Finite Group Databasebeta

← #27355353

Subgroups — #27355353

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284795C211yesmaximal, characteristic
2#20283871C211yesmaximal, characteristic
3#20283804C211yesmaximal, characteristic, complemented
4#20273621C211yesmaximal, characteristic, complemented
5#20273620C211yesmaximal, characteristic, complemented
6#20273568C211yesmaximal, characteristic, complemented
7#20273562C211yesmaximal, characteristic, complemented
8#20273555C211yesmaximal, characteristic, complemented
9#20273554C211yesmaximal, characteristic, complemented
10#20273531C211yesmaximal, characteristic, complemented
11#20273530C211yesmaximal, characteristic, complemented
12#20273498C211yesmaximal, characteristic, complemented
13#20273176C211yesmaximal, characteristic
14#20273007C211yesmaximal, characteristic, complemented
15C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17#16810953C411yescharacteristic
18#16810936C411yescharacteristic
19#16810930C411yescharacteristic
20#16810930C411yescharacteristic
21#16810845C411yescharacteristic
22#16810845C411yescharacteristic
23#16810779C411yescharacteristic
24#16810762C411yescharacteristic
25#16810754—21no—
26#16810754—21no—
27#16810754C2211yescharacteristic
28#16810754C2211yescharacteristic
29#16810750C2211yescharacteristic
30Q8×#88322C2211yescharacteristic
31Q8×#88322C2211yescharacteristic
32#16810747C2211yescharacteristic
33C4×#11719925C2211yescharacteristic
34C4×#11719925C2211yescharacteristic
35C4×#11719924C2211yescharacteristic
36C4×#11719924C2211yescharacteristic
37#16810700C2211yescharacteristic
38#16810694—21no—
39#16810688C2211yescharacteristic
40C8×C22⋊D53—21no—
41#16810683—21no—
42#16810680—21no—
43#16810680C2211yescharacteristic
44#16810676—21no—
45#16810676C2211yescharacteristic
46#16810676C2211yescharacteristic
47#16810648—21no—
48#16810375C2211yescharacteristic
49#16810375C2211yescharacteristic
50#16810373C2211yescharacteristic
51#16810373C2211yescharacteristic
52#16810355C2211yescharacteristic
53#16810355C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810345C2211yescharacteristic
57#16810345C2211yescharacteristic
58#16810345C2211yescharacteristic
59#16810345C2211yescharacteristic
60C2×C4×Q8×C53C2211yescharacteristic
61C2×C4⋊C8×C53C2211yescharacteristic
62C2×C4×C8×C53C2211yescharacteristic
63C53×Q8⋊C8C2211yescharacteristic
64C53×Q8⋊C8C2211yescharacteristic
65C53×Q8⋊C8C2211yescharacteristic
66C53×Q8⋊C8C2211yescharacteristic
67C2×#11719925C2×C411yescharacteristic
68C2×#11719925C2×C411yescharacteristic
69C2×#11719924C2×C411yescharacteristic
70C2×#11719924C2×C411yescharacteristic
71C2×#11719912—21no—
72C2×C8×D53—21no—
73#13133211—21no—
74#13133209C2×C411yescharacteristic
75#13133205C2×C411yescharacteristic
76#13133200—21no—
77#13133196C2×C411yescharacteristic
78#13133191C2×C411yescharacteristic
79C4×Dic106C2311yescharacteristic
80C4×D212C2311yescharacteristic
81C4×C22⋊D53—21no—
82#13133174—21no—
83#13133150C2×C411yescharacteristic
84#13133150C2×C411yescharacteristic
85#13133148C2×C411yescharacteristic
86#13133148C2×C411yescharacteristic
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2×C411yescharacteristic
92C4⋊#88322C2×C411yescharacteristic
93C4⋊#88322D411yescharacteristic
94C4⋊#88322D411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133141C2×C411yescharacteristic
100#13133141C2×C411yescharacteristic
101#13133141C2×C411yescharacteristic
102#13133141C2×C411yescharacteristic
103#13133140C2×C411yescharacteristic
104#13133140C2×C411yescharacteristic
105#13133137C2×C411yescharacteristic
106#13133137C2×C411yescharacteristic
107C22⋊#88322—21no—
108C22⋊#88322C2×C411yescharacteristic
109C22⋊#88322C2×C411yescharacteristic
110C22⋊#88322D411yescharacteristic
111C22⋊#88322D411yescharacteristic
112#13133129—21no—
113#13133129—21no—
114#13133128—21no—
115#13133128—21no—
116#13133128—21no—
117#13133127—21no—
118C4×#88322—21no—
119C4×#88322—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C8×Dic53—21no—
125C22×Q8×C53C2×C411yescharacteristic
126C22×C8×C53—21no—
127C22×C8×C53—21no—
128C4×Q8×C53—21no—
129C4×Q8×C53—21no—
130C4×Q8×C53C2×C411yescharacteristic
131C4×Q8×C53C2×C411yescharacteristic
132C4×Q8×C53C2311yescharacteristic
133C4×Q8×C53C2311yescharacteristic
134C2×C4⋊C4×C53—21no—
135C2×C4⋊C4×C53C2×C411yescharacteristic
136C2×C42×C53—21no—
137C2×C42×C53C2311yescharacteristic
138C4⋊C8×C53—21no—
139C4⋊C8×C53C2311yescharacteristic
140C4⋊C8×C53C2311yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×C22⋊D53—21no—
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×Dic106C22×C411yescharacteristic
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—42no—
163C2×C8×C53—81no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22×C411yescharacteristic
170C4⋊C4×C53C22×C411yescharacteristic
171C42×C53—21no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53C22×C411yescharacteristic
176C42×C53C2×D411yescharacteristic
177C42×C53#4111yescharacteristic
178C42×C53C2411yescharacteristic
179C22⋊Dic53—21no—
180#11719925—22no—
181#11719925—41no—
182#11719924—22no—
183#11719924—41no—
184#11719921—21no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188C4⋊Dic53C22×C411yescharacteristic
189C4⋊Dic53C22×C411yescharacteristic
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194C4×Dic53—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322C22×C411yescharacteristic
198C2×#88322C22×C411yescharacteristic
199C2×#88322C22×C411yescharacteristic
200C2×#88322C22×C411yescharacteristic
201C2×#88322C2×D411yescharacteristic
202C2×#88322C2×D411yescharacteristic
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205#11719912—81no—
206C8×D53—81no—
207C23×C53C4×D411yescharacteristic
208C22×D53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—42no—
213Q8×C53#13112yes—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C22⋊D53—42no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235D212—21no—
236D212—21no—
237C4×D53—41no—
238C4×D53—41no—
239Dic106—21no—
240Dic106—21no—
241C8×C53—41no—
242C8×C53—41no—
243C8×C53—41no—
244C8×C53—41no—
245#88322—21no—
246#88322—21no—
247#88322—21no—
248#88322—21no—
249#88322—21no—
250#88322—21no—
251#88322—21no—
252#88322—21no—
253#88322—41no—
254#88322—41no—
255#18023—531nomaximal, Sylow
256C22×C53—21no—
257C22×C53—21no—
258C22×C53#44412yes—
259C22×C53C2×C4×D411yescharacteristic
260C2×D53—41no—
261C2×D53—41no—
262C2×D53—41no—
263C4×C53—21no—
264C4×C53—22no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53C2×#13111yescharacteristic
272C4×C53#56811yescharacteristic
273C4×C53#57711yescharacteristic
274C4×C53#57811yescharacteristic
275Dic53—41no—
276Dic53—41no—
277Dic53—41no—
278#3108—531no—
279#2994—531no—
280#2984—531no—
281#1593—531no—
282#1592—531no—
283#1584—531no—
284#1584—531no—
285#1583—531no—
286#1582—531no—
287#1579—531no—
288SD16⋊C8—531no—
289#1573—531no—
290#1494—531no—
291C2×Q8⋊C8D5311yescharacteristic, complemented
292#1452—531no—
293C2×C53—22no—
294C2×C53#293311yescharacteristic
295C2×C53#294711yescharacteristic
296C2×C53C2×#44411yescharacteristic
297D53—81no—
298#529—531no—
299C2×C4×Q8C2×D5311yescharacteristic
300#483—531no—
301#480—531no—
302#479—531no—
303#479—531no—
304#464—531no—
305#464—531no—
306#451—531no—
307#448—531no—
308#446—531no—
309#446—531no—
310#446—531no—
311#446—1061no—
312#446—1061no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×SD16—531no—
319C4×SD16—531no—
320#436—531no—
321#435—1061no—
322C8×D4—531no—
323C8×D4—1061no—
324#433—1061no—
325#432—531no—
326#432—1061no—
327#431—531no—
328#431—531no—
329#431—1061no—
330#424—1061no—
331C2×C4⋊C8C2×D5311yescharacteristic
332C2×C4×C8C2×D5311yescharacteristic
333#335—531no—
334#335—531no—
335#334—531no—
336#334—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8C2×D5311yescharacteristic
340Q8⋊C8C2×D5311yescharacteristic
341Q8⋊C8C2×D5311yescharacteristic
342Q8⋊C8C2×D5311yescharacteristic
343D4⋊C8—531no—
344D4⋊C8—531no—
345#322—531no—
346#322—531no—
347#322—531no—
348#322—531no—
349C53#1802311yescharacteristic, complemented, Sylow
350C22×Q8C4×D5311yescharacteristic
351C2×Q16—531no—
352C2×Q16—531no—
353C2×SD16—531no—
354C2×SD16—531no—
355C2×M16—1061no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—1061no—
359#126—1061no—
360#125—531no—
361#124—531no—
362#123—1061no—
363C22⋊Q8—531no—
364#121—531no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—531no—
368C4×Q8C4×D5311yescharacteristic
369C4×Q8C4×D5311yescharacteristic
370C4×Q8C22×D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372C4×D4—531no—
373C4×D4—1061no—
374#117—1061no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C4×D5311yescharacteristic
377C2×C42—21no—
378C2×C42C22×D5311yescharacteristic
379#107—531no—
380#107—531no—
381#106—531no—
382#106—531no—
383C4⋊C8—21no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#102—531no—
405#102—531no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416#97—1061no—
417C4×C8—21no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8C22×D5311yescharacteristic
426C4×C8C22×D5311yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8—531no—
431C2×Q8C2×C4×D5311yescharacteristic
432C2×Q8C2×C4×D5311yescharacteristic
433C2×D4—531no—
434C2×D4—1061no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1061no—
438C22×C4C2×C4×D5311yescharacteristic
439C22×C4D4×D5311yescharacteristic
440C22×C4#1171994711yescharacteristic
441Q16—1062no—
442Q16—2121no—
443SD16—1062no—
444SD16—2121no—
445M16—4241no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—42no—
451C2×C8—81no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—531no—
456C2×C8—531no—
457C2×C8—531no—
458C2×C8—531no—
459C2×C8—531no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—4241no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4C2×C4×D5311yescharacteristic
475C4⋊C4C2×C4×D5311yescharacteristic
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C22⋊C4—1061no—
480C22⋊C4—1061no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—1061no—
486C42C2×C4×D5311yescharacteristic
487C42D4×D5311yescharacteristic
488C42#1171994711yescharacteristic
489C42C23×D5311yescharacteristic
490C23—1061no—
491C23#1313317711yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—42no—
496Q8—1061no—
497Q8—1061no—
498Q8#1313322312yes—
499D4—1061no—
500D4—1061no—
501D4—2122no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4#1313317711yescharacteristic
519C2×C4#1313317711yescharacteristic
520C2×C4#1313317711yescharacteristic
521C2×C4C22×C4×D5311yescharacteristic
522C2×C4C2×D4×D5311yescharacteristic
523C2×C4C2×#1171994711yescharacteristic
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—1061no—
529C8—1061no—
530C8—1061no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C22—21no—
539C22—21no—
540C22—2121no—
541C22—2121no—
542C22—2121no—
543C22#1681074212yes—
544C22C2×#1313317711yescharacteristic
545C4—21no—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4C2×#1313322311yescharacteristic
557C4#1681138911yescharacteristic
558C4#1681142911yescharacteristic
559C4#1681143611yescharacteristic
560C2—22no—
561C2—4241no—
562C2#2028343611yescharacteristic
563C2#2028355211yescharacteristic
564C2C2×#1681074211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×Q8⋊C8C2
socleC22×C53C2×C4×D4
prime radical#13133191C2×C4
cosocleC42×C53C24
hypercenterC2×Q8⋊C8D53
hypocenterC53#18023
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20283804C2semidirect
4. #20273621C2semidirect
5. #20273620C2semidirect
6. #20273568C2semidirect
7. #20273562C2semidirect
8. #20273555C2semidirect
9. #20273554C2semidirect
10. #20273531C2semidirect
11. #20273530C2semidirect
12. #20273498C2semidirect
14. #20273007C2semidirect
15. C2×C53×Q8⋊C8C2semidirect
291. C2×Q8⋊C8D53semidirect
349. C53#18023semidirect

Maximal subgroups

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

subgroupindex
#202847952
#202838712
#202838042
#202736212
#202736202
#202735682
#202735622
#202735552
#202735542
#202735312
#202735302
#202734982
#202731762
#202730072
C2×C53×Q8⋊C82
#1802353