Finite Group Databasebeta

← #27366950

Subgroups — #27366950

Order 13568.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285168C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic, complemented
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274197C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10#20273382C211yesmaximal, characteristic, complemented
11C53×#3103C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133242—21no—
14C22×#11719924—21no—
15C2×#13133214C2211yescharacteristic
16#16811013—22no—
17#16810999C2212yes—
18#16810997C2212yes—
19#16810823—21no—
20#16810823—21no—
21#16810823C2212yes—
22#16810823C2212yes—
23#16810754—22no—
24#16810754C2212yescomplemented
25Q8×#88322C2212yescomplemented
26#16810647—21no—
27#16810647C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133137—21no—
30C4×#11719917—21no—
31C4×#11719917C2211yescharacteristic
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#532C2211yescharacteristic
39C2×C4×Q8×C53C2211yescharacteristic
40C53×#481—21no—
41C53×#475C2211yescharacteristic, complemented
42C53×#475C2211yescharacteristic, complemented
43C53×#460C2211yescharacteristic, complemented
44C53×#460C2211yescharacteristic, complemented
45C53×#448—21no—
46C4×SD16×C53—21no—
47C4×SD16×C53—21no—
48C53×#431C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53#13133242—44no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719924—22no—
57C2×#11719924—22no—
58C2×#11719924—44no—
59C2×#11719922—22no—
60C2×#11719922—22no—
61C2×#11719917—21no—
62C2×#11719917D411yescharacteristic
63C2×#11719917D411yescharacteristic
64C22×#88322—21no—
65#13133214—21no—
66#13133214—21no—
67#13133214—21no—
68#13133214C2312yes—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2312yes—
75#13133137—22no—
76#13133137—22no—
77#13133128—22no—
78#13133128D412yes—
79C4×#88322—22no—
80C4×#88322C2312yes—
81C2×#41×C53—21no—
82C22×Q8×C53D411yescharacteristic
83C2×SD16×C53—21no—
84C2×SD16×C53—21no—
85C4⋊Q8×C53C2311yescharacteristic
86#127×C53C2311yescharacteristic
87#124×C53—21no—
88#121×C53—21no—
89#121×C53—21no—
90C4×Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53D411yescharacteristic
94C4×Q8×C53C2311yescharacteristic
95C4×Q8×C53C2311yescharacteristic
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102#106×C53—21no—
103#106×C53—21no—
104C4⋊C8×C53C2311yescharacteristic
105C4⋊C8×C53C2311yescharacteristic
106#103×C53—21no—
107#103×C53—21no—
108#102×C53—21no—
109#102×C53—21no—
110C22⋊C8×C53—21no—
111C4×C8×C53—21no—
112C23×D53—41no—
113#41×C53—42no—
114#41×C53—42no—
115C2×Q8×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—21no—
118C2×Q8×C53—41no—
119C2×Q8×C53C2×D411yescharacteristic
120C2×Q8×C53C2×D411yescharacteristic
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—41no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C2×D212—22no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—42no—
135C2×D212—42no—
136C2×D212—42no—
137C2×D212—42no—
138C2×D212—41no—
139C2×D212—41no—
140C2×D212—41no—
141C2×D212—82no—
142SD16×C53—42no—
143SD16×C53—42no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53C2×D411yescharacteristic
154C4⋊C4×C53C2×D411yescharacteristic
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53C2×D411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53#4111yescharacteristic
164C42×C53C2411yescharacteristic
165#11719924—44no—
166#11719924—44no—
167#11719924—44no—
168#11719924—44no—
169#11719922—44no—
170#11719922—44no—
171#11719917—22no—
172#11719917—42no—
173#11719917SD1614yescomplemented
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—42no—
177C2×#88322C2×D412yes—
178C2×#88322C2×D412yes—
179C23×C53—41no—
180C23×C53#12111yescharacteristic
181C22×D53—42no—
182C22×D53—82no—
183C22×D53—82no—
184C22×D53—82no—
185Q8×C53—21no—
186Q8×C53—21no—
187Q8×C53—21no—
188Q8×C53—42no—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53AGL(1,ℤ/8)12yescomplemented
192D4×C53—41no—
193D4×C53—41no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53—81no—
197D4×C53—82no—
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—41no—
205C2×C4×C53—42no—
206C2×C4×C53—41no—
207C2×C4×C53—81no—
208C2×C4×C53—82no—
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53C2×SD1612yes—
213C2×C4×C53AGL(1,ℤ/8)12yes—
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217D212—42no—
218D212—42no—
219D212—42no—
220D212—42no—
221D212—82no—
222D212—82no—
223D212—82no—
224D212—82no—
225C8×C53—41no—
226C8×C53—41no—
227#88322—22no—
228#88322—42no—
229#88322—42no—
230#88322C2×SD1614yescomplemented
231#19170—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53#46012yes—
238C22×C53C2×#12111yescharacteristic
239C2×D53—82no—
240C2×D53—82no—
241C2×D53—82no—
242C2×D53—164no—
243C4×C53—21no—
244C4×C53—22no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—41no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—81no—
251C4×C53#46012yes—
252C4×C53C22×SD1611yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C53#57511yescharacteristic
256#3151—531no—
257#3103D5311yescharacteristic, complemented
258C2×#460—531no—
259Q8×M16—531no—
260#1688—532no—
261#1686—532no—
262#1668—532no—
263#1666—532no—
264#1565—531no—
265#1547—531no—
266#1545—531no—
267C2×C53—22no—
268C2×C53—161no—
269C2×C53C2×#46011yescharacteristic
270C2×C53#305511yescharacteristic
271C2×C53#306811yescharacteristic
272D53—162no—
273C2×AGL(1,ℤ/8)—1061no—
274C2×AGL(1,ℤ/8)—1061no—
275C22×SD16—1061no—
276#532C2×D5311yescharacteristic
277C2×#127—531no—
278C2×C4×Q8C2×D5311yescharacteristic
279#496—1062no—
280#493—532no—
281#492—532no—
282#481—21no—
283#475C2×D5311yescharacteristic, complemented
284#475C2×D5311yescharacteristic, complemented
285#460—532no—
286#460—532no—
287#460—1061no—
288#460—1061no—
289#460C2×D5311yescharacteristic, complemented
290#460C2×D5311yescharacteristic, complemented
291#448—21no—
292#446—532no—
293#446—1062no—
294C8×Q8—532no—
295C4×SD16—21no—
296C4×SD16—21no—
297#431C2×D5311yescharacteristic
298#423—531no—
299#423—1061no—
300C2×C4⋊C8—531no—
301C2×#102—1061no—
302C4×M16—531no—
303C4×M16—1061no—
304#335—532no—
305#334—532no—
306#332—532no—
307#331—532no—
308#331—532no—
309#331—532no—
310C53#1917011yescharacteristic, complemented, Sylow
311C2×#41—21no—
312C22×Q8C22⋊D5311yescharacteristic
313C22×D4—1061no—
314C22×D4—1061no—
315C22×D4—1061no—
316AGL(1,ℤ/8)—2124no—
317AGL(1,ℤ/8)—2124no—
318C2×SD16—21no—
319C2×SD16—21no—
320C2×SD16—1062no—
321C2×SD16—1062no—
322C2×SD16—1062no—
323C2×SD16—1062no—
324C2×SD16—2124no—
325C2×D8—1062no—
326C2×D8—1062no—
327C2×M16—531no—
328C2×M16—531no—
329C2×M16—1061no—
330C22×C8—1061no—
331C4⋊Q8C22×D5311yescharacteristic
332#127—532no—
333#127—1061no—
334#127—1061no—
335#127—1061no—
336#127C22×D5311yescharacteristic
337#124—21no—
338#121—21no—
339#121—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D5311yescharacteristic
343C4×Q8C22⋊D5311yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×Q8C22×D5311yescharacteristic
346C4×D4—21no—
347C4×D4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D5311yescharacteristic
350C2×C42—21no—
351C2×C42C22×D5311yescharacteristic
352#106—21no—
353#106—21no—
354C4⋊C8—532no—
355C4⋊C8—532no—
356C4⋊C8—1062no—
357C4⋊C8—1062no—
358C4⋊C8—1061no—
359C4⋊C8—1061no—
360C4⋊C8C22×D5311yescharacteristic
361C4⋊C8C22×D5311yescharacteristic
362#103—21no—
363#103—21no—
364#102—21no—
365#102—21no—
366#102—1062no—
367#102—1062no—
368C22⋊C8—21no—
369#97—532no—
370#97—1062no—
371C4×C8—21no—
372C4×C8—532no—
373C4×C8—1062no—
374C24—2121no—
375#41—42no—
376#41—42no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8C2×C22⋊D5311yescharacteristic
382C2×Q8C2×C22⋊D5311yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—1062no—
388C2×D4—1062no—
389C2×D4—1062no—
390C2×D4—4242no—
391C2×D4—2122no—
392C2×D4—2122no—
393C2×D4—2121no—
394C2×D4—2121no—
395C2×D4—2121no—
396C2×D4—2122no—
397C2×D4—2122no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4D4×D5311yescharacteristic
402C22×C4#1171994711yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404SD16—42no—
405SD16—42no—
406SD16—2124no—
407SD16—2124no—
408SD16—2124no—
409SD16—2124no—
410D8—2124no—
411D8—2124no—
412M16—534no—
413M16—1062no—
414M16—2122no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—532no—
418C2×C8—532no—
419C2×C8—1062no—
420C2×C8—1062no—
421C2×C8—2122no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4C2×C22⋊D5311yescharacteristic
430C4⋊C4C2×C22⋊D5311yescharacteristic
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D5311yescharacteristic
438C42#1171994711yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—41no—
442C23—4242no—
443C23—4242no—
444C23—4242no—
445C23—2122no—
446C23#1313318911yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8#1313324212yescomplemented
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—81no—
459D4—82no—
460D4—4242no—
461D4—4242no—
462D4—4242no—
463D4—4242no—
464D4—2122no—
465D4—2122no—
466D4—2122no—
467D4—2122no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4#1313318911yescharacteristic
480C2×C4#1313318911yescharacteristic
481C2×C4#1313318911yescharacteristic
482C2×C4SD16×D5312yes—
483C2×C4#1313324412yes—
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4C22×C22⋊D5311yescharacteristic
486C2×C4C2×#1171994711yescharacteristic
487C8—41no—
488C8—41no—
489C8—534no—
490C8—1062no—
491C8—2122no—
492C8—2122no—
493C22—21no—
494C22—21no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—8484no—
499C22—4242no—
500C22—4242no—
501C22—4242no—
502C22#1681082112yes—
503C22C2×#1313318911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4#1681082112yes—
513C4C2×SD16×D5311yescharacteristic
514C4C2×#1313324211yescharacteristic
515C4C2×#1313324411yescharacteristic
516C4#1681142111yescharacteristic
517C2—22no—
518C2—161no—
519C2—8482no—
520C2C2×#1681082111yescharacteristic
521C2#2028433611yescharacteristic
522C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3103C2
socleC22×C53C2×#121
prime radical#20285168C2
hypercenter#3103D53
hypocenterC53#19170
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
2. C2×#16810823C2semidirect
3. Q8×#11719917C2semidirect
4. #20274197C2semidirect
5. #20274189C2semidirect
6. #20274073C2semidirect
7. #20274065C2semidirect
8. #20273468C2semidirect
9. #20273392C2semidirect
10. #20273382C2semidirect
11. C53×#3103C2semidirect
24. #16810754C22semidirect
25. Q8×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
41. C53×#475C22semidirect
42. C53×#475C22semidirect
43. C53×#460C22semidirect
44. C53×#460C22semidirect
173. #11719917SD16semidirect
191. Q8×C53AGL(1,ℤ/8)semidirect
230. #88322C2×SD16semidirect
257. #3103D53semidirect
283. #475C2×D53semidirect
284. #475C2×D53semidirect
289. #460C2×D53semidirect
290. #460C2×D53semidirect
310. C53#19170semidirect
453. Q8#13133242semidirect

Maximal subgroups

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

subgroupindex
#202851682
C2×#168108232
Q8×#117199172
#202741972
#202741892
#202740732
#202740652
#202734682
#202733922
#202733822
C53×#31032
#1917053