Finite Group Databasebeta

← #27290494

Subgroups — #27290494

Order 13568.

904 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810641C211yesmaximal, characteristic
2#20283232C211yesmaximal, characteristic
3#20283159C211yesmaximal, characteristic
4#20283134C211yesmaximal, characteristic, complemented
5C2×#16810393C211yesmaximal, characteristic, complemented
6C2×#16810381C211yesmaximal, characteristic, complemented
7C2×C53×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16810641—22no—
13#16810641C414yescomplemented
14C2×#13133145—22no—
15#16810631C412yescomplemented
16#16810627C412yescomplemented
17#16810622—22no—
18#16810617—22no—
19#16810591—22no—
20#16810589C414yescomplemented
21#16810586—22no—
22#16810579—22no—
23#16810579—22no—
24C4×#11719917—22no—
25#16810393—22no—
26#16810381—22no—
27C22×M16×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C2×#117×C53C2211yescharacteristic
30C53×H(3,ℤ/4)—22no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945C2×C412yes—
42C2×#11719945C2×C412yes—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917C2×C412yes—
61C2×#11719917C2×C412yes—
62C2×#11719917C2×C412yes—
63C2×#11719917D412yes—
64C2×#11719917D412yes—
65C22×#88322—21no—
66C22×#88322D412yes—
67C22×#88322D412yes—
68#13133145—28no—
69#13133140—24no—
70#13133137—24no—
71#13133129—28no—
72#13133128—24no—
73C4×#88322—24no—
74C23×C4×C53C2311yescharacteristic
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C22×C8×C53C2×C412yes—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53—21no—
86#117×C53—21no—
87#117×C53C2×C412yes—
88#117×C53C2×C412yes—
89#117×C53D412yes—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C22⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C42×C53—21no—
95C2×C42×C53D412yes—
96C24×C53#4111yescharacteristic
97C23×D53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C2×C22⋊D53—22no—
115C2×C22⋊D53—22no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C22×Dic53—21no—
121#11719945—22no—
122#11719945—22no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—41no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212C4212yes—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106C4212yes—
155M16×C53—41no—
156M16×C53—41no—
157M16×C53—41no—
158M16×C53—41no—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—41no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—41no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—41no—
173C42×C53C22⋊C414yes—
174#11719917—22no—
175#11719917—22no—
176#11719917—22no—
177#11719917—22no—
178#11719917—22no—
179#11719917—22no—
180#11719917—22no—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—41no—
185#11719917—41no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322C22⋊C414yes—
194C2×#88322C22⋊C414yes—
195C23×C53—81no—
196C23×C53#11711yescharacteristic
197C23×C53C4×D411yescharacteristic
198C23×C53C4×D411yescharacteristic
199C23×C53#12111yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12411yescharacteristic
202C23×C53#12611yescharacteristic
203C22×D53—22no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—82no—
207C2×C4×C53—24no—
208C2×C4×C53—42no—
209C2×C4×C53—82no—
210C2×C4×C53—82no—
211C2×C4×C53C2×C4211yescharacteristic
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53C2×C22⋊C411yescharacteristic
214C2×C4×C53C2×C22⋊C411yescharacteristic
215C2×C4×C53#11711yescharacteristic
216C2×C4×C53#11711yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12311yescharacteristic
235C2×C4×C53#12411yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12611yescharacteristic
238C2×C4×C53#12711yescharacteristic
239C22⋊D53—42no—
240C22⋊D53—42no—
241C22⋊D53—42no—
242C22⋊D53—42no—
243C2×Dic53—22no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246D212—22no—
247D212—22no—
248D212—41no—
249D212—41no—
250C4×D53—42no—
251C4×D53—42no—
252C4×D53—42no—
253C4×D53—42no—
254Dic106—22no—
255Dic106—22no—
256Dic106—41no—
257Dic106—41no—
258C8×C53—42no—
259#88322—24no—
260#88322—24no—
261#88322—42no—
262#10950—531nomaximal, Sylow
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53C4×C22⋊C411yescharacteristic
271C22×C53#38611yescharacteristic
272C22×C53#38811yescharacteristic
273C22×C53#38811yescharacteristic
274C22×C53#38811yescharacteristic
275C22×C53#38811yescharacteristic
276C22×C53#39011yescharacteristic
277C2×D53—44no—
278C2×D53—42no—
279C2×D53—82no—
280C2×D53—82no—
281C4×C53—44no—
282C4×C53—82no—
283C4×C53C4×C22⋊C411yescharacteristic
284C4×C53C4×C22⋊C411yescharacteristic
285C4×C53#38611yescharacteristic
286C4×C53#38611yescharacteristic
287C4×C53#38811yescharacteristic
288C4×C53#38811yescharacteristic
289C4×C53#39011yescharacteristic
290C4×C53#39011yescharacteristic
291Dic53—42no—
292C2×#421—531no—
293#2896—531no—
294#2879—531no—
295#2873—531no—
296C2×#344—531no—
297C2×#338—531no—
298C2×H(3,ℤ/4)D5311yescharacteristic, complemented
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—21no—
302C2×C53—21no—
303C2×C53—21no—
304C2×C53—21no—
305C2×C53—21no—
306C2×C53#143711yescharacteristic
307D53—82no—
308C22×#41—531no—
309C22×M16—531no—
310C22×M16—531no—
311C22×M16—531no—
312C22×M16C2×D5311yescharacteristic
313C2×#117C2×D5311yescharacteristic
314C2×#117C2×D5311yescharacteristic
315#421—534no—
316#421—1062no—
317C2×#104—1062no—
318#419—532no—
319#418—532no—
320#417—1062no—
321#416—1062no—
322#408—534no—
323#408—1062no—
324#407—1062no—
325#405—1062no—
326#405—1062no—
327C4×M16—1062no—
328#344—1062no—
329#338—1062no—
330H(3,ℤ/4)—22no—
331C53#1095011yescharacteristic, complemented, Sylow
332C2×#41—532no—
333C2×#41—532no—
334C2×#41—1061no—
335C2×#41—1061no—
336C2×#41—1061no—
337C2×#41—1061no—
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C2×#41—1061no—
342C2×#41—1061no—
343C2×#41—1061no—
344C22×Q8—531no—
345C22×D4—531no—
346C22×D4—1061no—
347C23×C4—1061no—
348C23×C4C22×D5311yescharacteristic
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—532no—
356C2×M16—532no—
357C2×M16—532no—
358C2×M16—532no—
359C2×M16—532no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C22×C8—532no—
374C22×C8—532no—
375C22×C8—1061no—
376C22×C8C4×D5312yes—
377#117—21no—
378#117—21no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117C4×D5312yes—
383#117C4×D5312yes—
384#117C22⋊D5312yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C42—21no—
390C2×C42D21212yes—
391#104—1068no—
392#103—1064no—
393#102—1064no—
394C22⋊C8—1068no—
395#97—1064no—
396C4×C8—1064no—
397C24—1061no—
398C24#1171994711yescharacteristic
399#41—1062no—
400#41—1062no—
401#41—1062no—
402#41—1062no—
403#41—1062no—
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—2121no—
408#41—2121no—
409#41—2121no—
410#41—2121no—
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415C2×Q8—532no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×D4—532no—
422C2×D4—1062no—
423C2×D4—1062no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C2×D4—2121no—
430C2×D4—2121no—
431C2×D4—2121no—
432C2×D4—2121no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4—1061no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—2121no—
439C22×C4—2121no—
440C22×C4—2121no—
441C22×C4—2121no—
442C22×C4C2×C4×D5311yescharacteristic
443C22×C4C2×C4×D5311yescharacteristic
444C22×C4C2×C4×D5311yescharacteristic
445C22×C4C2×D21211yescharacteristic
446C22×C4#1171994511yescharacteristic
447C22×C4#1171994511yescharacteristic
448C22×C4D4×D5311yescharacteristic
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5311yescharacteristic
452C22×C4#1171994711yescharacteristic
453C22×C4#1171994911yescharacteristic
454C22×C4#1171994911yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—1062no—
461M16—1062no—
462M16—1062no—
463M16—1062no—
464M16—1062no—
465M16—1062no—
466M16—1062no—
467M16—1062no—
468M16—1062no—
469M16—1062no—
470M16—2121no—
471M16—2121no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—534no—
476C2×C8—534no—
477C2×C8—1062no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—2121no—
483C2×C8—2121no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—41no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—41no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42#1171992114yes—
496C23—81no—
497C23—1062no—
498C23—4242no—
499C23—2121no—
500C23—2121no—
501C23#1313317511yescharacteristic
502C23#1313317711yescharacteristic
503C23#1313317711yescharacteristic
504C23#1313318911yescharacteristic
505C23#1313320111yescharacteristic
506C23#1313320311yescharacteristic
507C23#1313321211yescharacteristic
508Q8—1062no—
509Q8—1062no—
510Q8—2121no—
511Q8—2121no—
512D4—1062no—
513D4—1062no—
514D4—2122no—
515D4—2122no—
516D4—2122no—
517D4—2122no—
518D4—2121no—
519D4—2121no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—1062no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2122no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4C42×D5311yescharacteristic
532C2×C4C22⋊C4×D5311yescharacteristic
533C2×C4C22⋊C4×D5311yescharacteristic
534C2×C4C2×#1171992111yescharacteristic
535C2×C4#1313317411yescharacteristic
536C2×C4#1313317511yescharacteristic
537C2×C4#1313317711yescharacteristic
538C2×C4#1313317711yescharacteristic
539C2×C4C4×C22⋊D5311yescharacteristic
540C2×C4C4×C22⋊D5311yescharacteristic
541C2×C4#1313317911yescharacteristic
542C2×C4#1313317911yescharacteristic
543C2×C4#1313317911yescharacteristic
544C2×C4#1313317911yescharacteristic
545C2×C4C4×D21211yescharacteristic
546C2×C4C4×D21211yescharacteristic
547C2×C4#1313318711yescharacteristic
548C2×C4#1313318911yescharacteristic
549C2×C4#1313319011yescharacteristic
550C2×C4#1313319011yescharacteristic
551C2×C4#1313319211yescharacteristic
552C2×C4#1313319211yescharacteristic
553C2×C4#1313320211yescharacteristic
554C2×C4#1313320211yescharacteristic
555C2×C4#1313320311yescharacteristic
556C2×C4#1313320611yescharacteristic
557C2×C4#1313321211yescharacteristic
558C2×C4#1313321311yescharacteristic
559C8—42no—
560C8—1064no—
561C8—1064no—
562C8—2122no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—4242no—
571C22—4242no—
572C22—2124no—
573C22—2122no—
574C22#1681047011yescharacteristic
575C22#1681050011yescharacteristic
576C22#1681051211yescharacteristic
577C22#1681051311yescharacteristic
578C22#1681051511yescharacteristic
579C22#1681051511yescharacteristic
580C22#1681052311yescharacteristic
581C4—44no—
582C4—82no—
583C4—2122no—
584C4#1681047011yescharacteristic
585C4C4×#1171992111yescharacteristic
586C4#1681050111yescharacteristic
587C4#1681050111yescharacteristic
588C4#1681051211yescharacteristic
589C4#1681051311yescharacteristic
590C4#1681052311yescharacteristic
591C4#1681052411yescharacteristic
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—4242no—
600C2#2027295111yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×H(3,ℤ/4)C2
socleC2×C53#1437
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×H(3,ℤ/4)D53
hypocenterC53#10950
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
4. #20283134C2semidirect
5. C2×#16810393C2semidirect
6. C2×#16810381C2semidirect
7. C2×C53×H(3,ℤ/4)C2semidirect
13. #16810641C4semidirect
15. #16810631C4semidirect
16. #16810627C4semidirect
20. #16810589C4semidirect
298. C2×H(3,ℤ/4)D53semidirect
331. C53#10950semidirect

Maximal subgroups

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

subgroupindex
C2×#168106412
#202832322
#202831592
#202831342
C2×#168103932
C2×#168103812
C2×C53×H(3,ℤ/4)2
#1095053