Finite Group Databasebeta

← #28317580

Subgroups — #28317580

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291556C211yesmaximal, characteristic
2Q8×#11719940C211yesmaximal, characteristic, complemented
3#20290360C211yesmaximal, characteristic, complemented
4#20290353C211yesmaximal, characteristic, complemented
5#20290338C211yesmaximal, characteristic, complemented
6#20290333C211yesmaximal, characteristic, complemented
7#20290326C211yesmaximal, characteristic, complemented
8#20290321C211yesmaximal, characteristic, complemented
9#20290305C211yesmaximal, characteristic, complemented
10#20290299C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290057C211yesmaximal, characteristic, complemented
13#20290052C211yesmaximal, characteristic
14#20289938C211yesmaximal, characteristic, complemented
15#20283530C211yesmaximal, characteristic
16#16811674—21no—
17#16811674—21no—
18#16811674C2211yescharacteristic
19#16811674C2211yescharacteristic
20#16811673C2211yescharacteristic
21#16811673C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811655—21no—
25#16811655—21no—
26#16811651C2211yescharacteristic
27#16811651C2211yescharacteristic
28#16811645—21no—
29#16811644C2211yescharacteristic
30#16811626—21no—
31#16811626C2211yescharacteristic
32C4×#11719940—21no—
33C4×#11719940C2211yescharacteristic
34C4×#11719940C2211yescharacteristic
35#16811582—21no—
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46#16811471C2211yescharacteristic
47#16811471C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52C2×#13133248C411yescharacteristic
53#16811091C2211yescharacteristic
54C2×Q8×Dic53C2211yescharacteristic
55#16810727C411yescharacteristic
56#16810727C411yescharacteristic
57#16810726C411yescharacteristic
58#16810726C411yescharacteristic
59#16810715C2211yescharacteristic
60#16810715C2211yescharacteristic
61#16810708C2211yescharacteristic
62#16810708C2211yescharacteristic
63#16810666C411yescharacteristic
64#16810625C411yescharacteristic
65M16×Dic53C2211yescharacteristic
66C53×#419C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×#11719940D411yescharacteristic
70C2×#11719940D411yescharacteristic
71C2×D53⋊C8—21no—
72C2×D53⋊C8—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84#13133274—21no—
85#13133274—21no—
86C22⋊#88324—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93#13133272D411yescharacteristic
94#13133272D411yescharacteristic
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719945C2×C411yescharacteristic
102#13133248—24no—
103C2×#11719925C2×C411yescharacteristic
104C2×#11719925C2×C411yescharacteristic
105C2×#11719924C2×C411yescharacteristic
106C2×#11719924C2×C411yescharacteristic
107C2×#11719917C2×C411yescharacteristic
108Q8×Dic53—21no—
109Q8×Dic53—21no—
110Q8×Dic53C2×C411yescharacteristic
111Q8×Dic53C2×C411yescharacteristic
112Q8×Dic53C2311yescharacteristic
113Q8×Dic53C2311yescharacteristic
114#13133182C2×C411yescharacteristic
115#13133182C2311yescharacteristic
116#13133179C2×C411yescharacteristic
117#13133179C2311yescharacteristic
118#13133177—21no—
119#13133177—21no—
120#13133174—21no—
121C2×C4⋊Dic53—21no—
122C2×C4⋊Dic53C2×C411yescharacteristic
123C2×C4×Dic53—21no—
124C2×C4×Dic53C2311yescharacteristic
125C42×D53—21no—
126#13133150C2×C411yescharacteristic
127#13133150C2×C411yescharacteristic
128#13133148C2×C411yescharacteristic
129#13133148C2×C411yescharacteristic
130#13133142C2×C411yescharacteristic
131#13133142C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133139C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135#13133127C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138C22×Q8×C53C2×C411yescharacteristic
139C2×M16×C53C2×C411yescharacteristic
140#117×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143#103×C53C2×C411yescharacteristic
144#103×C53C2×C411yescharacteristic
145C2×Q8×C53—21no—
146C2×Q8×C53—22no—
147C2×Q8×C53—41no—
148C2×Q8×C53C22×C411yescharacteristic
149C2×Q8×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C2×C22⋊D53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53#4111yescharacteristic
156#11719945—22no—
157#11719945—41no—
158C2×D212C22×C411yescharacteristic
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×Dic106C22×C411yescharacteristic
163#11719940—22no—
164#11719940—41no—
165#11719940—42no—
166#11719940—42no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175C2×#88324C2×D411yescharacteristic
176C2×#88324C2×D411yescharacteristic
177D53⋊C8—42no—
178D53⋊C8—42no—
179M16×C53—41no—
180C2×C8×C53C22×C411yescharacteristic
181C2×C8×C53C22×C411yescharacteristic
182C4⋊C4×C53C22×C411yescharacteristic
183C4⋊C4×C53C22×C411yescharacteristic
184C22⋊C4×C53—21no—
185C42×C53—21no—
186#11719925—22no—
187#11719925—22no—
188#11719924—22no—
189#11719924M1614yescomplemented
190#11719921—21no—
191#11719921—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199#11719919—21no—
200#11719919—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211#11719917—22no—
212C2×#88322C22×C411yescharacteristic
213C2×#88322C22×C411yescharacteristic
214C23×C53C4×D411yescharacteristic
215C22×D53—21no—
216Q8×C53—21no—
217Q8×C53—21no—
218Q8×C53—21no—
219Q8×C53—42no—
220Q8×C53C2×M1612yes—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—42no—
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C23×C411yescharacteristic
230C22⋊D53—42no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—41no—
237C2×Dic53—81no—
238C2×Dic53C4×D411yescharacteristic
239C2×Dic53C4×D411yescharacteristic
240C2×Dic53C22×D411yescharacteristic
241C2×Dic53C2×#4111yescharacteristic
242D212—21no—
243D212C2×M1612yes—
244C4×D53—42no—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—81no—
248Dic106—21no—
249Dic106—21no—
250#88324—22no—
251#88324—21no—
252#88324—21no—
253#88324—21no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259#88324#13712yescomplemented
260C8×C53—21no—
261C8×C53—21no—
262#88322—21no—
263#88322C2×M1612yes—
264#18047—531nomaximal, Sylow
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53C2×C4×D411yescharacteristic
269C2×D53—22no—
270C2×D53—41no—
271C2×D53—41no—
272C4×C53—21no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53C22×M1611yescharacteristic
278C4×C53#56811yescharacteristic
279Dic53—21no—
280Dic53—21no—
281Dic53—21no—
282Dic53—41no—
283Dic53—41no—
284Dic53—41no—
285Dic53C2×#13711yescharacteristic
286Dic53#57511yescharacteristic
287#2970—531no—
288Q8×M16—531no—
289C4×#137—531no—
290#1593—531no—
291#1591—531no—
292#1587—531no—
293#1585—531no—
294#1584—531no—
295#1582—531no—
296SD16⋊C8—531no—
297C8×SD16—531no—
298#1569—531no—
299#1491—531no—
300#1489—531no—
301#1455—531no—
302C2×C53—41no—
303C2×C53D4×M1611yescharacteristic
304C2×C53C4×#13711yescharacteristic
305C2×C53#295611yescharacteristic
306D53—42no—
307C2×#137—531no—
308C4×#41—531no—
309C2×C4×Q8—531no—
310#446—531no—
311#446—531no—
312#446—531no—
313#446—531no—
314#446—1061no—
315#446—1061no—
316C8×Q8—531no—
317C8×Q8—531no—
318#441—531no—
319#441—531no—
320#440—531no—
321#440—531no—
322C4×Q16—531no—
323C4×Q16—531no—
324C4×SD16—531no—
325C4×SD16—531no—
326#435—1061no—
327#435—1061no—
328C8×D4—531no—
329C8×D4—531no—
330#432—531no—
331#432—1061no—
332#428—531no—
333#423—531no—
334#423—1061no—
335#419#615711yescharacteristic
336#417—531no—
337C4×M16—531no—
338C4×M16—531no—
339C4×M16—531no—
340C4×M16—1061no—
341C2×#97—1061no—
342#335—531no—
343#335—531no—
344#334—531no—
345#334—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352D4⋊C8—531no—
353D4⋊C8—531no—
354#322—531no—
355#322—531no—
356#322—531no—
357C82—531no—
358C53#1804711yescharacteristic, complemented, Sylow
359C2×#41—531no—
360C22×Q8C2×#615711yescharacteristic
361#137—1064no—
362C2×Q16—531no—
363C2×Q16—531no—
364C2×SD16—531no—
365C2×SD16—531no—
366C2×M16—531no—
367C2×M16—531no—
368C2×M16—531no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16C2×#615711yescharacteristic
372C22×C8—1061no—
373C22×C8—1061no—
374C4×Q8—531no—
375C4×Q8—531no—
376C4×Q8—531no—
377C4×Q8—531no—
378C4×Q8—531no—
379C4×Q8—531no—
380C4×Q8—1061no—
381C4×Q8—1061no—
382C4×D4—531no—
383C4×D4—531no—
384C4×D4—1061no—
385C4×D4—1061no—
386#117—1061no—
387#117C2×#615711yescharacteristic
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C42—531no—
391C2×C42—1061no—
392C2×C42—1061no—
393#107—531no—
394#107—531no—
395#106—531no—
396#106—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405C4⋊C8—1061no—
406C4⋊C8—1061no—
407C4⋊C8—1061no—
408#103—531no—
409#103—531no—
410#103C2×#615711yescharacteristic
411#103C2×#615711yescharacteristic
412#103C2×#615711yescharacteristic
413#103C2×#615711yescharacteristic
414#102—531no—
415#102—531no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—531no—
420#97—531no—
421#97—531no—
422#97—531no—
423#97—1061no—
424#97—1061no—
425#97—1061no—
426#97—1061no—
427#97—1061no—
428#97—1061no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—531no—
435C4×C8—1061no—
436C4×C8—1061no—
437#41—1062no—
438#41—2121no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—531no—
443C2×Q8C22×#615711yescharacteristic
444C2×Q8C22×#615711yescharacteristic
445C2×D4—531no—
446C2×D4—1061no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4C22×#615711yescharacteristic
455Q16—1062no—
456Q16—1062no—
457SD16—534no—
458SD16—1062no—
459M16—41no—
460M16—1062no—
461M16—1062no—
462M16—2121no—
463M16—2122no—
464M16—2122no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—531no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—531no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—2122no—
478C2×C8—2122no—
479C2×C8C22×#615711yescharacteristic
480C2×C8C22×#615711yescharacteristic
481C4⋊C4—531no—
482C4⋊C4—531no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4C22×#615711yescharacteristic
491C4⋊C4C22×#615711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1061no—
494C22⋊C4—1061no—
495C42—21no—
496C42—531no—
497C42—531no—
498C42—531no—
499C42—531no—
500C42—1061no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C23—1061no—
507C23D4×#615711yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—42no—
512Q8—1061no—
513Q8—1061no—
514Q8D53⋊M1612yes—
515D4—532no—
516D4—1061no—
517D4—2122no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—531no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—531no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—4241no—
535C2×C4—4241no—
536C2×C4—2122no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4D4×#615711yescharacteristic
541C2×C4C23×#615711yescharacteristic
542C8—21no—
543C8—21no—
544C8—532no—
545C8—532no—
546C8—1061no—
547C8—1061no—
548C8—1061no—
549C8—1062no—
550C8—1061no—
551C8—2121no—
552C8—2121no—
553C8—2121no—
554C8—2121no—
555C8—2121no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—1062no—
560C22—2121no—
561C22—2121no—
562C22C2×D4×#615711yescharacteristic
563C4—21no—
564C4—22no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—531no—
569C4—531no—
570C4—1061no—
571C4—1061no—
572C4—1061no—
573C4—2121no—
574C4—2121no—
575C4—2121no—
576C4C2×D53⋊M1611yescharacteristic
577C4#1681170111yescharacteristic
578C2—41no—
579C2—2122no—
580C2#2029145311yescharacteristic
581C2#137×#615711yescharacteristic
582C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#419C4
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C2×C4
cosocleC4×Dic53C24
hypercenter#419#6157
hypocenterC53#18047
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. Q8×#11719940C2semidirect
3. #20290360C2semidirect
4. #20290353C2semidirect
5. #20290338C2semidirect
6. #20290333C2semidirect
7. #20290326C2semidirect
8. #20290321C2semidirect
9. #20290305C2semidirect
10. #20290299C2semidirect
11. #20290279C2semidirect
12. #20290057C2semidirect
14. #20289938C2semidirect
189. #11719924M16semidirect
259. #88324#137semidirect
358. C53#18047semidirect

Maximal subgroups

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

subgroupindex
#202915562
Q8×#117199402
#202903602
#202903532
#202903382
#202903332
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