Finite Group Databasebeta

← #27368431

Subgroups — #27368431

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285460C211yesmaximal, characteristic
2#20284845C211yesmaximal, characteristic
3#20284520C211yesmaximal, characteristic
4#20274297C211yesmaximal, characteristic, complemented
5#20274269C211yesmaximal, characteristic, complemented
6#20274221C211yesmaximal, characteristic, complemented
7#20274209C211yesmaximal, characteristic, complemented
8#20274173C211yesmaximal, characteristic, complemented
9#20274145C211yesmaximal, characteristic, complemented
10#20274097C211yesmaximal, characteristic, complemented
11#20274085C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic
13#20273452C211yesmaximal, characteristic
14#20273443C211yesmaximal, characteristic
15Q8×M16×C53C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811180C2211yescharacteristic
18#16811021C2211yescharacteristic
19#16811003C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810974C2211yescharacteristic
22#16810953—21no—
23#16810930—21no—
24#16810930—21no—
25#16810918C2211yescharacteristic
26#16810918C2211yescharacteristic
27#16810908C2211yescharacteristic
28#16810908C2211yescharacteristic
29#16810908C2211yescharacteristic
30#16810908C2211yescharacteristic
31#16810881—21no—
32#16810881—21no—
33#16810833C2211yescharacteristic
34#16810833C2211yescharacteristic
35#16810779—21no—
36#16810731—21no—
37#16810731—21no—
38#16810728—21no—
39#16810728—21no—
40#16810680C2211yescharacteristic
41#16810663—21no—
42#16810647C2211yescharacteristic
43#16810632—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C2×C4×Q8×C53C2211yescharacteristic
57C53×#446—21no—
58C53×#446—21no—
59C53×#446C2211yescharacteristic
60C53×#446C2211yescharacteristic
61C8×Q8×C53C2211yescharacteristic
62C8×Q8×C53C2211yescharacteristic
63C53×#423—21no—
64C53×#423C2211yescharacteristic
65C4×M16×C53—21no—
66C4×M16×C53C2211yescharacteristic
67C2×Dic212—21no—
68C2×Dic212—21no—
69C2×#11719925—21no—
70C2×#11719925—21no—
71C2×#11719917—21no—
72#13133217C2311yescharacteristic
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133209—21no—
77#13133209—21no—
78#13133196—21no—
79#13133196—21no—
80#13133196—21no—
81#13133196—21no—
82C4×Dic106—21no—
83C4×Dic106—21no—
84C4×Dic106—21no—
85C4×Dic106—21no—
86#13133176—21no—
87#13133176—21no—
88#13133150—21no—
89#13133150—21no—
90#13133149—21no—
91#13133149—21no—
92#13133148—21no—
93#13133148—21no—
94#13133148—21no—
95#13133148—21no—
96C4⋊#88322—21no—
97C4⋊#88322C2311yescharacteristic
98C4⋊#88322C2311yescharacteristic
99C4⋊#88322C2311yescharacteristic
100C4⋊#88322C2311yescharacteristic
101#13133142—21no—
102#13133142—21no—
103#13133142—21no—
104#13133142—21no—
105#13133141—21no—
106#13133141—21no—
107#13133141—21no—
108#13133141—21no—
109#13133141—21no—
110#13133141—21no—
111#13133140—21no—
112#13133140—21no—
113C22⋊#88322—21no—
114#13133128—21no—
115C22×Q8×C53D411yescharacteristic
116C2×M16×C53—21no—
117C2×M16×C53D411yescharacteristic
118C2×M16×C53D411yescharacteristic
119C4×Q8×C53—21no—
120C4×Q8×C53—21no—
121C4×Q8×C53D411yescharacteristic
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53C2311yescharacteristic
124C4×Q8×C53C2311yescharacteristic
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53D411yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53—21no—
130C4⋊C8×C53—21no—
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C4⋊C8×C53—21no—
134C4⋊C8×C53C2311yescharacteristic
135C4⋊C8×C53C2311yescharacteristic
136#97×C53—21no—
137#97×C53—21no—
138#97×C53D411yescharacteristic
139#97×C53D411yescharacteristic
140C4×C8×C53—21no—
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C2×D411yescharacteristic
148C2×Q8×C53C2×D411yescharacteristic
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53#4111yescharacteristic
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158M16×C53—22no—
159M16×C53—41no—
160M16×C53—42no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C2×D411yescharacteristic
164C2×C8×C53C2×D411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53C2×D411yescharacteristic
173C4⋊C4×C53C2×D411yescharacteristic
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53#4111yescharacteristic
181C42×C53C2411yescharacteristic
182C22⋊Dic53—41no—
183C22⋊Dic53—41no—
184#11719925—42no—
185#11719925—42no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194#11719919—41no—
195#11719919—41no—
196#11719919—41no—
197#11719919—41no—
198C4×Dic53—41no—
199C4×Dic53—41no—
200#11719917—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205Dic212—42no—
206Dic212—42no—
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—42no—
212Q8×C53#13712yes—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—81no—
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234Dic106—41no—
235Dic106—41no—
236Dic106—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—41no—
240Dic106—81no—
241C8×C53—21no—
242C8×C53—21no—
243C8×C53—21no—
244C8×C53—41no—
245C8×C53—41no—
246C8×C53#13712yes—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#19311—531nomaximal, Sylow
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53C2×#12111yescharacteristic
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—22no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53C2×#13711yescharacteristic
265C4×C53C2×#13711yescharacteristic
266C4×C53#57511yescharacteristic
267C4×C53#57511yescharacteristic
268Dic53—81no—
269Dic53—81no—
270Dic53—81no—
271Dic53—81no—
272#3185—531no—
273#3113—531no—
274#3078—531no—
275Q8×M16D5311yescharacteristic
276#1704—531no—
277#1700—531no—
278#1692—531no—
279#1690—531no—
280#1684—531no—
281#1680—531no—
282#1672—531no—
283#1670—531no—
284#1569—531no—
285#1561—531no—
286#1559—531no—
287C2×C53—41no—
288C2×C53#303711yescharacteristic
289C2×C53#305611yescharacteristic
290C2×C53#306911yescharacteristic
291#533—531no—
292#533—531no—
293C2×C4×Q8C2×D5311yescharacteristic
294#498—531no—
295#494—531no—
296#487—531no—
297#487—531no—
298#483—1061no—
299#479—1061no—
300#479—1061no—
301#477—531no—
302#477—531no—
303#475—531no—
304#475—531no—
305#475—531no—
306#475—531no—
307#470—1061no—
308#470—1061no—
309#462—531no—
310#462—531no—
311#451—1061no—
312#446—21no—
313#446—21no—
314#446C2×D5311yescharacteristic
315#446C2×D5311yescharacteristic
316C8×Q8C2×D5311yescharacteristic
317C8×Q8C2×D5311yescharacteristic
318#441—1061no—
319#441—1061no—
320#441—1061no—
321#441—1061no—
322#432—531no—
323#428—1061no—
324#423—21no—
325#423—531no—
326#423C2×D5311yescharacteristic
327#419—1061no—
328C4×M16—21no—
329C4×M16C2×D5311yescharacteristic
330#335—531no—
331#335—531no—
332#334—531no—
333#334—531no—
334#333—531no—
335#333—531no—
336#332—531no—
337#332—531no—
338#332—531no—
339#332—531no—
340#332—531no—
341#332—531no—
342C53#1931111yescharacteristic, complemented, Sylow
343C22×Q8D21211yescharacteristic
344C2×Q16—1061no—
345C2×Q16—1061no—
346C2×Q16—1061no—
347C2×Q16—1061no—
348C2×M16—21no—
349C2×M16—1061no—
350C2×M16C22⋊D5311yescharacteristic
351C2×M16C22⋊D5311yescharacteristic
352C4⋊Q8—531no—
353C4⋊Q8—531no—
354C4⋊Q8—531no—
355C4⋊Q8—531no—
356#125—1061no—
357#125—1061no—
358C22⋊Q8—1061no—
359C22⋊Q8—1061no—
360C22⋊Q8—1061no—
361C22⋊Q8—1061no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—1061no—
365C4×Q8—1061no—
366C4×Q8—1061no—
367C4×Q8—1061no—
368C4×Q8D21211yescharacteristic
369C4×Q8D21211yescharacteristic
370C4×Q8C22×D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372#117—1061no—
373#117—1061no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D21211yescharacteristic
376C2×C42—21no—
377C2×C42C22×D5311yescharacteristic
378#107—1061no—
379#107—1061no—
380#106—1061no—
381#106—1061no—
382#106—1061no—
383#106—1061no—
384#106—1061no—
385#106—1061no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—1061no—
399#103—1061no—
400#103—1061no—
401#103—1061no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410C22⋊C8—1061no—
411#97—21no—
412#97—21no—
413#97—1061no—
414#97C22⋊D5311yescharacteristic
415#97C22⋊D5311yescharacteristic
416C4×C8—21no—
417C4×C8—21no—
418C4×C8C22×D5311yescharacteristic
419C4×C8C22×D5311yescharacteristic
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8C2×D21211yescharacteristic
428C2×Q8C2×D21211yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×D21211yescharacteristic
432C22×C4#1171994511yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434Q16—2122no—
435Q16—2122no—
436Q16—2122no—
437Q16—2122no—
438M16—22no—
439M16—41no—
440M16—42no—
441M16—2122no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
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—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—2121no—
466C4⋊C4—2121no—
467C4⋊C4—2121no—
468C4⋊C4—2121no—
469C4⋊C4C2×D21211yescharacteristic
470C4⋊C4C2×D21211yescharacteristic
471C22⋊C4—2121no—
472C22⋊C4—2121no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—2121no—
478C42—2121no—
479C42C2×D21211yescharacteristic
480C42#1171994511yescharacteristic
481C42C2×C22⋊D5311yescharacteristic
482C42C23×D5311yescharacteristic
483C23#1313319111yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—42no—
488Q8—4241no—
489Q8—2121no—
490Q8—2121no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494Q8—2121no—
495Q8#1313325012yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—81no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4#1313319111yescharacteristic
512C2×C4#1313319111yescharacteristic
513C2×C4#1313319111yescharacteristic
514C2×C4C22×C22⋊D5311yescharacteristic
515C2×C4C22×D21211yescharacteristic
516C2×C4C2×#1171994511yescharacteristic
517C8—21no—
518C8—21no—
519C8—21no—
520C8—41no—
521C8—41no—
522C8—2121no—
523C8—2121no—
524C8—2121no—
525C8—2121no—
526C8#1313324812yes—
527C22—21no—
528C22—21no—
529C22—21no—
530C22C2×#1313319111yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—4241no—
540C4—4241no—
541C4—4241no—
542C4—4241no—
543C4C2×#1313324811yescharacteristic
544C4C2×#1313325011yescharacteristic
545C4#1681142111yescharacteristic
546C4#1681142311yescharacteristic
547C2—41no—
548C2#2028418711yescharacteristic
549C2#2028434511yescharacteristic
550C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radicalC23×C53#121
hypercenterQ8×M16D53
hypocenterC53#19311
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
4. #20274297C2semidirect
5. #20274269C2semidirect
6. #20274221C2semidirect
7. #20274209C2semidirect
8. #20274173C2semidirect
9. #20274145C2semidirect
10. #20274097C2semidirect
11. #20274085C2semidirect
342. C53#19311semidirect

Maximal subgroups

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

subgroupindex
#202854602
#202848452
#202845202
#202742972
#202742692
#202742212
#202742092
#202741732
#202741452
#202740972
#202740852
#202734822
#202734522
#202734432
Q8×M16×C532
#1931153