Finite Group Databasebeta

← #27291080

Subgroups — #27291080

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283278C211yesmaximal, characteristic, complemented
2C2×#16810641C211yesmaximal, characteristic
3C2×#16810639C211yesmaximal, characteristic
4#20277205C211yesmaximal, characteristic
5#20274447C211yesmaximal, characteristic, complemented
6C2×#16810383C211yesmaximal, characteristic, complemented
7C2×C53×#339C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810651—22no—
13#16810649—22no—
14#16810646—22no—
15#16810641—22no—
16#16810641—22no—
17#16810639—22no—
18#16810639—22no—
19C2×#13133145—22no—
20C2×#13133143—22no—
21#16810529C412yescomplemented
22#16810525C412yescomplemented
23#16810502—22no—
24C4×#11719921—22no—
25#16810389—22no—
26#16810383—22no—
27C22×M16×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C22×C42×C53C2211yescharacteristic
30C53×#339—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×#11719945D412yes—
42C2×#11719945Q812yes—
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×#11719917D412yes—
53C22×#88322—21no—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176D412yes—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C412yes—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53D412yes—
66C2×#11719921—24no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53D412yes—
69#13133146—28no—
70#13133145—28no—
71#13133143—28no—
72#13133124—24no—
73C23×C4×C53—21no—
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×C53D412yes—
81C22×C8×C53—21no—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53D412yes—
86C2×C4⋊C4×C53—21no—
87C2×C22⋊C4×C53—21no—
88C2×C42×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53—21no—
94C2×C42×C53C2×C412yes—
95C24×C53C2×D411yescharacteristic
96C23×D53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—44no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
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×C53C2×Q811yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C2×C22⋊D53—22no—
117C2×C22⋊D53—22no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C22×Dic53—21no—
123C22×Dic53—41no—
124C22×Dic53C22⋊C414yes—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212C4⋊C412yes—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×C4×D53—41no—
152C2×C4×D53—41no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106C4⋊C412yes—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—41no—
162M16×C53—41no—
163C2×C8×C53—22no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—41no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—41no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—41no—
174C42×C53—41no—
175C42×C53—41no—
176C22⋊Dic53—22no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—41no—
180#11719921—48no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—41no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—41no—
189#11719917—22no—
190#11719917—22no—
191#11719917—41no—
192#11719917—41no—
193C2×#88322—22no—
194C2×#88322—41no—
195C2×#88322—41no—
196C23×C53—42no—
197C23×C53C2×C22⋊C411yescharacteristic
198C23×C53#12111yescharacteristic
199C23×C53#12111yescharacteristic
200C23×C53#12111yescharacteristic
201C23×C53#12111yescharacteristic
202C23×C53#12111yescharacteristic
203C23×C53C22⋊Q811yescharacteristic
204C22×D53—22no—
205C22×D53—44no—
206C22×D53—41no—
207C22×D53—41no—
208C2×C4×C53—24no—
209C2×C4×C53—42no—
210C2×C4×C53—44no—
211C2×C4×C53—44no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—82no—
215C2×C4×C53C2×C4⋊C411yescharacteristic
216C2×C4×C53#11711yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53#12011yescharacteristic
222C2×C4×C53#12011yescharacteristic
223C2×C4×C53#12011yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53C22⋊Q811yescharacteristic
228C2×C4×C53C22⋊Q811yescharacteristic
229C2×C4×C53C22⋊Q811yescharacteristic
230C2×C4×C53C22⋊Q811yescharacteristic
231C2×C4×C53C22⋊Q811yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12311yescharacteristic
235C2×C4×C53#12311yescharacteristic
236C2×C4×C53#12311yescharacteristic
237C2×C4×C53#12411yescharacteristic
238C2×C4×C53#12411yescharacteristic
239C2×C4×C53#12411yescharacteristic
240C2×C4×C53#12611yescharacteristic
241C2×C4×C53#12611yescharacteristic
242C2×C4×C53#12711yescharacteristic
243C22⋊D53—42no—
244C22⋊D53—42no—
245C22⋊D53—42no—
246C22⋊D53—42no—
247C2×Dic53—22no—
248C2×Dic53—24no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—44no—
252C2×Dic53—42no—
253C2×Dic53—82no—
254D212—22no—
255D212—22no—
256D212—41no—
257D212—41no—
258C4×D53—42no—
259C4×D53—42no—
260C4×D53—42no—
261C4×D53—42no—
262Dic106—22no—
263Dic106—22no—
264Dic106—41no—
265Dic106—41no—
266C8×C53—42no—
267#88322—42no—
268#11056—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53#38011yescharacteristic
277C22×C53#38611yescharacteristic
278C22×C53#39011yescharacteristic
279C22×C53#39411yescharacteristic
280C22×C53#39411yescharacteristic
281C22×C53#39611yescharacteristic
282C22×C53#39611yescharacteristic
283C2×D53—44no—
284C2×D53—42no—
285C2×D53—82no—
286C2×D53—82no—
287C4×C53—44no—
288C4×C53—82no—
289C4×C53#38511yescharacteristic
290C4×C53#38511yescharacteristic
291C4×C53#38811yescharacteristic
292C4×C53#38811yescharacteristic
293C4×C53#39211yescharacteristic
294C4×C53#39311yescharacteristic
295C4×C53#39711yescharacteristic
296C4×C53#39911yescharacteristic
297Dic53—44no—
298Dic53—42no—
299Dic53—82no—
300#2905—531no—
301C2×#421—531no—
302C2×#421—531no—
303#2297—531no—
304#1740—531no—
305C2×#339—531no—
306C2×#339D5311yescharacteristic, complemented
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53#143911yescharacteristic
314C2×C53#188412yes—
315D53—82no—
316C22×#41—531no—
317C22×M16—531no—
318C22×M16C2×D5311yescharacteristic
319C2×#117—531no—
320C2×#117—531no—
321C2×#117C2×D5311yescharacteristic
322C22×C42C2×D5311yescharacteristic
323#424—1062no—
324#424—1062no—
325#423—1062no—
326#421—1062no—
327#421—1062no—
328#421—1062no—
329#421—1062no—
330C2×#104—1062no—
331C2×#104—1062no—
332#391—532no—
333#390—532no—
334#386—1062no—
335C4×C22⋊C4—1062no—
336#342—1062no—
337#339—22no—
338#339—1062no—
339C53#1105611yescharacteristic, complemented, Sylow
340C2×#41—532no—
341C2×#41—532no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C22×Q8—531no—
353C22×D4—531no—
354C22×D4—1061no—
355C23×C4—21no—
356C23×C4—1061no—
357C23×C4C22×D5311yescharacteristic
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—532no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16D21212yes—
370C22×C8—21no—
371C22×C8—1061no—
372#117—21no—
373#117—21no—
374#117—21no—
375#117—532no—
376#117—1061no—
377#117—1061no—
378#117—1061no—
379#117—1061no—
380#117—1061no—
381#117—1061no—
382#117—1061no—
383#117C22⋊D5312yes—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—532no—
386C2×C4⋊C4—1061no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—532no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1064no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—532no—
398C2×C42—1061no—
399C2×C42C4×D5312yes—
400C4⋊C8—1068no—
401#104—1068no—
402#104—1068no—
403#95—1064no—
404C24—1061no—
405C24D4×D5311yescharacteristic
406#41—1062no—
407#41—1062no—
408#41—1062no—
409#41—1062no—
410#41—1062no—
411#41—1062no—
412#41—1062no—
413#41—1062no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419#41—2121no—
420#41—2121no—
421#41—2121no—
422C2×Q8—532no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×D4—532no—
429C2×D4—1062no—
430C2×D4—1062no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—2121no—
437C2×D4—2121no—
438C2×D4—2121no—
439C2×D4—2121no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—534no—
446C22×C4—1061no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4—2121no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4C2×C4×D5311yescharacteristic
455C22×C4C2×D21211yescharacteristic
456C22×C4#1171994511yescharacteristic
457C22×C4#1171994511yescharacteristic
458C22×C4#1171994511yescharacteristic
459C22×C4#1171994511yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4D4×D5311yescharacteristic
462C22×C4D4×D5311yescharacteristic
463C22×C4D4×D5311yescharacteristic
464C22×C4#1171994711yescharacteristic
465C22×C4Q8×D5311yescharacteristic
466C22×C4#1171994911yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468M16—22no—
469M16—22no—
470M16—41no—
471M16—41no—
472M16—1062no—
473M16—1062no—
474M16—2121no—
475M16—2121no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—1062no—
480C2×C8—2121no—
481C2×C8—2121no—
482C4⋊C4—22no—
483C4⋊C4—41no—
484C4⋊C4—1062no—
485C4⋊C4—1062no—
486C4⋊C4—1062no—
487C4⋊C4—2121no—
488C22⋊C4—22no—
489C22⋊C4—41no—
490C22⋊C4—1062no—
491C22⋊C4—1062no—
492C22⋊C4—1062no—
493C22⋊C4—2128no—
494C22⋊C4—2121no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—1062no—
502C42—1062no—
503C42—1062no—
504C42—2121no—
505C23—42no—
506C23—1062no—
507C23—2124no—
508C23—2121no—
509C23—2121no—
510C23C22⋊C4×D5311yescharacteristic
511C23#1313318911yescharacteristic
512C23#1313319011yescharacteristic
513C23#1313319011yescharacteristic
514C23#1313319011yescharacteristic
515C23#1313319211yescharacteristic
516C23#1313319511yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521D4—1062no—
522D4—1062no—
523D4—2122no—
524D4—2122no—
525D4—2122no—
526D4—2122no—
527D4—2121no—
528D4—2121no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—1064no—
537C2×C4—1062no—
538C2×C4—4242no—
539C2×C4—2122no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2124no—
547C2×C4C4⋊C4×D5311yescharacteristic
548C2×C4#1313317411yescharacteristic
549C2×C4#1313317711yescharacteristic
550C2×C4C4×C22⋊D5311yescharacteristic
551C2×C4#1313317911yescharacteristic
552C2×C4C4×D21211yescharacteristic
553C2×C4#1313318511yescharacteristic
554C2×C4#1313318711yescharacteristic
555C2×C4#1313318711yescharacteristic
556C2×C4#1313319011yescharacteristic
557C2×C4#1313319011yescharacteristic
558C2×C4#1313319111yescharacteristic
559C2×C4#1313319311yescharacteristic
560C2×C4#1313319511yescharacteristic
561C2×C4#1313319511yescharacteristic
562C2×C4#1313319511yescharacteristic
563C2×C4#1313319711yescharacteristic
564C2×C4#1313319911yescharacteristic
565C2×C4#1313320011yescharacteristic
566C2×C4#1313320211yescharacteristic
567C2×C4#1313320211yescharacteristic
568C2×C4#1313320211yescharacteristic
569C2×C4#1313320311yescharacteristic
570C2×C4#1313320311yescharacteristic
571C2×C4#1313320511yescharacteristic
572C2×C4#1313321111yescharacteristic
573C2×C4#1313321211yescharacteristic
574C2×C4#1313321311yescharacteristic
575C8—42no—
576C8—2122no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—4242no—
585C22—4242no—
586C22—2124no—
587C22—2122no—
588C22#1681048211yescharacteristic
589C22#1681050111yescharacteristic
590C22#1681052311yescharacteristic
591C22#1681053811yescharacteristic
592C22#1681054111yescharacteristic
593C22#1681054511yescharacteristic
594C22#1681054711yescharacteristic
595C4—44no—
596C4—82no—
597C4—4242no—
598C4—2122no—
599C4—2124no—
600C4#1681049411yescharacteristic
601C4#1681049711yescharacteristic
602C4#1681051211yescharacteristic
603C4#1681051411yescharacteristic
604C4#1681053211yescharacteristic
605C4#1681053611yescharacteristic
606C4#1681055211yescharacteristic
607C4#1681055811yescharacteristic
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—4242no—
615C2#2027295911yescharacteristic
616C2#2027509712yes—
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#339C2
socleC22×C53#386
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#339D53
hypocenterC53#11056
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. #20283278C2semidirect
5. #20274447C2semidirect
6. C2×#16810383C2semidirect
7. C2×C53×#339C2semidirect
21. #16810529C4semidirect
22. #16810525C4semidirect
306. C2×#339D53semidirect
339. C53#11056semidirect

Maximal subgroups

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

subgroupindex
#202832782
C2×#168106412
C2×#168106392
#202772052
#202744472
C2×#168103832
C2×C53×#3392
#1105653