Finite Group Databasebeta

← #28314080

Subgroups — #28314080

Order 13568.

1350 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#20291566C212yesmaximal, complemented
2#20289975C212yesmaximal, complemented
3#20289971C212yesmaximal, complemented
4#20289966C212yesmaximal, complemented
5#20289960C212yesmaximal, complemented
6#20289951C212yesmaximal, complemented
7C2×#16811466C212yesmaximal, complemented
8C2×#16811098C211yesmaximal, characteristic
9#16811654—28no—
10#16811643C2214yescomplemented
11#16811593C2214yescomplemented
12C2×C22⋊#88324C2214yescomplemented
13#16811468C2214yescomplemented
14#16811466C2214yes—
15#16811466C2214yescomplemented
16#16811466C2214yescomplemented
17#16811098—28no—
18C2×D4×Dic53C411yescharacteristic
19C2×D4×Dic53C411yescharacteristic
20C2×D4×Dic53C411yescharacteristic
21C2×D4×Dic53C411yescharacteristic
22C2×D4×Dic53C2211yescharacteristic
23C2×D4×Dic53C2211yescharacteristic
24C2×D4×Dic53C2211yescharacteristic
25C2×D4×Dic53C2211yescharacteristic
26C2×#13133176C411yescharacteristic
27C2×#13133176C2211yescharacteristic
28C22×C22⋊Dic53C411yescharacteristic
29C22×C22⋊Dic53C411yescharacteristic
30C22×C22⋊Dic53C2211yescharacteristic
31C22×C22⋊Dic53C2211yescharacteristic
32C23×D4×C53C411yescharacteristic
33C2×#11719940—24no—
34C22×#88324—24no—
35#13133284—24no—
36C22⋊#88324—24no—
37C22⋊#88324—24no—
38C22⋊#88324C2318yes—
39#13133272—24no—
40C23×Dic53C2×C411yescharacteristic
41C23×Dic53C2×C411yescharacteristic
42C23×Dic53C2×C411yescharacteristic
43C23×Dic53C2×C411yescharacteristic
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48D4×Dic53—24no—
49D4×Dic53—24no—
50D4×Dic53—24no—
51D4×Dic53—24no—
52D4×Dic53—24no—
53D4×Dic53—24no—
54D4×Dic53—24no—
55D4×Dic53D418yes—
56#13133176—24no—
57#13133176—24no—
58C2×C4⋊Dic53C2×C411yescharacteristic
59C2×C4⋊Dic53C2×C411yescharacteristic
60C2×C4⋊Dic53C2×C411yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53—24no—
64C2×C22⋊Dic53—24no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53C2×C411yescharacteristic
67C2×C22⋊Dic53C2×C411yescharacteristic
68C2×C22⋊Dic53C2×C411yescharacteristic
69C2×C22⋊Dic53C2×C411yescharacteristic
70C2×C22⋊Dic53C2×C411yescharacteristic
71C2×C22⋊Dic53C2×C411yescharacteristic
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×C22⋊Dic53C2×C411yescharacteristic
75C2×C22⋊Dic53C2311yescharacteristic
76C2×C22⋊Dic53C2311yescharacteristic
77C2×C22⋊Dic53C2311yescharacteristic
78C2×C4×Dic53C2×C411yescharacteristic
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C25×C53C2×C411yescharacteristic
83C25×C53C2×C411yescharacteristic
84C22×D4×C53—24no—
85C22×D4×C53—24no—
86C22×D4×C53—24no—
87C22×D4×C53C2×C411yescharacteristic
88C22×D4×C53C2×C411yescharacteristic
89C22×D4×C53C2×C411yescharacteristic
90C22×D4×C53C2×C411yescharacteristic
91C23×C4×C53C2×C411yescharacteristic
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—81no—
98C24×C53—81no—
99C24×C53—81no—
100C24×C53—81no—
101C24×C53M1614yescomplemented
102C24×C53C22×C411yescharacteristic
103C24×C53C22×C411yescharacteristic
104C24×C53C22×C411yescharacteristic
105C24×C53C22×C411yescharacteristic
106C24×C53C22×C411yescharacteristic
107C2×D4×C53—232no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—22no—
114C2×D4×C53—22no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53C22⋊C414yes—
119C2×D4×C53M1614yescomplemented
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—81no—
132C22×Dic53—81no—
133C22×Dic53—81no—
134C22×Dic53—81no—
135C22×Dic53—81no—
136C22×Dic53—81no—
137C22×Dic53—81no—
138C22×Dic53—81no—
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C2×D414yes—
147C22×Dic53C2×D414yes—
148C22×Dic53C2411yescharacteristic
149#11719940—84no—
150C2×#88324—28no—
151C2×#88324—84no—
152C22⋊Dic53—22no—
153C22⋊Dic53—22no—
154C22⋊Dic53—22no—
155C22⋊Dic53—22no—
156C22⋊Dic53—22no—
157C22⋊Dic53—22no—
158C22⋊Dic53—22no—
159C22⋊Dic53—22no—
160C22⋊Dic53—22no—
161C22⋊Dic53—22no—
162C22⋊Dic53C22⋊C414yes—
163C22⋊Dic53C22⋊C414yes—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53C22⋊C414yes—
168C4×Dic53—22no—
169C4×Dic53—22no—
170C4×Dic53—22no—
171C4×Dic53C2×D414yes—
172C23×C53—28no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—21no—
180C23×C53—21no—
181C23×C53—28no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—82no—
193C23×C53—82no—
194C23×C53—82no—
195C23×C53—82no—
196C23×C53—84no—
197C23×C53—84no—
198C23×C53—84no—
199C23×C53—84no—
200C23×C53C2×C22⋊C412yes—
201C23×C53C2×C22⋊C412yes—
202C23×C53C2×M1612yes—
203C23×C53C2×M1612yes—
204C23×C53C2×M1612yes—
205C23×C53C2×M1612yes—
206C23×C53C2×M1612yes—
207C23×C53C23×C411yescharacteristic
208D4×C53—28no—
209D4×C53—28no—
210D4×C53—28no—
211D4×C53—28no—
212C2×C4×C53—28no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53C2×C22⋊C412yes—
218C2×C4×C53C2×M1612yes—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—44no—
224C2×Dic53—44no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—82no—
234C2×Dic53—82no—
235C2×Dic53—82no—
236C2×Dic53—82no—
237C2×Dic53—82no—
238C2×Dic53—82no—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C22⋊C412yes—
241C2×Dic53C2×C22⋊C412yes—
242C2×Dic53C22×D412yes—
243#88324—48no—
244#17367—531nomaximal, Sylow
245C22×C53—21no—
246C22×C53—22no—
247C22×C53—21no—
248C22×C53—22no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—88no—
272C22×C53—88no—
273C22×C53#40714yes—
274C22×C53C22×C22⋊C411yescharacteristic
275C22×C53C22×M1611yescharacteristic
276C22×C53C22×M1611yescharacteristic
277C4×C53—22no—
278C4×C53—22no—
279Dic53—22no—
280Dic53—22no—
281Dic53—22no—
282Dic53—22no—
283Dic53—41no—
284Dic53—41no—
285Dic53—41no—
286Dic53—41no—
287C2×#518—531no—
288#2973—532no—
289#1466—532no—
290#1465—532no—
291#1464—532no—
292#1462—532no—
293#1459—532no—
294C2×#323—532no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53—24no—
298C2×C53—42no—
299C2×C53—42no—
300C2×C53—42no—
301C2×C53—42no—
302C2×C53C2×#40711yescharacteristic
303C2×C53#288311yescharacteristic
304C2×C53#288811yescharacteristic
305C23×D4#615711yescharacteristic
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—531no—
315C2×#117—531no—
316C2×#117—531no—
317C22×C22⋊C4—531no—
318C22×C22⋊C4—531no—
319C22×C22⋊C4—531no—
320C22×C22⋊C4—531no—
321#435—1068no—
322#432—534no—
323#407—534no—
324C2×C22⋊C8—534no—
325#324—534no—
326#323—534no—
327#323—534no—
328#323—534no—
329C53#1736711yescharacteristic, complemented, Sylow
330C25C2×#615711yescharacteristic
331C25C2×#615711yescharacteristic
332C22×D4—24no—
333C22×D4—24no—
334C22×D4—24no—
335C22×D4C2×#615711yescharacteristic
336C22×D4C2×#615711yescharacteristic
337C22×D4C2×#615711yescharacteristic
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×C4—531no—
347C23×C4C2×#615711yescharacteristic
348C2×M16—1064no—
349C22×C8—1064no—
350C4×D4—538no—
351C4×D4—1064no—
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⋊C4—531no—
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—531no—
375C2×C22⋊C4—531no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C42—531no—
381C2×C42—531no—
382C2×C42—531no—
383C2×C42—531no—
384C4⋊C8—1064no—
385C22⋊C8—538no—
386C22⋊C8—1064no—
387C22⋊C8—1064no—
388#97—1064no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—81no—
395C24—81no—
396C24—81no—
397C24—81no—
398C24#1171994014yescomplemented
399C24C22×#615711yescharacteristic
400C24C22×#615711yescharacteristic
401C24C22×#615711yescharacteristic
402C24C22×#615711yescharacteristic
403C24C22×#615711yescharacteristic
404C2×D4—232no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4#1171994014yescomplemented
416C2×D4C22⋊#615714yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—534no—
421C22×C4—534no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—4241no—
437C22×C4—4241no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—4241no—
441C22×C4—4241no—
442C22×C4—4241no—
443C22×C4—4241no—
444C22×C4C22×#615711yescharacteristic
445C22×C4C22×#615711yescharacteristic
446M16—4244no—
447C2×C8—1068no—
448C2×C8—4244no—
449C4⋊C4—534no—
450C4⋊C4—1062no—
451C4⋊C4—1062no—
452C4⋊C4—1062no—
453C22⋊C4—534no—
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⋊C4—1062no—
465C42—534no—
466C42—1062no—
467C42—1062no—
468C42—1062no—
469C23—28no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—28no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23C2×C22⋊#615712yes—
498C23C2×C22⋊#615712yes—
499C23C2×#1171994012yes—
500C23C2×#1171994012yes—
501C23C2×#1171994012yes—
502C23C2×#1171994012yes—
503C23C2×#1171994012yes—
504C23C23×#615711yescharacteristic
505D4—28no—
506D4—28no—
507D4—28no—
508D4—28no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—532no—
515C2×C4—532no—
516C2×C4—532no—
517C2×C4—532no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—4242no—
523C2×C4—4242no—
524C2×C4—4242no—
525C2×C4—4242no—
526C2×C4—4242no—
527C2×C4—4242no—
528C2×C4—2124no—
529C2×C4—2124no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4C2×C22⋊#615712yes—
539C2×C4C2×#1171994012yes—
540C8—2128no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—88no—
568C22—88no—
569C22#1681159314yes—
570C22C22×C22⋊#615711yescharacteristic
571C22C22×#1171994011yescharacteristic
572C22C22×#1171994011yescharacteristic
573C4—22no—
574C4—22no—
575C4—1062no—
576C4—1062no—
577C4—1062no—
578C4—1062no—
579C4—2121no—
580C4—2121no—
581C4—2121no—
582C4—2121no—
583C2—21no—
584C2—21no—
585C2—24no—
586C2—42no—
587C2—42no—
588C2—42no—
589C2—42no—
590C2C2×#1681159311yescharacteristic
591C2#2029131411yescharacteristic
592C2#2029132511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
Fitting, prime radicalC23×D4×C53C4
socleC22×C53C22×C22⋊C4
cosocleC22×Dic53C24
hypercenterC23×D4#6157
hypocenterC53#17367
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×D4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #20291566C2semidirect
2. #20289975C2semidirect
3. #20289971C2semidirect
4. #20289966C2semidirect
5. #20289960C2semidirect
6. #20289951C2semidirect
7. C2×#16811466C2semidirect
10. #16811643C22semidirect
11. #16811593C22semidirect
12. C2×C22⋊#88324C22semidirect
13. #16811468C22semidirect
15. #16811466C22semidirect
16. #16811466C22semidirect
101. C24×C53M16semidirect
119. C2×D4×C53M16semidirect
329. C53#17367semidirect
398. C24#11719940semidirect
415. C2×D4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202915662
#202899752
#202899712
#202899662
#202899602
#202899512
C2×#168114662
C2×#168110982
#1736753