Finite Group Databasebeta

← #27468207

Subgroups — #27468207

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#20279150C211yesmaximal, characteristic
2#20279133C211yesmaximal, characteristic
3#20276280C211yesmaximal, characteristic, complemented
4#20276276C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic
6#20275442C211yesmaximal, characteristic, complemented
7#20275442C211yesmaximal, characteristic, complemented
8#20275299C211yesmaximal, characteristic, complemented
9#20275295C211yesmaximal, characteristic, complemented
10#20274870C211yesmaximal, characteristic, complemented
11#20274738C211yesmaximal, characteristic
12#20274738C211yesmaximal, characteristic
13#20274603C211yesmaximal, characteristic, complemented
14#20274489C211yesmaximal, characteristic
15C53×#2077C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133210—21no—
18C2×#13133149C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24C2×C8×Dic53C2211yescharacteristic
25#16810564—21no—
26#16810564—21no—
27#16810551—21no—
28#16810546—21no—
29#16810522C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810517—21no—
34#16810510—21no—
35#16810508—21no—
36#16810492C2211yescharacteristic
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133210—44no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53C2311yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85#13133149—22no—
86#13133148—22no—
87#13133148—22no—
88#13133148—22no—
89#13133148—22no—
90C8×Dic53—22no—
91C8×Dic53—22no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124D411yescharacteristic
101#13133124D411yescharacteristic
102C23×C4×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111#106×C53—22no—
112C22⋊C8×C53—22no—
113#95×C53—21no—
114#95×C53—21no—
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×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53#4112yes—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C22⋊C4×C53—42no—
151C22⋊C4×C53—81no—
152C22⋊Dic53—22no—
153C22⋊Dic53—42no—
154C22⋊Dic53—81no—
155C22⋊Dic53SD1614yes—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—42no—
165C4⋊Dic53—42no—
166#11719919—42no—
167#11719919—42no—
168#11719919—42no—
169#11719919—81no—
170#11719919—81no—
171#11719919—81no—
172C4×Dic53—21no—
173C4×Dic53—21no—
174C4×Dic53—21no—
175C4×Dic53—21no—
176C4×Dic53—42no—
177C4×Dic53—81no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—44no—
182C2×#88322#4112yes—
183C2×#88322#4112yes—
184C2×#88322#4112yes—
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—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
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—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
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×Dic53C2×SD1612yes—
240C2×Dic53C2×SD1612yes—
241C2×Dic53#13512yes—
242C8×C53—22no—
243C8×C53—41no—
244#88322—41no—
245#88322#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—41no—
262C4×C53—81no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53C2×#12611yescharacteristic
267C4×C53#54111yescharacteristic
268C4×C53#55211yescharacteristic
269C4×C53#55611yescharacteristic
270Dic53—22no—
271Dic53—22no—
272Dic53—22no—
273Dic53—41no—
274Dic53—81no—
275Dic53—81no—
276Dic53—81no—
277Dic53—81no—
278#2497—531no—
279#2496—531no—
280#2100—531no—
281#2099—531no—
282#2077—531no—
283#2077D5311yescharacteristic
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⋊C4—531no—
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×C8—531no—
302C2×#126—1061no—
303C2×#106—531no—
304C2×#106—531no—
305C2×#106—531no—
306C2×#106—531no—
307C2×#106—531no—
308C2×#106C2×D5311yescharacteristic
309C2×C22⋊C8C2×D5311yescharacteristic
310C2×C4×C8—531no—
311C2×C4×C8—531no—
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#380C2×D5311yescharacteristic
328#380C2×D5311yescharacteristic
329C4×C4⋊C4—531no—
330C4×C4⋊C4—531no—
331C4×C22⋊C4—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340C2×D5311yescharacteristic
340#340C2×D5311yescharacteristic
341#340C2×D5311yescharacteristic
342#336—531no—
343#336—531no—
344C53#2783511yescharacteristic, complemented, Sylow
345C23×C4C22×D5311yescharacteristic
346C22×C8—531no—
347C22×C8—531no—
348C22×C8—1062no—
349C22×C8—1062no—
350C22×C8—1062no—
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—1061no—
359C2×C4⋊C4—1061no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1061no—
362C2×C4⋊C4—1061no—
363C2×C4⋊C4—1061no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C4⋊C4C22×D5311yescharacteristic
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—531no—
373C2×C22⋊C4—531no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—1061no—
376C2×C42—531no—
377C2×C42—531no—
378C2×C42—1061no—
379C2×C42—1061no—
380#106—22no—
381#106—1062no—
382#106—1062no—
383#106—1062no—
384#106—1062no—
385#106—1062no—
386C22⋊C8—22no—
387C4×C8—1062no—
388C4×C8—1062no—
389#95—21no—
390#95—21no—
391#95—531no—
392#95—531no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401C24D4×D5311yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4D4×D5311yescharacteristic
421C22×C4C23×D5311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—532no—
428C2×C8—532no—
429C2×C8—532no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—2124no—
434C2×C8#1171994712yes—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
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—42no—
460C22⋊C4—81no—
461C22⋊C4—534no—
462C22⋊C4—1062no—
463C22⋊C4—4241no—
464C22⋊C4—2122no—
465C42—1061no—
466C42—1061no—
467C42—1061no—
468C42—1061no—
469C42—4241no—
470C42—2122no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23SD16×D5312yes—
478C23C2×D4×D5311yescharacteristic
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—532no—
499C2×C4—532no—
500C2×C4—532no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
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×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4#41×D5311yescharacteristic
523C2×C4#41×D5311yescharacteristic
524C2×C4#41×D5311yescharacteristic
525C2×C4C2×#1171994711yescharacteristic
526C2×C4#1313326411yescharacteristic
527C2×C4#1313326411yescharacteristic
528C8—22no—
529C8—41no—
530C8—534no—
531C8—2121no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22#1681116811yescharacteristic
540C22#1681127111yescharacteristic
541C22#1681127111yescharacteristic
542C22C2×SD16×D5311yescharacteristic
543C22#135×D5311yescharacteristic
544C22#1681143211yescharacteristic
545C22#1681143311yescharacteristic
546C4—41no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—1062no—
552C4—1062no—
553C4—1062no—
554C4—4241no—
555C4—4241no—
556C4—4241no—
557C4—4241no—
558C4—2121no—
559C4#126×D5311yescharacteristic
560C4#1681123011yescharacteristic
561C4#1681131011yescharacteristic
562C4#1681134211yescharacteristic
563C2—42no—
564C2#2027913011yescharacteristic
565C2#2028526911yescharacteristic
566C2#2028533811yescharacteristic
567C2#2028706511yescharacteristic
568C2#2028707011yescharacteristic
569C2#2028709711yescharacteristic
570C2#2028712611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2077C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2077D53
hypocenterC53#27835
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
3. #20276280C2semidirect
4. #20276276C2semidirect
6. #20275442C2semidirect
7. #20275442C2semidirect
8. #20275299C2semidirect
9. #20275295C2semidirect
10. #20274870C2semidirect
13. #20274603C2semidirect
245. #88322#126semidirect
344. C53#27835semidirect

Maximal subgroups

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

subgroupindex
#202791502
#202791332
#202762802
#202762762
#202761482
#202754422
#202752992
#202752952
#202748702
#202747382
#202746032
#202744892
C53×#20772
#2783553