Finite Group Databasebeta

← #27366871

Subgroups — #27366871

Order 13568.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285174C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3C2×#16810903C211yesmaximal, characteristic, complemented
4#20274204C212yesmaximal, complemented
5#20274180C212yesmaximal, complemented
6#20274080C212yesmaximal, complemented
7#20274056C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10#20273388C211yesmaximal, characteristic, complemented
11D4×M16×C53C211yesmaximal, characteristic, complemented
12C2×#13133250—21no—
13C2×#13133250—21no—
14#16811173C2211yescharacteristic
15C2×#13133217C2211yescharacteristic
16#16811017—22no—
17#16811003C2212yescomplemented
18#16810996C2212yescomplemented
19#16810936—21no—
20#16810903—21no—
21#16810903—21no—
22#16810903C2212yes—
23#16810903C2211yescharacteristic
24#16810903C2211yescharacteristic
25#16810903C2212yes—
26#16810818C2211yescharacteristic
27#16810818C2211yescharacteristic
28#16810757—21no—
29C4×#11719922—21no—
30C4×#11719922—21no—
31#16810676C2211yescharacteristic
32C2×#13133150—21no—
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133138—21no—
35#16810376C2212yescomplemented
36#16810374C2212yescomplemented
37#16810370C2212yescomplemented
38#16810370C2212yescomplemented
39#16810370C2212yescomplemented
40#16810366C2212yescomplemented
41C22×M16×C53—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C53×#436C2212yes—
44C53×#435—22no—
45C8×D4×C53C2212yes—
46C53×#423C2211yescharacteristic
47C53×#407—21no—
48C4×M16×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×Dic106—21no—
51#13133250—44no—
52#13133250—44no—
53C2×Dic212—22no—
54C2×Dic212—22no—
55C2×#11719913—22no—
56C2×#11719913—22no—
57C2×#11719922—21no—
58C2×#11719922—21no—
59C22×#88322—21no—
60#13133217—21no—
61#13133217—21no—
62#13133217—21no—
63#13133217C2312yes—
64#13133217C2311yescharacteristic
65#13133214C2311yescharacteristic
66#13133205—21no—
67#13133191—21no—
68#13133191—21no—
69C4×D212—21no—
70C4×D212—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73#13133150—22no—
74#13133150—22no—
75#13133150—21no—
76#13133150—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2311yescharacteristic
79C4⋊#88322C2311yescharacteristic
80C4⋊#88322C2312yes—
81#13133138—21no—
82#13133138—21no—
83#13133138—22no—
84#13133138—22no—
85#13133137—21no—
86#13133137—21no—
87C22⋊#88322—21no—
88C4×#88322—21no—
89C22×D4×C53D411yescharacteristic
90C23×C4×C53—21no—
91C2×M16×C53—22no—
92C2×M16×C53—21no—
93C2×M16×C53—44no—
94C2×M16×C53D411yescharacteristic
95C2×M16×C53D411yescharacteristic
96C22×C8×C53—22no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53D411yescharacteristic
100C4×D4×C53D411yescharacteristic
101C4×D4×C53C2311yescharacteristic
102C4×D4×C53C2311yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106C4⋊C8×C53—21no—
107C4⋊C8×C53C2312yes—
108C22⋊C8×C53—22no—
109C22⋊C8×C53—22no—
110#97×C53D412yes—
111C4×C8×C53C2312yes—
112C24×C53—21no—
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53C2×D411yescharacteristic
117C2×D4×C53C2×D411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C2×C22⋊D53—41no—
129C22×Dic53—41no—
130#11719945—42no—
131#11719945—42no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×C4×D53—41no—
136C2×Dic106—21no—
137C2×Dic106—22no—
138C2×Dic106—42no—
139C2×Dic106—42no—
140C2×Dic106—41no—
141M16×C53—22no—
142M16×C53—42no—
143M16×C53—44no—
144M16×C53D814yescomplemented
145C2×C8×C53—22no—
146C2×C8×C53—44no—
147C2×C8×C53C2×D412yes—
148C2×C8×C53C2×D412yes—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C2×D411yescharacteristic
151C4⋊C4×C53C2×D411yescharacteristic
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—21no—
155C22⋊C4×C53—21no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160#11719922—42no—
161#11719922—42no—
162#11719921—41no—
163#11719921—41no—
164C4⋊Dic53—22no—
165C4⋊Dic53—21no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—41no—
169C4⋊Dic53—41no—
170C2×#88322—21no—
171C2×#88322—21no—
172C2×#88322—22no—
173C2×#88322—42no—
174Dic212—44no—
175Dic212—44no—
176#11719913—44no—
177#11719913—44no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—81no—
183C23×C53#12111yescharacteristic
184C22×D53—41no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—21no—
188D4×C53—42no—
189D4×C53#13712yescomplemented
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—41no—
195C2×C4×C53—42no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53C2×D812yes—
206C2×C4×C53AGL(1,ℤ/8)12yes—
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C22⋊D53—82no—
211C2×Dic53—42no—
212C2×Dic53—41no—
213C2×Dic53—82no—
214D212—41no—
215D212—41no—
216D212—41no—
217D212—41no—
218D212—81no—
219C4×D53—82no—
220Dic106—42no—
221Dic106—41no—
222Dic106—42no—
223Dic106—41no—
224Dic106—82no—
225C8×C53—22no—
226C8×C53—42no—
227C8×C53C2×D814yes—
228#88322—41no—
229#88322—42no—
230#88322—41no—
231#19163—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—22no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53—84no—
241C22×C53#45912yes—
242C22×C53C2×#12111yescharacteristic
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53#45912yes—
252C4×C53C22×D811yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53C2×#13711yescharacteristic
255C4×C53#57511yescharacteristic
256Dic53—82no—
257Dic53—81no—
258#3152—531no—
259#3100—531no—
260C2×#474—531no—
261D4×M16D5311yescharacteristic, complemented
262#1689—532no—
263#1685—532no—
264#1669—532no—
265#1665—532no—
266#1565—531no—
267#1547—531no—
268#1546—531no—
269C2×C53—22no—
270C2×C53—42no—
271C2×C53—81no—
272C2×C53C2×#45911yescharacteristic
273C2×C53#305611yescharacteristic
274C2×C53#306911yescharacteristic
275D53—161no—
276C2×#137—1061no—
277C2×#137—1061no—
278C22×M16—21no—
279#532—531no—
280C2×C4⋊Q8—531no—
281C2×C4×D4C2×D5311yescharacteristic
282#497—1062no—
283#494—532no—
284#492—532no—
285#480—1061no—
286#474—531no—
287#474—531no—
288#474—532no—
289#474—532no—
290#474—1061no—
291#474—1061no—
292#459—531no—
293#459—531no—
294#447—1061no—
295C4×D8—1061no—
296C4×D8—1061no—
297#436C2×D5312yes—
298#435—22no—
299C8×D4C2×D5312yes—
300#431—531no—
301C2×#107—1061no—
302#423C2×D5311yescharacteristic
303C2×C4⋊C8—531no—
304C2×#102—1061no—
305#407—21no—
306C4×M16C2×D5311yescharacteristic
307#335—532no—
308#334—532no—
309#332—532no—
310#332—532no—
311#332—532no—
312#331—532no—
313C53#1916311yescharacteristic, complemented, Sylow
314C2×#41—1061no—
315C22×Q8—1061no—
316C22×D4D21211yescharacteristic
317C23×C4—21no—
318#137—2124no—
319#137—2124no—
320C2×Q16—1062no—
321C2×Q16—1062no—
322C2×SD16—1062no—
323C2×SD16—1062no—
324C2×D8—1061no—
325C2×D8—1061no—
326C2×M16—21no—
327C2×M16—22no—
328C2×M16—44no—
329C2×M16C22⋊D5311yescharacteristic
330C2×M16C22⋊D5311yescharacteristic
331C22×C8—22no—
332C22×C8—1061no—
333C4⋊Q8—532no—
334C4⋊Q8—531no—
335C4⋊Q8—1061no—
336C4⋊Q8—1061no—
337C4⋊Q8—1061no—
338#127—531no—
339#124—1061no—
340#121—1061no—
341#121—1061no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—1061no—
345C4×D4—1061no—
346C4×D4D21211yescharacteristic
347C4×D4D21211yescharacteristic
348C4×D4C22×D5311yescharacteristic
349C4×D4C22×D5311yescharacteristic
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—1061no—
352C2×C4⋊C4D21211yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D5311yescharacteristic
355#107—1062no—
356#107—1062no—
357#107—1061no—
358#107—1061no—
359C4⋊C8—21no—
360C4⋊C8—532no—
361C4⋊C8—531no—
362C4⋊C8—531no—
363C4⋊C8—1061no—
364C4⋊C8C22×D5312yes—
365#102—1062no—
366#102—1062no—
367#102—1061no—
368#102—1061no—
369#102—1061no—
370#102—1061no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—1061no—
374#97C22⋊D5312yes—
375C4×C8—1061no—
376C4×C8C22×D5312yes—
377C24—21no—
378#41—2122no—
379#41—2122no—
380C2×Q8—1061no—
381C2×Q8—1062no—
382C2×Q8—2122no—
383C2×Q8—2122no—
384C2×Q8—2121no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—2121no—
392C2×D4C2×D21211yescharacteristic
393C2×D4C2×D21211yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4C2×D21211yescharacteristic
404C22×C4#1171994511yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406Q16—2124no—
407Q16—2124no—
408SD16—2124no—
409SD16—2124no—
410D8—2122no—
411D8—2122no—
412M16—22no—
413M16—42no—
414M16—44no—
415M16#1171992214yescomplemented
416C2×C8—22no—
417C2×C8—44no—
418C2×C8—1061no—
419C2×C8—1061no—
420C2×C8—1062no—
421C2×C8—2122no—
422C2×C8C2×C22⋊D5312yes—
423C2×C8C2×C22⋊D5312yes—
424C4⋊C4—21no—
425C4⋊C4—1062no—
426C4⋊C4—1061no—
427C4⋊C4—1062no—
428C4⋊C4—1061no—
429C4⋊C4—2121no—
430C4⋊C4—2121no—
431C4⋊C4C2×D21211yescharacteristic
432C4⋊C4C2×D21211yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C42C2×D21211yescharacteristic
440C42#1171994511yescharacteristic
441C42C2×C22⋊D5311yescharacteristic
442C42C23×D5311yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—2121no—
449C23#1313319111yescharacteristic
450Q8—4242no—
451Q8—2122no—
452Q8—2121no—
453Q8—2122no—
454Q8—2121no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—4241no—
460D4—4242no—
461D4—2121no—
462D4—2121no—
463D4—2121no—
464D4—2121no—
465D4#1313325012yescomplemented
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—4242no—
479C2×C4—4242no—
480C2×C4—2121no—
481C2×C4—2122no—
482C2×C4#1313319111yescharacteristic
483C2×C4#1313319111yescharacteristic
484C2×C4#1313319111yescharacteristic
485C2×C4C2×#1171992212yes—
486C2×C4#1313324212yes—
487C2×C4C22×C22⋊D5311yescharacteristic
488C2×C4C22×D21211yescharacteristic
489C2×C4C2×#1171994511yescharacteristic
490C8—22no—
491C8—42no—
492C8—2121no—
493C8—2122no—
494C8—2121no—
495C8C2×#1171992214yes—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—84no—
505C22—4241no—
506C22—4241no—
507C22—4241no—
508C22#1681081812yes—
509C22C2×#1313319111yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—4241no—
516C4—4242no—
517C4#1681081812yes—
518C4C22×#1171992211yescharacteristic
519C4C2×#1313324211yescharacteristic
520C4C2×#1313325011yescharacteristic
521C4#1681142311yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—8481no—
526C2C2×#1681081811yescharacteristic
527C2#2028434511yescharacteristic
528C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingD4×M16×C53C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenterD4×M16D53
hypocenterC53#19163
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #20285174C2semidirect
3. C2×#16810903C2semidirect
4. #20274204C2semidirect
5. #20274180C2semidirect
6. #20274080C2semidirect
7. #20274056C2semidirect
8. #20273467C2semidirect
9. #20273393C2semidirect
10. #20273388C2semidirect
11. D4×M16×C53C2semidirect
17. #16811003C22semidirect
18. #16810996C22semidirect
35. #16810376C22semidirect
36. #16810374C22semidirect
37. #16810370C22semidirect
38. #16810370C22semidirect
39. #16810370C22semidirect
40. #16810366C22semidirect
144. M16×C53D8semidirect
189. D4×C53#137semidirect
261. D4×M16D53semidirect
313. C53#19163semidirect
415. M16#11719922semidirect
465. D4#13133250semidirect

Maximal subgroups

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

subgroupindex
#202851742
#202847152
C2×#168109032
#202742042
#202741802
#202740802
#202740562
#202734672
#202733932
#202733882
D4×M16×C532
#1916353