Finite Group Databasebeta

← #27291920

Subgroups — #27291920

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133145C211yesmaximal, characteristic
2C2×#16810620C211yesmaximal, characteristic
3C2×#16810589C211yesmaximal, characteristic
4C2×#16810388C211yesmaximal, characteristic, complemented
5C2×#16810383C211yesmaximal, characteristic, complemented
6#20274411C211yesmaximal, characteristic, complemented
7C2×C53×#336C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12C2×#13133145—22no—
13C2×#13133145—22no—
14C2×#13133145—22no—
15#16810620—22no—
16#16810620—22no—
17C2×#13133142C412yescomplemented
18C2×#13133139C412yescomplemented
19#16810589—22no—
20#16810589—22no—
21C2×#13133129—22no—
22#16810388—22no—
23#16810383—22no—
24#16810377—22no—
25C23×C8×C53C2211yescharacteristic
26C23×C8×C53C2211yescharacteristic
27C22×C42×C53C2211yescharacteristic
28C53×#336—22no—
29C2×#11719945—21no—
30C2×#11719945—21no—
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×#11719945D412yes—
40C2×#11719945D412yes—
41C22×Dic106C2×C411yescharacteristic
42C22×D212C2×C411yescharacteristic
43C22×C22⋊D53—21no—
44C22×C4×D53—21no—
45C2×#11719917—21no—
46C2×#11719917—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×#11719917D412yes—
56C2×#11719917D412yes—
57C22×#88322—21no—
58C22×#88322—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176—21no—
62#13133176—21no—
63C2×C4⋊Dic53C2×C412yes—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—21no—
66#13133145—28no—
67#13133142—24no—
68#13133139—24no—
69#13133129—28no—
70C23×C4×C53—21no—
71C23×C4×C53C2311yescharacteristic
72C22×C8×C53—21no—
73C22×C8×C53—21no—
74C22×C8×C53—21no—
75C22×C8×C53—21no—
76C22×C8×C53—21no—
77C22×C8×C53—21no—
78C22×C8×C53—21no—
79C22×C8×C53—21no—
80C22×C8×C53—21no—
81C22×C8×C53—21no—
82C22×C8×C53—21no—
83C22×C8×C53—21no—
84C22×C8×C53C2×C412yes—
85C22×C8×C53D412yes—
86C2×C42×C53—21no—
87C2×C42×C53—21no—
88C2×C42×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53D412yes—
92C24×C53#4111yescharacteristic
93C23×D53—21no—
94C22×C4×C53—24no—
95C22×C4×C53—22no—
96C22×C4×C53—42no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C2×C22⋊D53—22no—
114C2×C22⋊D53—22no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C22×Dic53—21no—
120C22×Dic53—22no—
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×D212C22⋊C412yes—
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×Dic106C22⋊C412yes—
155C2×C8×C53—22no—
156C2×C8×C53—22no—
157C2×C8×C53—22no—
158C2×C8×C53—22no—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53—41no—
162C2×C8×C53—41no—
163C2×C8×C53—41no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C2×C8×C53—41no—
167C2×C8×C53—41no—
168C2×C8×C53—41no—
169C42×C53—22no—
170C42×C53—22no—
171C42×C53—41no—
172C42×C53—41no—
173C22⋊Dic53—41no—
174C22⋊Dic53—41no—
175C4⋊Dic53—22no—
176C4⋊Dic53—22no—
177C4×Dic53—41no—
178C4×Dic53—41no—
179#11719917—22no—
180#11719917—22no—
181#11719917—22no—
182#11719917—22no—
183#11719917—41no—
184#11719917—41no—
185#11719917—41no—
186#11719917—41no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53C4×D411yescharacteristic
196C23×C53#12111yescharacteristic
197C23×C53#12311yescharacteristic
198C23×C53#12311yescharacteristic
199C23×C53#12411yescharacteristic
200C23×C53#12611yescharacteristic
201C23×C53#12611yescharacteristic
202C22×D53—22no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—42no—
206C22×D53—42no—
207C2×C4×C53—24no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53C2×C22⋊C411yescharacteristic
213C2×C4×C53#11711yescharacteristic
214C2×C4×C53#11711yescharacteristic
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12111yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12311yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12311yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12311yescharacteristic
235C2×C4×C53#12411yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12411yescharacteristic
238C2×C4×C53#12611yescharacteristic
239C2×C4×C53#12611yescharacteristic
240C22⋊D53—42no—
241C22⋊D53—42no—
242C22⋊D53—42no—
243C22⋊D53—42no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250D212—22no—
251D212—22no—
252D212—41no—
253D212—41no—
254C4×D53—42no—
255C4×D53—42no—
256C4×D53—42no—
257C4×D53—42no—
258Dic106—22no—
259Dic106—22no—
260Dic106—41no—
261Dic106—41no—
262C8×C53—42no—
263C8×C53—42no—
264#88322—42no—
265#88322—42no—
266#11191—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—21no—
272C22×C53—21no—
273C22×C53—21no—
274C22×C53—21no—
275C22×C53#38611yescharacteristic
276C22×C53#38811yescharacteristic
277C22×C53#38811yescharacteristic
278C22×C53#39411yescharacteristic
279C22×C53#39711yescharacteristic
280C22×C53#39711yescharacteristic
281C22×C53#39711yescharacteristic
282C22×C53#44312yes—
283C22×C53#44412yes—
284C2×D53—44no—
285C2×D53—42no—
286C2×D53—82no—
287C2×D53—81no—
288C2×D53—81no—
289C4×C53—44no—
290C4×C53#38111yescharacteristic
291C4×C53#38611yescharacteristic
292C4×C53#38811yescharacteristic
293C4×C53#38811yescharacteristic
294C4×C53#39411yescharacteristic
295C4×C53#39411yescharacteristic
296C4×C53#39411yescharacteristic
297C4×C53#39711yescharacteristic
298Dic53—42no—
299Dic53—82no—
300C22×#104—531no—
301C2×#416—531no—
302C2×#408—531no—
303C2×#341—531no—
304C2×#339—531no—
305#1729—531no—
306C2×#336D5311yescharacteristic, complemented
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53#144011yescharacteristic
312C2×C53#187612yes—
313C2×C53#192912yes—
314C2×C53#193711yescharacteristic
315C2×C53#193811yescharacteristic
316D53—82no—
317C22×#41—531no—
318C22×M16—531no—
319C22×M16—531no—
320C23×C8C2×D5311yescharacteristic
321C23×C8C2×D5311yescharacteristic
322C2×#117—531no—
323C22×C42C2×D5311yescharacteristic
324C2×#104—1062no—
325C2×#104—1062no—
326C2×#104—1062no—
327#416—1062no—
328#416—1062no—
329C2×#103—532no—
330C2×#102—532no—
331#408—1062no—
332#408—1062no—
333C2×C22⋊C8—1062no—
334#341—1062no—
335#339—1062no—
336#336—22no—
337#336—1062no—
338C53#1119111yescharacteristic, complemented, Sylow
339C2×#41—532no—
340C2×#41—532no—
341C2×#41—1061no—
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—
351C22×Q8—531no—
352C22×D4—531no—
353C22×D4—1061no—
354C23×C4—21no—
355C23×C4—1061no—
356C23×C4C22×D5311yescharacteristic
357C2×M16—532no—
358C2×M16—532no—
359C2×M16—1061no—
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—
369C22×C8—21no—
370C22×C8—21no—
371C22×C8—21no—
372C22×C8—21no—
373C22×C8—21no—
374C22×C8—21no—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—1061no—
382C22×C8—1061no—
383C22×C8C4×D5312yes—
384C22×C8D21212yes—
385#117—1061no—
386#117—1061no—
387#117—1061no—
388#117—1061no—
389C2×C4⋊C4—532no—
390C2×C22⋊C4—1061no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—1061no—
397C2×C42C22⋊D5312yes—
398#104—1068no—
399#103—1064no—
400#102—1064no—
401C22⋊C8—1068no—
402C24—1061no—
403C24#1171994511yescharacteristic
404#41—1062no—
405#41—1062no—
406#41—1062no—
407#41—1062no—
408#41—1062no—
409#41—1062no—
410#41—1062no—
411#41—1062no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419#41—2121no—
420C2×Q8—532no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×D4—532no—
427C2×D4—1062no—
428C2×D4—1062no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—2121no—
435C2×D4—2121no—
436C2×D4—2121no—
437C2×D4—2121no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—1061no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4C2×C4×D5311yescharacteristic
452C22×C4C2×D21211yescharacteristic
453C22×C4#1171994511yescharacteristic
454C22×C4#1171994511yescharacteristic
455C22×C4#1171994511yescharacteristic
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4#1171994711yescharacteristic
461C22×C4#1171994711yescharacteristic
462C22×C4#1171994911yescharacteristic
463C22×C4#1171994911yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465M16—1062no—
466M16—1062no—
467M16—1062no—
468M16—1062no—
469M16—2121no—
470M16—2121no—
471M16—2121no—
472M16—2121no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—1062no—
488C2×C8—1062no—
489C2×C8—2121no—
490C2×C8—2121no—
491C2×C8—2121no—
492C2×C8—2121no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C22⋊C4—2121no—
496C22⋊C4—2121no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—2121no—
502C42—2121no—
503C23—22no—
504C23—22no—
505C23—1062no—
506C23—2121no—
507C23—2121no—
508C23—2122no—
509C23—2122no—
510C23C4×D21211yescharacteristic
511C23#1313319011yescharacteristic
512C23#1313320011yescharacteristic
513C23#1313320211yescharacteristic
514C23#1313320311yescharacteristic
515C23#1313321111yescharacteristic
516C23#1313321211yescharacteristic
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—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—1062no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4C22⋊C4×D5311yescharacteristic
545C2×C4#1313317411yescharacteristic
546C2×C4#1313317511yescharacteristic
547C2×C4#1313317711yescharacteristic
548C2×C4C4×C22⋊D5311yescharacteristic
549C2×C4#1313317911yescharacteristic
550C2×C4#1313318511yescharacteristic
551C2×C4#1313318711yescharacteristic
552C2×C4#1313318811yescharacteristic
553C2×C4#1313318911yescharacteristic
554C2×C4#1313319011yescharacteristic
555C2×C4#1313319011yescharacteristic
556C2×C4#1313319111yescharacteristic
557C2×C4#1313319211yescharacteristic
558C2×C4#1313319211yescharacteristic
559C2×C4#1313319911yescharacteristic
560C2×C4#1313319911yescharacteristic
561C2×C4#1313319911yescharacteristic
562C2×C4#1313320111yescharacteristic
563C2×C4#1313320211yescharacteristic
564C2×C4#1313320211yescharacteristic
565C2×C4#1313320211yescharacteristic
566C2×C4#1313320211yescharacteristic
567C2×C4#1313320311yescharacteristic
568C2×C4#1313320511yescharacteristic
569C2×C4#1313320611yescharacteristic
570C2×C4#1313321211yescharacteristic
571C2×C4#1313321211yescharacteristic
572C8—42no—
573C8—42no—
574C8—2122no—
575C8—2122no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—4242no—
585C22—4241no—
586C22—4241no—
587C22—2122no—
588C22—2124no—
589C22#1681050111yescharacteristic
590C22#1681051411yescharacteristic
591C22#1681051511yescharacteristic
592C22#1681053811yescharacteristic
593C22#1681054911yescharacteristic
594C22#1681055211yescharacteristic
595C22#1681055211yescharacteristic
596C22#1681074112yes—
597C22#1681074612yes—
598C4—44no—
599C4—4242no—
600C4—2122no—
601C4#1681048511yescharacteristic
602C4#1681049911yescharacteristic
603C4#1681051211yescharacteristic
604C4#1681051311yescharacteristic
605C4#1681053911yescharacteristic
606C4#1681054011yescharacteristic
607C4#1681054111yescharacteristic
608C4#1681055011yescharacteristic
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—4242no—
614C2#2027296311yescharacteristic
615C2#2027504112yes—
616C2#2027536412yes—
617C2#2027541911yescharacteristic
618C2#2027542611yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#336C2
socleC23×C53C4×D4
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#336D53
hypocenterC53#11191
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C8 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
4. C2×#16810388C2semidirect
5. C2×#16810383C2semidirect
6. #20274411C2semidirect
7. C2×C53×#336C2semidirect
17. C2×#13133142C4semidirect
18. C2×#13133139C4semidirect
306. C2×#336D53semidirect
338. C53#11191semidirect

Maximal subgroups

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

subgroupindex
C22×#131331452
C2×#168106202
C2×#168105892
C2×#168103882
C2×#168103832
#202744112
C2×C53×#3362
#1119153