Finite Group Databasebeta

← #28314100

Subgroups — #28314100

Order 13568.

1346 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291579C212yesmaximal, complemented
2#20289972C212yesmaximal, complemented
3#20289971C212yesmaximal, complemented
4#20289962C212yesmaximal, complemented
5#20289961C212yesmaximal, complemented
6#20289954C212yesmaximal, complemented
7#20289949C212yesmaximal, complemented
8C2×#16811095C211yesmaximal, characteristic
9#16811656—28no—
10#16811640C2214yescomplemented
11#16811594C2214yescomplemented
12#16811593C2214yescomplemented
13#16811468C2214yescomplemented
14#16811467C2214yes—
15#16811467C2214yescomplemented
16#16811466C2214yescomplemented
17C22×D4×D53C411yescharacteristic
18#16811095—28no—
19C2×C4×D212C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133179C411yescharacteristic
22C2×C4×C22⋊D53C411yescharacteristic
23C2×C4×C22⋊D53C2211yescharacteristic
24C2×#13133177C411yescharacteristic
25C2×#13133177C2211yescharacteristic
26C2×#13133175C2211yescharacteristic
27C2×#13133174C411yescharacteristic
28C22×#11719921C411yescharacteristic
29C22×#11719921C2211yescharacteristic
30C2×C22⋊C4×D53C411yescharacteristic
31C2×C22⋊C4×D53C2211yescharacteristic
32C2×C4×D4×C53C411yescharacteristic
33C2×#11719936—24no—
34C2×#11719940—24no—
35#13133284—24no—
36#13133274—24no—
37C22⋊#88324—24no—
38C22⋊#88324C2318yes—
39C4×#88324—24no—
40C24×D53C2×C411yescharacteristic
41C24×D53C2×C411yescharacteristic
42C22×D212C2×C411yescharacteristic
43C22×C22⋊D53C2×C411yescharacteristic
44C22×C22⋊D53C2×C411yescharacteristic
45C2×D4×D53—24no—
46C2×D4×D53—24no—
47C2×D4×D53—24no—
48C23×Dic53C2×C411yescharacteristic
49C23×Dic53C2311yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C4×D212—24no—
56D4×Dic53—24no—
57#13133179—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60#13133177—24no—
61#13133177D418yes—
62#13133175—24no—
63#13133174—24no—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×#11719919C2×C411yescharacteristic
66C2×#11719919C2×C411yescharacteristic
67C2×C22⋊Dic53C2×C411yescharacteristic
68C2×C22⋊Dic53C2311yescharacteristic
69C2×#11719921—24no—
70C2×#11719921—24no—
71C2×#11719921C2×C411yescharacteristic
72C2×#11719921C2×C411yescharacteristic
73C2×#11719921C2×C411yescharacteristic
74C2×#11719921C2×C411yescharacteristic
75C2×#11719921C2×C411yescharacteristic
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2311yescharacteristic
78C2×#11719921C2311yescharacteristic
79C22⋊C4×D53—24no—
80C22⋊C4×D53—24no—
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C22×D4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C411yescharacteristic
87C4×D4×C53—24no—
88C2×C4⋊C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53C2×C411yescharacteristic
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C42×C53C2×C411yescharacteristic
92C24×C53C22×C411yescharacteristic
93C24×C53C22×C411yescharacteristic
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—81no—
101C23×D53—81no—
102C23×D53—81no—
103C23×D53—81no—
104C23×D53C22×C411yescharacteristic
105C23×D53C22×C411yescharacteristic
106C23×D53C22×C411yescharacteristic
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—81no—
113C22×C4×C53—81no—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C22×C411yescharacteristic
116C22×C4×C53C22×C411yescharacteristic
117C22×C4×C53C22×C411yescharacteristic
118C22×C4×C53C22×C411yescharacteristic
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C2×C22⋊D53C22⋊C414yes—
124C2×C22⋊D53M1614yescomplemented
125C22×Dic53—22no—
126C22×Dic53—81no—
127C22×Dic53—81no—
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C2×D414yes—
133C22×Dic53C2411yescharacteristic
134D4×D53—232no—
135C2×D212—22no—
136C2×D212—22no—
137C2×D212—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—81no—
144C2×C4×D53—81no—
145C2×C4×D53—81no—
146C2×C4×D53—81no—
147C2×C4×D53M1614yescomplemented
148C2×C4×D53C2×D414yes—
149#11719940—84no—
150C2×#88324—28no—
151#11719936—84no—
152C4⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53C22⋊C414yes—
155C42×C53—22no—
156C22⋊Dic53—22no—
157C22⋊Dic53—22no—
158#11719921—22no—
159#11719921—22no—
160#11719921—22no—
161#11719921—22no—
162#11719921—22no—
163#11719921—22no—
164#11719921—22no—
165#11719921C22⋊C414yes—
166C4⋊Dic53—22no—
167#11719919—22no—
168#11719919C22⋊C414yes—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53C2×D414yes—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53C2×C22⋊C412yes—
181C23×C53C2×M1612yes—
182C23×C53C23×C411yescharacteristic
183C22×D53—21no—
184C22×D53—21no—
185C22×D53—21no—
186C22×D53—21no—
187C22×D53—21no—
188C22×D53—21no—
189C22×D53—28no—
190C22×D53—28no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—82no—
198C22×D53—82no—
199C22×D53—82no—
200C22×D53—82no—
201C22×D53—84no—
202C22×D53—84no—
203C22×D53—84no—
204C22×D53—84no—
205C22×D53C2×C22⋊C412yes—
206C22×D53C2×M1612yes—
207C22×D53C2×M1612yes—
208D4×C53—28no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—82no—
218C2×C4×C53—82no—
219C2×C4×C53C2×C22⋊C412yes—
220C2×C4×C53C2×C22⋊C412yes—
221C2×C4×C53C2×M1612yes—
222C22⋊D53—28no—
223C22⋊D53—28no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—44no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—82no—
232C2×Dic53C2×C22⋊C412yes—
233C2×Dic53C2×C22⋊C412yes—
234C2×Dic53C2×M1612yes—
235C2×Dic53C2×M1612yes—
236C2×Dic53C22×D412yes—
237D212—28no—
238C4×D53—28no—
239C4×D53—44no—
240C4×D53—82no—
241C4×D53—82no—
242C4×D53—82no—
243#88324—48no—
244#17369—531nomaximal, Sylow
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53C22×C22⋊C411yescharacteristic
258C22×C53C22×M1611yescharacteristic
259C22×C53C22×M1611yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—42no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—42no—
269C2×D53—41no—
270C2×D53—41no—
271C2×D53—41no—
272C2×D53—41no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—88no—
276C2×D53—88no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282Dic53—22no—
283Dic53—22no—
284Dic53—41no—
285Dic53—41no—
286Dic53#40714yes—
287C2×#518—531no—
288#2975—532no—
289#1465—532no—
290#1465—532no—
291#1462—532no—
292#1462—532no—
293#1460—532no—
294#1458—532no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—42no—
298C2×C53—42no—
299C2×C53C2×#40711yescharacteristic
300C2×C53#288311yescharacteristic
301C2×C53#288811yescharacteristic
302D53—42no—
303D53—42no—
304D53—42no—
305C23×D4—531no—
306#518—1068no—
307C2×C4×D4—531no—
308C2×C4×D4—531no—
309C2×C4×D4—531no—
310C2×C4×D4—531no—
311C2×C4×D4—531no—
312C2×C4×D4—531no—
313C2×C4×D4—531no—
314C2×C4×D4#615711yescharacteristic
315C2×#117—531no—
316C2×#117—531no—
317C22×C22⋊C4—531no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320C22×C22⋊C4—531no—
321#436—1068no—
322#431—534no—
323#407—534no—
324#407—534no—
325#324—534no—
326#323—534no—
327#323—534no—
328#323—534no—
329C53#1736911yescharacteristic, complemented, Sylow
330C25—531no—
331C25—531no—
332C22×D4—531no—
333C22×D4—531no—
334C22×D4—531no—
335C22×D4—1064no—
336C22×D4—1064no—
337C22×D4—1064no—
338C22×D4C2×#615711yescharacteristic
339C23×C4—531no—
340C23×C4—531no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4C2×#615711yescharacteristic
347C23×C4C2×#615711yescharacteristic
348C2×M16—1064no—
349C2×M16—1064no—
350C4×D4—24no—
351C4×D4—538no—
352C4×D4—1064no—
353C4×D4—1064no—
354C4×D4—1064no—
355C4×D4—1064no—
356C4×D4—1064no—
357C4×D4—1064no—
358#117—1064no—
359#117—1064no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4C2×#615711yescharacteristic
364C2×C22⋊C4—531no—
365C2×C22⋊C4—531no—
366C2×C22⋊C4—531no—
367C2×C22⋊C4—531no—
368C2×C22⋊C4—531no—
369C2×C22⋊C4—531no—
370C2×C22⋊C4—531no—
371C2×C22⋊C4—531no—
372C2×C22⋊C4—531no—
373C2×C22⋊C4—531no—
374C2×C22⋊C4—1064no—
375C2×C22⋊C4—1064no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4C2×#615711yescharacteristic
379C2×C22⋊C4C2×#615711yescharacteristic
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42C2×#615711yescharacteristic
384C4⋊C8—1064no—
385C22⋊C8—538no—
386C22⋊C8—1064no—
387C22⋊C8—1064no—
388C4×C8—1064no—
389C24—531no—
390C24—531no—
391C24—531no—
392C24—1062no—
393C24—1062no—
394C24—1062no—
395C24—1062no—
396C24—1062no—
397C24—1062no—
398C24—4241no—
399C24—4241no—
400C24—4241no—
401C24—4241no—
402C24C22×#615711yescharacteristic
403C24C22×#615711yescharacteristic
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—534no—
408C2×D4—534no—
409C2×D4—1062no—
410C2×D4—1062no—
411C2×D4—1062no—
412C2×D4—1062no—
413C2×D4—1062no—
414C2×D4—1062no—
415C2×D4—10632no—
416C2×D4—1062no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—531no—
422C22×C4—534no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—534no—
426C22×C4—534no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—4241no—
436C22×C4—4241no—
437C22×C4—4241no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—4241no—
441C22×C4C22×#615711yescharacteristic
442C22×C4C22×#615711yescharacteristic
443C22×C4C22×#615711yescharacteristic
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446M16—4244no—
447M16—4244no—
448C2×C8—1068no—
449C4⋊C4—22no—
450C4⋊C4—534no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C22⋊C4—22no—
454C22⋊C4—534no—
455C22⋊C4—1062no—
456C22⋊C4—1062no—
457C22⋊C4—1062no—
458C22⋊C4—1062no—
459C22⋊C4—1062no—
460C22⋊C4—1062no—
461C22⋊C4—1062no—
462C22⋊C4—1062no—
463C22⋊C4—1062no—
464C22⋊C4C22⋊#615714yes—
465C42—22no—
466C42—534no—
467C42—1062no—
468C42—1062no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—532no—
478C23—532no—
479C23—532no—
480C23—1068no—
481C23—1068no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23—4242no—
489C23—4242no—
490C23—4242no—
491C23—4242no—
492C23—4244no—
493C23—4244no—
494C23—4244no—
495C23—4244no—
496C23—2121no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23C2×C22⋊#615712yes—
503C23D53⋊M1612yes—
504C23C23×#615711yescharacteristic
505D4—28no—
506D4—1068no—
507D4—1068no—
508D4—1068no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—1068no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—4242no—
528C2×C4—4242no—
529C2×C4—4242no—
530C2×C4—4242no—
531C2×C4—2124no—
532C2×C4—2124no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4C2×C22⋊#615712yes—
538C2×C4C2×C22⋊#615712yes—
539C2×C4D53⋊M1612yes—
540C8—2128no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—41no—
553C22—1062no—
554C22—1062no—
555C22—1062no—
556C22—1062no—
557C22—1062no—
558C22—4248no—
559C22—4248no—
560C22—2122no—
561C22—2122no—
562C22—2122no—
563C22—2122no—
564C22—2122no—
565C22—2122no—
566C22—2121no—
567C22—2121no—
568C22—2121no—
569C22—2121no—
570C22C22×C22⋊#615711yescharacteristic
571C22C2×D53⋊M1611yescharacteristic
572C22C2×D53⋊M1611yescharacteristic
573C4—22no—
574C4—22no—
575C4—22no—
576C4—41no—
577C4—41no—
578C4—534no—
579C4—1062no—
580C4—1062no—
581C4—2121no—
582C4—2121no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—42no—
587C2—2122no—
588C2—2122no—
589C2—2122no—
590C2D53⋊#40711yescharacteristic
591C2#2029131511yescharacteristic
592C2#2029132511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C4
cosocleC22×Dic53C24
hypercenterC2×C4×D4#6157
hypocenterC53#17369
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20291579C2semidirect
2. #20289972C2semidirect
3. #20289971C2semidirect
4. #20289962C2semidirect
5. #20289961C2semidirect
6. #20289954C2semidirect
7. #20289949C2semidirect
10. #16811640C22semidirect
11. #16811594C22semidirect
12. #16811593C22semidirect
13. #16811468C22semidirect
15. #16811467C22semidirect
16. #16811466C22semidirect
124. C2×C22⋊D53M16semidirect
147. C2×C4×D53M16semidirect
329. C53#17369semidirect

Maximal subgroups

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

subgroupindex
#202915792
#202899722
#202899712
#202899622
#202899612
#202899542
#202899492
C2×#168110952
#1736953