Finite Group Databasebeta

← #27366867

Subgroups — #27366867

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#20285172C211yesmaximal, characteristic, complemented
2#20284715C211yesmaximal, characteristic
3D4×#11719917C211yesmaximal, characteristic, complemented
4#20274202C212yesmaximal, complemented
5#20274182C212yesmaximal, complemented
6#20274078C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic, complemented
10#20273387C211yesmaximal, characteristic, complemented
11C2×C53×#474C211yesmaximal, characteristic, complemented
12C2×#13133248—21no—
13C2×#13133248—21no—
14C22×#11719917—21no—
15#16811173C2211yescharacteristic
16#16811015—22no—
17#16811002C2212yescomplemented
18#16810997C2212yescomplemented
19#16810936—21no—
20#16810903C2211yescharacteristic
21#16810903C2211yescharacteristic
22#16810818C2211yescharacteristic
23#16810818C2211yescharacteristic
24#16810757—21no—
25C4×#11719922—21no—
26C4×#11719922—21no—
27#16810701C2212yes—
28#16810695—22no—
29D4×#88322C2212yes—
30#16810676C2211yescharacteristic
31#16810647C2211yescharacteristic
32#16810587—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810369C2212yescomplemented
37#16810369C2212yescomplemented
38#16810368C2212yescomplemented
39#16810366C2212yescomplemented
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4×D4×C53C2211yescharacteristic
42C53×#474—21no—
43C53×#474—21no—
44C53×#474C2212yes—
45C53×#474C2212yes—
46C2×#107×C53—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#102×C53—21no—
49C2×#11719945—21no—
50#13133248—44no—
51#13133248—44no—
52C2×#11719925—22no—
53C2×#11719925—22no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719922—21no—
57C2×#11719922—21no—
58C2×#11719917—22no—
59C2×#11719917—21no—
60C2×#11719917—44no—
61C2×#11719917D411yescharacteristic
62C2×#11719917D411yescharacteristic
63C22×#88322—22no—
64#13133217C2311yescharacteristic
65#13133214C2311yescharacteristic
66#13133205—21no—
67#13133191—21no—
68#13133191—21no—
69C4×D212—21no—
70C4×D212—21no—
71#13133150—21no—
72#13133150—21no—
73C4⋊#88322—21no—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76C4⋊#88322C2312yes—
77#13133138—21no—
78#13133138—21no—
79#13133137—21no—
80#13133137—21no—
81C22⋊#88322—22no—
82C22⋊#88322—22no—
83C22⋊#88322—21no—
84#13133128D412yes—
85C4×#88322—21no—
86C4×#88322C2312yes—
87C22×Q8×C53—21no—
88C22×D4×C53D411yescharacteristic
89C23×C4×C53—21no—
90C22×C8×C53—21no—
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53C2312yes—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53D411yescharacteristic
99C4×D4×C53C2311yescharacteristic
100C4×D4×C53C2311yescharacteristic
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C42×C53C2311yescharacteristic
106#107×C53—22no—
107#107×C53—22no—
108C4⋊C8×C53—21no—
109C4⋊C8×C53C2312yes—
110#102×C53—22no—
111#102×C53—22no—
112C24×C53—21no—
113C2×Q8×C53—22no—
114C2×Q8×C53—41no—
115C2×Q8×C53—42no—
116C2×Q8×C53—42no—
117C2×D4×C53—22no—
118C2×D4×C53—21no—
119C2×D4×C53—41no—
120C2×D4×C53C2×D411yescharacteristic
121C2×D4×C53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—41no—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53#4111yescharacteristic
133C2×C22⋊D53—41no—
134#11719945—42no—
135#11719945—42no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×C4×D53—41no—
140C2×Dic106—21no—
141C2×C8×C53—22no—
142C2×C8×C53—42no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53C2×D411yescharacteristic
149C4⋊C4×C53C2×D411yescharacteristic
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C42×C53#4111yescharacteristic
157C42×C53C2411yescharacteristic
158#11719925—44no—
159#11719925—44no—
160#11719924—44no—
161#11719924—44no—
162#11719922—42no—
163#11719922—42no—
164#11719921—41no—
165#11719921—41no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168#11719917—22no—
169#11719917—42no—
170#11719917—44no—
171#11719917D814yescomplemented
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—22no—
175C2×#88322—44no—
176C2×#88322C2×D412yes—
177C2×#88322C2×D412yes—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—81no—
183C23×C53#12111yescharacteristic
184C22×D53—41no—
185Q8×C53—42no—
186Q8×C53—42no—
187Q8×C53—82no—
188D4×C53—21no—
189D4×C53—21no—
190D4×C53—21no—
191D4×C53—42no—
192D4×C53#13712yescomplemented
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—42no—
206C2×C4×C53—82no—
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53C2×D812yes—
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×D814yes—
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—82no—
252C4×C53#45912yes—
253C4×C53C22×D811yescharacteristic
254C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C53C2×#13711yescharacteristic
256C4×C53#57511yescharacteristic
257Dic53—81no—
258#3152—531no—
259#3100—531no—
260C2×#474D5311yescharacteristic, complemented
261D4×M16—531no—
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—1061no—
279#532—531no—
280C2×C4⋊Q8C2×D5311yescharacteristic
281C2×C4×D4C2×D5311yescharacteristic
282#497—1062no—
283#494—532no—
284#492—532no—
285#480—1061no—
286#474—21no—
287#474—21no—
288#474—531no—
289#474—531no—
290#474C2×D5312yes—
291#474C2×D5312yes—
292#459—531no—
293#459—531no—
294#447—1061no—
295C4×D8—1061no—
296C4×D8—1061no—
297#436—532no—
298#435—1062no—
299C8×D4—532no—
300#431—531no—
301C2×#107—21no—
302#423—531no—
303C2×C4⋊C8C2×D5311yescharacteristic
304C2×#102—21no—
305#407—1061no—
306C4×M16—531no—
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—21no—
316C22×D4C22⋊D5311yescharacteristic
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—531no—
327C2×M16—531no—
328C2×M16—1062no—
329C2×M16—1061no—
330C2×M16—2124no—
331C22×C8—21no—
332C22×C8—1062no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8—531no—
337C4⋊Q8C22×D5312yes—
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×D4C22⋊D5311yescharacteristic
347C4×D4C22⋊D5311yescharacteristic
348C4×D4C22×D5311yescharacteristic
349C4×D4C22×D5311yescharacteristic
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D5311yescharacteristic
353C2×C22⋊C4—21no—
354C2×C42C22×D5311yescharacteristic
355#107—22no—
356#107—22no—
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—22no—
366#102—22no—
367#102—1061no—
368#102—1061no—
369#102—1061no—
370#102—1061no—
371C22⋊C8—1062no—
372C22⋊C8—1062no—
373C22⋊C8—1061no—
374#97—532no—
375C4×C8—532no—
376C4×C8—1061no—
377C24—21no—
378#41—2122no—
379#41—2122no—
380C2×Q8—22no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—1061no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—2121no—
392C2×D4C2×C22⋊D5311yescharacteristic
393C2×D4C2×C22⋊D5311yescharacteristic
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—2121no—
403C22×C4D4×D5311yescharacteristic
404C22×C4#1171994711yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406Q16—2124no—
407Q16—2124no—
408SD16—2124no—
409SD16—2124no—
410D8—2122no—
411D8—2122no—
412M16—534no—
413M16—1062no—
414M16—2124no—
415M16—2122no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—532no—
419C2×C8—532no—
420C2×C8—1062no—
421C2×C8—1061no—
422C2×C8—1061no—
423C2×C8—2124no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4C2×C22⋊D5311yescharacteristic
432C4⋊C4C2×C22⋊D5311yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—2121no—
438C22⋊C4—2121no—
439C42D4×D5311yescharacteristic
440C42#1171994711yescharacteristic
441C42C2×C22⋊D5311yescharacteristic
442C42C23×D5311yescharacteristic
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—81no—
448C23—2121no—
449C23#1313318911yescharacteristic
450Q8—42no—
451Q8—42no—
452Q8—82no—
453Q8—2121no—
454Q8—2121no—
455D4—21no—
456D4—21no—
457D4—21no—
458D4—42no—
459D4—4241no—
460D4—4242no—
461D4—2121no—
462D4—2121no—
463D4—2121no—
464D4—2121no—
465D4#1313324812yescomplemented
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—42no—
479C2×C4—82no—
480C2×C4—4242no—
481C2×C4—2121no—
482C2×C4#1313318911yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4#1313318911yescharacteristic
485C2×C4D8×D5312yes—
486C2×C4#1313324112yes—
487C2×C4C2×D4×D5311yescharacteristic
488C2×C4C22×C22⋊D5311yescharacteristic
489C2×C4C2×#1171994711yescharacteristic
490C8—42no—
491C8—534no—
492C8—1062no—
493C8—2122no—
494C8—2121no—
495C8—2121no—
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#1681081612yes—
509C22C2×#1313318911yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—4241no—
517C4#1681081612yes—
518C4C2×D8×D5311yescharacteristic
519C4C2×#1313324111yescharacteristic
520C4C2×#1313324811yescharacteristic
521C4#1681142111yescharacteristic
522C2—22no—
523C2—42no—
524C2—81no—
525C2—8481no—
526C2C2×#1681081611yescharacteristic
527C2#2028434111yescharacteristic
528C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#474C2
socleC22×C53C2×#121
prime radical#20284715C2
hypercenterC2×#474D53
hypocenterC53#19163
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #20285172C2semidirect
3. D4×#11719917C2semidirect
4. #20274202C2semidirect
5. #20274182C2semidirect
6. #20274078C2semidirect
7. #20274058C2semidirect
8. #20273465C2semidirect
9. #20273390C2semidirect
10. #20273387C2semidirect
11. C2×C53×#474C2semidirect
17. #16811002C22semidirect
18. #16810997C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
36. #16810369C22semidirect
37. #16810369C22semidirect
38. #16810368C22semidirect
39. #16810366C22semidirect
171. #11719917D8semidirect
192. D4×C53#137semidirect
260. C2×#474D53semidirect
313. C53#19163semidirect
465. D4#13133248semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202847152
D4×#117199172
#202742022
#202741822
#202740782
#202740582
#202734652
#202733902
#202733872
C2×C53×#4742
#1916353