Finite Group Databasebeta

← #27369504

Subgroups — #27369504

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#20285630C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic, complemented
3#20283826C211yesmaximal, characteristic, complemented
4#20274161C212yesmaximal, complemented
5#20274154C212yesmaximal, complemented
6#20274073C212yesmaximal, complemented
7#20274068C212yesmaximal, complemented
8#20273489C211yesmaximal, characteristic, complemented
9#20273392C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#3035C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133214C2211yescharacteristic
14#16810999C2212yes—
15#16810971C2212yes—
16#16810823—21no—
17#16810823C2212yes—
18#16810818—21no—
19#16810818—21no—
20#16810818—21no—
21#16810818C2212yes—
22#16810756—22no—
23Q8×#88322C2212yescomplemented
24D4×#88322—22no—
25D4×#88322C2212yescomplemented
26#16810676—21no—
27#16810676C2211yescharacteristic
28#16810647C2211yescharacteristic
29C2×C4⋊#88322—21no—
30C2×C4⋊#88322C2211yescharacteristic
31#16810627—21no—
32#16810375C2212yescomplemented
33#16810375C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36#16810366C2212yes—
37#16810366C2212yes—
38C2×#135×C53—21no—
39C53×#532C2211yescharacteristic
40C4×#41×C53C2211yescharacteristic
41C53×#464—21no—
42C53×#463—21no—
43C53×#462C2211yescharacteristic, complemented
44C53×#461C2211yescharacteristic, complemented
45C53×#460C2211yescharacteristic, complemented
46C53×#459C2211yescharacteristic, complemented
47C2×C4⋊C8×C53C2211yescharacteristic
48C53×#416—21no—
49C22×D212—21no—
50C22×D212—21no—
51C22×D212—21no—
52#13133242—44no—
53C2×#11719924—22no—
54C2×#11719922—22no—
55C2×#11719922—22no—
56C2×#11719922—22no—
57C2×#11719917—21no—
58C22×#88322—22no—
59C22×#88322—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214—21no—
63#13133214C2312yes—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—22no—
67C4⋊#88322D411yescharacteristic
68C4⋊#88322D411yescharacteristic
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133137—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—22no—
76C22⋊#88322D412yes—
77C4×#88322—22no—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C2×#41×C53D411yescharacteristic
81#135×C53—44no—
82C2×Q16×C53—21no—
83C2×SD16×C53—21no—
84C2×SD16×C53—21no—
85C2×D8×C53—21no—
86C22×C8×C53—21no—
87C4⋊Q8×C53C2311yescharacteristic
88#127×C53C2311yescharacteristic
89#124×C53—21no—
90#121×C53—21no—
91#121×C53—21no—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53D411yescharacteristic
99C4×D4×C53C2311yescharacteristic
100#117×C53—21no—
101#117×C53D411yescharacteristic
102C2×C42×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53C2311yescharacteristic
106C4⋊C8×C53C2311yescharacteristic
107#103×C53—21no—
108#103×C53—21no—
109#102×C53—21no—
110#102×C53—21no—
111C23×D53—41no—
112#41×C53—22no—
113#41×C53—42no—
114#41×C53—42no—
115#41×C53D814yescomplemented
116C2×Q8×C53—21no—
117C2×Q8×C53C2×D411yescharacteristic
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—41no—
123C2×D4×C53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—41no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C2×D212—22no—
132C2×D212—22no—
133C2×D212—22no—
134C2×D212—42no—
135C2×D212—42no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—81no—
140C2×D212—81no—
141C2×D212—82no—
142Q16×C53—42no—
143SD16×C53—42no—
144SD16×C53—42no—
145D8×C53—42no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—42no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53C2×D411yescharacteristic
154C4⋊C4×C53C2×D411yescharacteristic
155C22⋊C4×C53—21no—
156C22⋊C4×C53—21no—
157C22⋊C4×C53—21no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53C2×D411yescharacteristic
164C42×C53C2×D411yescharacteristic
165C42×C53#4111yescharacteristic
166C42×C53C2411yescharacteristic
167#11719924—44no—
168#11719922—44no—
169#11719922—82no—
170#11719922—82no—
171#11719917—42no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—82no—
175C2×#88322—81no—
176C2×#88322C2×D412yes—
177C2×#88322C2×D412yes—
178C23×C53—21no—
179C23×C53—41no—
180C23×C53#12111yescharacteristic
181C22×D53—42no—
182C22×D53—82no—
183C22×D53—82no—
184C22×D53—82no—
185Q8×C53—21no—
186Q8×C53—41no—
187Q8×C53—41no—
188Q8×C53C2×D812yes—
189D4×C53—24no—
190D4×C53—21no—
191D4×C53—41no—
192D4×C53—41no—
193D4×C53—41no—
194D4×C53—41no—
195D4×C53—81no—
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—41no—
208C2×C4×C53—81no—
209C2×C4×C53—82no—
210C2×C4×C53#12111yescharacteristic
211C2×C4×C53#12111yescharacteristic
212C2×C4×C53#12111yescharacteristic
213C2×C4×C53C2×D812yes—
214C2×C4×C53#13512yes—
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218D212—42no—
219D212—42no—
220D212—42no—
221D212—42no—
222D212—82no—
223D212—82no—
224D212—82no—
225D212—82no—
226C8×C53—41no—
227C8×C53—41no—
228#88322—22no—
229#88322—22no—
230#88322—42no—
231#88322—42no—
232#19410—531nomaximal, Sylow
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—22no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53#46612yes—
242C22×C53C2×#12111yescharacteristic
243C2×D53—82no—
244C2×D53—82no—
245C2×D53—82no—
246C2×D53—164no—
247C4×C53—21no—
248C4×C53—22no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53—81no—
254C4×C53#46612yes—
255C4×C53C22×D811yescharacteristic
256C4×C53C2×#13511yescharacteristic
257C4×C53#57611yescharacteristic
258C4×C53#57711yescharacteristic
259#3205—531no—
260#3038—531no—
261#3035D5311yescharacteristic, complemented
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—161no—
273C2×C53#304511yescharacteristic
274C2×C53C2×#46611yescharacteristic
275C2×C53#306811yescharacteristic
276D53—162no—
277C2×AGL(1,ℤ/8)—1061no—
278C2×#135—21no—
279#532C2×D5311yescharacteristic
280C2×#127—531no—
281C4×#41C2×D5311yescharacteristic
282#493—532no—
283#486—532no—
284#464—21no—
285#463—21no—
286#462C2×D5311yescharacteristic, complemented
287#461C2×D5311yescharacteristic, complemented
288#460—532no—
289#460—1061no—
290#460C2×D5311yescharacteristic, complemented
291#459—532no—
292#459—1061no—
293#459—1061no—
294#459—1061no—
295#459C2×D5311yescharacteristic, complemented
296#447—1062no—
297C8×Q8—532no—
298C8×D4—532no—
299C8×D4—1062no—
300#431—531no—
301#431—1061no—
302#423—531no—
303C2×C4⋊C8—531no—
304C2×C4⋊C8—1061no—
305C2×C4⋊C8C2×D5311yescharacteristic
306#418—1061no—
307#416—21no—
308#335—532no—
309#335—532no—
310#332—532no—
311#331—532no—
312#331—532no—
313#331—532no—
314C53#1941011yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41C22⋊D5311yescharacteristic
317C22×D4—1061no—
318C22×D4—1061no—
319C22×D4—1061no—
320AGL(1,ℤ/8)—2124no—
321#135—44no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—1062no—
326C2×D8—21no—
327C2×D8—1062no—
328C2×D8—1062no—
329C2×D8—1062no—
330C2×M16—1061no—
331C22×C8—21no—
332C22×C8—1062no—
333C22×C8—1061no—
334C4⋊Q8C22×D5311yescharacteristic
335#127—532no—
336#127—1061no—
337#127—1061no—
338#127—1061no—
339#127C22×D5311yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8C22⋊D5311yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
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—532no—
360C4⋊C8—1062no—
361C4⋊C8—1061no—
362C4⋊C8—1061no—
363C4⋊C8C22×D5311yescharacteristic
364C4⋊C8C22×D5311yescharacteristic
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369#102—1062no—
370#102—1062no—
371#102—1062no—
372#102—1062no—
373C22⋊C8—532no—
374C22⋊C8—1062no—
375C4×C8—532no—
376C4×C8—1062no—
377C24—2121no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41#1171992214yescomplemented
382C2×Q8—21no—
383C2×Q8C2×C22⋊D5311yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—1062no—
390C2×D4—1062no—
391C2×D4—1062no—
392C2×D4—4241no—
393C2×D4—4241no—
394C2×D4—4242no—
395C2×D4—2121no—
396C2×D4—2121no—
397C2×D4—2121no—
398C2×D4—2122no—
399C2×D4—2122no—
400C2×D4C2×C22⋊D5311yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4D4×D5311yescharacteristic
406C22×C4#1171994711yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408Q16—42no—
409SD16—42no—
410SD16—42no—
411SD16—2124no—
412D8—42no—
413D8—4242no—
414D8—4242no—
415D8—2124no—
416M16—2122no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—532no—
421C2×C8—532no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—4242no—
425C2×C8—4241no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4C2×C22⋊D5311yescharacteristic
431C4⋊C4C2×C22⋊D5311yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D5311yescharacteristic
441C42#1171994711yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C42C23×D5311yescharacteristic
444C23—21no—
445C23—41no—
446C23—4242no—
447C23—4242no—
448C23—4242no—
449C23—2122no—
450C23#1313318911yescharacteristic
451Q8—21no—
452Q8—41no—
453Q8—41no—
454Q8C2×#1171992212yes—
455D4—24no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—81no—
462D4—82no—
463D4—4242no—
464D4—4242no—
465D4—4242no—
466D4—4242no—
467D4—2122no—
468D4—2122no—
469D4—2122no—
470D4—2122no—
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—41no—
482C2×C4—81no—
483C2×C4—82no—
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—41no—
493C8—41no—
494C8—1062no—
495C8—1062no—
496C8—2122no—
497C8—2122no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—8484no—
507C22—4242no—
508C22—4242no—
509C22—4242no—
510C22#1681085712yes—
511C22C2×#1313318911yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—81no—
519C4#1681085712yes—
520C4C22×#1171992211yescharacteristic
521C4C2×#1313323711yescharacteristic
522C4#1681142511yescharacteristic
523C4#1681143011yescharacteristic
524C2—22no—
525C2—42no—
526C2—161no—
527C2—8482no—
528C2#2028425211yescharacteristic
529C2C2×#1681085711yescharacteristic
530C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3035C2
socleC22×C53C2×#121
hypercenter#3035D53
hypocenterC53#19410
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20285630C2semidirect
2. #20284195C2semidirect
3. #20283826C2semidirect
4. #20274161C2semidirect
5. #20274154C2semidirect
6. #20274073C2semidirect
7. #20274068C2semidirect
8. #20273489C2semidirect
9. #20273392C2semidirect
10. C2×#16810366C2semidirect
11. C53×#3035C2semidirect
23. Q8×#88322C22semidirect
25. D4×#88322C22semidirect
32. #16810375C22semidirect
33. #16810375C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
43. C53×#462C22semidirect
44. C53×#461C22semidirect
45. C53×#460C22semidirect
46. C53×#459C22semidirect
115. #41×C53D8semidirect
261. #3035D53semidirect
286. #462C2×D53semidirect
287. #461C2×D53semidirect
290. #460C2×D53semidirect
295. #459C2×D53semidirect
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
#202856302
#202841952
#202838262
#202741612
#202741542
#202740732
#202740682
#202734892
#202733922
C2×#168103662
C53×#30352
#1941053