Finite Group Databasebeta

← #28317282

Subgroups — #28317282

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291546C211yesmaximal, characteristic, complemented
2#20291455C211yesmaximal, characteristic, complemented
3#20290363C211yesmaximal, characteristic, complemented
4#20290350C211yesmaximal, characteristic, complemented
5#20290343C211yesmaximal, characteristic, complemented
6#20290337C211yesmaximal, characteristic, complemented
7#20290319C211yesmaximal, characteristic, complemented
8#20290314C211yesmaximal, characteristic, complemented
9#20290307C211yesmaximal, characteristic, complemented
10#20290295C211yesmaximal, characteristic, complemented
11#20290291C211yesmaximal, characteristic, complemented
12#20290070C211yesmaximal, characteristic, complemented
13#20290016C211yesmaximal, characteristic, complemented
14#20289932C211yesmaximal, characteristic, complemented
15#20284759C211yesmaximal, characteristic
16C2×D53⋊M16—21no—
17#16811673C2211yescharacteristic, complemented
18#16811672C2211yescharacteristic, complemented
19#16811658—21no—
20#16811658C2211yescharacteristic, complemented
21#16811657C2211yescharacteristic
22#16811657C2211yescharacteristic
23#16811654—21no—
24#16811653—21no—
25#16811653—21no—
26#16811651C2211yescharacteristic
27#16811651C2211yescharacteristic
28D4×#88324C2211yescharacteristic, complemented
29#16811647—21no—
30#16811639C2211yescharacteristic
31#16811639C2211yescharacteristic
32#16811626C2211yescharacteristic
33#16811595—21no—
34#16811587—21no—
35C4×#11719940C2211yescharacteristic
36#16811487C2211yescharacteristic, complemented
37#16811486C2211yescharacteristic, complemented
38#16811485C2211yescharacteristic, complemented
39#16811484C2211yescharacteristic, complemented
40#16811473C2211yescharacteristic, complemented
41#16811473C2211yescharacteristic, complemented
42#16811471C2211yescharacteristic
43#16811471C2211yescharacteristic
44#16811471C2211yescharacteristic
45#16811471C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic, complemented
47D4⋊#88324C2211yescharacteristic, complemented
48#88324⋊C8C2211yescharacteristic, complemented
49#88324⋊C8C2211yescharacteristic, complemented
50#16811464C2211yescharacteristic, complemented
51C8×#88324C2211yescharacteristic, complemented
52#16811154C2211yescharacteristic
53C2×#13133179C2211yescharacteristic
54#16810958C411yescharacteristic
55#16810922C411yescharacteristic
56#16810922C411yescharacteristic
57#16810837C411yescharacteristic
58#16810837C411yescharacteristic
59#16810776C411yescharacteristic
60#16810756C411yescharacteristic
61#16810710C2211yescharacteristic
62#16810710C2211yescharacteristic
63#16810703C2211yescharacteristic
64#16810703C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#481C411yescharacteristic
67C2×#11719936—21no—
68C2×#11719936—21no—
69D53⋊M16—44no—
70C2×#11719940—21no—
71C2×#11719940—21no—
72C2×#11719940D411yescharacteristic
73C2×#11719940D411yescharacteristic
74C2×D53⋊C8—21no—
75C2×D53⋊C8—21no—
76C22×#88324—21no—
77#13133284—21no—
78#13133284—21no—
79C4⋊#88324—21no—
80C4⋊#88324—21no—
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85#13133274—21no—
86#13133274—21no—
87#13133274—21no—
88#13133274—21no—
89C22⋊#88324—21no—
90C22⋊#88324—21no—
91C22⋊#88324—21no—
92C22⋊#88324—21no—
93#13133272—21no—
94#13133272D411yescharacteristic
95#13133272D411yescharacteristic
96C4×#88324—21no—
97C4×#88324—21no—
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C4×#88324C2311yescharacteristic
102C22×D212C2×C411yescharacteristic
103C22×C4×D53—21no—
104C2×#11719913C2×C411yescharacteristic
105C2×#11719913C2×C411yescharacteristic
106C2×#11719922C2×C411yescharacteristic
107C2×#11719922C2×C411yescharacteristic
108#13133210—21no—
109#13133208C2×C411yescharacteristic
110#13133204C2×C411yescharacteristic
111#13133198—21no—
112#13133196C2×C411yescharacteristic
113#13133182C2311yescharacteristic
114D4×Dic53—21no—
115D4×Dic53C2311yescharacteristic
116#13133179—21no—
117#13133179—21no—
118#13133179C2×C411yescharacteristic
119#13133179C2×C411yescharacteristic
120#13133179C2311yescharacteristic
121#13133179C2311yescharacteristic
122#13133173—21no—
123C2×#11719921—21no—
124C2×C4×Dic53C2311yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133150C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133146C2×C411yescharacteristic
129#13133140C2×C411yescharacteristic
130#13133140C2×C411yescharacteristic
131#13133139C2×C411yescharacteristic
132#13133139C2×C411yescharacteristic
133#13133137C2×C411yescharacteristic
134#13133137C2×C411yescharacteristic
135C22⋊#88322C2×C411yescharacteristic
136C8×Dic53C2311yescharacteristic
137C8×Dic53C2311yescharacteristic
138#121×C53C2×C411yescharacteristic
139C2×C4⋊C4×C53C2×C411yescharacteristic
140#106×C53C2×C411yescharacteristic
141#106×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143#102×C53C2×C411yescharacteristic
144C22⋊C8×C53C2×C411yescharacteristic
145C23×D53—21no—
146C2×D4×C53—21no—
147C2×D4×C53C22×C411yescharacteristic
148C22×C4×C53—21no—
149C22×C4×C53C22×C411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53#4111yescharacteristic
153C2×D212—22no—
154C2×D212—21no—
155C2×D212—41no—
156C2×D212C22×C411yescharacteristic
157C2×D212C22×C411yescharacteristic
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—21no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×Dic106C22×C411yescharacteristic
165#11719940—22no—
166#11719940—41no—
167#11719940—42no—
168#11719940—42no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324—21no—
173C2×#88324—21no—
174C2×#88324—21no—
175C2×#88324—42no—
176C2×#88324C2×D411yescharacteristic
177C2×#88324C2×D411yescharacteristic
178C2×#88324C2×D411yescharacteristic
179C2×#88324C2×D411yescharacteristic
180#11719936—42no—
181#11719936—42no—
182D53⋊C8—42no—
183D53⋊C8—42no—
184C2×C8×C53C22×C411yescharacteristic
185C2×C8×C53C22×C411yescharacteristic
186C4⋊C4×C53—21no—
187C4⋊C4×C53C22×C411yescharacteristic
188C4⋊C4×C53C22×C411yescharacteristic
189C4⋊C4×C53C22×C411yescharacteristic
190C22⋊C4×C53—21no—
191C22⋊Dic53—21no—
192C22⋊Dic53—21no—
193C22⋊Dic53—21no—
194C22⋊Dic53—21no—
195#11719922—22no—
196#11719922—41no—
197#11719921—21no—
198#11719921—21no—
199#11719921—21no—
200#11719921—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53C22×C411yescharacteristic
203#11719919—21no—
204#11719919—21no—
205#11719919—21no—
206C4×Dic53—21no—
207C4×Dic53C22×C411yescharacteristic
208C4×Dic53C2×D411yescharacteristic
209C4×Dic53#4111yescharacteristic
210C4×Dic53C2411yescharacteristic
211C2×#88322C22×C411yescharacteristic
212C2×#88322C22×C411yescharacteristic
213#11719913—22no—
214#11719913—41no—
215C23×C53—21no—
216C23×C53C4×D411yescharacteristic
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—81no—
222D4×C53—21no—
223D4×C53—21no—
224D4×C53—81no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—42no—
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C23×C411yescharacteristic
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53C4×D411yescharacteristic
241C2×Dic53C4×D411yescharacteristic
242C2×Dic53C22×D411yescharacteristic
243C2×Dic53C2×#4111yescharacteristic
244D212—21no—
245D212—21no—
246D212—21no—
247D212—42no—
248D212#13112yes—
249C4×D53—42no—
250C4×D53—42no—
251C4×D53—41no—
252C4×D53—41no—
253C4×D53—41no—
254C4×D53—41no—
255Dic106—21no—
256Dic106—21no—
257#88324—21no—
258#88324—21no—
259#88324—21no—
260#88324—41no—
261#88324—41no—
262#88324—41no—
263#88324—41no—
264#88324—41no—
265#88324—41no—
266#88324AGL(1,ℤ/8)12yes—
267C8×C53—21no—
268C8×C53—21no—
269#88322—21no—
270#88322—21no—
271#18006—531nomaximal, Sylow
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×C4×D411yescharacteristic
279C2×D53—22no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—84no—
286C4×C53—21no—
287C4×C53—41no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53C2×#13111yescharacteristic
291C4×C53#56811yescharacteristic
292Dic53—21no—
293Dic53—21no—
294Dic53—21no—
295Dic53—41no—
296Dic53—41no—
297Dic53C2×AGL(1,ℤ/8)11yescharacteristic
298Dic53#57511yescharacteristic
299#3105—531no—
300#2969—531no—
301D4×M16—531no—
302#1594—531no—
303#1590—531no—
304#1588—531no—
305#1586—531no—
306#1582—531no—
307#1581—531no—
308SD16⋊C8—531no—
309C8×D8—531no—
310#1574—531no—
311#1495—531no—
312#1478—531no—
313#1453—531no—
314C2×C53—41no—
315C2×C53—81no—
316C2×C53#293311yescharacteristic
317C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C53#295711yescharacteristic
319D53—42no—
320D53—81no—
321C22×M16—1061no—
322#529—531no—
323C2×C4×D4—531no—
324#484—531no—
325#481#615711yescharacteristic
326#478—531no—
327#478—531no—
328#463—531no—
329#463—531no—
330#450—531no—
331#447—531no—
332#446—531no—
333C8×Q8—531no—
334C4×SD16—531no—
335C4×SD16—531no—
336C4×D8—531no—
337C4×D8—531no—
338#436—531no—
339#436—531no—
340#436—531no—
341#436—1061no—
342#435—1061no—
343#435—1061no—
344#435—1061no—
345C8×D4—531no—
346C8×D4—531no—
347C8×D4—531no—
348#433—1061no—
349#431—531no—
350#431—531no—
351#431—531no—
352#423—531no—
353#407—1061no—
354#405—1061no—
355C4×M16—531no—
356#335—531no—
357#335—531no—
358#334—531no—
359#334—531no—
360Q8⋊C8—531no—
361Q8⋊C8—531no—
362D4⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8—531no—
365D4⋊C8—531no—
366D4⋊C8—531no—
367D4⋊C8—531no—
368#322—531no—
369#322—531no—
370#322—531no—
371C82—531no—
372C53#1800611yescharacteristic, complemented, Sylow
373C22×D4—531no—
374C23×C4—1061no—
375C2×SD16—531no—
376C2×SD16—531no—
377C2×D8—531no—
378C2×D8—531no—
379C2×M16—531no—
380C2×M16—531no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—2124no—
386C22×C8—1061no—
387C22×C8—1061no—
388C22×C8—1061no—
389#126—1061no—
390#125—531no—
391#124—531no—
392#123—1061no—
393C22⋊Q8—531no—
394#121C2×#615711yescharacteristic
395C4×Q8—531no—
396C4×D4—531no—
397C4×D4—531no—
398C4×D4—531no—
399C4×D4—531no—
400C4×D4—531no—
401C4×D4—1061no—
402C4×D4—1061no—
403C4×D4—1061no—
404#117—1061no—
405C2×C4⋊C4C2×#615711yescharacteristic
406C2×C22⋊C4—1061no—
407C2×C42—531no—
408#107—531no—
409#107—531no—
410#106C2×#615711yescharacteristic
411#106C2×#615711yescharacteristic
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—531no—
416C4⋊C8—531no—
417C4⋊C8—531no—
418C4⋊C8—1061no—
419C4⋊C8—1061no—
420C4⋊C8—1061no—
421C4⋊C8—1061no—
422#103—531no—
423#103—531no—
424#102—531no—
425#102—531no—
426#102—531no—
427#102—531no—
428#102C2×#615711yescharacteristic
429#102C2×#615711yescharacteristic
430C22⋊C8—531no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8—1061no—
437C22⋊C8—1061no—
438C22⋊C8—1061no—
439C22⋊C8C2×#615711yescharacteristic
440#97—531no—
441#97—531no—
442#97—1061no—
443C4×C8—531no—
444C4×C8—531no—
445C4×C8—531no—
446C4×C8—531no—
447C4×C8—531no—
448C4×C8—531no—
449C4×C8—1061no—
450C4×C8—1061no—
451C24—1061no—
452C2×Q8—531no—
453C2×D4—21no—
454C2×D4—531no—
455C2×D4—531no—
456C2×D4—1062no—
457C2×D4—1061no—
458C2×D4—2121no—
459C2×D4C22×#615711yescharacteristic
460C22×C4—21no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—2122no—
469C22×C4—2122no—
470C22×C4C22×#615711yescharacteristic
471SD16—1062no—
472SD16—2121no—
473D8—1062no—
474D8—2121no—
475M16—1062no—
476M16—2122no—
477M16—2121no—
478M16—2122no—
479M16—2122no—
480M16—2122no—
481C2×C8—531no—
482C2×C8—531no—
483C2×C8—531no—
484C2×C8—531no—
485C2×C8—531no—
486C2×C8—531no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—2122no—
494C2×C8—2122no—
495C2×C8—2122no—
496C2×C8C22×#615711yescharacteristic
497C2×C8C22×#615711yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—531no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4C22×#615711yescharacteristic
505C4⋊C4C22×#615711yescharacteristic
506C4⋊C4C22×#615711yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C22⋊C4—1061no—
513C22⋊C4—1061no—
514C22⋊C4—1061no—
515C22⋊C4—1061no—
516C42—531no—
517C42—531no—
518C42—531no—
519C42—531no—
520C42—1061no—
521C23—21no—
522C23—1061no—
523C23—1061no—
524C23—4241no—
525C23—2122no—
526C23—2122no—
527C23D4×#615711yescharacteristic
528Q8—1061no—
529Q8—1061no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—532no—
534D4—1061no—
535D4—1061no—
536D4—1061no—
537D4—2122no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—42no—
544C2×C4—531no—
545C2×C4—531no—
546C2×C4—531no—
547C2×C4—531no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4D4×#615711yescharacteristic
562C2×C4C23×#615711yescharacteristic
563C8—21no—
564C8—21no—
565C8—532no—
566C8—1061no—
567C8—1061no—
568C8—1061no—
569C8—1061no—
570C8—1061no—
571C8—2121no—
572C8—2121no—
573C8—2121no—
574C8—2121no—
575C8—2121no—
576C8—2121no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—1062no—
584C22—4244no—
585C22—2121no—
586C22—2121no—
587C22—2121no—
588C22—2121no—
589C22—2121no—
590C22C2×D4×#615711yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—531no—
596C4—531no—
597C4—1061no—
598C4—1061no—
599C4—1061no—
600C4—2121no—
601C4—2121no—
602C4D53⋊#13111yescharacteristic
603C4#1681170211yescharacteristic
604C2—41no—
605C2—81no—
606C2—4241no—
607C2—2122no—
608C2#2029144911yescharacteristic
609C2AGL(1,ℤ/8)×#615711yescharacteristic
610C2#2029149311yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#481C4
socleC22×C53C2×C4×D4
prime radical#16810756C4
cosocleC4×Dic53C24
hypercenter#481#6157
hypocenterC53#18006
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #20291546C2semidirect
2. #20291455C2semidirect
3. #20290363C2semidirect
4. #20290350C2semidirect
5. #20290343C2semidirect
6. #20290337C2semidirect
7. #20290319C2semidirect
8. #20290314C2semidirect
9. #20290307C2semidirect
10. #20290295C2semidirect
11. #20290291C2semidirect
12. #20290070C2semidirect
13. #20290016C2semidirect
14. #20289932C2semidirect
17. #16811673C22semidirect
18. #16811672C22semidirect
20. #16811658C22semidirect
28. D4×#88324C22semidirect
36. #16811487C22semidirect
37. #16811486C22semidirect
38. #16811485C22semidirect
39. #16811484C22semidirect
40. #16811473C22semidirect
41. #16811473C22semidirect
46. D4⋊#88324C22semidirect
47. D4⋊#88324C22semidirect
48. #88324⋊C8C22semidirect
49. #88324⋊C8C22semidirect
50. #16811464C22semidirect
51. C8×#88324C22semidirect
372. C53#18006semidirect

Maximal subgroups

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

subgroupindex
#202915462
#202914552
#202903632
#202903502
#202903432
#202903372
#202903192
#202903142
#202903072
#202902952
#202902912
#202900702
#202900162
#202899322
#202847592
#1800653