Finite Group Databasebeta

← #27369510

Subgroups — #27369510

Order 13568.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285633C211yesmaximal, characteristic, complemented
2#20284170C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274157C212yesmaximal, complemented
5#20274150C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274068C212yesmaximal, complemented
8#20273491C211yesmaximal, characteristic, complemented
9#20273393C211yesmaximal, characteristic, complemented
10C2×#16810367C211yesmaximal, characteristic, complemented
11C53×#3038C211yesmaximal, characteristic, complemented
12C2×#13133237—21no—
13#16811173C2211yescharacteristic
14#16810999C2212yes—
15#16810969C2212yes—
16#16810845—21no—
17#16810838—21no—
18#16810833C2211yescharacteristic, complemented
19#16810828C2211yescharacteristic, complemented
20#16810823C2211yescharacteristic, complemented
21#16810818C2211yescharacteristic, complemented
22#16810815—21no—
23#16810815—21no—
24#16810757—22no—
25Q8×#88322C2212yescomplemented
26D4×#88322—22no—
27D4×#88322C2212yescomplemented
28#16810676—21no—
29#16810676C2211yescharacteristic
30C2×C4⋊#88322—21no—
31C2×C4⋊#88322C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810617—21no—
34#16810375C2212yescomplemented
35#16810375C2212yescomplemented
36#16810370C2212yescomplemented
37#16810367C2212yes—
38#16810367C2212yes—
39#16810366C2212yescomplemented
40C2×AGL(1,ℤ/8)×C53—21no—
41C2×#127×C53C2211yescharacteristic
42C4×#41×C53C2211yescharacteristic
43C53×#460—21no—
44C53×#460C2212yes—
45C53×#459—21no—
46C53×#459C2212yes—
47C53×#423C2211yescharacteristic
48C53×#418—21no—
49C2×#11719945—21no—
50#13133237—44no—
51C2×#11719925—21no—
52C2×#11719923—21no—
53C2×#11719924—21no—
54C2×#11719922—22no—
55C2×#11719922—22no—
56C2×#11719922—21no—
57C22×#88322—22no—
58C22×#88322—21no—
59C22×#88322—21no—
60#13133217C2311yescharacteristic
61#13133214C2311yescharacteristic
62#13133205—21no—
63#13133191—21no—
64#13133191—21no—
65C4×D212—21no—
66C4×D212—21no—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—22no—
70C4⋊#88322D411yescharacteristic
71C4⋊#88322D411yescharacteristic
72C4⋊#88322C2311yescharacteristic
73C4⋊#88322C2311yescharacteristic
74C4⋊#88322C2312yes—
75#13133140—21no—
76#13133140—21no—
77#13133138—22no—
78#13133138—22no—
79#13133137—21no—
80#13133137—21no—
81C22⋊#88322—22no—
82C22⋊#88322D412yes—
83C4×#88322—22no—
84C4×#88322C2312yes—
85C2×#41×C53D411yescharacteristic
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53—21no—
89AGL(1,ℤ/8)×C53—44no—
90C2×SD16×C53—22no—
91C2×D8×C53—22no—
92C2×M16×C53—21no—
93#127×C53—21no—
94#127×C53—21no—
95#127×C53—21no—
96#127×C53C2312yes—
97C4×Q8×C53D411yescharacteristic
98C4×Q8×C53C2311yescharacteristic
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53D411yescharacteristic
102C4×D4×C53C2311yescharacteristic
103#117×C53—21no—
104#117×C53D411yescharacteristic
105C2×C42×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#102×C53—22no—
110#102×C53—22no—
111C24×C53—41no—
112#41×C53—22no—
113#41×C53D814yescomplemented
114C2×Q8×C53C2×D411yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—42no—
123C2×D4×C53—42no—
124C2×D4×C53—81no—
125C2×D4×C53—81no—
126C2×D4×C53—82no—
127C2×D4×C53C2×D411yescharacteristic
128C22×C4×C53—21no—
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—
143D8×C53—44no—
144M16×C53—42no—
145C2×C8×C53—22no—
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—
153C42×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53#4111yescharacteristic
159C42×C53C2411yescharacteristic
160#11719925—42no—
161#11719924—42no—
162#11719923—42no—
163#11719922—42no—
164#11719922—82no—
165#11719922—82no—
166#11719921—41no—
167#11719921—41no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C2×#88322—22no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—42no—
174C2×#88322—82no—
175C2×#88322—81no—
176C2×#88322C2×D412yes—
177C2×#88322C2×D412yes—
178C23×C53—21no—
179C23×C53—42no—
180C23×C53—82no—
181C23×C53—82no—
182C23×C53—82no—
183C23×C53#12111yescharacteristic
184C22×D53—41no—
185Q8×C53—21no—
186Q8×C53C2×D812yes—
187D4×C53—24no—
188D4×C53—21no—
189D4×C53—42no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—42no—
193D4×C53—82no—
194D4×C53—82no—
195D4×C53—82no—
196D4×C53—82no—
197D4×C53C2×D812yes—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—24no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—81no—
208C2×C4×C53#12111yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53C2×D812yes—
212C2×C4×C53#13512yes—
213C2×C4×C53C22×D411yescharacteristic
214C2×C4×C53C22×D411yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C22⋊D53—82no—
217C2×Dic53—41no—
218D212—41no—
219D212—41no—
220D212—41no—
221D212—41no—
222D212—81no—
223C4×D53—82no—
224Dic106—41no—
225Dic106—41no—
226C8×C53—42no—
227#88322—22no—
228#88322—22no—
229#88322—41no—
230#88322—41no—
231#88322—42no—
232#19410—531nomaximal, Sylow
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—22no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—82no—
239C22×C53—82no—
240C22×C53—82no—
241C22×C53—164no—
242C22×C53#46612yes—
243C22×C53C2×#12111yescharacteristic
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C4×C53—21no—
248C4×C53—22no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53#46612yes—
254C4×C53C22×D811yescharacteristic
255C4×C53C2×#13511yescharacteristic
256C4×C53#57611yescharacteristic
257C4×C53#57711yescharacteristic
258Dic53—81no—
259#3205—531no—
260#3038D5311yescharacteristic, complemented
261#3035—531no—
262#2988—531no—
263#1682—532no—
264#1681—532no—
265#1668—532no—
266#1667—532no—
267#1571—531no—
268#1547—531no—
269C2×#331—531no—
270C2×C53—22no—
271C2×C53—42no—
272C2×C53—162no—
273C2×C53#304511yescharacteristic
274C2×C53C2×#46611yescharacteristic
275C2×C53#306811yescharacteristic
276D53—161no—
277C2×AGL(1,ℤ/8)—21no—
278C2×#135—1061no—
279#532—531no—
280C2×#127C2×D5311yescharacteristic
281C4×#41C2×D5311yescharacteristic
282#493—532no—
283#486—532no—
284#464—1061no—
285#463—1061no—
286#462—531no—
287#461—531no—
288#460—21no—
289#460—531no—
290#460C2×D5312yes—
291#459—21no—
292#459—531no—
293#459—1061no—
294#459—1061no—
295#459C2×D5312yes—
296#447—1062no—
297C8×Q8—532no—
298C8×D4—532no—
299C8×D4—1062no—
300#431—531no—
301#431—1061no—
302#423C2×D5311yescharacteristic
303C2×C4⋊C8—531no—
304C2×C4⋊C8—531no—
305C2×C4⋊C8—1061no—
306#418—21no—
307#416—1061no—
308#335—532no—
309#335—532no—
310#332—532no—
311#331—532no—
312#331—532no—
313#331—532no—
314C53#1941011yescharacteristic, complemented, Sylow
315C2×#41—1061no—
316C2×#41C22⋊D5311yescharacteristic
317C22×D4—21no—
318C22×D4—21no—
319C22×D4—21no—
320AGL(1,ℤ/8)—44no—
321#135—2124no—
322C2×Q16—1061no—
323C2×SD16—22no—
324C2×SD16—1061no—
325C2×SD16—1061no—
326C2×D8—22no—
327C2×D8—1062no—
328C2×D8—1062no—
329C2×D8—1061no—
330C2×M16—21no—
331C22×C8—1062no—
332C22×C8—1061no—
333C22×C8—1061no—
334C4⋊Q8—531no—
335#127—21no—
336#127—21no—
337#127—21no—
338#127—531no—
339#127C22×D5312yes—
340#124—1061no—
341#121—1061no—
342#121—1061no—
343C4×Q8C22⋊D5311yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—1061no—
348C4×D4—1061no—
349C4×D4C22⋊D5311yescharacteristic
350C4×D4C22×D5311yescharacteristic
351#117—21no—
352#117C22⋊D5311yescharacteristic
353C2×C42—21no—
354C2×C42C22×D5311yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—531no—
357C4⋊C8—531no—
358C4⋊C8—532no—
359C4⋊C8—531no—
360C4⋊C8—531no—
361C4⋊C8—1062no—
362C4⋊C8—1061no—
363C4⋊C8—1061no—
364C4⋊C8C22×D5312yes—
365#103—1061no—
366#103—1061no—
367#102—22no—
368#102—22no—
369#102—1062no—
370#102—1062no—
371#102—1061no—
372#102—1061no—
373C22⋊C8—532no—
374C22⋊C8—1062no—
375C4×C8—532no—
376C4×C8—1062no—
377C24—41no—
378#41—22no—
379#41—2122no—
380#41—2122no—
381#41#1171992214yescomplemented
382C2×Q8—1061no—
383C2×Q8C2×C22⋊D5311yescharacteristic
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4—82no—
396C2×D4—1061no—
397C2×D4—1061no—
398C2×D4—1061no—
399C2×D4—2121no—
400C2×D4C2×C22⋊D5311yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—2121no—
405C22×C4D4×D5311yescharacteristic
406C22×C4#1171994711yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408Q16—2122no—
409SD16—44no—
410SD16—2122no—
411SD16—2122no—
412D8—44no—
413D8—4242no—
414D8—4242no—
415D8—2122no—
416M16—42no—
417C2×C8—22no—
418C2×C8—532no—
419C2×C8—532no—
420C2×C8—1062no—
421C2×C8—1061no—
422C2×C8—1061no—
423C2×C8—4242no—
424C2×C8—4241no—
425C2×C8—2122no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4C2×C22⋊D5311yescharacteristic
431C4⋊C4C2×C22⋊D5311yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D5311yescharacteristic
441C42#1171994711yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C42C23×D5311yescharacteristic
444C23—21no—
445C23—42no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—2121no—
450C23#1313318911yescharacteristic
451Q8—21no—
452Q8—2121no—
453Q8—2121no—
454Q8C2×#1171992212yes—
455D4—24no—
456D4—21no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—4241no—
466D4—4242no—
467D4—2121no—
468D4—2121no—
469D4—2121no—
470D4—2121no—
471D4C2×#1171992212yes—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—4242no—
483C2×C4—2121no—
484C2×C4#1313318911yescharacteristic
485C2×C4#1313318911yescharacteristic
486C2×C4#1313318911yescharacteristic
487C2×C4C2×#1171992212yes—
488C2×C4#1313323712yes—
489C2×C4C2×D4×D5311yescharacteristic
490C2×C4C22×C22⋊D5311yescharacteristic
491C2×C4C2×#1171994711yescharacteristic
492C8—42no—
493C8—1062no—
494C8—1062no—
495C8—2121no—
496C8—2121no—
497C8—2122no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—164no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22#1681085712yes—
511C22C2×#1313318911yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—4241no—
519C4#1681085712yes—
520C4C22×#1171992211yescharacteristic
521C4C2×#1313323711yescharacteristic
522C4#1681142711yescharacteristic
523C4#1681143311yescharacteristic
524C2—22no—
525C2—42no—
526C2—162no—
527C2—8481no—
528C2#2028425811yescharacteristic
529C2C2×#1681085711yescharacteristic
530C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3038C2
socleC22×C53C2×#121
hypercenter#3038D53
hypocenterC53#19410
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20285633C2semidirect
2. #20284170C2semidirect
3. #20283826C2semidirect
4. #20274157C2semidirect
5. #20274150C2semidirect
6. #20274073C2semidirect
7. #20274068C2semidirect
8. #20273491C2semidirect
9. #20273393C2semidirect
10. C2×#16810367C2semidirect
11. C53×#3038C2semidirect
18. #16810833C22semidirect
19. #16810828C22semidirect
20. #16810823C22semidirect
21. #16810818C22semidirect
25. Q8×#88322C22semidirect
27. D4×#88322C22semidirect
34. #16810375C22semidirect
35. #16810375C22semidirect
36. #16810370C22semidirect
39. #16810366C22semidirect
113. #41×C53D8semidirect
260. #3038D53semidirect
314. C53#19410semidirect
381. #41#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856332
#202841702
#202838262
#202741572
#202741502
#202740732
#202740682
#202734912
#202733932
C2×#168103672
C53×#30382
#1941053