C2×C4⋊C4—533no—
432C2×C42—533no—
433C2×C42—533no—
434C2×C42—533no—
435C4⋊C8—533no—
436C4⋊C8—533no—
437C4⋊C8—533no—
438C4⋊C8—533no—
439C4⋊C8—536no—
440C4⋊C8—536no—
441C4⋊C8—533no—
442C4⋊C8—533no—
443C4⋊C8—533no—
444C4⋊C8—531no—
445C4⋊C8—533no—
446C4⋊C8—533no—
447C22⋊C8—531no—
448C22⋊C8—531no—
449C22⋊C8—533no—
450C22⋊C8—533no—
451C22⋊C8—536no—
452C22⋊C8—533no—
453C22⋊C8—533no—
454#97—533no—
455#97—533no—
456#97—533no—
457#97—533no—
458C4×C8—531no—
459C4×C8—533no—
460C4×C8—533no—
461C4×C8—533no—
462C4×C8—533no—
463C4×C8—533no—
464C4×C8—536no—
465C4×C8—536no—
466#41—24no—
467#41—2126no—
468C2×Q8—24no—
469C2×Q8—533no—
470C2×Q8—533no—
471C2×Q8—1064no—
472C2×Q8C22×#615713yes—
473C2×Q8C22×#615711yescharacteristic
474C2×D4—531no—
475C2×D4—533no—
476C2×D4C22×#615713yes—
477C2×D4C22×#615713yes—
478C22×C4—533no—
479C22×C4—531no—
480C22×C4—533no—
481C22×C4—533no—
482C22×C4—533no—
483C22×C4—531no—
484C22×C4—533no—
485C22×C4C22×#615713yes—
486C22×C4C22×#615713yes—
487M16—1062no—
488M16—1062no—
489M16—2123no—
490M16—2123no—
491M16—2126no—
492C2×C8—531no—
493C2×C8—533no—
494C2×C8—531no—
495C2×C8—531no—
496C2×C8—531no—
497C2×C8—533no—
498C2×C8—533no—
499C2×C8—533no—
500C2×C8—1066no—
501C2×C8—2123no—
502C4⋊C4—533no—
503C4⋊C4—533no—
504C4⋊C4—536no—
505C4⋊C4—536no—
506C4⋊C4—531no—
507C4⋊C4—533no—
508C4⋊C4—533no—
509C4⋊C4—533no—
510C4⋊C4—533no—
511C4⋊C4—533no—
512C4⋊C4—536no—
513C4⋊C4—536no—
514C4⋊C4C22×#615713yes—
515C4⋊C4C22×#615713yes—
516C22⋊C4—533no—
517C22⋊C4—536no—
518C22⋊C4—533no—
519C22⋊C4—536no—
520C22⋊C4—531no—
521C22⋊C4—533no—
522C22⋊C4C22×#615711yescharacteristic
523C22⋊C4C22×#615713yes—
524C22⋊C4C22×#615716yes—
525C42—531no—
526C42—533no—
527C42—533no—
528C42—533no—
529C42—533no—
530C42—533no—
531C42—533no—
532C42—533no—
533C42—533no—
534C42C22×#615713yes—
535C23—531no—
536C23C23×#615713yes—
537C23C23×#615711yescharacteristic
538Q8—26no—
539Q8—1066no—
540Q8—1066no—
541Q8#1313330414yes—
542D4—26no—
543D4—43no—
544D4—1062no—
545D4—2123no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—531no—
549C2×C4—531no—
550C2×C4—531no—
551C2×C4—533no—
552C2×C4—533no—
553C2×C4—533no—
554C2×C4—533no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1063no—
558C2×C4—1063no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1066no—
562C2×C4—1066no—
563C2×C4—2123no—
564C2×C4—2123no—
565C2×C4C23×#615711yescharacteristic
566C2×C4C23×#615711yescharacteristic
567C2×C4C23×#615713yes—
568C2×C4C23×#615713yes—
569C2×C4C23×#615713yes—
570C8—532no—
571C8—1063no—
572C8—1061no—
573C8—1063no—
574C8—1061no—
575C8—1061no—
576C8—1063no—
577C8—1063no—
578C22—23no—
579C22—21no—
580C22—21no—
581C22—23no—
582C22—23no—
583C22—21no—
584C22—1061no—
585C22—1061no—
586C22—1061no—
587C22C24×#615711yescharacteristic
588C4—21no—
589C4—21no—
590C4—23no—
591C4—23no—
592C4—531no—
593C4—531no—
594C4—532no—
595C4—1061no—
596C4—1063no—
597C4—1063no—
598C4—1063no—
599C4—1063no—
600C4C2×#1313330416yes—
601C2—41no—
602C2—43no—
603C2—2121no—
604C22−1+4×#615711yescharacteristic
605C2C22×#1313330411yescharacteristic
606C2D53⋊#56811yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#548C4
prime radical#20288331C2
cosocleC2×Dic53C25
hypercenter#548#6157
hypocenterC53#60345
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20291658C2semidirect
2. #20291657C2semidirect
3. #20291647C2semidirect
4. #20291646C2semidirect
5. #20291624C2semidirect
6. #20291609C2semidirect
7. #20291608C2semidirect
8. #20291547C2semidirect
9. #20291546C2semidirect
10. #20291525C2semidirect
11. #41×#88324C2semidirect
12. Q8×#11719936C2semidirect
13. Q8×#11719940C2semidirect
14. #20291503C2semidirect
377. C53#60345semidirect

Maximal subgroups

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

subgroupindex
#202916582
#202916572
#202916472
#202916462
#202916242
#202916092
#202916082
#202915472
#202915462
#202915252
#41×#883242
Q8×#117199362
Q8×#117199402
#202915032
#202883312
#6034553