Finite Group Databasebeta

← #28316801

Subgroups — #28316801

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291608C211yesmaximal, characteristic, complemented
2#20291605C211yesmaximal, characteristic
3#20290344C211yesmaximal, characteristic, complemented
4#20290342C211yesmaximal, characteristic, complemented
5#20290324C211yesmaximal, characteristic, complemented
6#20290322C211yesmaximal, characteristic, complemented
7#20290305C211yesmaximal, characteristic, complemented
8SD16⋊#88324C211yesmaximal, characteristic, complemented
9#20290299C211yesmaximal, characteristic, complemented
10SD16×#88324C211yesmaximal, characteristic, complemented
11#20290286C211yesmaximal, characteristic, complemented
12#20290057C211yesmaximal, characteristic, complemented
13#20290034C211yesmaximal, characteristic
14#20289932C211yesmaximal, characteristic, complemented
15#20284741C211yesmaximal, characteristic
16Q8×#88324—21no—
17Q8×#88324—21no—
18Q8×#88324C2211yescharacteristic
19Q8×#88324C2211yescharacteristic
20Q8×#88324C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22#16811658C2211yescharacteristic
23#16811657C2211yescharacteristic
24#16811655—21no—
25#16811652—21no—
26#16811651C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811644C2211yescharacteristic
29#16811639—21no—
30#16811639—21no—
31#16811639C2211yescharacteristic
32#16811639C2211yescharacteristic
33#16811597—21no—
34#16811596—21no—
35C4×#11719940C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811168C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810942C411yescharacteristic
55#16810939C411yescharacteristic
56#16810906C411yescharacteristic
57#16810906C411yescharacteristic
58#16810821C411yescharacteristic
59#16810821C411yescharacteristic
60#16810762C411yescharacteristic
61SD16×Dic53C2211yescharacteristic
62SD16×Dic53C2211yescharacteristic
63#16810708C2211yescharacteristic
64#16810708C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#448C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719940—21no—
69C2×#11719940—21no—
70C2×D53⋊C8—21no—
71C22×#88324—21no—
72C4⋊#88324—21no—
73C4⋊#88324—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324D411yescharacteristic
77C4⋊#88324D411yescharacteristic
78C4⋊#88324C2311yescharacteristic
79C4⋊#88324C2311yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82#13133274—21no—
83#13133274—21no—
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324D411yescharacteristic
89C22⋊#88324D411yescharacteristic
90#13133272—21no—
91#13133272—21no—
92C4×#88324—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719947—21no—
102C2×#11719924C2×C411yescharacteristic
103C2×#11719924C2×C411yescharacteristic
104#13133216C2×C411yescharacteristic
105#13133213C2×C411yescharacteristic
106#13133203—21no—
107#13133191C2×C411yescharacteristic
108#13133189—21no—
109Q8×Dic53—21no—
110Q8×Dic53—21no—
111Q8×Dic53C2×C411yescharacteristic
112Q8×Dic53C2×C411yescharacteristic
113Q8×Dic53C2311yescharacteristic
114Q8×Dic53C2311yescharacteristic
115D4×Dic53C2311yescharacteristic
116#13133179C2311yescharacteristic
117#13133177—21no—
118C2×C4⋊Dic53—21no—
119C2×C4⋊Dic53C2×C411yescharacteristic
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133149C2×C411yescharacteristic
123#13133149C2×C411yescharacteristic
124#13133148C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133139C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22×Q8×C53C2×C411yescharacteristic
138C2×SD16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140#121×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×Q8×C53—22no—
145C2×Q8×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C2×D4×C53—21no—
150C2×D4×C53C22×C411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53C22×C411yescharacteristic
153C2×C22⋊D53—21no—
154C2×C22⋊D53—21no—
155C2×C22⋊D53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53#4111yescharacteristic
160#11719947—81no—
161#11719947—81no—
162C2×D212C22×C411yescharacteristic
163C2×C4×D53—21no—
164C2×Dic106—21no—
165#11719940—42no—
166#11719940—81no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—81no—
174C2×#88324C2×D411yescharacteristic
175C2×#88324C2×D411yescharacteristic
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178#11719936—81no—
179D53⋊C8—81no—
180SD16×C53—22no—
181SD16×C53—22no—
182C2×C8×C53C22×C411yescharacteristic
183C2×C8×C53C22×C411yescharacteristic
184C4⋊C4×C53C22×C411yescharacteristic
185C22⋊C4×C53—21no—
186C22⋊Dic53—21no—
187#11719924—22no—
188#11719924—22no—
189#11719921—21no—
190#11719921—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199#11719919—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C22×C411yescharacteristic
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53#4111yescharacteristic
207C4×Dic53C2411yescharacteristic
208C2×#88322C22×C411yescharacteristic
209C2×#88322C22×C411yescharacteristic
210C23×C53—21no—
211C23×C53C4×D411yescharacteristic
212C22×D53—21no—
213Q8×C53—24no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53#13112yes—
217Q8×C53#13112yes—
218D4×C53—21no—
219D4×C53—21no—
220D4×C53—81no—
221C2×C4×C53—24no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C22⋊D53—81no—
230C22⋊D53—81no—
231C22⋊D53—81no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—81no—
241C2×Dic53C4×D411yescharacteristic
242C2×Dic53C4×D411yescharacteristic
243C2×Dic53C22×D411yescharacteristic
244C2×Dic53C2×#4111yescharacteristic
245D212—21no—
246D212—21no—
247C4×D53—41no—
248C4×D53—41no—
249Dic106—81no—
250#88324—22no—
251#88324—21no—
252#88324—21no—
253#88324—21no—
254#88324—21no—
255#88324—41no—
256#88324—41no—
257#88324—41no—
258#88324—41no—
259#88324—41no—
260C8×C53—21no—
261C8×C53—21no—
262#88322—21no—
263#88322—21no—
264#17923—531nomaximal, Sylow
265C22×C53—21no—
266C22×C53—21no—
267C22×C53—21no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×C4×D411yescharacteristic
272C2×D53—41no—
273C2×D53—41no—
274C2×D53—41no—
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—21no—
278C4×C53—41no—
279C4×C53C2×#13111yescharacteristic
280C4×C53C2×#13111yescharacteristic
281Dic53—21no—
282Dic53—21no—
283Dic53—21no—
284Dic53—41no—
285Dic53—41no—
286Dic53—41no—
287Dic53#57711yescharacteristic
288Dic53#57711yescharacteristic
289#3103—531no—
290#2982—531no—
291#2981—531no—
292#1588—531no—
293#1588—531no—
294#1583—531no—
295#1583—531no—
296SD16⋊C8—531no—
297SD16⋊C8—531no—
298C8×SD16—531no—
299C8×SD16—531no—
300#1572—531no—
301#1491—531no—
302#1484—531no—
303#1453—531no—
304C2×C53—41no—
305C2×C53—81no—
306C2×C53#292911yescharacteristic
307C2×C53#294911yescharacteristic
308C2×C53#295611yescharacteristic
309D53—81no—
310#532—531no—
311C2×C4×Q8—531no—
312#481—531no—
313#481—531no—
314#475—531no—
315#475—531no—
316#460—531no—
317#460—531no—
318#448—531no—
319#448#615711yescharacteristic
320C8×Q8—531no—
321C8×Q8—531no—
322C8×Q8—531no—
323C8×Q8—531no—
324C8×Q8—1061no—
325C8×Q8—1061no—
326C4×SD16—531no—
327C4×SD16—531no—
328C4×SD16—531no—
329C4×SD16—531no—
330#436—531no—
331#436—531no—
332#435—1061no—
333#435—1061no—
334C8×D4—531no—
335C8×D4—531no—
336#432—531no—
337#431—531no—
338#431—531no—
339#431—531no—
340#431—1061no—
341#431—1061no—
342#408—1061no—
343#408—1061no—
344C4×M16—531no—
345#335—531no—
346#335—531no—
347#334—531no—
348#334—531no—
349Q8⋊C8—531no—
350Q8⋊C8—531no—
351Q8⋊C8—531no—
352Q8⋊C8—531no—
353D4⋊C8—531no—
354D4⋊C8—531no—
355D4⋊C8—531no—
356D4⋊C8—531no—
357#322—531no—
358#322—531no—
359#322—531no—
360C82—531no—
361C53#1792311yescharacteristic, complemented, Sylow
362C2×#41—1061no—
363C22×Q8C2×#615711yescharacteristic
364C2×SD16—531no—
365C2×SD16—531no—
366C2×SD16C2×#615711yescharacteristic
367C2×SD16C2×#615711yescharacteristic
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C22×C8—1061no—
372C22×C8—1061no—
373C4⋊Q8—531no—
374#127—531no—
375#124—1061no—
376#121—531no—
377#121—1061no—
378#121C2×#615711yescharacteristic
379C4×Q8—531no—
380C4×Q8—531no—
381C4×Q8—531no—
382C4×Q8—531no—
383C4×Q8—1061no—
384C4×Q8—1061no—
385C4×D4—531no—
386C4×D4—531no—
387C4×D4—1061no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C42—531no—
391C2×C42—1061no—
392#106—531no—
393#106—531no—
394#106—531no—
395#106—531no—
396C4⋊C8—531no—
397C4⋊C8—531no—
398C4⋊C8—531no—
399C4⋊C8—531no—
400C4⋊C8—531no—
401C4⋊C8—531no—
402C4⋊C8—531no—
403C4⋊C8—531no—
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#102—531no—
413#102—531no—
414#102—531no—
415#102—531no—
416C22⋊C8—531no—
417C22⋊C8—531no—
418C22⋊C8—531no—
419C22⋊C8—1061no—
420C22⋊C8—1061no—
421C22⋊C8—1061no—
422C22⋊C8—1061no—
423C22⋊C8—1061no—
424C22⋊C8—1061no—
425C22⋊C8C2×#615711yescharacteristic
426#97—1061no—
427#97—1061no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—531no—
431C4×C8—531no—
432C4×C8—531no—
433C4×C8—531no—
434C4×C8—1061no—
435C4×C8—1061no—
436C4×C8—1061no—
437C4×C8—1061no—
438C4×C8—1061no—
439#41—4241no—
440#41—4241no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—1061no—
445C2×Q8C22×#615711yescharacteristic
446C2×Q8C22×#615711yescharacteristic
447C2×D4—21no—
448C2×D4—531no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4C22×#615711yescharacteristic
453C22×C4—21no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4C22×#615711yescharacteristic
460SD16—22no—
461SD16—22no—
462SD16—1062no—
463SD16—1062no—
464M16—4241no—
465M16—4241no—
466M16—2122no—
467C2×C8—531no—
468C2×C8—531no—
469C2×C8—531no—
470C2×C8—531no—
471C2×C8—531no—
472C2×C8—531no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—4241no—
480C2×C8—4241no—
481C2×C8C22×#615711yescharacteristic
482C2×C8C22×#615711yescharacteristic
483C4⋊C4—531no—
484C4⋊C4—531no—
485C4⋊C4—531no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4C22×#615711yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—1061no—
495C22⋊C4—1061no—
496C22⋊C4—1061no—
497C42—531no—
498C42—531no—
499C42—531no—
500C42—531no—
501C42—1061no—
502C42—1061no—
503C42—1061no—
504C42—1061no—
505C23—21no—
506C23—1061no—
507C23D4×#615711yescharacteristic
508Q8—24no—
509Q8—21no—
510Q8—21no—
511Q8—4241no—
512Q8#1313330412yes—
513Q8#1313330412yes—
514D4—21no—
515D4—21no—
516D4—81no—
517D4—1061no—
518D4—1061no—
519D4—4241no—
520D4—4241no—
521D4—4241no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—531no—
529C2×C4—531no—
530C2×C4—531no—
531C2×C4—531no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—4241no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4D4×#615711yescharacteristic
544C2×C4C23×#615711yescharacteristic
545C8—21no—
546C8—21no—
547C8—1061no—
548C8—1061no—
549C8—1061no—
550C8—1061no—
551C8—1062no—
552C8—1061no—
553C8—1061no—
554C8—2121no—
555C8—2121no—
556C8—2121no—
557C8—2121no—
558C8—2121no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—2121no—
566C22—2121no—
567C22—2121no—
568C22C2×D4×#615711yescharacteristic
569C4—22no—
570C4—22no—
571C4—21no—
572C4—41no—
573C4—531no—
574C4—531no—
575C4—1061no—
576C4—1061no—
577C4—1061no—
578C4—2121no—
579C4—2121no—
580C4—2121no—
581C4C2×#1313330411yescharacteristic
582C4C2×#1313330411yescharacteristic
583C2—41no—
584C2—81no—
585C2—4241no—
586C2#2029143011yescharacteristic
587C2#2029147511yescharacteristic
588C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#448C4
socleC22×C53C2×C4×D4
prime radical#16811168C22
cosocleC4×Dic53C24
hypercenter#448#6157
hypocenterC53#17923
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #20291608C2semidirect
3. #20290344C2semidirect
4. #20290342C2semidirect
5. #20290324C2semidirect
6. #20290322C2semidirect
7. #20290305C2semidirect
8. SD16⋊#88324C2semidirect
9. #20290299C2semidirect
10. SD16×#88324C2semidirect
11. #20290286C2semidirect
12. #20290057C2semidirect
14. #20289932C2semidirect
361. C53#17923semidirect

Maximal subgroups

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

subgroupindex
#202916082
#202916052
#202903442
#202903422
#202903242
#202903222
#202903052
SD16⋊#883242
#202902992
SD16×#883242
#202902862
#202900572
#202900342
#202899322
#202847412
#1792353