Finite Group Databasebeta

← #27369173

Subgroups — #27369173

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286082C211yesmaximal, characteristic
2#20285936C211yesmaximal, characteristic
3#20283719C211yesmaximal, characteristic
4#20274240C211yesmaximal, characteristic, complemented
5#20274235C211yesmaximal, characteristic, complemented
6#20274218C211yesmaximal, characteristic, complemented
7#20274211C211yesmaximal, characteristic, complemented
8#20274116C211yesmaximal, characteristic, complemented
9#20274111C211yesmaximal, characteristic, complemented
10#20274094C211yesmaximal, characteristic, complemented
11#20274087C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273425C211yesmaximal, characteristic, complemented
14#20273396C211yesmaximal, characteristic, complemented
15C53×#3165C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17#16811029—21no—
18#16810985—21no—
19#16810953—21no—
20#16810936—21no—
21#16810930—21no—
22#16810915C2211yescharacteristic
23#16810915C2211yescharacteristic
24#16810915C2211yescharacteristic
25#16810910C2211yescharacteristic
26#16810910C2211yescharacteristic
27#16810910C2211yescharacteristic
28#16810879—21no—
29#16810871—21no—
30#16810845—21no—
31#16810830C2211yescharacteristic
32#16810825C2211yescharacteristic
33#16810753C2211yescharacteristic
34C8×Dic106C2211yescharacteristic
35#16810702C2211yescharacteristic
36#16810699—21no—
37#16810692—21no—
38C8×D212C2211yescharacteristic
39#16810685—21no—
40#16810680C2211yescharacteristic
41#16810676C2211yescharacteristic
42#16810647C2211yescharacteristic
43#16810580—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810372C2211yescharacteristic
51#16810372C2211yescharacteristic
52#16810372C2211yescharacteristic
53#16810372C2211yescharacteristic
54#16810369C2211yescharacteristic
55#16810369C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C2×C4⋊Q8×C53C2211yescharacteristic
58C53×#501—21no—
59C53×#501—21no—
60C53×#500C2211yescharacteristic
61C53×#500C2211yescharacteristic
62C53×#498C2211yescharacteristic
63C53×#498C2211yescharacteristic
64C53×#428—21no—
65C53×#428—21no—
66C4×M16×C53C2211yescharacteristic
67C2×#11719925—21no—
68C2×#11719924—21no—
69C2×#11719917—21no—
70C2×#11719912—21no—
71C2×C8×D53—21no—
72#13133211—21no—
73#13133209D411yescharacteristic
74#13133205D411yescharacteristic
75#13133200—21no—
76#13133196D411yescharacteristic
77#13133191D411yescharacteristic
78C4×Dic106C2311yescharacteristic
79C4×D212C2311yescharacteristic
80C4×C22⋊D53—21no—
81#13133174—21no—
82#13133150—21no—
83#13133150—21no—
84#13133150—21no—
85#13133148—21no—
86#13133148—21no—
87#13133148—21no—
88C4⋊#88322—21no—
89C4⋊#88322C2311yescharacteristic
90C4⋊#88322C2311yescharacteristic
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133146—21no—
96#13133146—21no—
97#13133141—21no—
98#13133141—21no—
99#13133140—21no—
100#13133140—21no—
101#13133140—21no—
102#13133137—21no—
103#13133137—21no—
104#13133137—21no—
105C22⋊#88322—21no—
106C22⋊#88322—21no—
107#13133129—21no—
108#13133129—21no—
109#13133128—21no—
110#13133127—21no—
111C4×#88322—21no—
112C8×Dic53—21no—
113C22×Q8×C53—21no—
114C2×M16×C53D411yescharacteristic
115C2×M16×C53D411yescharacteristic
116C4⋊Q8×C53—21no—
117C4⋊Q8×C53—21no—
118C4⋊Q8×C53—21no—
119C4⋊Q8×C53C2311yescharacteristic
120C4⋊Q8×C53C2311yescharacteristic
121C4⋊Q8×C53C2311yescharacteristic
122C4⋊Q8×C53C2311yescharacteristic
123#125×C53—21no—
124C22⋊Q8×C53—21no—
125C22⋊Q8×C53—21no—
126C4×Q8×C53—21no—
127C4×Q8×C53—21no—
128#117×C53—21no—
129C2×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53—21no—
131C2×C42×C53C2311yescharacteristic
132#107×C53—21no—
133#107×C53—21no—
134#107×C53—21no—
135#107×C53—21no—
136#106×C53—21no—
137#106×C53—21no—
138#106×C53—21no—
139#106×C53—21no—
140#97×C53D411yescharacteristic
141#97×C53D411yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—41no—
149C2×Q8×C53—82no—
150C22×C4×C53—41no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×C22⋊D53—21no—
155C2×D212C2×D411yescharacteristic
156C2×C4×D53—21no—
157C2×Dic106C2×D411yescharacteristic
158M16×C53—22no—
159M16×C53—22no—
160C2×C8×C53C2×D411yescharacteristic
161C2×C8×C53C2×D411yescharacteristic
162C2×C8×C53C2×D411yescharacteristic
163C2×C8×C53C2×D411yescharacteristic
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176C22⋊C4×C53—41no—
177C42×C53—41no—
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53#4111yescharacteristic
181C42×C53C2411yescharacteristic
182C22⋊Dic53—21no—
183#11719925—81no—
184#11719924—81no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189C4⋊Dic53C2×D411yescharacteristic
190C4⋊Dic53C2×D411yescharacteristic
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195C4×Dic53—21no—
196#11719917—81no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203#11719912—42no—
204C8×D53—42no—
205C23×C53#12111yescharacteristic
206C22×D53—21no—
207Q8×C53—41no—
208Q8×C53—41no—
209Q8×C53—82no—
210Q8×C53—81no—
211Q8×C53—81no—
212Q8×C53—81no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—82no—
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C22⋊D53—81no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233D212—21no—
234D212—21no—
235C4×D53—41no—
236C4×D53—41no—
237Dic106—21no—
238Dic106—21no—
239C8×C53—21no—
240C8×C53—21no—
241C8×C53#13712yes—
242C8×C53#13712yes—
243#88322—41no—
244#88322—41no—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
249#19379—531nomaximal, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53C2×#12111yescharacteristic
254C2×D53—41no—
255C2×D53—41no—
256C2×D53—41no—
257C4×C53—21no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—81no—
262C4×C53—81no—
263C4×C53—81no—
264C4×C53—81no—
265C4×C53C2×#13711yescharacteristic
266C4×C53C2×#13711yescharacteristic
267C4×C53#57711yescharacteristic
268C4×C53#57811yescharacteristic
269Dic53—41no—
270Dic53—41no—
271Dic53—41no—
272#3252—531no—
273#3238—531no—
274#3165D5311yescharacteristic, complemented
275#2969—531no—
276#1696—531no—
277#1695—531no—
278#1692—531no—
279#1691—531no—
280#1676—531no—
281#1675—531no—
282#1672—531no—
283#1671—531no—
284#1569—531no—
285#1554—531no—
286#1548—531no—
287C2×C53—41no—
288C2×C53#303911yescharacteristic
289C2×C53#306311yescharacteristic
290C2×C53#306811yescharacteristic
291D53—81no—
292#533C2×D5311yescharacteristic
293C2×C4⋊Q8C2×D5311yescharacteristic
294#529—531no—
295#501—21no—
296#501—21no—
297#501—1061no—
298#500C2×D5311yescharacteristic
299#500C2×D5311yescharacteristic
300#498C2×D5311yescharacteristic
301#498C2×D5311yescharacteristic
302#489—1061no—
303#483—1061no—
304#480—1061no—
305#479—1061no—
306#477—531no—
307#477—531no—
308#477—531no—
309#476—531no—
310#476—531no—
311#476—531no—
312#470—1061no—
313#468—1061no—
314#464—1061no—
315#462—531no—
316#461—531no—
317#446—531no—
318C8×Q8—531no—
319#436—531no—
320#436—1061no—
321#435—1061no—
322C8×D4—531no—
323#433—1061no—
324#432—531no—
325#431—531no—
326#428—21no—
327#428—21no—
328#423—531no—
329#405—1061no—
330C4×M16C2×D5311yescharacteristic
331#335—531no—
332#335—531no—
333#334—531no—
334#334—531no—
335#333—531no—
336#333—531no—
337#333—531no—
338#333—531no—
339#333—531no—
340#333—531no—
341#332—531no—
342#332—531no—
343C53#1937911yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C2×Q16—1061no—
346C2×SD16—1061no—
347C2×M16—1061no—
348C2×M16—1061no—
349C2×M16C22⋊D5311yescharacteristic
350C2×M16C22⋊D5311yescharacteristic
351C22×C8—1061no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8C22×D5311yescharacteristic
356C4⋊Q8C22×D5311yescharacteristic
357C4⋊Q8C22×D5311yescharacteristic
358C4⋊Q8C22×D5311yescharacteristic
359#126—1061no—
360#125—21no—
361#125—531no—
362#124—531no—
363#123—1061no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—531no—
367#121—531no—
368C4×Q8—21no—
369C4×Q8—21no—
370C4×Q8—531no—
371C4×D4—531no—
372C4×D4—1061no—
373#117—21no—
374#117—1061no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C42C22×D5311yescharacteristic
378#107—21no—
379#107—21no—
380#107—21no—
381#107—21no—
382#107—1061no—
383#107—1061no—
384#107—1061no—
385#106—21no—
386#106—21no—
387#106—21no—
388#106—21no—
389#106—1061no—
390#106—1061no—
391#106—1061no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#102—1061no—
407#102—1061no—
408#102—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414#97—1061no—
415#97C22⋊D5311yescharacteristic
416#97C22⋊D5311yescharacteristic
417C4×C8—1061no—
418C4×C8—1061no—
419C4×C8C22×D5311yescharacteristic
420C4×C8C22×D5311yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—41no—
426C2×Q8—82no—
427C2×Q8—531no—
428C2×D4—531no—
429C2×D4—1061no—
430C22×C4—41no—
431C22×C4—1061no—
432C22×C4D4×D5311yescharacteristic
433C22×C4#1171994711yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435Q16—4241no—
436SD16—4241no—
437M16—22no—
438M16—22no—
439M16—4241no—
440M16—2122no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—2122no—
448C2×C8C2×C22⋊D5311yescharacteristic
449C2×C8C2×C22⋊D5311yescharacteristic
450C2×C8C2×C22⋊D5311yescharacteristic
451C2×C8C2×C22⋊D5311yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—531no—
465C4⋊C4—531no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C22⋊C4—41no—
471C22⋊C4—1061no—
472C22⋊C4—1061no—
473C22⋊C4—1061no—
474C22⋊C4—1061no—
475C22⋊C4—1061no—
476C42—41no—
477C42—1061no—
478C42D4×D5311yescharacteristic
479C42#1171994711yescharacteristic
480C42C2×C22⋊D5311yescharacteristic
481C42C23×D5311yescharacteristic
482C23—1061no—
483C23#1313318811yescharacteristic
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—81no—
488Q8—81no—
489Q8—82no—
490Q8—1061no—
491Q8—1061no—
492D4—1061no—
493D4—1061no—
494D4—4241no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4#1313318811yescharacteristic
511C2×C4#1313318811yescharacteristic
512C2×C4#1313318811yescharacteristic
513C2×C4C2×D4×D5311yescharacteristic
514C2×C4C22×C22⋊D5311yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C8—21no—
517C8—21no—
518C8—2121no—
519C8—2121no—
520C8—2121no—
521C8—2121no—
522C8—2121no—
523C8—2121no—
524C8#1313324812yes—
525C8#1313324812yes—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—2121no—
530C22—2121no—
531C22—2121no—
532C22C2×#1313318811yescharacteristic
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—2121no—
542C4—2121no—
543C4—2121no—
544C4C2×#1313324811yescharacteristic
545C4C2×#1313324811yescharacteristic
546C4#1681143211yescharacteristic
547C4#1681143911yescharacteristic
548C2—41no—
549C2—4241no—
550C2#2028419811yescharacteristic
551C2#2028440011yescharacteristic
552C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radical#13133191D4
hypercenter#3165D53
hypocenterC53#19379
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20274240C2semidirect
5. #20274235C2semidirect
6. #20274218C2semidirect
7. #20274211C2semidirect
8. #20274116C2semidirect
9. #20274111C2semidirect
10. #20274094C2semidirect
11. #20274087C2semidirect
12. #20273482C2semidirect
13. #20273425C2semidirect
14. #20273396C2semidirect
15. C53×#3165C2semidirect
274. #3165D53semidirect
343. C53#19379semidirect

Maximal subgroups

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

subgroupindex
#202860822
#202859362
#202837192
#202742402
#202742352
#202742182
#202742112
#202741162
#202741112
#202740942
#202740872
#202734822
#202734252
#202733962
C53×#31652
#1937953