Finite Group Databasebeta

← #27366944

Subgroups — #27366944

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#20285165C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic, complemented
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274197C212yesmaximal, complemented
5#20274185C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274061C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic, complemented
10#20273383C211yesmaximal, characteristic, complemented
11C2×C53×#460C211yesmaximal, characteristic, complemented
12C2×#13133243—21no—
13C2×#13133243—21no—
14#16811173C2211yescharacteristic
15#16811010—22no—
16#16810999C2212yes—
17#16810995C2212yes—
18#16810941—21no—
19#16810908C2211yescharacteristic, complemented
20#16810908C2211yescharacteristic, complemented
21#16810823C2211yescharacteristic, complemented
22#16810823C2211yescharacteristic, complemented
23#16810762—21no—
24#16810754—22no—
25#16810754C2212yescomplemented
26Q8×#88322C2212yescomplemented
27C4×#11719924—21no—
28C4×#11719924—21no—
29#16810676C2211yescharacteristic
30#16810647—21no—
31#16810647C2211yescharacteristic
32C4×#11719917—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810368C2212yescomplemented
37#16810367C2212yes—
38#16810367C2212yes—
39#16810366C2212yescomplemented
40C22×SD16×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460—21no—
45C53×#460C2212yes—
46C53×#460C2212yes—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719945—21no—
50#13133243—44no—
51#13133243—44no—
52C2×#11719923—22no—
53C2×#11719923—22no—
54C2×#11719924—21no—
55C2×#11719924—21no—
56C2×#11719922—22no—
57C2×#11719922—22no—
58C2×#11719917—21no—
59C2×#11719917D411yescharacteristic
60C2×#11719917D411yescharacteristic
61#13133217C2311yescharacteristic
62#13133214C2311yescharacteristic
63#13133205—21no—
64#13133191—21no—
65#13133191—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133148—21no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76#13133141—21no—
77#13133141—21no—
78#13133137—21no—
79#13133137—21no—
80C22⋊#88322—21no—
81#13133128—22no—
82#13133128D412yes—
83C4×#88322—21no—
84C4×#88322—22no—
85C4×#88322C2312yes—
86C22×Q8×C53D411yescharacteristic
87C22×D4×C53—21no—
88C22×D4×C53—21no—
89C22×D4×C53—21no—
90C2×SD16×C53—22no—
91C2×SD16×C53—22no—
92C2×SD16×C53—44no—
93C22×C8×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53—21no—
97#127×C53C2312yes—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53D411yescharacteristic
101C4×Q8×C53D411yescharacteristic
102C4×Q8×C53C2311yescharacteristic
103C4×Q8×C53C2311yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—21no—
109C4⋊C8×C53C2312yes—
110#102×C53—22no—
111#102×C53—22no—
112C24×C53—41no—
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—41no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×Q8×C53C2×D411yescharacteristic
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53—22no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53—82no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53#4111yescharacteristic
134C2×C22⋊D53—41no—
135#11719945—42no—
136#11719945—42no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×C4×D53—41no—
141C2×Dic106—21no—
142SD16×C53—44no—
143SD16×C53—44no—
144C2×C8×C53—22no—
145C2×C8×C53—42no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53C2×D411yescharacteristic
152C4⋊C4×C53C2×D411yescharacteristic
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719924—42no—
162#11719924—42no—
163#11719923—44no—
164#11719923—44no—
165#11719922—44no—
166#11719922—44no—
167#11719921—41no—
168#11719921—41no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171#11719917—22no—
172#11719917—42no—
173#11719917SD1614yescomplemented
174C2×#88322—22no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322C2×D412yes—
178C2×#88322C2×D412yes—
179C23×C53—42no—
180C23×C53—82no—
181C23×C53—82no—
182C23×C53—82no—
183C23×C53#12111yescharacteristic
184C22×D53—41no—
185Q8×C53—21no—
186Q8×C53—21no—
187Q8×C53—21no—
188Q8×C53—42no—
189Q8×C53AGL(1,ℤ/8)12yescomplemented
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—42no—
194D4×C53—82no—
195D4×C53—82no—
196D4×C53—82no—
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—81no—
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53C2×SD1612yes—
211C2×C4×C53AGL(1,ℤ/8)12yes—
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C22⋊D53—82no—
216C2×Dic53—41no—
217D212—41no—
218D212—41no—
219D212—41no—
220D212—41no—
221D212—81no—
222C4×D53—82no—
223Dic106—41no—
224Dic106—41no—
225C8×C53—42no—
226#88322—22no—
227#88322—42no—
228#88322—41no—
229#88322—41no—
230#88322C2×SD1614yescomplemented
231#19170—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—82no—
235C22×C53—82no—
236C22×C53—82no—
237C22×C53—164no—
238C22×C53#46012yes—
239C22×C53C2×#12111yescharacteristic
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C4×C53—21no—
244C4×C53—22no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—41no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53#46012yes—
251C4×C53C22×SD1611yescharacteristic
252C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
254C4×C53#57511yescharacteristic
255Dic53—81no—
256#3151—531no—
257#3103—531no—
258C2×#460D5311yescharacteristic, complemented
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—162no—
269C2×C53C2×#46011yescharacteristic
270C2×C53#305511yescharacteristic
271C2×C53#306811yescharacteristic
272D53—161no—
273C2×AGL(1,ℤ/8)—1061no—
274C2×AGL(1,ℤ/8)—1061no—
275C22×SD16—21no—
276#532—531no—
277C2×#127C2×D5311yescharacteristic
278C2×C4×Q8C2×D5311yescharacteristic
279#496—1062no—
280#493—532no—
281#492—532no—
282#481—1061no—
283#475—531no—
284#475—531no—
285#460—21no—
286#460—21no—
287#460—531no—
288#460—531no—
289#460C2×D5312yes—
290#460C2×D5312yes—
291#448—1061no—
292#446—532no—
293#446—1062no—
294C8×Q8—532no—
295C4×SD16—1061no—
296C4×SD16—1061no—
297#431—531no—
298#423—531no—
299#423—1061no—
300C2×C4⋊C8C2×D5311yescharacteristic
301C2×#102—21no—
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—1061no—
312C22×Q8C22⋊D5311yescharacteristic
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—2124no—
317AGL(1,ℤ/8)—2124no—
318C2×SD16—22no—
319C2×SD16—22no—
320C2×SD16—44no—
321C2×SD16—1062no—
322C2×SD16—1062no—
323C2×SD16—1061no—
324C2×SD16—1061no—
325C2×D8—1062no—
326C2×D8—1062no—
327C2×M16—531no—
328C2×M16—531no—
329C2×M16—1061no—
330C22×C8—21no—
331C4⋊Q8—531no—
332#127—21no—
333#127—21no—
334#127—21no—
335#127—531no—
336#127C22×D5312yes—
337#124—1061no—
338#121—1061no—
339#121—1061no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8C22⋊D5311yescharacteristic
343C4×Q8C22⋊D5311yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×Q8C22×D5311yescharacteristic
346C4×D4—1061no—
347C4×D4—1061no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D5311yescharacteristic
350C2×C42—21no—
351C2×C42C22×D5311yescharacteristic
352#106—1061no—
353#106—1061no—
354C4⋊C8—21no—
355C4⋊C8—532no—
356C4⋊C8—531no—
357C4⋊C8—531no—
358C4⋊C8—1062no—
359C4⋊C8—1062no—
360C4⋊C8—1061no—
361C4⋊C8C22×D5312yes—
362#103—1061no—
363#103—1061no—
364#102—22no—
365#102—22no—
366#102—1061no—
367#102—1061no—
368C22⋊C8—1061no—
369#97—532no—
370#97—1062no—
371C4×C8—532no—
372C4×C8—1062no—
373C4×C8—1061no—
374C24—41no—
375#41—2122no—
376#41—2122no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—1061no—
381C2×Q8C2×C22⋊D5311yescharacteristic
382C2×Q8C2×C22⋊D5311yescharacteristic
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—1061no—
395C2×D4—1061no—
396C2×D4—1061no—
397C2×D4—2121no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—2121no—
401C22×C4D4×D5311yescharacteristic
402C22×C4#1171994711yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404SD16—44no—
405SD16—44no—
406SD16—2124no—
407SD16—2124no—
408SD16—2122no—
409SD16—2122no—
410D8—2124no—
411D8—2124no—
412M16—534no—
413M16—1062no—
414M16—2122no—
415C2×C8—22no—
416C2×C8—42no—
417C2×C8—532no—
418C2×C8—532no—
419C2×C8—1062no—
420C2×C8—1061no—
421C2×C8—1061no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C4⋊C4C2×C22⋊D5311yescharacteristic
430C4⋊C4C2×C22⋊D5311yescharacteristic
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42D4×D5311yescharacteristic
438C42#1171994711yescharacteristic
439C42C2×C22⋊D5311yescharacteristic
440C42C23×D5311yescharacteristic
441C23—42no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—2121no—
446C23#1313318911yescharacteristic
447Q8—21no—
448Q8—21no—
449Q8—21no—
450Q8—42no—
451Q8—2121no—
452Q8—2121no—
453Q8#1313324312yescomplemented
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—4241no—
463D4—4242no—
464D4—2121no—
465D4—2121no—
466D4—2121no—
467D4—2121no—
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—81no—
477C2×C4—4242no—
478C2×C4—2121no—
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—42no—
488C8—534no—
489C8—1062no—
490C8—2121no—
491C8—2122no—
492C8—2121no—
493C22—21no—
494C22—21no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—164no—
499C22—4241no—
500C22—4241no—
501C22—4241no—
502C22#1681082112yes—
503C22C2×#1313318911yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—4241no—
512C4#1681082112yes—
513C4C2×SD16×D5311yescharacteristic
514C4C2×#1313324311yescharacteristic
515C4C2×#1313324411yescharacteristic
516C4#1681142111yescharacteristic
517C2—22no—
518C2—162no—
519C2—8481no—
520C2C2×#1681082111yescharacteristic
521C2#2028433011yescharacteristic
522C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#460C2
socleC22×C53C2×#121
prime radical#20285165C2
hypercenterC2×#460D53
hypocenterC53#19170
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20284746C2semidirect
3. Q8×#11719917C2semidirect
4. #20274197C2semidirect
5. #20274185C2semidirect
6. #20274073C2semidirect
7. #20274061C2semidirect
8. #20273465C2semidirect
9. #20273390C2semidirect
10. #20273383C2semidirect
11. C2×C53×#460C2semidirect
19. #16810908C22semidirect
20. #16810908C22semidirect
21. #16810823C22semidirect
22. #16810823C22semidirect
25. #16810754C22semidirect
26. Q8×#88322C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810368C22semidirect
39. #16810366C22semidirect
173. #11719917SD16semidirect
189. Q8×C53AGL(1,ℤ/8)semidirect
230. #88322C2×SD16semidirect
258. C2×#460D53semidirect
310. C53#19170semidirect
453. Q8#13133243semidirect

Maximal subgroups

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

subgroupindex
#202851652
#202847462
Q8×#117199172
#202741972
#202741852
#202740732
#202740612
#202734652
#202733902
#202733832
C2×C53×#4602
#1917053