Finite Group Databasebeta

← #27368401

Subgroups — #27368401

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285433C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3#20284520C211yesmaximal, characteristic, complemented
4#20274297C211yesmaximal, characteristic, complemented
5#20274255C211yesmaximal, characteristic, complemented
6#20274243C211yesmaximal, characteristic, complemented
7#20274187C211yesmaximal, characteristic, complemented
8#20274173C211yesmaximal, characteristic, complemented
9#20274131C211yesmaximal, characteristic, complemented
10#20274119C211yesmaximal, characteristic, complemented
11#20274063C211yesmaximal, characteristic, complemented
12#20273482C211yesmaximal, characteristic, complemented
13#20273448C211yesmaximal, characteristic
14#20273443C211yesmaximal, characteristic, complemented
15Q8×M16×C53C211yesmaximal, characteristic, complemented
16#16811180C2211yescharacteristic
17#16811173C2211yescharacteristic
18#16811028C2211yescharacteristic
19#16810996C2211yescharacteristic
20#16810974C2211yescharacteristic
21#16810970C2211yescharacteristic
22#16810936—21no—
23#16810930—21no—
24#16810930—21no—
25#16810918C2211yescharacteristic
26#16810918C2211yescharacteristic
27#16810918C2211yescharacteristic
28#16810918C2211yescharacteristic
29#16810908C2211yescharacteristic
30#16810908C2211yescharacteristic
31#16810869—21no—
32#16810869—21no—
33#16810823C2211yescharacteristic
34#16810823C2211yescharacteristic
35#16810762—21no—
36#16810725—21no—
37#16810725—21no—
38#16810721—21no—
39#16810721—21no—
40#16810680C2211yescharacteristic
41#16810663—21no—
42#16810647C2211yescharacteristic
43#16810632—21no—
44#16810376C2211yescharacteristic
45#16810376C2211yescharacteristic
46#16810374C2211yescharacteristic
47#16810374C2211yescharacteristic
48#16810371C2211yescharacteristic
49#16810371C2211yescharacteristic
50#16810371C2211yescharacteristic
51#16810371C2211yescharacteristic
52#16810370C2211yescharacteristic
53#16810370C2211yescharacteristic
54#16810366C2211yescharacteristic
55#16810366C2211yescharacteristic
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×#11719945—21no—
68C2×#11719913—21no—
69C2×#11719913—21no—
70C2×#11719924—21no—
71C2×#11719924—21no—
72C2×#11719917—21no—
73#13133217C2311yescharacteristic
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133214C2311yescharacteristic
77#13133209—21no—
78#13133205—21no—
79#13133196—21no—
80#13133196—21no—
81#13133191—21no—
82#13133191—21no—
83C4×Dic106—21no—
84C4×Dic106—21no—
85C4×D212—21no—
86C4×D212—21no—
87#13133176—21no—
88#13133151—21no—
89#13133151—21no—
90#13133150—21no—
91#13133150—21no—
92#13133150—21no—
93#13133150—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#13133141—21no—
104#13133141—21no—
105#13133141—21no—
106#13133141—21no—
107#13133141—21no—
108#13133141—21no—
109#13133139—21no—
110#13133139—21no—
111#13133137—21no—
112#13133137—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×C22⋊D53—41no—
155#11719945—82no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×C4×D53—41no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163M16×C53—22no—
164M16×C53—41no—
165M16×C53—42no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C2×C8×C53C2×D411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53C2×D411yescharacteristic
178C4⋊C4×C53C2×D411yescharacteristic
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C42×C53#4111yescharacteristic
186C42×C53C2411yescharacteristic
187C22⋊Dic53—41no—
188#11719924—42no—
189#11719924—42no—
190#11719921—41no—
191#11719921—41no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198#11719919—41no—
199#11719919—41no—
200C4×Dic53—41no—
201#11719917—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206#11719913—42no—
207#11719913—42no—
208C23×C53#12111yescharacteristic
209C22×D53—41no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—42no—
214Q8×C53#13712yes—
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—21no—
222C2×C4×C53—21no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—81no—
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C22⋊D53—82no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236D212—41no—
237D212—41no—
238D212—41no—
239D212—41no—
240D212—81no—
241C4×D53—82no—
242Dic106—41no—
243Dic106—41no—
244Dic106—81no—
245Dic106—81no—
246C8×C53—21no—
247C8×C53—21no—
248C8×C53—21no—
249C8×C53—41no—
250C8×C53—41no—
251C8×C53#13712yes—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#19309—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53C2×#12111yescharacteristic
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—22no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53C2×#13711yescharacteristic
273C4×C53C2×#13711yescharacteristic
274C4×C53#57511yescharacteristic
275C4×C53#57511yescharacteristic
276Dic53—81no—
277Dic53—81no—
278Dic53—81no—
279#3182—531no—
280#3112—531no—
281#3078—531no—
282Q8×M16D5311yescharacteristic, complemented
283#1704—531no—
284#1698—531no—
285#1696—531no—
286#1686—531no—
287#1684—531no—
288#1678—531no—
289#1676—531no—
290#1666—531no—
291#1569—531no—
292#1560—531no—
293#1559—531no—
294C2×C53—41no—
295C2×C53#303711yescharacteristic
296C2×C53#305611yescharacteristic
297C2×C53#306811yescharacteristic
298D53—161no—
299#533—531no—
300#532—531no—
301C2×C4×Q8C2×D5311yescharacteristic
302#500—531no—
303#492—531no—
304#487—531no—
305#486—531no—
306#480—1061no—
307#479—1061no—
308#479—1061no—
309#477—531no—
310#477—531no—
311#477—531no—
312#477—531no—
313#475—531no—
314#475—531no—
315#468—1061no—
316#468—1061no—
317#460—531no—
318#460—531no—
319#448—1061no—
320#446—21no—
321#446—21no—
322#446C2×D5311yescharacteristic
323#446C2×D5311yescharacteristic
324C8×Q8C2×D5311yescharacteristic
325C8×Q8C2×D5311yescharacteristic
326#440—1061no—
327#440—1061no—
328#440—1061no—
329#440—1061no—
330#432—531no—
331#428—1061no—
332#423—21no—
333#423—531no—
334#423C2×D5311yescharacteristic
335#419—1061no—
336C4×M16—21no—
337C4×M16C2×D5311yescharacteristic
338#335—531no—
339#335—531no—
340#334—531no—
341#334—531no—
342#333—531no—
343#333—531no—
344#333—531no—
345#333—531no—
346#332—531no—
347#332—531no—
348#331—531no—
349#331—531no—
350C53#1930911yescharacteristic, complemented, Sylow
351C2×#41—1061no—
352C22×Q8D21211yescharacteristic
353C2×SD16—1061no—
354C2×SD16—1061no—
355C2×SD16—1061no—
356C2×SD16—1061no—
357C2×M16—21no—
358C2×M16—1061no—
359C2×M16C22⋊D5311yescharacteristic
360C2×M16C22⋊D5311yescharacteristic
361C4⋊Q8—531no—
362C4⋊Q8—531no—
363C4⋊Q8—531no—
364#127—531no—
365#125—1061no—
366#124—1061no—
367C22⋊Q8—1061no—
368C22⋊Q8—1061no—
369#121—1061no—
370#121—1061no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8—1061no—
374C4×Q8—1061no—
375C4×Q8D21211yescharacteristic
376C4×Q8D21211yescharacteristic
377C4×Q8C22×D5311yescharacteristic
378C4×Q8C22×D5311yescharacteristic
379C4×D4—1061no—
380C4×D4—1061no—
381#117—1061no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4D21211yescharacteristic
384C2×C42—21no—
385C2×C42C22×D5311yescharacteristic
386#107—1061no—
387#107—1061no—
388#107—1061no—
389#107—1061no—
390#107—1061no—
391#107—1061no—
392#106—1061no—
393#106—1061no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—1061no—
404C4⋊C8C22×D5311yescharacteristic
405C4⋊C8C22×D5311yescharacteristic
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410#103—1061no—
411#103—1061no—
412#103—1061no—
413#103—1061no—
414#102—1061no—
415#102—1061no—
416#102—1061no—
417#102—1061no—
418C22⋊C8—1061no—
419#97—21no—
420#97—21no—
421#97—1061no—
422#97C22⋊D5311yescharacteristic
423#97C22⋊D5311yescharacteristic
424C4×C8—21no—
425C4×C8—21no—
426C4×C8C22×D5311yescharacteristic
427C4×C8C22×D5311yescharacteristic
428#41—4242no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—41no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8C2×D21211yescharacteristic
436C2×Q8C2×D21211yescharacteristic
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—2121no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—2121no—
444C22×C4C2×D21211yescharacteristic
445C22×C4#1171994511yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447SD16—2122no—
448SD16—2122no—
449SD16—2122no—
450SD16—2122no—
451M16—22no—
452M16—41no—
453M16—42no—
454M16—2122no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8C2×C22⋊D5311yescharacteristic
462C2×C8C2×C22⋊D5311yescharacteristic
463C2×C8C2×C22⋊D5311yescharacteristic
464C2×C8C2×C22⋊D5311yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—2121no—
477C4⋊C4—2121no—
478C4⋊C4C2×D21211yescharacteristic
479C4⋊C4C2×D21211yescharacteristic
480C22⋊C4—2121no—
481C22⋊C4—2121no—
482C22⋊C4—2121no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—2121no—
488C42C2×D21211yescharacteristic
489C42#1171994511yescharacteristic
490C42C2×C22⋊D5311yescharacteristic
491C42C23×D5311yescharacteristic
492C23—2121no—
493C23#1313319111yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—4241no—
499Q8—4241no—
500Q8—2121no—
501Q8—2121no—
502Q8#1313325012yes—
503D4—4241no—
504D4—4242no—
505D4—2121no—
506D4—2121no—
507D4—2121no—
508D4—2121no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—4242no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4#1313319111yescharacteristic
525C2×C4#1313319111yescharacteristic
526C2×C4#1313319111yescharacteristic
527C2×C4C22×C22⋊D5311yescharacteristic
528C2×C4C22×D21211yescharacteristic
529C2×C4C2×#1171994511yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8#1313324812yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—4241no—
544C22—4241no—
545C22—4241no—
546C22C2×#1313319111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—4241no—
556C4—4241no—
557C4—4241no—
558C4C2×#1313324811yescharacteristic
559C4C2×#1313325011yescharacteristic
560C4#1681142111yescharacteristic
561C4#1681142311yescharacteristic
562C2—41no—
563C2—8481no—
564C2#2028418711yescharacteristic
565C2#2028434511yescharacteristic
566C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingQ8×M16×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterQ8×M16D53
hypocenterC53#19309
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
3. #20284520C2semidirect
4. #20274297C2semidirect
5. #20274255C2semidirect
6. #20274243C2semidirect
7. #20274187C2semidirect
8. #20274173C2semidirect
9. #20274131C2semidirect
10. #20274119C2semidirect
11. #20274063C2semidirect
12. #20273482C2semidirect
14. #20273443C2semidirect
15. Q8×M16×C53C2semidirect
282. Q8×M16D53semidirect
350. C53#19309semidirect

Maximal subgroups

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

subgroupindex
#202854332
#202848372
#202845202
#202742972
#202742552
#202742432
#202741872
#202741732
#202741312
#202741192
#202740632
#202734822
#202734482
#202734432
Q8×M16×C532
#1930953