Finite Group Databasebeta

← #27468214

Subgroups — #27468214

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279148C211yesmaximal, characteristic
2#20279137C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276274C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic
6#20276148C211yesmaximal, characteristic
7#20275443C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275301C211yesmaximal, characteristic, complemented
10#20275294C211yesmaximal, characteristic, complemented
11#20274872C211yesmaximal, characteristic, complemented
12#20274740C211yesmaximal, characteristic
13#20274740C211yesmaximal, characteristic
14C4×#13133149C211yesmaximal, characteristic, complemented
15C8×C22⋊C4×C53C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×#13133149C2211yescharacteristic
18C2×#13133149C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23#16810565—21no—
24#16810563—21no—
25#16810548—21no—
26#16810546—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810516—21no—
32#16810510—21no—
33#16810508—21no—
34#16810493C2211yescharacteristic
35#16810493C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C4×C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59#13133210—44no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133149—22no—
83#13133149—22no—
84#13133149—22no—
85#13133149—22no—
86#13133149—22no—
87#13133149—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—22no—
100C22×C8×C53—22no—
101C22×C8×C53—22no—
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C22⋊C8×C53—22no—
111C4×C8×C53—22no—
112C4×C8×C53—22no—
113#95×C53D411yescharacteristic
114#95×C53D411yescharacteristic
115C24×C53C2×D411yescharacteristic
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—44no—
145C2×C8×C53#4112yes—
146C2×C8×C53#4112yes—
147C2×C8×C53#4112yes—
148C2×C8×C53#4112yes—
149C22⋊C4×C53—22no—
150C22⋊C4×C53SD1614yes—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—21no—
154C42×C53—21no—
155C22⋊Dic53—42no—
156C22⋊Dic53—42no—
157C22⋊Dic53—81no—
158C22⋊Dic53—81no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4⋊Dic53—81no—
178C4⋊Dic53—81no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—81no—
183C4×Dic53—42no—
184C4×Dic53—81no—
185C23×C53—21no—
186C23×C53—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53C2×SD1612yes—
192C23×C53C22×D411yescharacteristic
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C2×SD1612yes—
210C2×C4×C53C2×SD1612yes—
211C2×C4×C53#13512yes—
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C532−1+411yescharacteristic
217C2×C4×C532−1+411yescharacteristic
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C8×C53—22no—
243C8×C53—41no—
244C8×C53—41no—
245C8×C53#12614yescomplemented
246#27835—531nomaximal, Sylow
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53#53211yescharacteristic
255C22×C53#54711yescharacteristic
256C22×C53#54711yescharacteristic
257C22×C53C22×SD1611yescharacteristic
258C22×C53C2×#13511yescharacteristic
259C22×C53#57711yescharacteristic
260C22×C53#57711yescharacteristic
261C4×C53—22no—
262C4×C53—22no—
263C4×C53—22no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#12611yescharacteristic
267C4×C53#54111yescharacteristic
268C4×C53#55211yescharacteristic
269C4×C53#55611yescharacteristic
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273Dic53—81no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278#2497—531no—
279#2496—531no—
280#2100—531no—
281#2099—531no—
282#2077—531no—
283#2077—531no—
284#1941—531no—
285#1941—531no—
286#1920—531no—
287#1919—531no—
288#1844—531no—
289#1807—531no—
290#1807—531no—
291C4×#106—531no—
292C8×C22⋊C4D5311yescharacteristic
293C2×C53—42no—
294C2×C53#249611yescharacteristic
295C2×C53#316711yescharacteristic
296C2×C53#317411yescharacteristic
297C2×C53#333511yescharacteristic
298C2×C53#333511yescharacteristic
299C2×C53#333711yescharacteristic
300C2×C53#334011yescharacteristic
301C23×C8C2×D5311yescharacteristic
302C2×#126—1061no—
303C2×#106—531no—
304C2×#106—531no—
305C2×#106—531no—
306C2×#106—531no—
307C2×#106—531no—
308C2×#106—531no—
309C2×C22⋊C8C2×D5311yescharacteristic
310C2×C4×C8C2×D5311yescharacteristic
311C2×C4×C8C2×D5311yescharacteristic
312#400—1061no—
313#400—1061no—
314#397—1061no—
315#396—1061no—
316#389—531no—
317#389—531no—
318#389—531no—
319#389—531no—
320#388—1061no—
321#387—1061no—
322#387—1061no—
323#384—531no—
324#384—531no—
325#380—531no—
326#380—531no—
327#380—531no—
328#380—531no—
329C4×C4⋊C4—531no—
330C4×C4⋊C4—531no—
331C4×C22⋊C4C2×D5311yescharacteristic
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#336C2×D5311yescharacteristic
343#336C2×D5311yescharacteristic
344C53#2783511yescharacteristic, complemented, Sylow
345C23×C4C22×D5311yescharacteristic
346C22×C8—22no—
347C22×C8—22no—
348C22×C8—22no—
349C22×C8C22×D5311yescharacteristic
350C22×C8C22×D5311yescharacteristic
351C22×C8C22×D5311yescharacteristic
352C22×C8C22×D5311yescharacteristic
353#126—2124no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—531no—
357C2×C4⋊C4—531no—
358C2×C4⋊C4—531no—
359C2×C4⋊C4—531no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C22⋊C4—1061no—
371C2×C22⋊C4—1061no—
372C2×C22⋊C4—1061no—
373C2×C22⋊C4—1061no—
374C2×C22⋊C4D21211yescharacteristic
375C2×C22⋊C4D21211yescharacteristic
376C2×C42—1061no—
377C2×C42—1061no—
378C2×C42C22×D5311yescharacteristic
379C2×C42C22×D5311yescharacteristic
380#106—1062no—
381#106—1062no—
382#106—1062no—
383#106—1062no—
384#106—1062no—
385#106—1062no—
386C22⋊C8—22no—
387C4×C8—22no—
388C4×C8—22no—
389#95—1061no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95D21211yescharacteristic
400#95D21211yescharacteristic
401C24C2×D21211yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4C2×D21211yescharacteristic
417C22×C4C2×D21211yescharacteristic
418C22×C4C2×D21211yescharacteristic
419C22×C4C2×D21211yescharacteristic
420C22×C4C2×D21211yescharacteristic
421C22×C4C23×D5311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—44no—
431C2×C8#1171994512yes—
432C2×C8#1171994512yes—
433C2×C8#1171994712yes—
434C2×C8#1171994712yes—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—1061no—
438C4⋊C4—1061no—
439C4⋊C4—1061no—
440C4⋊C4—1061no—
441C4⋊C4—1061no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—1061no—
446C4⋊C4—1061no—
447C4⋊C4—1061no—
448C4⋊C4—1061no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—4241no—
452C4⋊C4—4241no—
453C4⋊C4—4241no—
454C4⋊C4—2122no—
455C4⋊C4—2122no—
456C4⋊C4—2122no—
457C4⋊C4—2122no—
458C4⋊C4—2122no—
459C22⋊C4—22no—
460C22⋊C4—4241no—
461C22⋊C4—4241no—
462C22⋊C4—2122no—
463C22⋊C4—2122no—
464C22⋊C4#1171991314yes—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—4241no—
470C42—2122no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23C2×#1171991312yes—
478C23C22×D21211yescharacteristic
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4C2×#1171991312yes—
520C2×C4C2×#1171991312yes—
521C2×C4#1313323912yes—
522C2×C4C2×#1171994711yescharacteristic
523C2×C4C2×#1171994711yescharacteristic
524C2×C4C2×#1171994511yescharacteristic
525C2×C4C2×#1171994511yescharacteristic
526C2×C4#1313326411yescharacteristic
527C2×C4#1313326411yescharacteristic
528C8—22no—
529C8—41no—
530C8—41no—
531C8#1313321014yescomplemented
532C22—22no—
533C22—22no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22#1681117311yescharacteristic
540C22#1681127711yescharacteristic
541C22#1681127711yescharacteristic
542C22C22×#1171991311yescharacteristic
543C22C2×#1313323911yescharacteristic
544C22#1681143511yescharacteristic
545C22#1681143511yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—41no—
551C4—4241no—
552C4—4241no—
553C4—4241no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4—4241no—
558C4—4241no—
559C4C2×#1313321011yescharacteristic
560C4#1681123511yescharacteristic
561C4#1681131311yescharacteristic
562C4#1681134311yescharacteristic
563C2—42no—
564C2#2027913711yescharacteristic
565C2#2028527411yescharacteristic
566C2#2028534311yescharacteristic
567C2#2028707311yescharacteristic
568C2#2028707311yescharacteristic
569C2#2028709611yescharacteristic
570C2#2028712911yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC8×C22⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC8×C22⋊C4D53
hypocenterC53#27835
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276274C2semidirect
7. #20275443C2semidirect
8. #20275443C2semidirect
9. #20275301C2semidirect
10. #20275294C2semidirect
11. #20274872C2semidirect
14. C4×#13133149C2semidirect
245. C8×C53#126semidirect
344. C53#27835semidirect
531. C8#13133210semidirect

Maximal subgroups

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

subgroupindex
#202791482
#202791372
#202762792
#202762742
#202761482
#202754432
#202753012
#202752942
#202748722
#202747402
C4×#131331492
C8×C22⋊C4×C532
#2783553