Finite Group Databasebeta

← #28316747

Subgroups — #28316747

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291529C211yesmaximal, characteristic
2Q8×D53⋊C8C211yesmaximal, characteristic, complemented
3#20290329C211yesmaximal, characteristic, complemented
4#20290328C211yesmaximal, characteristic, complemented
5#20290325C211yesmaximal, characteristic, complemented
6#20290324C211yesmaximal, characteristic, complemented
7#20290309C211yesmaximal, characteristic, complemented
8#20290308C211yesmaximal, characteristic, complemented
9#20290306C211yesmaximal, characteristic, complemented
10SD16⋊#88324C211yesmaximal, characteristic, complemented
11#20290289C211yesmaximal, characteristic, complemented
12#20290058C211yesmaximal, characteristic, complemented
13#20290031C211yesmaximal, characteristic
14#20289927C211yesmaximal, characteristic, complemented
15#20284794C211yesmaximal, characteristic
16#16811674C2211yescharacteristic
17#16811672—21no—
18#16811672—21no—
19#16811672C2211yescharacteristic
20#16811672C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658C2211yescharacteristic
25#16811655—21no—
26#16811650—21no—
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811645C2211yescharacteristic
30D53⋊(C4⋊C8)—21no—
31D53⋊(C4⋊C8)C2211yescharacteristic
32#16811587—21no—
33C4×#11719936C2211yescharacteristic
34C4×D53⋊C8—21no—
35C4×D53⋊C8C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51#16811464C2211yescharacteristic
52#16811152C2211yescharacteristic
53C4×Q8×D53C2211yescharacteristic
54#16810952C411yescharacteristic
55#16810935C411yescharacteristic
56#16810931C411yescharacteristic
57#16810930C411yescharacteristic
58#16810842C411yescharacteristic
59#16810777C411yescharacteristic
60#16810760C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810714C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65#16810675C2211yescharacteristic
66C53×#464C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×D53⋊C8—21no—
70C2×D53⋊C8D411yescharacteristic
71C2×D53⋊C8D411yescharacteristic
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—
90C4×#88324—21no—
91C4×#88324—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324D411yescharacteristic
96C4×#88324D411yescharacteristic
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×Q8×D53C2×C411yescharacteristic
102C2×#11719925C2×C411yescharacteristic
103C2×#11719923C2×C411yescharacteristic
104#13133210—21no—
105#13133209C2×C411yescharacteristic
106#13133199—21no—
107#13133193C2×C411yescharacteristic
108#13133188C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110Q8×Dic53C2311yescharacteristic
111C4×Dic106—21no—
112C4×Dic106C2×C411yescharacteristic
113#13133182—21no—
114#13133182C2311yescharacteristic
115D4×Dic53C2311yescharacteristic
116#13133177—21no—
117#13133173—21no—
118C4⋊C4×D53—21no—
119C4⋊C4×D53C2×C411yescharacteristic
120C42×D53—21no—
121C42×D53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126C4⋊#88322C2×C411yescharacteristic
127#13133142C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133129C2×C411yescharacteristic
134#13133129C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×Q16×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139#124×C53C2×C411yescharacteristic
140C4×Q8×C53C2×C411yescharacteristic
141C4⋊C8×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C2×Q8×C53C22×C411yescharacteristic
145C2×Q8×C53C22×C411yescharacteristic
146C2×D4×C53C22×C411yescharacteristic
147C2×C22⋊D53—21no—
148C22×Dic53—21no—
149Q8×D53—22no—
150Q8×D53—22no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53C22×C411yescharacteristic
154C2×C4×D53C2×D411yescharacteristic
155C2×C4×D53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106C22×C411yescharacteristic
158#11719940—42no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—42no—
166C2×#88324C2×D411yescharacteristic
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170#11719936—42no—
171D53⋊C8—22no—
172D53⋊C8—41no—
173D53⋊C8—42no—
174Q16×C53—22no—
175SD16×C53—22no—
176C2×C8×C53C22×C411yescharacteristic
177C2×C8×C53C22×C411yescharacteristic
178C4⋊C4×C53—21no—
179C4⋊C4×C53C22×C411yescharacteristic
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C42×C53—21no—
183C42×C53C22×C411yescharacteristic
184C22⋊Dic53—21no—
185C22⋊Dic53—21no—
186#11719925—22no—
187#11719923—22no—
188#11719921—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193C4⋊Dic53C22×C411yescharacteristic
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197#11719919—21no—
198#11719919—21no—
199#11719919—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C2×D411yescharacteristic
205C4×Dic53#4111yescharacteristic
206C4×Dic53C2411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C23×C53—21no—
210C22×D53C4×D411yescharacteristic
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53#13112yes—
215D4×C53—21no—
216D4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C23×C411yescharacteristic
223C22⋊D53—42no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53C4×D411yescharacteristic
231C2×Dic53C22×D411yescharacteristic
232C2×Dic53C2×#4111yescharacteristic
233C4×D53—24no—
234C4×D53—22no—
235C4×D53—21no—
236C4×D53—21no—
237C4×D53—21no—
238C4×D53—21no—
239C4×D53—42no—
240Dic106—24no—
241Dic106—21no—
242Dic106#13112yes—
243#88324—21no—
244#88324—21no—
245#88324—21no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250#88324—41no—
251#88324—41no—
252#88324#13712yes—
253C8×C53—21no—
254C8×C53—21no—
255#88322—21no—
256#88322—21no—
257#17915—531nomaximal, Sylow
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C2×D53—21no—
263C2×D53—21no—
264C2×D53#44412yes—
265C4×C53—22no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—41no—
269C4×C53—41no—
270C4×C53C2×#13111yescharacteristic
271C4×C53C2×#13111yescharacteristic
272Dic53—21no—
273Dic53—22no—
274Dic53—21no—
275Dic53—41no—
276Dic53—41no—
277Dic53—41no—
278Dic53C2×#13711yescharacteristic
279Dic53#57511yescharacteristic
280#3108—531no—
281#2966—531no—
282C2×C8×Q8—531no—
283#1584—531no—
284#1584—531no—
285#1583—531no—
286#1583—531no—
287#1579—531no—
288#1579—531no—
289SD16⋊C8—531no—
290SD16⋊C8—531no—
291#1573—531no—
292#1491—531no—
293#1483—531no—
294#1451—531no—
295C2×C53—81no—
296C2×C53#292911yescharacteristic
297C2×C53C4×#13711yescharacteristic
298C2×C53C2×#44411yescharacteristic
299D53—22no—
300#529—531no—
301C2×C4×Q8—531no—
302#483—531no—
303#480—531no—
304#479—531no—
305#479—531no—
306#464—531no—
307#464#615711yescharacteristic
308#451—531no—
309#448—531no—
310#446—531no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C8×Q8—1061no—
317C8×Q8—1061no—
318C4×Q16—531no—
319C4×Q16—531no—
320C4×SD16—531no—
321C4×SD16—531no—
322#436—531no—
323#435—1061no—
324C8×D4—531no—
325C8×D4—1061no—
326#433—1061no—
327#432—531no—
328#431—531no—
329C2×C4⋊C8—531no—
330C2×C4⋊C8—1061no—
331#405—1061no—
332C4×M16—531no—
333C2×C4×C8—531no—
334C2×C4×C8—1061no—
335#335—531no—
336#335—531no—
337#334—531no—
338#334—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342Q8⋊C8—531no—
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345D4⋊C8—531no—
346D4⋊C8—531no—
347#322—531no—
348#322—531no—
349#322—531no—
350#322—531no—
351C53#1791511yescharacteristic, complemented, Sylow
352C22×Q8—531no—
353C2×Q16—531no—
354C2×Q16C2×#615711yescharacteristic
355C2×SD16—531no—
356C2×SD16C2×#615711yescharacteristic
357C2×M16—1061no—
358C2×M16—1061no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—1061no—
362C22×C8—1061no—
363#126—1061no—
364#125—531no—
365#124C2×#615711yescharacteristic
366#123—1061no—
367C22⋊Q8—531no—
368#121—531no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—531no—
373C4×Q8—1061no—
374C4×Q8—1061no—
375C4×Q8C2×#615711yescharacteristic
376C4×D4—531no—
377C4×D4—1061no—
378#117—1061no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—1061no—
381C2×C42—531no—
382C2×C42—1061no—
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—531no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8C2×#615711yescharacteristic
400#103—531no—
401#103—531no—
402#103—531no—
403#103—531no—
404#103—531no—
405#103C2×#615711yescharacteristic
406#102—531no—
407#102C2×#615711yescharacteristic
408C22⋊C8—531no—
409C22⋊C8—531no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8—1061no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—531no—
423C4×C8—531no—
424C4×C8—1061no—
425C4×C8—1061no—
426C4×C8—1061no—
427C4×C8—1061no—
428C4×C8—1061no—
429C2×Q8—531no—
430C2×Q8—1061no—
431C2×Q8—1062no—
432C2×Q8—1062no—
433C2×Q8C22×#615711yescharacteristic
434C2×Q8C22×#615711yescharacteristic
435C2×D4—1061no—
436C2×D4C22×#615711yescharacteristic
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443Q16—22no—
444Q16—1062no—
445SD16—22no—
446SD16—1062no—
447M16—2122no—
448M16—2122no—
449C2×C8—531no—
450C2×C8—531no—
451C2×C8—531no—
452C2×C8—531no—
453C2×C8—531no—
454C2×C8—531no—
455C2×C8—1062no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—2121no—
463C2×C8—2122no—
464C2×C8—2122no—
465C2×C8C22×#615711yescharacteristic
466C2×C8C22×#615711yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—531no—
469C4⋊C4—531no—
470C4⋊C4—531no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4C22×#615711yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—1061no—
483C22⋊C4—1061no—
484C22⋊C4—1061no—
485C42—21no—
486C42—531no—
487C42—531no—
488C42—531no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1061no—
493C42C22×#615711yescharacteristic
494C23—21no—
495C23—531no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—532no—
500Q8—1064no—
501Q8—1061no—
502Q8#1313330412yes—
503D4—21no—
504D4—21no—
505D4—2122no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—531no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1062no—
515C2×C4—1064no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4D4×#615711yescharacteristic
526C2×C4D4×#615711yescharacteristic
527C2×C4C23×#615711yescharacteristic
528C8—21no—
529C8—21no—
530C8—532no—
531C8—1061no—
532C8—1061no—
533C8—1061no—
534C8—1061no—
535C8—1061no—
536C8—2121no—
537C8—2121no—
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—532no—
546C22—1061no—
547C22—1061no—
548C22C2×D4×#615711yescharacteristic
549C4—22no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—531no—
555C4—531no—
556C4—1061no—
557C4—1062no—
558C4—1061no—
559C4—2121no—
560C4—2121no—
561C4—2121no—
562C4C2×#1313330411yescharacteristic
563C4D53⋊#13111yescharacteristic
564C2—81no—
565C2—1062no—
566C2#2029142911yescharacteristic
567C2#137×#615711yescharacteristic
568C2D53⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#464C4
socleC22×C53C2×C4×D4
prime radical#13133188C2×C4
cosocleC4×Dic53C24
hypercenter#464#6157
hypocenterC53#17915
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×D53⋊C8C2semidirect
3. #20290329C2semidirect
4. #20290328C2semidirect
5. #20290325C2semidirect
6. #20290324C2semidirect
7. #20290309C2semidirect
8. #20290308C2semidirect
9. #20290306C2semidirect
10. SD16⋊#88324C2semidirect
11. #20290289C2semidirect
12. #20290058C2semidirect
14. #20289927C2semidirect
351. C53#17915semidirect

Maximal subgroups

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

subgroupindex
#202915292
Q8×D53⋊C82
#202903292
#202903282
#202903252
#202903242
#202903092
#202903082
#202903062
SD16⋊#883242
#202902892
#202900582
#202900312
#202899272
#202847942
#1791553