Finite Group Databasebeta

← #27369506

Subgroups — #27369506

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#20285635C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic, complemented
3#20284170C211yesmaximal, characteristic, complemented
4#20274159C212yesmaximal, complemented
5#20274152C212yesmaximal, complemented
6#20274075C212yesmaximal, complemented
7#20274070C212yesmaximal, complemented
8#20273490C211yesmaximal, characteristic, complemented
9#20273390C211yesmaximal, characteristic, complemented
10C2×#16810366C211yesmaximal, characteristic, complemented
11C53×#2988C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C2×#13133237—21no—
14#16811173C2211yescharacteristic
15C2×#13133214C2211yescharacteristic
16#16811000C2212yes—
17#16810970C2212yes—
18#16810845—21no—
19#16810838—21no—
20#16810833C2211yescharacteristic, complemented
21#16810828C2211yescharacteristic, complemented
22#16810823—21no—
23#16810823C2212yes—
24#16810823C2211yescharacteristic, complemented
25#16810819—21no—
26#16810819—21no—
27#16810818—21no—
28#16810818C2212yes—
29#16810818C2211yescharacteristic, complemented
30#16810759—22no—
31#16810647C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810627—21no—
34#16810617—21no—
35#16810376C2212yescomplemented
36#16810376C2212yescomplemented
37#16810368C2212yescomplemented
38#16810366C2212yes—
39#16810366C2212yes—
40#16810366C2212yescomplemented
41C4×#41×C53C2211yescharacteristic
42C8×Q8×C53C2212yescomplemented
43C8×D4×C53—22no—
44C8×D4×C53C2212yescomplemented
45C53×#431—21no—
46C53×#431C2211yescharacteristic
47C2×C4⋊C8×C53—21no—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C22×D212—21no—
51C22×D212—21no—
52C22×D212—21no—
53#13133242—44no—
54#13133237—44no—
55C2×#11719925—21no—
56C2×#11719923—21no—
57C2×#11719924—22no—
58C2×#11719924—21no—
59C2×D424—22no—
60C2×D424—22no—
61C2×#11719922—22no—
62C2×#11719922—21no—
63C2×#11719917—21no—
64C22×#88322—21no—
65#13133217C2311yescharacteristic
66#13133214—21no—
67#13133214—21no—
68#13133214—21no—
69#13133214C2312yes—
70#13133214C2311yescharacteristic
71#13133205—21no—
72#13133191—21no—
73#13133191—21no—
74C4×D212—21no—
75C4×D212—21no—
76C4⋊#88322—21no—
77C4⋊#88322—21no—
78C4⋊#88322C2312yes—
79C4⋊#88322C2311yescharacteristic
80C4⋊#88322C2311yescharacteristic
81#13133140—21no—
82#13133140—21no—
83#13133139—22no—
84#13133139—22no—
85#13133137—22no—
86#13133137—22no—
87#13133137—21no—
88#13133137—21no—
89C2×#41×C53D411yescharacteristic
90C22×C8×C53—22no—
91C22×C8×C53—21no—
92C4×Q8×C53D411yescharacteristic
93C4×Q8×C53C2311yescharacteristic
94C4×D4×C53—21no—
95C4×D4×C53—21no—
96C4×D4×C53D411yescharacteristic
97C4×D4×C53C2311yescharacteristic
98#117×C53—21no—
99#117×C53D411yescharacteristic
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102C4⋊C8×C53—22no—
103C4⋊C8×C53—21no—
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53C2312yes—
107C22⋊C8×C53—22no—
108C22⋊C8×C53D412yes—
109C4×C8×C53—22no—
110C4×C8×C53C2312yes—
111C23×D53—41no—
112#41×C53—22no—
113#41×C53D814yescomplemented
114C2×Q8×C53C2×D411yescharacteristic
115C2×D4×C53—21no—
116C2×D4×C53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—42no—
125#11719945—42no—
126C2×D212—21no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—22no—
132C2×D212—42no—
133C2×D212—42no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—81no—
138C2×D212—81no—
139C2×D212—82no—
140C2×C4×D53—41no—
141C2×Dic106—21no—
142C2×C8×C53—22no—
143C2×C8×C53—82no—
144C2×C8×C53—81no—
145C2×C8×C53C2×D412yes—
146C2×C8×C53C2×D412yes—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53C2×D411yescharacteristic
150C4⋊C4×C53C2×D411yescharacteristic
151C22⋊C4×C53—21no—
152C22⋊C4×C53—21no—
153C22⋊C4×C53—21no—
154C42×C53—21no—
155C42×C53—21no—
156C42×C53—21no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161#11719925—42no—
162#11719924—42no—
163#11719924—44no—
164#11719923—42no—
165#11719922—42no—
166#11719922—44no—
167#11719921—41no—
168#11719921—41no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171#11719917—42no—
172C2×#88322—22no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—42no—
176D424—82no—
177D424—82no—
178C23×C53—21no—
179C23×C53#12111yescharacteristic
180C22×D53—41no—
181C22×D53—42no—
182C22×D53—82no—
183C22×D53—82no—
184C22×D53—82no—
185Q8×C53—21no—
186Q8×C53C2×D812yes—
187D4×C53—24no—
188D4×C53—21no—
189D4×C53C2×D812yes—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—24no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—81no—
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53C2×D812yes—
204C2×C4×C53#13512yes—
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C2×#4111yescharacteristic
208C22⋊D53—82no—
209C2×Dic53—41no—
210D212—42no—
211D212—41no—
212D212—42no—
213D212—41no—
214D212—42no—
215D212—41no—
216D212—42no—
217D212—41no—
218D212—81no—
219D212—82no—
220D212—82no—
221D212—82no—
222D212—82no—
223C4×D53—82no—
224Dic106—41no—
225Dic106—41no—
226C8×C53—22no—
227C8×C53—22no—
228C8×C53—42no—
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#46612yes—
239C22×C53C2×#12111yescharacteristic
240C2×D53—82no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—82no—
244C2×D53—82no—
245C2×D53—81no—
246C2×D53—164no—
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#3038—531no—
261#3035—531no—
262#2988D5311yescharacteristic, complemented
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#304511yescharacteristic
273C2×C53C2×#46611yescharacteristic
274C2×C53#306811yescharacteristic
275D53—161no—
276D53—162no—
277C2×AGL(1,ℤ/8)—1061no—
278C2×#135—1061no—
279#532—531no—
280C2×#127—531no—
281C4×#41C2×D5311yescharacteristic
282#493—532no—
283#486—532no—
284#464—1061no—
285#463—1061no—
286#462—531no—
287#461—531no—
288#460—532no—
289#460—531no—
290#460—1061no—
291#459—532no—
292#459—531no—
293#459—1061no—
294#459—1061no—
295#459—1061no—
296#447—1062no—
297C8×Q8C2×D5312yescomplemented
298C8×D4—22no—
299C8×D4C2×D5312yescomplemented
300#431—21no—
301#431C2×D5311yescharacteristic
302#423—531no—
303C2×C4⋊C8—21no—
304C2×C4⋊C8—531no—
305C2×C4⋊C8C2×D5311yescharacteristic
306#418—1061no—
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×#41D21211yescharacteristic
317C22×D4—1061no—
318C22×D4—1061no—
319C22×D4—1061no—
320AGL(1,ℤ/8)—2124no—
321#135—2124no—
322C2×Q16—1061no—
323C2×SD16—1061no—
324C2×SD16—1062no—
325C2×SD16—1061no—
326C2×D8—1061no—
327C2×D8—1062no—
328C2×D8—1062no—
329C2×D8—1062no—
330C2×M16—1061no—
331C22×C8—21no—
332C22×C8—22no—
333C22×C8—1061no—
334C4⋊Q8—531no—
335#127—532no—
336#127—531no—
337#127—1061no—
338#127—1061no—
339#127—1061no—
340#124—1061no—
341#121—1061no—
342#121—1061no—
343C4×Q8D21211yescharacteristic
344C4×Q8C22×D5311yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—1061no—
348C4×D4—1061no—
349C4×D4D21211yescharacteristic
350C4×D4C22×D5311yescharacteristic
351#117—21no—
352#117D21211yescharacteristic
353C2×C42—21no—
354C2×C42C22×D5311yescharacteristic
355C4⋊C8—22no—
356C4⋊C8—21no—
357C4⋊C8—531no—
358C4⋊C8—531no—
359C4⋊C8—532no—
360C4⋊C8—1061no—
361C4⋊C8—1061no—
362C4⋊C8C22⋊D5311yescharacteristic
363C4⋊C8C22⋊D5311yescharacteristic
364C4⋊C8C22×D5312yes—
365#103—1061no—
366#103—1061no—
367#102—1062no—
368#102—1062no—
369#102—1061no—
370#102—1061no—
371#102—1062no—
372#102—1062no—
373C22⋊C8—22no—
374C22⋊C8C22⋊D5312yes—
375C4×C8—22no—
376C4×C8C22×D5312yes—
377C24—2121no—
378#41—22no—
379#41—2122no—
380#41—2122no—
381#41D42414yescomplemented
382C2×Q8—1061no—
383C2×Q8C2×D21211yescharacteristic
384C2×D4—21no—
385C2×D4—1062no—
386C2×D4—1061no—
387C2×D4—1061no—
388C2×D4—1062no—
389C2×D4—1061no—
390C2×D4—1062no—
391C2×D4—4241no—
392C2×D4—4241no—
393C2×D4—4242no—
394C2×D4—2121no—
395C2×D4—2121no—
396C2×D4—2121no—
397C2×D4—2121no—
398C2×D4—2122no—
399C2×D4—2122no—
400C2×D4C2×D21211yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—2121no—
405C22×C4C2×D21211yescharacteristic
406C22×C4#1171994511yescharacteristic
407C22×C4C2×C22⋊D5311yescharacteristic
408Q16—2122no—
409SD16—2122no—
410SD16—2124no—
411SD16—2122no—
412D8—4242no—
413D8—4242no—
414D8—2122no—
415D8—2124no—
416M16—2122no—
417C2×C8—22no—
418C2×C8—82no—
419C2×C8—81no—
420C2×C8—1062no—
421C2×C8—1061no—
422C2×C8—1061no—
423C2×C8—2122no—
424C2×C8C2×C22⋊D5312yes—
425C2×C8C2×C22⋊D5312yes—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1061no—
429C4⋊C4—1061no—
430C4⋊C4C2×D21211yescharacteristic
431C4⋊C4C2×D21211yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2121no—
436C22⋊C4—2121no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42C2×D21211yescharacteristic
441C42#1171994511yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C42C23×D5311yescharacteristic
444C23—21no—
445C23—4242no—
446C23—4242no—
447C23—4242no—
448C23—2121no—
449C23—2122no—
450C23#1313319111yescharacteristic
451Q8—21no—
452Q8—2121no—
453Q8—2121no—
454Q8C2×D42412yes—
455D4—24no—
456D4—21no—
457D4—4241no—
458D4—4242no—
459D4—4242no—
460D4—4242no—
461D4—4242no—
462D4—4242no—
463D4—2122no—
464D4—2121no—
465D4—2122no—
466D4—2121no—
467D4—2121no—
468D4—2122no—
469D4—2121no—
470D4—2122no—
471D4C2×D42412yes—
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#1313319111yescharacteristic
485C2×C4#1313319111yescharacteristic
486C2×C4#1313319111yescharacteristic
487C2×C4C2×D42412yes—
488C2×C4#1313323912yes—
489C2×C4C22×C22⋊D5311yescharacteristic
490C2×C4C22×D21211yescharacteristic
491C2×C4C2×#1171994511yescharacteristic
492C8—22no—
493C8—22no—
494C8—42no—
495C8—2121no—
496C8—2121no—
497C8—2122no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—8484no—
504C22—4242no—
505C22—4241no—
506C22—4242no—
507C22—4241no—
508C22—4241no—
509C22—4242no—
510C22#1681085912yes—
511C22C2×#1313319111yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—4241no—
519C4#1681085912yes—
520C4C22×D42411yescharacteristic
521C4C2×#1313323911yescharacteristic
522C4#1681142611yescharacteristic
523C4#1681143111yescharacteristic
524C2—22no—
525C2—42no—
526C2—8482no—
527C2—8481no—
528C2#2028425411yescharacteristic
529C2C2×#1681085911yescharacteristic
530C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2988C2
socleC22×C53C2×#121
hypercenter#2988D53
hypocenterC53#19410
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2988

Complements

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

NH ≅ G/Ntype
1. #20285635C2semidirect
2. #20284195C2semidirect
3. #20284170C2semidirect
4. #20274159C2semidirect
5. #20274152C2semidirect
6. #20274075C2semidirect
7. #20274070C2semidirect
8. #20273490C2semidirect
9. #20273390C2semidirect
10. C2×#16810366C2semidirect
11. C53×#2988C2semidirect
20. #16810833C22semidirect
21. #16810828C22semidirect
24. #16810823C22semidirect
29. #16810818C22semidirect
35. #16810376C22semidirect
36. #16810376C22semidirect
37. #16810368C22semidirect
40. #16810366C22semidirect
42. C8×Q8×C53C22semidirect
44. C8×D4×C53C22semidirect
113. #41×C53D8semidirect
262. #2988D53semidirect
297. C8×Q8C2×D53semidirect
299. C8×D4C2×D53semidirect
314. C53#19410semidirect
381. #41D424semidirect

Maximal subgroups

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

subgroupindex
#202856352
#202841952
#202841702
#202741592
#202741522
#202740752
#202740702
#202734902
#202733902
C2×#168103662
C53×#29882
#1941053