Finite Group Databasebeta

← #27391034

Subgroups — #27391034

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284431C211yesmaximal, characteristic
2#20283614C212yesmaximal, complemented
3C2×#16810740C211yesmaximal, characteristic, complemented
4#20283137C211yesmaximal, characteristic, complemented
5#20275505C211yesmaximal, characteristic, complemented
6#20275387C212yesmaximal, complemented
7#20275213C212yesmaximal, complemented
8#20274766C211yesmaximal, characteristic, complemented
9#20274596C211yesmaximal, characteristic, complemented
10#20274517C212yesmaximal, complemented
11C53×#1976C211yesmaximal, characteristic, complemented
12#16811421C2212yes—
13C2×#13133225C2211yescharacteristic
14C2×#13133224C2212yes—
15C2×#13133223C2211yescharacteristic
16#16811091C2211yescharacteristic
17#16810895C412yescomplemented
18#16810890C412yescomplemented
19#16810883C414yescomplemented
20#16810742—24no—
21#16810740—21no—
22#16810740—21no—
23#16810740—21no—
24#16810740—21no—
25#16810737—24no—
26#16810671C2212yes—
27C2×#13133153C2211yescharacteristic
28C2×#13133153C2211yescharacteristic
29#16810641C2212yes—
30C2×#13133144C2212yes—
31#16810607C2212yes—
32#16810580—21no—
33#16810579C2211yescharacteristic
34#16810579C2212yes—
35#16810577—21no—
36#16810575—21no—
37M16×Dic53C2211yescharacteristic
38C2×C8×Dic53C2211yescharacteristic
39C4×C8×D53—21no—
40#16810393C2212yes—
41#16810393C2212yes—
42#16810382C2212yes—
43C2×#135×C53C2211yescharacteristic
44C53×#427C2211yescharacteristic
45C53×#421C2212yes—
46C53×#410C2212yes—
47C53×#405C2211yescharacteristic
48C2×#11719945C2311yescharacteristic
49#13133248C2×C414yes—
50#13133247—22no—
51#13133243C2×C414yes—
52#13133242—22no—
53#13133237—24no—
54#13133225—21no—
55#13133225—21no—
56#13133225D412yes—
57#13133225D412yes—
58#13133224—22no—
59#13133224—22no—
60#13133224—22no—
61#13133224—22no—
62#13133223—21no—
63#13133223—21no—
64#13133223C2×C412yes—
65#13133223C2×C412yes—
66C2×#11719917—22no—
67C2×#11719917C2312yes—
68C2×#11719917C2311yescharacteristic
69C2×#11719917C2312yes—
70C2×#11719912—21no—
71C2×#11719912—21no—
72C22×#88322—22no—
73C22×#88322C2311yescharacteristic
74C2×C8×D53—21no—
75C2×C8×D53—21no—
76#13133182—21no—
77#13133179—21no—
78#13133177—22no—
79#13133174—21no—
80C2×C4×Dic53C2311yescharacteristic
81C42×D53—21no—
82#13133153—21no—
83#13133153—21no—
84#13133153C2×C412yescomplemented
85#13133153C2×C412yescomplemented
86#13133152—24no—
87#13133145—24no—
88#13133144—22no—
89#13133144—22no—
90#13133136C2×C414yescomplemented
91#13133134C2×C414yescomplemented
92#13133128—21no—
93#13133128—22no—
94#13133127—21no—
95C4×#88322—21no—
96C4×#88322—22no—
97C8×Dic53—21no—
98C8×Dic53D411yescharacteristic
99C8×Dic53D411yescharacteristic
100C8×Dic53D411yescharacteristic
101C8×Dic53D411yescharacteristic
102C2×#41×C53C2312yes—
103#135×C53—21no—
104#135×C53—21no—
105#135×C53—21no—
106#135×C53—21no—
107C2×Q16×C53C2×C411yescharacteristic, complemented
108C2×SD16×C53C2×C412yescomplemented
109C2×D8×C53C2×C411yescharacteristic, complemented
110C2×M16×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112#117×C53C2312yes—
113#117×C53C2311yescharacteristic
114#107×C53—21no—
115#106×C53—21no—
116#104×C53—24no—
117#99×C53—24no—
118#97×C53—21no—
119C4×C8×C53—21no—
120#41×C53—22no—
121#41×C53—22no—
122#41×C53—22no—
123#41×C53—22no—
124C2×Q8×C53C22×C412yes—
125C2×D4×C53—22no—
126C2×D4×C53C22×C412yes—
127C22×C4×C53—22no—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—21no—
130C22×Dic53—21no—
131#11719945—21no—
132#11719945—21no—
133#11719945C22×C412yes—
134#11719945C2×D412yes—
135C2×D212#4111yescharacteristic
136C2×C4×D53—21no—
137C2×C4×D53—22no—
138C2×Dic106#4111yescharacteristic
139Q16×C53—21no—
140Q16×C53—21no—
141SD16×C53—22no—
142SD16×C53—22no—
143D8×C53—21no—
144D8×C53—21no—
145M16×C53—21no—
146M16×C53C2×D412yes—
147C2×C8×C53—21no—
148C2×C8×C53C22×C411yescharacteristic
149C2×C8×C53C22×C411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53#4111yescharacteristic
153C2×C8×C53#4111yescharacteristic
154C4⋊C4×C53—21no—
155C4⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53#4112yes—
158C42×C53—21no—
159C42×C53—22no—
160#11719925—24no—
161#11719924—24no—
162#11719923—24no—
163#11719922—24no—
164#11719921—22no—
165#11719919—22no—
166C4×Dic53—22no—
167C4×Dic53—21no—
168C4×Dic53C2×D411yescharacteristic
169C4×Dic53C2×D411yescharacteristic
170#11719917—21no—
171#11719917—22no—
172#11719917—22no—
173#11719917—22no—
174#11719917—42no—
175#11719917—42no—
176#11719917C22×C412yes—
177#11719917C22×C414yes—
178C2×#88322—21no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—42no—
182C2×#88322—42no—
183C2×#88322C2×D411yescharacteristic
184C2×#88322C2×D411yescharacteristic
185C2×#88322C2×D411yescharacteristic
186C2×#88322C2×D411yescharacteristic
187C2×#88322#4111yescharacteristic
188C2×#88322#4111yescharacteristic
189#11719912—22no—
190#11719912—22no—
191#11719912—22no—
192#11719912—22no—
193C8×D53—22no—
194C8×D53—22no—
195C8×D53—22no—
196C8×D53—22no—
197C23×C53—22no—
198C23×C53C2×#4111yescharacteristic
199C22×D53—21no—
200Q8×C53—22no—
201Q8×C53—22no—
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—42no—
205D4×C53—42no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53C23×C411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C22⋊D53—22no—
218C22⋊D53—22no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—42no—
223D212—21no—
224D212C4×D412yes—
225C4×D53—22no—
226C4×D53—22no—
227C4×D53—44no—
228Dic106—21no—
229Dic106C4×D412yes—
230C8×C53—22no—
231C8×C53C4×D411yescharacteristic
232C8×C53C4×D411yescharacteristic
233C8×C53C4×D411yescharacteristic
234C8×C53C4×D411yescharacteristic
235#88322—22no—
236#88322—24no—
237#88322—42no—
238#88322#12111yescharacteristic
239#88322#12111yescharacteristic
240#88322#12111yescharacteristic
241#88322#12111yescharacteristic
242#21558—531nomaximal, Sylow
243C22×C53—41no—
244C22×C53—42no—
245C22×C53—82no—
246C22×C53C4×#4111yescharacteristic
247C22×C53#52511yescharacteristic
248C22×C53#53211yescharacteristic
249C2×D53—22no—
250C2×D53—42no—
251C4×C53—42no—
252C4×C53—42no—
253C4×C53—82no—
254C4×C53C2×C4×D411yescharacteristic
255C4×C53C2×C4×D411yescharacteristic
256C4×C53C2×#12111yescharacteristic
257C4×C53#52911yescharacteristic
258Dic53—22no—
259Dic53—22no—
260Dic53—41no—
261#3067—531no—
262#2955—532no—
263C2×#443—531no—
264C4×#131—531no—
265#1976D5311yescharacteristic, complemented
266#1953—531no—
267#1932—532no—
268#1906—532no—
269#1814—531no—
270#1772—531no—
271#1757—532no—
272C2×C53—21no—
273C2×C53—21no—
274C2×C53—21no—
275C2×C53—82no—
276C2×C53C4×#12111yescharacteristic
277D53—42no—
278#575—532no—
279C2×#135C2×D5311yescharacteristic
280C2×#131—532no—
281C2×#131—531no—
282C2×#131—531no—
283C4×#41—531no—
284#473—532no—
285#472—532no—
286#471—534no—
287#444—1064no—
288#443—1061no—
289#443—1061no—
290#443—1061no—
291#443—1061no—
292#443—1064no—
293#430—532no—
294C2×#108—531no—
295C2×#108—531no—
296#427C2×D5311yescharacteristic
297#421—532no—
298#421C2×D5312yes—
299C2×#104—532no—
300#413—532no—
301#410C2×D5312yes—
302#405—531no—
303#405—532no—
304#405—1061no—
305#405—1061no—
306#405C2×D5311yescharacteristic
307C4×M16—531no—
308C4×M16—1061no—
309C2×C4×C8—531no—
310C2×C4×C8—1061no—
311#344—532no—
312#344—532no—
313#339—532no—
314C53#2155811yescharacteristic, complemented, Sylow
315C2×#41—531no—
316C2×#41C22×D5312yes—
317#137—534no—
318#137—1062no—
319AGL(1,ℤ/8)—534no—
320AGL(1,ℤ/8)—1062no—
321#135—21no—
322#135—21no—
323#135—21no—
324#135—21no—
325#135—1064no—
326C2×Q16C4×D5311yescharacteristic, complemented
327C2×SD16C4×D5312yescomplemented
328C2×D8C4×D5311yescharacteristic, complemented
329#131—532no—
330#131—532no—
331#131—532no—
332#131—532no—
333#131—1061no—
334#131—1061no—
335#131—1062no—
336#131—1062no—
337#131—1062no—
338#131—1062no—
339#131—1061no—
340#131—1061no—
341C2×M16—531no—
342C2×M16—532no—
343C2×M16—532no—
344C2×M16—1061no—
345C2×M16—1062no—
346C2×M16—1061no—
347C2×M16C22×D5311yescharacteristic
348C22×C8—531no—
349C22×C8—1061no—
350C22×C8—1062no—
351C22×C8—1061no—
352C22×C8C22×D5311yescharacteristic
353C4×Q8—1061no—
354C4×D4—1061no—
355C4×D4—1062no—
356#117—1061no—
357#117C22×D5312yes—
358#117C22×D5311yescharacteristic
359C2×C42—531no—
360C2×C42—1061no—
361#108—532no—
362#108—532no—
363#108—1064no—
364#108—1061no—
365#108—1061no—
366#107—21no—
367#106—21no—
368#104—24no—
369#104—1064no—
370#104—1062no—
371#104—1062no—
372#101—534no—
373#100—534no—
374#99—24no—
375#97—21no—
376#97—1061no—
377#97—1062no—
378#97—1061no—
379C4×C8—21no—
380C4×C8—531no—
381C4×C8—531no—
382C4×C8—531no—
383C4×C8—531no—
384C4×C8—1061no—
385C4×C8—1062no—
386C4×C8—1061no—
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391#41—532no—
392#41—532no—
393#41—1061no—
394#41—1061no—
395C2×Q8—531no—
396C2×Q8C2×C4×D5312yes—
397C2×D4—22no—
398C2×D4—531no—
399C2×D4—1061no—
400C2×D4C2×C4×D5312yes—
401C22×C4—22no—
402C22×C4—1062no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4C23×D5311yescharacteristic
406Q16—21no—
407Q16—21no—
408Q16—1064no—
409SD16—22no—
410SD16—22no—
411SD16—1064no—
412SD16—1064no—
413D8—21no—
414D8—21no—
415D8—1064no—
416M16—21no—
417M16—532no—
418M16—534no—
419M16—1062no—
420M16—1062no—
421M16—1062no—
422M16—1061no—
423M16—1062no—
424M16—1062no—
425M16—1062no—
426M16—1062no—
427M16—2122no—
428M16—2122no—
429M16D4×D5312yes—
430C2×C8—21no—
431C2×C8—531no—
432C2×C8—531no—
433C2×C8—531no—
434C2×C8—531no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1061no—
440C2×C8—1062no—
441C2×C8—1062no—
442C2×C8—1062no—
443C2×C8—1062no—
444C2×C8—2122no—
445C2×C8—2122no—
446C2×C8C2×C4×D5311yescharacteristic
447C2×C8C2×C4×D5311yescharacteristic
448C2×C8D4×D5311yescharacteristic
449C2×C8D4×D5311yescharacteristic
450C2×C8#1171994711yescharacteristic
451C2×C8#1171994711yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—1062no—
455C22⋊C4—22no—
456C22⋊C4—1062no—
457C22⋊C4#1171994712yes—
458C42—21no—
459C42—22no—
460C42—531no—
461C42—531no—
462C42—1062no—
463C42—1061no—
464C23—22no—
465C23—1061no—
466C23C2×#1171994711yescharacteristic
467Q8—22no—
468Q8—22no—
469Q8—532no—
470Q8—1061no—
471D4—22no—
472D4—22no—
473D4—42no—
474D4—42no—
475D4—532no—
476D4—1062no—
477D4—1062no—
478D4—1061no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—1061no—
485C2×C4—1061no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1061no—
489C2×C4—2124no—
490C2×C4—2122no—
491C2×C4C22×C4×D5311yescharacteristic
492C2×C4C2×D4×D5311yescharacteristic
493C2×C4C2×D4×D5311yescharacteristic
494C2×C4#41×D5311yescharacteristic
495C2×C4#41×D5311yescharacteristic
496C2×C4C2×#1171994711yescharacteristic
497C8—22no—
498C8—531no—
499C8—531no—
500C8—531no—
501C8—531no—
502C8—1064no—
503C8—1062no—
504C8—2122no—
505C8#1313317711yescharacteristic
506C8#1313317711yescharacteristic
507C8#1313317711yescharacteristic
508C8#1313317711yescharacteristic
509C22—41no—
510C22—42no—
511C22—82no—
512C22—1062no—
513C22—2122no—
514C22C4×#1171994711yescharacteristic
515C22#1681113311yescharacteristic
516C22#1681117111yescharacteristic
517C4—42no—
518C4—42no—
519C4—82no—
520C4—1062no—
521C4—1062no—
522C4—2121no—
523C4C4×D4×D5311yescharacteristic
524C4C2×#1313317711yescharacteristic
525C4#121×D5311yescharacteristic
526C4#1681115211yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—82no—
531C2—2122no—
532C2#2027723111yescharacteristic
subgroupG/H
centerC4#121×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1976C2
socleC2×C53C4×#121
prime radical#20284431C2
cosocleC22×C4×C53C24
hypercenter#1976D53
hypocenterC53#21558
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #20283614C2semidirect
3. C2×#16810740C2semidirect
4. #20283137C2semidirect
5. #20275505C2semidirect
6. #20275387C2semidirect
7. #20275213C2semidirect
8. #20274766C2semidirect
9. #20274596C2semidirect
10. #20274517C2semidirect
11. C53×#1976C2semidirect
17. #16810895C4semidirect
18. #16810890C4semidirect
19. #16810883C4semidirect
84. #13133153C2×C4semidirect
85. #13133153C2×C4semidirect
90. #13133136C2×C4semidirect
91. #13133134C2×C4semidirect
107. C2×Q16×C53C2×C4semidirect
108. C2×SD16×C53C2×C4semidirect
109. C2×D8×C53C2×C4semidirect
265. #1976D53semidirect
314. C53#21558semidirect
326. C2×Q16C4×D53semidirect
327. C2×SD16C4×D53semidirect
328. C2×D8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202844312
#202836142
C2×#168107402
#202831372
#202755052
#202753872
#202752132
#202747662
#202745962
#202745172
C53×#19762
#2155853