Finite Group Databasebeta

← #27384871

Subgroups — #27384871

Order 13568.

1462 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
1C2×#16811104C211yesmaximal, characteristic
2C2×#16810663C211yesmaximal, characteristic, complemented
3C2×#16810661C211yesmaximal, characteristic, complemented
4C2×#16810589C211yesmaximal, characteristic
5#20277308C211yesmaximal, characteristic
6#20274917C212yesmaximal, complemented
7#20274915C212yesmaximal, complemented
8#20274909C212yesmaximal, complemented
9#20274904C212yesmaximal, complemented
10#20274434C211yesmaximal, characteristic, complemented
11C2×C53×#340C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811104—24no—
15#16811104—24no—
16C2×#13133182—21no—
17C2×#13133179—21no—
18C2×C4×C22⋊D53—22no—
19C2×#13133176C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133175C2212yes—
22C2×C4⋊C4×D53C2212yes—
23#16810663—24no—
24#16810661—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133150C2212yes—
28C2×#13133149C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133148C2212yes—
31#16810589C418yescomplemented
32C2×#13133129C2212yes—
33C2×#13133129C2212yes—
34#16810528—21no—
35#16810524—21no—
36#16810502—22no—
37#16810480C2212yes—
38#16810480C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C23×C8×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C22×C4⋊C4×C53C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945C2×C414yes—
51C2×#11719945Q814yes—
52C22×Dic106D411yescharacteristic
53C22×D212D411yescharacteristic
54C22×C22⋊D53D412yes—
55C22×C4×D53—22no—
56C22×C4×D53C2312yes—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C2×#11719917C2×C414yescomplemented
60C22×#88322C2312yes—
61#13133182—28no—
62#13133179—28no—
63C4×C22⋊D53—216no—
64#13133176—22no—
65#13133176—22no—
66#13133176—22no—
67#13133176D414yes—
68#13133175—24no—
69#13133175—24no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—22no—
75C4⋊C4×D53—24no—
76C4⋊C4×D53—24no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—22no—
80C2×#11719921—22no—
81C2×#11719921—22no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—22no—
85#13133151—22no—
86#13133151—22no—
87#13133150—24no—
88#13133149—22no—
89#13133149—22no—
90#13133148—24no—
91#13133129C2×C418yescomplemented
92#13133129C2×C418yescomplemented
93#13133124—22no—
94#13133124—22no—
95C23×C4×C53—21no—
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53—22no—
99C22×C8×C53—22no—
100C22×C8×C53C2×C414yescomplemented
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53D414yes—
109C2×C4⋊C4×C53C2312yes—
110C2×C4⋊C4×C53C2312yes—
111C24×C53C2×D411yescharacteristic
112C23×D53C2×D412yes—
113C22×C4×C53—22no—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—42no—
117C22×C4×C53—81no—
118C22×C4×C53—81no—
119C22×C4×C53—81no—
120C22×C4×C53—81no—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×Q812yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—24no—
130C2×C22⋊D53—24no—
131C2×C22⋊D53C4⋊C418yes—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D412yes—
137#11719945—28no—
138#11719945—28no—
139#11719945—28no—
140#11719945—28no—
141C2×D212—22no—
142C2×D212—22no—
143C2×D212C4⋊C414yes—
144C2×C4×D53—22no—
145C2×C4×D53—22no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—28no—
149C2×C4×D53—82no—
150C2×C4×D53—82no—
151C2×C4×D53C22×C414yes—
152C2×C4×D53C2×Q814yes—
153C2×Dic106—22no—
154C2×Dic106—22no—
155C2×Dic106C4⋊C414yes—
156C2×C8×C53—28no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53C22×C412yes—
162C2×C8×C53C22×C412yes—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—42no—
168C4⋊C4×C53C2×D414yes—
169C22⋊Dic53—24no—
170C22⋊Dic53—42no—
171#11719921—28no—
172#11719921—82no—
173#11719921—82no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C2×D412yes—
181C4⋊Dic53C2×D412yes—
182#11719919—28no—
183C4×Dic53—28no—
184C4×Dic53—24no—
185C4×Dic53—42no—
186#11719917—28no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322C22×C414yes—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53C2×C4⋊C412yes—
195C23×C53#12012yes—
196C23×C53#12312yes—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53C2×C4⋊C414yes—
205C2×C4×C53—24no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—84no—
212C2×C4×C53—84no—
213C2×C4×C53C2×C4⋊C412yes—
214C2×C4×C53C4×D414yes—
215C2×C4×C53C4×D414yes—
216C2×C4×C53#12012yes—
217C2×C4×C53C22⋊Q814yes—
218C2×C4×C53C22⋊Q814yes—
219C2×C4×C53#12312yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C22×D411yescharacteristic
222C2×C4×C53C22×D411yescharacteristic
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C22⋊D53—216no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53C2×C4⋊C414yes—
242D212—28no—
243C4×D53—24no—
244C4×D53—24no—
245C4×D53—24no—
246C4×D53—24no—
247C4×D53—88no—
248Dic106—28no—
249C8×C53—22no—
250C8×C53—22no—
251#88322—24no—
252#20849—531nomaximal, Sylow
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53#38514yes—
260C22×C53C22×C4⋊C411yescharacteristic
261C22×C53C2×#12011yescharacteristic
262C22×C53C2×#12311yescharacteristic
263C22×C53#57611yescharacteristic
264C22×C53#57711yescharacteristic
265C22×C53#57711yescharacteristic
266C22×C53#57811yescharacteristic
267C2×D53—24no—
268C2×D53—24no—
269C2×D53—24no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C4×C53—44no—
275C4×C53—82no—
276C4×C53#38514yes—
277C4×C53C2×C4×D411yescharacteristic
278C4×C53C2×C4×D411yescharacteristic
279C4×C53C2×C22⋊Q811yescharacteristic
280C4×C53C2×C22⋊Q811yescharacteristic
281Dic53—24no—
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286C2×#520—531no—
287C2×#428—531no—
288C2×#427—531no—
289C2×#408—531no—
290#2311—531no—
291#1854—532no—
292#1854—532no—
293#1853—532no—
294#1852—532no—
295#1736—531no—
296C2×#340D5311yescharacteristic, complemented
297C2×C53—24no—
298C2×C53C2×#38511yescharacteristic
299C2×C53#291211yescharacteristic
300C2×C53#291311yescharacteristic
301C2×C53#301211yescharacteristic
302C2×C53#301311yescharacteristic
303C2×C53#309611yescharacteristic
304C2×C53#309711yescharacteristic
305D53—44no—
306C22×#41—531no—
307C22×M16—531no—
308C23×C8C2×D5311yescharacteristic
309#520—1064no—
310#520—1064no—
311C2×C4×Q8—1061no—
312C2×C4×D4—1061no—
313C2×C4×D4—1062no—
314C2×#117—531no—
315C2×#117—531no—
316C2×#117—532no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4C2×D5311yescharacteristic
319C22×C4⋊C4C2×D5311yescharacteristic
320#428—1064no—
321#427—1064no—
322C2×#107—531no—
323C2×#107—531no—
324C2×#107—532no—
325C2×#106—531no—
326C2×#106—531no—
327C2×#106—532no—
328#408—538no—
329C2×C22⋊C8—532no—
330C2×C22⋊C8—532no—
331#391—1061no—
332#390—1061no—
333#386—1062no—
334#380—532no—
335#380—532no—
336#340—532no—
337#340—532no—
338#340C2×D5312yes—
339#340C2×D5312yes—
340C53#2084911yescharacteristic, complemented, Sylow
341C2×#41—534no—
342C2×#41—534no—
343C2×#41—1062no—
344C2×#41—1062no—
345C2×#41—1062no—
346C2×#41—1062no—
347C22×Q8—531no—
348C22×D4—532no—
349C22×D4—531no—
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—532no—
353C23×C4—1062no—
354C23×C4C22×D5311yescharacteristic
355C2×M16—534no—
356C2×M16—1062no—
357C2×M16—1062no—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—532no—
361C22×C8C4×D5314yescomplemented
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C4×Q8—1068no—
365C4×D4—1068no—
366C4×D4—10616no—
367#117—534no—
368#117—1064no—
369#117—1062no—
370#117—1064no—
371#117—1062no—
372#117—1062no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4D21214yes—
386C2×C4⋊C4C22×D5312yes—
387C2×C4⋊C4C22×D5312yes—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1061no—
393C2×C42—1062no—
394C2×C42—1061no—
395C2×C42—1061no—
396#107—1062no—
397#107—1062no—
398#107—1064no—
399#106—1062no—
400#106—1062no—
401#106—1064no—
402C22⋊C8—538no—
403C22⋊C8—538no—
404#95—1062no—
405#95—1062no—
406C24—532no—
407C24D4×D5311yescharacteristic
408#41—1068no—
409#41—1068no—
410#41—1068no—
411#41—1068no—
412C2×Q8—534no—
413C2×Q8—1062no—
414C2×Q8—1062no—
415C2×D4—538no—
416C2×D4—534no—
417C2×D4—1064no—
418C2×D4—1064no—
419C2×D4—1062no—
420C2×D4—1062no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—532no—
430C22×C4—534no—
431C22×C4—534no—
432C22×C4—1068no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—4242no—
442C22×C4—4242no—
443C22×C4C2×C4×D5312yes—
444C22×C4C2×D21212yes—
445C22×C4#1171994512yes—
446C22×C4D4×D5311yescharacteristic
447C22×C4D4×D5312yes—
448C22×C4#1171994712yes—
449C22×C4Q8×D5312yes—
450C22×C4C23×D5311yescharacteristic
451M16—1068no—
452C2×C8—28no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—534no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8C2×C4×D5312yes—
461C2×C8C2×C4×D5312yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—532no—
468C4⋊C4—532no—
469C4⋊C4—1068no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4C2×D21214yes—
477C22⋊C4—1068no—
478C22⋊C4—1064no—
479C22⋊C4—4242no—
480C22⋊C4—4242no—
481C22⋊C4—2122no—
482C42—1068no—
483C42—1064no—
484C42—2122no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—534no—
490C23—1062no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—1062no—
495C23—1062no—
496C23C4⋊C4×D5312yes—
497C23#1313318712yes—
498C23#1313319912yes—
499C23C2×D4×D5311yescharacteristic
500Q8—1068no—
501D4—10616no—
502D4—1068no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—534no—
512C2×C4—1062no—
513C2×C4—1062no—
514C2×C4—1064no—
515C2×C4—1064no—
516C2×C4—1064no—
517C2×C4—1064no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—4248no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4C4⋊C4×D5312yes—
532C2×C4#1313317714yes—
533C2×C4C4×D21214yes—
534C2×C4#1313318712yes—
535C2×C4#1313319514yes—
536C2×C4#1313319714yes—
537C2×C4#1313319912yes—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4C2×D4×D5311yescharacteristic
540C2×C4C22×D21211yescharacteristic
541C2×C4C2×Q8×D5311yescharacteristic
542C2×C4C2×#1171994711yescharacteristic
543C2×C4C2×#1171994511yescharacteristic
544C8—22no—
545C8—22no—
546C8—1064no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—1064no—
554C22—1064no—
555C22—1064no—
556C22—2122no—
557C22—2122no—
558C22—2122no—
559C22—2122no—
560C22#1681049714yes—
561C22C2×C4⋊C4×D5311yescharacteristic
562C22C2×#1313318711yescharacteristic
563C22C2×#1313319911yescharacteristic
564C22#1681142411yescharacteristic
565C22#1681142911yescharacteristic
566C22#1681142911yescharacteristic
567C22#1681143611yescharacteristic
568C4—44no—
569C4—82no—
570C4—1064no—
571C4—4241no—
572C4—4241no—
573C4—2122no—
574C4—2122no—
575C4#1681049714yes—
576C4C2×#1313317711yescharacteristic
577C4C2×C4×D21211yescharacteristic
578C4C2×#1313319511yescharacteristic
579C4C2×#1313319711yescharacteristic
580C2—24no—
581C2—2124no—
582C2C2×#1681049711yescharacteristic
583C2#2028330611yescharacteristic
584C2#2028331111yescharacteristic
585C2#2028402411yescharacteristic
586C2#2028403511yescharacteristic
587C2#2028466911yescharacteristic
588C2#2028468011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53#20849
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. C2×#16810663C2semidirect
3. C2×#16810661C2semidirect
6. #20274917C2semidirect
7. #20274915C2semidirect
8. #20274909C2semidirect
9. #20274904C2semidirect
10. #20274434C2semidirect
11. C2×C53×#340C2semidirect
31. #16810589C4semidirect
59. C2×#11719917C2×C4semidirect
91. #13133129C2×C4semidirect
92. #13133129C2×C4semidirect
100. C22×C8×C53C2×C4semidirect
296. C2×#340D53semidirect
340. C53#20849semidirect
361. C22×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111042
C2×#168106632
C2×#168106612
C2×#168105892
#202773082
#202749172
#202749152
#202749092
#202749042
#202744342
C2×C53×#3402
#2084953