Finite Group Databasebeta

← #27384862

Subgroups — #27384862

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×#16811103C211yesmaximal, characteristic
2C2×#16810666C211yesmaximal, characteristic, complemented
3C2×#16810589C211yesmaximal, characteristic
4#20277303C211yesmaximal, characteristic
5#20274913C212yesmaximal, complemented
6#20274911C212yesmaximal, complemented
7#20274906C212yesmaximal, complemented
8#20274901C212yesmaximal, complemented
9#20274436C211yesmaximal, characteristic, complemented
10C2×#16810386C211yesmaximal, characteristic, complemented
11C2×C53×#427C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811103—24no—
15#16811103—24no—
16C2×C4×Dic106—21no—
17C2×C4×D212—21no—
18C2×#13133177—22no—
19C2×#13133175C2212yes—
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C4×D53C2212yes—
23#16810666—24no—
24C2×#13133150C2212yes—
25C2×#13133148C2212yes—
26#16810589C418yescomplemented
27C2×#13133129C2212yes—
28C2×#13133129C2212yes—
29#16810529—21no—
30#16810525—21no—
31#16810500—22no—
32#16810480C2212yes—
33#16810480C2212yes—
34#16810386C2212yes—
35#16810386C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38C23×C8×C53C2211yescharacteristic
39C2×#117×C53C2211yescharacteristic
40C2×#117×C53C2211yescharacteristic
41C53×#427—24no—
42C2×#107×C53C2211yescharacteristic
43C2×#107×C53C2211yescharacteristic
44C2×#106×C53C2211yescharacteristic
45C2×#106×C53C2211yescharacteristic
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945C2×C414yes—
51C2×#11719945Q814yes—
52C22×Dic106D411yescharacteristic
53C22×D212D411yescharacteristic
54C22×C22⋊D53D412yes—
55C23×Dic53—21no—
56C23×Dic53—21no—
57C22×C4×D53—22no—
58C22×C4×D53C2312yes—
59C2×#11719917—22no—
60C2×#11719917—22no—
61C2×#11719917C2×C414yescomplemented
62C22×#88322C2312yes—
63C4×Dic106—28no—
64C4×D212—28no—
65#13133177—216no—
66#13133175—24no—
67#13133175—24no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53D414yes—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2312yes—
76C2×#11719919—22no—
77C4⋊C4×D53—24no—
78C4⋊C4×D53—24no—
79C2×#11719921—22no—
80C2×#11719921—22no—
81C2×#11719921—22no—
82C2×C4×Dic53—22no—
83#13133150—24no—
84#13133148—24no—
85#13133129C2×C418yescomplemented
86#13133129C2×C418yescomplemented
87#13133124—22no—
88#13133124—22no—
89C23×C4×C53C2311yescharacteristic
90C22×C8×C53—22no—
91C22×C8×C53—22no—
92C22×C8×C53C2×C414yescomplemented
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#117×C53—22no—
96#117×C53—22no—
97#117×C53—22no—
98#117×C53D414yes—
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C42×C53—21no—
106C2×C42×C53—21no—
107#107×C53—22no—
108#107×C53—22no—
109#106×C53—22no—
110#106×C53—22no—
111C24×C53C2×D411yescharacteristic
112C23×D53C2×D412yes—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53C22×C412yes—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×Q812yes—
122C22×C4×C53#4112yes—
123C22×C4×C53#4112yes—
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—24no—
126C2×C22⋊D53—24no—
127C2×C22⋊D53C4⋊C418yes—
128C22×Dic53—22no—
129C22×Dic53—24no—
130C22×Dic53—22no—
131C22×Dic53—42no—
132C22×Dic53—81no—
133C22×Dic53—81no—
134C22×Dic53—81no—
135C22×Dic53—81no—
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—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C2×D412yes—
170C4⋊C4×C53C2×D412yes—
171C22⋊C4×C53—24no—
172C22⋊C4×C53—42no—
173C42×C53—24no—
174C42×C53—42no—
175#11719921—28no—
176#11719921—82no—
177#11719921—82no—
178C4⋊Dic53—24no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4⋊Dic53—22no—
182C4⋊Dic53—42no—
183C4⋊Dic53C2×D414yes—
184#11719919—28no—
185C4×Dic53—28no—
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—22no—
206C2×C4×C53—22no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53C2×C4⋊C412yes—
218C2×C4×C53C4×D414yes—
219C2×C4×C53C4×D414yes—
220C2×C4×C53#12012yes—
221C2×C4×C53C22⋊Q814yes—
222C2×C4×C53C22⋊Q814yes—
223C2×C4×C53#12312yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×Q811yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C22⋊D53—216no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—24no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—84no—
240C2×Dic53—84no—
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—42no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#38514yes—
279C4×C53C2×C4×D411yescharacteristic
280C4×C53C2×C4×D411yescharacteristic
281C4×C53C2×C22⋊Q811yescharacteristic
282C4×C53C2×C22⋊Q811yescharacteristic
283Dic53—24no—
284Dic53—44no—
285Dic53—82no—
286C2×#520—531no—
287C2×#428—531no—
288C2×#427D5311yescharacteristic, complemented
289C2×#408—531no—
290#2311—531no—
291#1854—532no—
292#1854—532no—
293#1853—532no—
294#1852—532no—
295#1736—531no—
296C2×#340—531no—
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—532no—
315C2×#117C2×D5311yescharacteristic
316C2×#117C2×D5311yescharacteristic
317C22×C4⋊C4—531no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—532no—
320#428—1064no—
321#427—24no—
322C2×#107—532no—
323C2×#107C2×D5311yescharacteristic
324C2×#107C2×D5311yescharacteristic
325C2×#106—532no—
326C2×#106C2×D5311yescharacteristic
327C2×#106C2×D5311yescharacteristic
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#340—532no—
339#340—532no—
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—532no—
351C23×C4—1062no—
352C23×C4—1061no—
353C23×C4—1061no—
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—22no—
368#117—22no—
369#117—22no—
370#117—1064no—
371#117—1064no—
372#117C22⋊D5314yes—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—532no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1062no—
391C2×C22⋊C4—1062no—
392C2×C22⋊C4—1062no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—1062no—
396#107—22no—
397#107—22no—
398#107—1064no—
399#106—22no—
400#106—22no—
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—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—532no—
426C22×C4—534no—
427C22×C4—534no—
428C22×C4—1068no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1062no—
433C22×C4—1064no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—4242no—
437C22×C4—4242no—
438C22×C4—4241no—
439C22×C4—4241no—
440C22×C4—4241no—
441C22×C4—4241no—
442C22×C4—2122no—
443C22×C4C2×C4×D5312yes—
444C22×C4#1171994512yes—
445C22×C4D4×D5311yescharacteristic
446C22×C4D4×D5312yes—
447C22×C4Q8×D5312yes—
448C22×C4#1171994912yes—
449C22×C4C2×C22⋊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—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—534no—
469C4⋊C4—1068no—
470C4⋊C4—1064no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—1062no—
474C4⋊C4—2122no—
475C4⋊C4C2×C22⋊D5312yes—
476C4⋊C4C2×C22⋊D5312yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—1068no—
480C22⋊C4—4242no—
481C22⋊C4—4242no—
482C42—24no—
483C42—42no—
484C42—1068no—
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#1313318512yes—
498C23#1313320212yes—
499C23C2×D4×D5311yescharacteristic
500Q8—1068no—
501D4—10616no—
502D4—1068no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—534no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1064no—
519C2×C4—1064no—
520C2×C4—1064no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—4248no—
524C2×C4—4244no—
525C2×C4—4244no—
526C2×C4—2122no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4C4⋊C4×D5312yes—
532C2×C4C4×C22⋊D5314yes—
533C2×C4#1313317914yes—
534C2×C4#1313318512yes—
535C2×C4#1313319314yes—
536C2×C4#1313319514yes—
537C2×C4#1313320212yes—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4C2×D4×D5311yescharacteristic
540C2×C4C22×C22⋊D5311yescharacteristic
541C2×C4C2×Q8×D5311yescharacteristic
542C2×C4C2×#1171994511yescharacteristic
543C2×C4C2×#1171994911yescharacteristic
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#1681049414yes—
561C22C2×C4⋊C4×D5311yescharacteristic
562C22C2×#1313318511yescharacteristic
563C22C2×#1313320211yescharacteristic
564C22#1681142511yescharacteristic
565C22#1681143011yescharacteristic
566C22#1681143011yescharacteristic
567C22#1681143711yescharacteristic
568C4—42no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—1064no—
573C4—4242no—
574C4—2124no—
575C4#1681049414yes—
576C4C2×C4×C22⋊D5311yescharacteristic
577C4C2×#1313317911yescharacteristic
578C4C2×#1313319311yescharacteristic
579C4C2×#1313319511yescharacteristic
580C2—24no—
581C2—2124no—
582C2C2×#1681049411yescharacteristic
583C2#2028330711yescharacteristic
584C2#2028331211yescharacteristic
585C2#2028401511yescharacteristic
586C2#2028402611yescharacteristic
587C2#2028467611yescharacteristic
588C2#2028468711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#427C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#427D53
hypocenterC53#20849
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. C2×#16810666C2semidirect
5. #20274913C2semidirect
6. #20274911C2semidirect
7. #20274906C2semidirect
8. #20274901C2semidirect
9. #20274436C2semidirect
10. C2×#16810386C2semidirect
11. C2×C53×#427C2semidirect
26. #16810589C4semidirect
61. C2×#11719917C2×C4semidirect
85. #13133129C2×C4semidirect
86. #13133129C2×C4semidirect
92. C22×C8×C53C2×C4semidirect
288. C2×#427D53semidirect
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×#168111032
C2×#168106662
C2×#168105892
#202773032
#202749132
#202749112
#202749062
#202749012
#202744362
C2×#168103862
C2×C53×#4272
#2084953