Finite Group Databasebeta

← #27388739

Subgroups — #27388739

Order 13568.

1338 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811248C211yesmaximal, characteristic
2#20277328C211yesmaximal, characteristic
3#20277067C211yesmaximal, characteristic
4#20275701C211yesmaximal, characteristic, complemented
5#20275700C211yesmaximal, characteristic, complemented
6#20275699C211yesmaximal, characteristic, complemented
7#20275698C214yesmaximal, complemented
8#342×D53C214yesmaximal
9C53×#1996C211yesmaximal, characteristic, complemented
10#16811248—22no—
11#16811248—22no—
12#16811248—22no—
13#16811248—22no—
14C2×#13133197C412yescomplemented
15C2×#13133196C2211yescharacteristic
16C2×#13133195C412yescomplemented
17C2×#13133195C412yescomplemented
18C2×#13133194C2212yes—
19C2×#13133193C412yescomplemented
20C22×#11719921C2211yescharacteristic
21C2×C22⋊C4×D53C2212yes—
22C2×C22⋊C4×D53C2212yes—
23C2×C22⋊C4×D53C2212yes—
24#16810497—22no—
25#16810494—22no—
26#16810483C2211yescharacteristic
27#16810481C2212yes—
28C22⋊C4×Dic53C2212yes—
29C4×#11719921—22no—
30C4×C22⋊Dic53C2211yescharacteristic
31#16810470—22no—
32#16810390C2214yes—
33#16810389C2214yes—
34#16810389C2214yes—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C4×C22⋊C4×C53C2211yescharacteristic
38C53×#342C2214yes—
39C24×D53C2311yescharacteristic
40C22×Dic106C2×C412yescomplemented
41C22×Dic106C2×C411yescharacteristic, complemented
42C23×Dic53C2311yescharacteristic
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53C2×C412yes—
46C22×C4×D53C2×C412yes—
47#13133197—24no—
48#13133197—24no—
49#13133196—21no—
50#13133196—21no—
51#13133196—21no—
52#13133196—21no—
53#13133195—24no—
54#13133195—24no—
55#13133195—24no—
56#13133195—24no—
57#13133194—24no—
58#13133194—24no—
59#13133193—24no—
60#13133193—24no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2×C412yescomplemented
63C2×C4⋊Dic53C2×C411yescharacteristic, complemented
64C2×#11719919—22no—
65C2×#11719919C2×C412yescomplemented
66C2×#11719919C2×C412yescomplemented
67C2×#11719919C2×C412yescomplemented
68C2×C22⋊Dic53C2312yes—
69C2×C22⋊Dic53C2312yes—
70C2×C22⋊Dic53C2312yes—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921C2×C412yes—
74C2×#11719921C2×C412yes—
75C2×#11719921C2×C412yes—
76C2×#11719921C2×C412yes—
77C2×#11719921D412yes—
78C2×#11719921D412yes—
79C2×#11719921D412yes—
80C2×#11719921D412yes—
81C22⋊C4×D53—24no—
82C22⋊C4×D53—24no—
83C22⋊C4×D53—24no—
84C22⋊C4×D53—28no—
85C22⋊C4×D53Q818yescomplemented
86C2×C4×Dic53—21no—
87C2×C4×Dic53—22no—
88#13133124—21no—
89#13133124—21no—
90#13133124—22no—
91#13133124—22no—
92C22×Q8×C53C2×C411yescharacteristic, complemented
93C23×C4×C53C2311yescharacteristic
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C22⋊Q8×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
100C2×C4⋊C4×C53C2×C412yescomplemented
101C2×C22⋊C4×C53C2312yes—
102C2×C22⋊C4×C53C2312yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C42×C53—21no—
105#95×C53—21no—
106#95×C53—21no—
107C24×C53C2411yescharacteristic
108C23×D53—81no—
109C23×D53—81no—
110C23×D53C22×C412yes—
111C23×D53C2×D412yes—
112C23×D53C2×D412yes—
113C23×D53C2×Q814yes—
114C23×D53#4114yes—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—81no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—81no—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—81no—
132C22×Dic53C22×C412yes—
133C22×Dic53C22×C412yes—
134C22×Dic53C22×C412yes—
135C22×Dic53C2×D412yes—
136C22×Dic53C2×D412yes—
137C2×C4×D53—24no—
138C2×C4×D53—24no—
139C2×C4×D53—28no—
140C2×C4×D53—44no—
141C2×C4×D53—82no—
142C2×C4×D53—82no—
143C2×C4×D53—82no—
144C2×C4×D53—82no—
145C2×C4×D53—82no—
146C2×C4×D53—82no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—24no—
150C2×Dic106—82no—
151C2×Dic106—81no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—81no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53C2×Q814yes—
162C42×C53—81no—
163C22⋊Dic53—22no—
164C22⋊Dic53—22no—
165C22⋊Dic53—22no—
166C22⋊Dic53—24no—
167C22⋊Dic53C2×Q814yes—
168#11719921—24no—
169#11719921—24no—
170#11719921—28no—
171#11719921—44no—
172#11719921C22⋊C414yes—
173#11719921C22⋊C414yes—
174#11719921C22⋊C414yes—
175#11719921C22⋊C414yes—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—24no—
179C4⋊Dic53—81no—
180#11719919—22no—
181#11719919—22no—
182#11719919—24no—
183#11719919—24no—
184#11719919—82no—
185C4×Dic53—82no—
186C4×Dic53—81no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53C23×C411yescharacteristic
190C23×C53C22×D411yescharacteristic
191C23×C53C22×D411yescharacteristic
192C23×C53C22×Q812yes—
193C23×C53C2×#4112yes—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—84no—
201C22×D53—84no—
202C22×D53C2×C22⋊C414yes—
203C22×D53C4×Q818yes—
204C22×D53C22⋊Q814yes—
205C22×D53C22⋊Q814yes—
206C22×D53#12414yes—
207C22×D53C4⋊Q814yes—
208Q8×C53—81no—
209Q8×C53—81no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—24no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—24no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53C2×C22⋊C414yes—
244C4×D53—84no—
245C4×D53—84no—
246C4×D53—88no—
247C4×D53—88no—
248Dic106—84no—
249Dic106—81no—
250Dic106—81no—
251#21333—531nomaximal, Sylow
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53C22×C22⋊C411yescharacteristic
261C22×C53C2×C4×Q812yes—
262C22×C53C2×C22⋊Q812yes—
263C22×C53C2×#12411yescharacteristic
264C22×C53C2×C4⋊Q811yescharacteristic
265C2×D53—24no—
266C2×D53—24no—
267C2×D53—44no—
268C2×D53—44no—
269C2×D53—48no—
270C2×D53—44no—
271C2×D53—44no—
272C2×D53#39118yes—
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—44no—
278C4×C53—42no—
279C4×C53—42no—
280C4×C53—82no—
281Dic53—24no—
282Dic53—44no—
283Dic53—42no—
284Dic53—42no—
285Dic53—82no—
286C2×#543—531no—
287#2315—531no—
288#2273—531no—
289#1996—531no—
290#1996—531no—
291#1996—534no—
292#1996—531no—
293#1996D5311yescharacteristic, complemented
294C2×#342—534no—
295C2×C53—22no—
296C2×C53—21no—
297C2×C53—22no—
298C2×C53—21no—
299C2×C53C2×#39111yescharacteristic
300C2×C53#288311yescharacteristic
301C2×C53#288311yescharacteristic
302D53—28no—
303#543—1062no—
304#543—1062no—
305#543—1062no—
306#543—1062no—
307C2×C22⋊Q8—532no—
308C2×C22⋊Q8—531no—
309C2×C22⋊Q8—532no—
310C2×C22⋊Q8—532no—
311C2×C22⋊Q8—532no—
312C2×C22⋊Q8—532no—
313C2×C22⋊Q8C2×D5311yescharacteristic
314C22×C22⋊C4—531no—
315C22×C22⋊C4—532no—
316C22×C22⋊C4—532no—
317C22×C22⋊C4—532no—
318#385—1062no—
319#385—1062no—
320#380—531no—
321#380—532no—
322#380C2×D5311yescharacteristic
323C4×C22⋊C4—531no—
324C4×C22⋊C4—532no—
325C4×C22⋊C4—1062no—
326C4×C22⋊C4—1062no—
327C4×C22⋊C4C2×D5311yescharacteristic
328#342—534no—
329#342—534no—
330#342—534no—
331#342C2×D5314yes—
332C53#2133311yescharacteristic, complemented, Sylow
333C25—531no—
334C22×Q8—532no—
335C22×Q8—531no—
336C22×Q8C4×D5311yescharacteristic, complemented
337C23×C4—531no—
338C23×C4—532no—
339C23×C4—532no—
340C23×C4—1062no—
341C23×C4—1062no—
342C23×C4C22×D5311yescharacteristic
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—1064no—
348C22⋊Q8—1064no—
349C22⋊Q8—1064no—
350C22⋊Q8—1064no—
351C22⋊Q8—1064no—
352C22⋊Q8—1064no—
353C22⋊Q8—1064no—
354C22⋊Q8—1064no—
355C22⋊Q8—1064no—
356C22⋊Q8—1064no—
357C22⋊Q8—1061no—
358C22⋊Q8—1061no—
359C22⋊Q8—1061no—
360C22⋊Q8—1061no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—532no—
363C2×C4⋊C4—532no—
364C2×C4⋊C4—532no—
365C2×C4⋊C4—532no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4C4×D5312yescomplemented
370C2×C4⋊C4C4×D5311yescharacteristic, complemented
371C2×C22⋊C4—532no—
372C2×C22⋊C4—532no—
373C2×C22⋊C4—532no—
374C2×C22⋊C4—532no—
375C2×C22⋊C4—532no—
376C2×C22⋊C4—532no—
377C2×C22⋊C4—532no—
378C2×C22⋊C4—538no—
379C2×C22⋊C4—532no—
380C2×C22⋊C4—532no—
381C2×C22⋊C4—532no—
382C2×C22⋊C4—532no—
383C2×C22⋊C4—1064no—
384C2×C22⋊C4—1064no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1068no—
387C2×C22⋊C4—1062no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4C22×D5312yes—
390C2×C22⋊C4C22×D5312yes—
391C2×C22⋊C4C22×D5312yes—
392C2×C42—21no—
393C2×C42—1062no—
394C2×C42—1061no—
395#95—21no—
396#95—21no—
397#95—1062no—
398#95—1062no—
399#95—1061no—
400#95—1061no—
401C24—534no—
402C24—534no—
403C24—532no—
404C24—532no—
405C24—532no—
406C24—4241no—
407C24—4241no—
408C24C23×D5311yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—81no—
412C2×Q8—1064no—
413C2×Q8—1061no—
414C2×Q8—1061no—
415C2×Q8—4241no—
416C2×Q8—4242no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—81no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—1068no—
426C22×C4—1064no—
427C22×C4—1064no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—4242no—
431C22×C4—4242no—
432C22×C4—4242no—
433C22×C4—4242no—
434C22×C4—4242no—
435C22×C4—4242no—
436C22×C4—4241no—
437C22×C4—2124no—
438C22×C4C2×C4×D5312yes—
439C22×C4C2×C4×D5312yes—
440C22×C4C2×C4×D5311yescharacteristic
441C22×C4C2×C4×D5311yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4D4×D5311yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—81no—
451C4⋊C4—1064no—
452C4⋊C4—1064no—
453C4⋊C4—1064no—
454C4⋊C4—1062no—
455C4⋊C4—1062no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—4242no—
459C4⋊C4—4241no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—534no—
465C22⋊C4—534no—
466C22⋊C4—534no—
467C22⋊C4—534no—
468C22⋊C4—534no—
469C22⋊C4—1064no—
470C22⋊C4—1064no—
471C22⋊C4—1068no—
472C22⋊C4—1062no—
473C22⋊C4—1062no—
474C22⋊C4—1062no—
475C22⋊C4—1064no—
476C22⋊C4—2124no—
477C22⋊C4Q8×D5314yes—
478C42—81no—
479C42—4241no—
480C42—4242no—
481C23—41no—
482C23—41no—
483C23—534no—
484C23—538no—
485C23—534no—
486C23—534no—
487C23—534no—
488C23—534no—
489C23—4244no—
490C23—4244no—
491C23—2122no—
492C23—2122no—
493C23—2122no—
494C23—2122no—
495C23—2122no—
496C23—2122no—
497C23C22×C4×D5311yescharacteristic
498C23C2×D4×D5311yescharacteristic
499C23C2×D4×D5311yescharacteristic
500C23C2×Q8×D5312yes—
501C23#41×D5312yes—
502Q8—81no—
503Q8—81no—
504Q8—4241no—
505Q8—4241no—
506Q8—4244no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—534no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1064no—
527C2×C4—4244no—
528C2×C4—4244no—
529C2×C4—4248no—
530C2×C4—4248no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4C22⋊C4×D5311yescharacteristic
542C2×C4C22⋊C4×D5311yescharacteristic
543C2×C4C22⋊C4×D5311yescharacteristic
544C2×C4C22⋊C4×D5311yescharacteristic
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—538no—
554C22—1064no—
555C22—1064no—
556C22—2124no—
557C22—2124no—
558C22—2124no—
559C22—2128no—
560C22—2124no—
561C22C2×C22⋊C4×D5311yescharacteristic
562C22C4×Q8×D5312yes—
563C22C22⋊Q8×D5312yes—
564C22#124×D5311yescharacteristic
565C22C4⋊Q8×D5311yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—1064no—
575C4—4242no—
576C4—2124no—
577C4—2122no—
578C4—2122no—
579C2—22no—
580C2—21no—
581C2—22no—
582C2—21no—
583C2—1068no—
584C2#391×D5311yescharacteristic
585C2#2028317311yescharacteristic
586C2#2028317411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1996C2
socleC22×C53C22×C22⋊C4
prime radicalC24×D53C23
cosocleC24×C53C24
hypercenter#1996D53
hypocenterC53#21333
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. #20275701C2semidirect
5. #20275700C2semidirect
6. #20275699C2semidirect
7. #20275698C2semidirect
9. C53×#1996C2semidirect
14. C2×#13133197C4semidirect
16. C2×#13133195C4semidirect
17. C2×#13133195C4semidirect
19. C2×#13133193C4semidirect
40. C22×Dic106C2×C4semidirect
41. C22×Dic106C2×C4semidirect
62. C2×C4⋊Dic53C2×C4semidirect
63. C2×C4⋊Dic53C2×C4semidirect
65. C2×#11719919C2×C4semidirect
66. C2×#11719919C2×C4semidirect
67. C2×#11719919C2×C4semidirect
85. C22⋊C4×D53Q8semidirect
92. C22×Q8×C53C2×C4semidirect
99. C2×C4⋊C4×C53C2×C4semidirect
100. C2×C4⋊C4×C53C2×C4semidirect
293. #1996D53semidirect
332. C53#21333semidirect
336. C22×Q8C4×D53semidirect
369. C2×C4⋊C4C4×D53semidirect
370. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112482
#202773282
#202770672
#202757012
#202757002
#202756992
#202756982
#342×D532
C53×#19962
#2133353