Finite Group Databasebeta

← #27368448

Subgroups — #27368448

Order 13568.

768 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285168C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic, complemented
3#20283729C211yesmaximal, characteristic, complemented
4#20274197C212yesmaximal, complemented
5#20274189C212yesmaximal, complemented
6#20274068C212yesmaximal, complemented
7#20274058C212yesmaximal, complemented
8#20273483C211yesmaximal, characteristic, complemented
9#20273437C211yesmaximal, characteristic, complemented
10#20273428C211yesmaximal, characteristic, complemented
11C53×#3115C211yesmaximal, characteristic, complemented
12C2×#13133242—22no—
13C2×#13133242—21no—
14C2×#13133214C2211yescharacteristic
15#16811013—21no—
16#16811013—21no—
17#16810999C2212yes—
18#16810997C2212yes—
19#16810823—21no—
20#16810823C2212yes—
21#16810818—21no—
22#16810818C2212yes—
23#16810754C2212yescomplemented
24#16810695—22no—
25D4×#88322C2212yescomplemented
26#16810680—21no—
27#16810680C2211yescharacteristic
28#16810647C2211yescharacteristic
29#16810627—21no—
30C4×#11719917C2211yescharacteristic
31C2×#13133128—21no—
32#16810375C2212yescomplemented
33#16810373C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C53×#532C2211yescharacteristic
39C4×#41×C53C2211yescharacteristic
40C53×#482—21no—
41C53×#475C2211yescharacteristic, complemented
42C53×#474C2211yescharacteristic, complemented
43C53×#460C2211yescharacteristic, complemented
44C53×#459C2211yescharacteristic, complemented
45C53×#449—21no—
46C53×#442—21no—
47C53×#440—21no—
48C53×#432C2211yescharacteristic
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53#13133242—48no—
54C2×#11719924—22no—
55C2×#11719924—22no—
56C2×#11719924—22no—
57C2×#11719922—22no—
58C2×#11719922—22no—
59C2×#11719922—22no—
60C2×#11719917—21no—
61C2×#11719917D412yes—
62C22×#88322—22no—
63#13133214—21no—
64#13133214—21no—
65#13133214—21no—
66#13133214C2312yes—
67C4⋊#88322—21no—
68C4⋊#88322—22no—
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133137—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75#13133128—21no—
76#13133128—22no—
77#13133128D411yescharacteristic
78#13133128D411yescharacteristic
79C4×#88322C2312yes—
80C2×#41×C53—21no—
81C2×#41×C53D411yescharacteristic
82C2×SD16×C53—21no—
83C2×D8×C53—21no—
84C4⋊Q8×C53C2311yescharacteristic
85#127×C53C2311yescharacteristic
86#124×C53—21no—
87#121×C53—21no—
88#121×C53—21no—
89C4×Q8×C53D411yescharacteristic
90C4×Q8×C53C2311yescharacteristic
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53D411yescharacteristic
96C4×D4×C53C2311yescharacteristic
97#117×C53—21no—
98#117×C53D411yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101#107×C53—21no—
102#106×C53—21no—
103C4⋊C8×C53C2311yescharacteristic
104C4⋊C8×C53C2311yescharacteristic
105#103×C53—21no—
106#102×C53—21no—
107#102×C53—21no—
108#102×C53—21no—
109C22⋊C8×C53—21no—
110#97×C53—21no—
111C23×D53—41no—
112#41×C53—22no—
113#41×C53—41no—
114#41×C53—44no—
115C2×Q8×C53—21no—
116C2×Q8×C53C2×D411yescharacteristic
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C2×D4×C53C2×D411yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—41no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53#4111yescharacteristic
130C2×D212—22no—
131C2×D212—22no—
132C2×D212—22no—
133C2×D212—42no—
134C2×D212—44no—
135C2×D212—42no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—82no—
140SD16×C53—42no—
141D8×C53—42no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53C2×D411yescharacteristic
149C4⋊C4×C53C2×D411yescharacteristic
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160C42×C53#4111yescharacteristic
161C42×C53C2411yescharacteristic
162#11719924—44no—
163#11719924—44no—
164#11719924—44no—
165#11719922—44no—
166#11719922—44no—
167#11719922—44no—
168#11719917—24no—
169#11719917—42no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—44no—
173C2×#88322C2×D412yes—
174C2×#88322C2×D412yes—
175C23×C53—21no—
176C23×C53—41no—
177C23×C53#12111yescharacteristic
178C22×D53—42no—
179C22×D53—82no—
180C22×D53—82no—
181C22×D53—82no—
182Q8×C53—21no—
183Q8×C53—21no—
184Q8×C53—42no—
185D4×C53—21no—
186D4×C53—42no—
187D4×C53—42no—
188D4×C53—41no—
189D4×C53—41no—
190D4×C53—81no—
191D4×C53—82no—
192D4×C53AGL(1,ℤ/8)12yescomplemented
193C2×C4×C53—21no—
194C2×C4×C53—22no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—41no—
203C2×C4×C53—81no—
204C2×C4×C53—82no—
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211D212—42no—
212D212—42no—
213D212—42no—
214D212—42no—
215D212—82no—
216D212—82no—
217D212—82no—
218D212—82no—
219C8×C53—41no—
220C8×C53—41no—
221#88322—22no—
222#88322—42no—
223#88322—42no—
224#88322AGL(1,ℤ/8)14yescomplemented
225#19312—531nomaximal, Sylow
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—22no—
229C22×C53—21no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53C2×#12111yescharacteristic
236C2×D53—82no—
237C2×D53—82no—
238C2×D53—82no—
239C2×D53—164no—
240C4×C53—21no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—21no—
244C4×C53—41no—
245C4×C53—41no—
246C4×C53—41no—
247C4×C53—81no—
248C4×C53C2×AGL(1,ℤ/8)12yes—
249C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C53#57511yescharacteristic
251#3151—531no—
252#3115D5311yescharacteristic, complemented
253#3038—531no—
254#2970—531no—
255#1688—532no—
256#1686—532no—
257#1667—532no—
258#1665—532no—
259#1569—531no—
260#1557—531no—
261#1555—531no—
262C2×C53—41no—
263C2×C53—42no—
264C2×C53—161no—
265C2×C53#303811yescharacteristic
266C2×C53#305511yescharacteristic
267C2×C53#306811yescharacteristic
268D53—162no—
269C2×AGL(1,ℤ/8)—1062no—
270C2×AGL(1,ℤ/8)—1061no—
271#532C2×D5311yescharacteristic
272C2×#127—531no—
273C4×#41C2×D5311yescharacteristic
274#496—1061no—
275#496—1061no—
276#493—532no—
277#492—532no—
278#482—21no—
279#475C2×D5311yescharacteristic, complemented
280#474C2×D5311yescharacteristic, complemented
281#460—532no—
282#460—1061no—
283#460C2×D5311yescharacteristic, complemented
284#459—532no—
285#459—1061no—
286#459C2×D5311yescharacteristic, complemented
287#449—21no—
288#446—532no—
289#442—21no—
290#440—21no—
291#435—1062no—
292C8×D4—532no—
293#432—531no—
294#432—1061no—
295#432C2×D5311yescharacteristic
296#423—531no—
297#418—1061no—
298C4×M16—531no—
299C2×#97—1061no—
300#335—532no—
301#334—532no—
302#332—532no—
303#331—532no—
304#331—532no—
305#331—532no—
306C53#1931211yescharacteristic, complemented, Sylow
307C2×#41—21no—
308C2×#41C22⋊D5311yescharacteristic
309C22×D4—1061no—
310C22×D4—1061no—
311C22×D4—1061no—
312AGL(1,ℤ/8)—2128no—
313AGL(1,ℤ/8)—2124no—
314C2×SD16—21no—
315C2×SD16—1062no—
316C2×SD16—1062no—
317C2×SD16—1062no—
318C2×D8—21no—
319C2×D8—1062no—
320C2×D8—1062no—
321C2×D8—1062no—
322C2×M16—532no—
323C2×M16—1061no—
324C22×C8—1062no—
325C4⋊Q8C22×D5311yescharacteristic
326#127—532no—
327#127—1061no—
328#127—1061no—
329#127—1061no—
330#127C22×D5311yescharacteristic
331#124—21no—
332#121—21no—
333#121—21no—
334C4×Q8C22⋊D5311yescharacteristic
335C4×Q8C22×D5311yescharacteristic
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D5311yescharacteristic
341C4×D4C22×D5311yescharacteristic
342#117—21no—
343#117C22⋊D5311yescharacteristic
344C2×C42—21no—
345C2×C42C22×D5311yescharacteristic
346#107—21no—
347#106—21no—
348C4⋊C8—532no—
349C4⋊C8—532no—
350C4⋊C8—1062no—
351C4⋊C8—1061no—
352C4⋊C8C22×D5311yescharacteristic
353C4⋊C8C22×D5311yescharacteristic
354#103—21no—
355#102—21no—
356#102—21no—
357#102—21no—
358#102—1062no—
359#102—1062no—
360C22⋊C8—21no—
361C22⋊C8—1062no—
362C22⋊C8—1061no—
363#97—21no—
364#97—531no—
365#97—531no—
366#97—1062no—
367#97—1061no—
368C4×C8—532no—
369C24—2121no—
370#41—22no—
371#41—41no—
372#41—44no—
373C2×Q8—21no—
374C2×Q8C2×C22⋊D5311yescharacteristic
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—1062no—
381C2×D4—1062no—
382C2×D4—1062no—
383C2×D4—4242no—
384C2×D4—2124no—
385C2×D4—2122no—
386C2×D4—2122no—
387C2×D4—2121no—
388C2×D4—2121no—
389C2×D4—2121no—
390C2×D4C2×C22⋊D5311yescharacteristic
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4D4×D5311yescharacteristic
396C22×C4#1171994711yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398SD16—42no—
399SD16—2124no—
400SD16—2124no—
401SD16—2124no—
402D8—42no—
403D8—2124no—
404D8—2124no—
405D8—2124no—
406M16—1064no—
407M16—2122no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—532no—
411C2×C8—532no—
412C2×C8—1062no—
413C2×C8—1062no—
414C2×C8—2124no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4C2×C22⋊D5311yescharacteristic
420C4⋊C4C2×C22⋊D5311yescharacteristic
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42D4×D5311yescharacteristic
430C42#1171994711yescharacteristic
431C42C2×C22⋊D5311yescharacteristic
432C42C23×D5311yescharacteristic
433C23—21no—
434C23—41no—
435C23—4242no—
436C23—4242no—
437C23—4242no—
438C23—2122no—
439C23#1313318911yescharacteristic
440Q8—21no—
441Q8—21no—
442Q8—42no—
443D4—21no—
444D4—42no—
445D4—42no—
446D4—41no—
447D4—41no—
448D4—81no—
449D4—82no—
450D4—4242no—
451D4—4242no—
452D4—4242no—
453D4—4242no—
454D4—2122no—
455D4—2122no—
456D4—2122no—
457D4—2122no—
458D4#1313324212yescomplemented
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4#1313318911yescharacteristic
472C2×C4#1313318911yescharacteristic
473C2×C4#1313318911yescharacteristic
474C2×C4C2×D4×D5311yescharacteristic
475C2×C4C22×C22⋊D5311yescharacteristic
476C2×C4C2×#1171994711yescharacteristic
477C8—41no—
478C8—41no—
479C8—534no—
480C8—1062no—
481C8—2122no—
482C8—2122no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—8484no—
493C22—4242no—
494C22—4242no—
495C22—4242no—
496C22C2×#1313318911yescharacteristic
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—81no—
505C4AGL(1,ℤ/8)×D5312yes—
506C4C2×#1313324211yescharacteristic
507C4#1681142111yescharacteristic
508C2—41no—
509C2—42no—
510C2—161no—
511C2—8482no—
512C2#2028419311yescharacteristic
513C2#2028433611yescharacteristic
514C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3115C2
socleC22×C53C2×#121
hypercenter#3115D53
hypocenterC53#19312
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #20285168C2semidirect
2. #20284195C2semidirect
3. #20283729C2semidirect
4. #20274197C2semidirect
5. #20274189C2semidirect
6. #20274068C2semidirect
7. #20274058C2semidirect
8. #20273483C2semidirect
9. #20273437C2semidirect
10. #20273428C2semidirect
11. C53×#3115C2semidirect
23. #16810754C22semidirect
25. D4×#88322C22semidirect
32. #16810375C22semidirect
33. #16810373C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
41. C53×#475C22semidirect
42. C53×#474C22semidirect
43. C53×#460C22semidirect
44. C53×#459C22semidirect
192. D4×C53AGL(1,ℤ/8)semidirect
224. #88322AGL(1,ℤ/8)semidirect
252. #3115D53semidirect
279. #475C2×D53semidirect
280. #474C2×D53semidirect
283. #460C2×D53semidirect
286. #459C2×D53semidirect
306. C53#19312semidirect
458. D4#13133242semidirect

Maximal subgroups

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

subgroupindex
#202851682
#202841952
#202837292
#202741972
#202741892
#202740682
#202740582
#202734832
#202734372
#202734282
C53×#31152
#1931253