Finite Group Databasebeta

← #28317777

Subgroups — #28317777

Order 13568.

588 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#20291547C211yesmaximal, characteristic
2#20291509C211yesmaximal, characteristic, complemented
3#20290358C211yesmaximal, characteristic, complemented
4#20290355C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290340C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290320C211yesmaximal, characteristic, complemented
9#20290308C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290267C211yesmaximal, characteristic, complemented
12#20290038C211yesmaximal, characteristic, complemented
13#20290031C211yesmaximal, characteristic
14#20289939C211yesmaximal, characteristic, complemented
15#20285318C211yesmaximal, characteristic
16#16811675—21no—
17#16811675—21no—
18#16811674C2211yescharacteristic
19#16811674C2211yescharacteristic
20#16811674C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658C2211yescharacteristic
25#16811656—21no—
26#16811652—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811642C2211yescharacteristic
30#16811630—21no—
31D53⋊(C4⋊C8)C2211yescharacteristic
32#16811588—21no—
33#16811587—21no—
34C4×#11719936C2211yescharacteristic
35C4×#11719936C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811178C2211yescharacteristic
53#16811152C2211yescharacteristic
54#16811032C411yescharacteristic
55#16810991C411yescharacteristic
56#16810982C411yescharacteristic
57#16810878C411yescharacteristic
58#16810878C411yescharacteristic
59#16810866C411yescharacteristic
60#16810866C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62Q16×Dic53C2211yescharacteristic
63SD16×Dic53C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65#16810575C2211yescharacteristic
66C53×#497C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719936D411yescharacteristic
69C2×#11719936D411yescharacteristic
70C2×#11719940—21no—
71C2×D53⋊C8—21no—
72C22×#88324—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272D411yescharacteristic
92#13133272D411yescharacteristic
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719912C2×C411yescharacteristic
102C2×#11719912C2×C411yescharacteristic
103#13133217C2×C411yescharacteristic
104#13133210—21no—
105#13133209C2×C411yescharacteristic
106#13133208—21no—
107#13133199—21no—
108#13133197—21no—
109#13133193C2×C411yescharacteristic
110#13133193C2×C411yescharacteristic
111#13133193C2×C411yescharacteristic
112#13133188C2×C411yescharacteristic
113Q8×Dic53C2311yescharacteristic
114Q8×Dic53C2311yescharacteristic
115Q8×Dic53C2311yescharacteristic
116#13133182—21no—
117D4×Dic53C2311yescharacteristic
118#13133177—21no—
119#13133175—21no—
120#13133173—21no—
121C42×D53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133151C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133149C2×C411yescharacteristic
126#13133141C2×C411yescharacteristic
127#13133141C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134#13133128C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×Q16×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140C2×SD16×C53C2×C411yescharacteristic
141C4⋊Q8×C53C2×C411yescharacteristic
142#124×C53C2×C411yescharacteristic
143#97×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×Q8×C53C22×C411yescharacteristic
147C2×D4×C53C22×C411yescharacteristic
148C2×C22⋊D53—21no—
149C22×Dic53—21no—
150C2×C4×D53C22×C411yescharacteristic
151C2×C4×D53C2×D411yescharacteristic
152C2×C4×D53#4111yescharacteristic
153C2×Dic106—21no—
154#11719940—42no—
155C2×#88324—21no—
156C2×#88324—21no—
157C2×#88324—21no—
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—42no—
162C2×#88324C2×D411yescharacteristic
163C2×#88324C2×D411yescharacteristic
164C2×#88324C2×D411yescharacteristic
165C2×#88324C2×D411yescharacteristic
166#11719936—22no—
167#11719936—41no—
168#11719936—42no—
169D53⋊C8—42no—
170Q16×C53—22no—
171Q16×C53—41no—
172SD16×C53—22no—
173SD16×C53—41no—
174C2×C8×C53C22×C411yescharacteristic
175C2×C8×C53C22×C411yescharacteristic
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C42×C53C22×C411yescharacteristic
181C22⋊Dic53—21no—
182C22⋊Dic53—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198#11719919—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53C2×D411yescharacteristic
203C4×Dic53#4111yescharacteristic
204C4×Dic53C2411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207#11719912—22no—
208#11719912—41no—
209C23×C53—21no—
210C22×D53C4×D411yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53—21no—
217D4×C53—21no—
218D4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C23×C411yescharacteristic
225C22⋊D53—81no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C22×D411yescharacteristic
234C2×Dic53C2×#4111yescharacteristic
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—41no—
240Dic106—81no—
241#88324—22no—
242#88324—21no—
243#88324—21no—
244#88324—21no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324#13712yes—
251C8×C53—21no—
252C8×C53#13112yes—
253#88322—21no—
254#88322—21no—
255#18077—531nomaximal, Sylow
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×C4×D411yescharacteristic
260C2×D53—21no—
261C2×D53—21no—
262C2×D53—21no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53C2×#13111yescharacteristic
269C4×C53#56811yescharacteristic
270Dic53—21no—
271Dic53—21no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53—41no—
276Dic53C2×#13711yescharacteristic
277Dic53#57511yescharacteristic
278#3172—531no—
279#2969—531no—
280#2962—531no—
281#1592—531no—
282#1591—531no—
283#1588—531no—
284#1587—531no—
285#1584—531no—
286#1582—531no—
287#1579—531no—
288C8×SD16—531no—
289#1565—531no—
290#1485—531no—
291#1483—531no—
292#1455—531no—
293C2×C53—81no—
294C2×C53#293511yescharacteristic
295C2×C53C4×#13711yescharacteristic
296C2×C53#295611yescharacteristic
297D53—41no—
298#533—531no—
299#529—531no—
300#501—531no—
301#497#615711yescharacteristic
302#491—531no—
303#489—531no—
304#470—531no—
305#470—531no—
306#468—531no—
307#468—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#436—1061no—
322#435—1061no—
323C8×D4—531no—
324#433—1061no—
325#431—531no—
326#424—1061no—
327C2×C4⋊C8—531no—
328#405—1061no—
329#405—1061no—
330C4×M16—531no—
331C4×M16—531no—
332C4×M16—531no—
333#335—531no—
334#335—531no—
335#334—531no—
336#334—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343D4⋊C8—531no—
344D4⋊C8—531no—
345#322—531no—
346#322—531no—
347#322—531no—
348C82—531no—
349C53#1807711yescharacteristic, complemented, Sylow
350C2×Q16C2×#615711yescharacteristic
351C2×Q16C2×#615711yescharacteristic
352C2×SD16C2×#615711yescharacteristic
353C2×SD16C2×#615711yescharacteristic
354C2×M16—531no—
355C2×M16—531no—
356C2×M16—531no—
357C2×M16—531no—
358C2×M16—1061no—
359C2×M16—1061no—
360C22×C8—1061no—
361C22×C8—1061no—
362C4⋊Q8—531no—
363C4⋊Q8C2×#615711yescharacteristic
364#126—1061no—
365#125—531no—
366#125—1061no—
367#124C2×#615711yescharacteristic
368#123—1061no—
369C22⋊Q8—531no—
370C22⋊Q8—531no—
371C22⋊Q8—531no—
372C22⋊Q8—1061no—
373#121—531no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—1061no—
378C4×D4—531no—
379C4×D4—1061no—
380#117—1061no—
381#117—1061no—
382C2×C42—531no—
383#107—531no—
384#107—531no—
385#106—531no—
386#106—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
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—1061no—
407C22⋊C8—1061no—
408C22⋊C8—1061no—
409#97—531no—
410#97—531no—
411#97—531no—
412#97—1061no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416#97C2×#615711yescharacteristic
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C2×Q8—1061no—
428C2×Q8C22×#615711yescharacteristic
429C2×Q8C22×#615711yescharacteristic
430C2×Q8C22×#615711yescharacteristic
431C2×D4—1061no—
432C2×D4C22×#615711yescharacteristic
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—1061no—
437Q16—22no—
438Q16—41no—
439SD16—22no—
440SD16—41no—
441M16—1062no—
442M16—1062no—
443M16—2121no—
444M16—2121no—
445M16—2122no—
446M16—2122no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—2122no—
460C2×C8—2122no—
461C2×C8C22×#615711yescharacteristic
462C2×C8C22×#615711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—531no—
466C4⋊C4—531no—
467C4⋊C4—531no—
468C4⋊C4—531no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1061no—
481C22⋊C4—1061no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—531no—
486C42—531no—
487C42—531no—
488C42—1061no—
489C42—1061no—
490C42—1061no—
491C42C22×#615711yescharacteristic
492C23—21no—
493C23—531no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—4241no—
501D4—21no—
502D4—21no—
503D4—4241no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4D4×#615711yescharacteristic
522C2×C4D4×#615711yescharacteristic
523C2×C4C23×#615711yescharacteristic
524C8—21no—
525C8—532no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1062no—
530C8—1061no—
531C8—1061no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8#1313330412yes—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—1061no—
542C22—1061no—
543C22—1061no—
544C22C2×D4×#615711yescharacteristic
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—531no—
551C4—531no—
552C4—1061no—
553C4—1061no—
554C4—2121no—
555C4—2121no—
556C4—2121no—
557C4—2121no—
558C4C2×#1313330411yescharacteristic
559C4#1681170111yescharacteristic
560C2—81no—
561C2—2121no—
562C2#2029145811yescharacteristic
563C2#137×#615711yescharacteristic
564C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#497C4
socleC22×C53C2×C4×D4
prime radical#13133188C2×C4
cosocleC4×Dic53C24
hypercenter#497#6157
hypocenterC53#18077
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #20291509C2semidirect
3. #20290358C2semidirect
4. #20290355C2semidirect
5. #20290344C2semidirect
6. #20290340C2semidirect
7. #20290328C2semidirect
8. #20290320C2semidirect
9. #20290308C2semidirect
10. SD16×#88324C2semidirect
11. #20290267C2semidirect
12. #20290038C2semidirect
14. #20289939C2semidirect
349. C53#18077semidirect

Maximal subgroups

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

subgroupindex
#202915472
#202915092
#202903582
#202903552
#202903442
#202903402
#202903282
#202903202
#202903082
SD16×#883242
#202902672
#202900382
#202900312
#202899392
#202853182
#1807753