Finite Group Databasebeta

← #27351901

Subgroups — #27351901

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288554C211yesmaximal, characteristic
2#20283744C211yesmaximal, characteristic, complemented
3#20273269C212yesmaximal, complemented
4#20273251C212yesmaximal, complemented
5#20273247C211yesmaximal, characteristic, complemented
6#20273243C211yesmaximal, characteristic, complemented
7#20273238C212yesmaximal, complemented
8#20273234C211yesmaximal, characteristic, complemented
9#20273230C211yesmaximal, characteristic, complemented
10#20273204C212yesmaximal, complemented
11C53×#2357C211yesmaximal, characteristic, complemented
12#16811322—22no—
13#16811319C412yescomplemented
14#16811313C412yescomplemented
15#16811302—22no—
16#16811282C412yescomplemented
17#16811269C412yescomplemented
18#16811220—21no—
19#16811211—21no—
20#16811182—21no—
21#16811157C2212yes—
22#16811134C2212yes—
23C2×#13133196C2211yescharacteristic
24C2×#13133191C2211yescharacteristic
25#16811096—21no—
26#16810695—22no—
27#16810695—22no—
28#16810680C2212yescomplemented
29#16810587C2211yescharacteristic
30#16810587C2211yescharacteristic
31C2×C22⋊#88322C2212yescomplemented
32#16810359C2212yescomplemented
33#16810359C2212yescomplemented
34#16810359C2212yescomplemented
35#16810359C2212yescomplemented
36#16810356C2212yes—
37#16810356C2212yes—
38#16810356C2212yes—
39#16810356C2212yes—
40C2×C4×D4×C53C2211yescharacteristic
41C53×#390—21no—
42C53×#388—21no—
43C53×#388—21no—
44C53×#382C2211yescharacteristic, complemented
45C53×#381C2211yescharacteristic
46C53×#381C2211yescharacteristic, complemented
47C4×C4⋊C4×C53—21no—
48C4×C4⋊C4×C53C2211yescharacteristic, complemented
49C4×C22⋊C4×C53C2211yescharacteristic
50C4×C22⋊C4×C53C2211yescharacteristic, complemented
51C2×#11719945C2×C412yes—
52C22×Dic106C2×C411yescharacteristic
53C22×D212C2×C411yescharacteristic
54C22×C22⋊D53—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C22×#88322—22no—
58#13133212—22no—
59#13133211—22no—
60#13133210—22no—
61#13133209C2×C412yes—
62#13133207—22no—
63#13133205C2×C412yes—
64#13133203—22no—
65#13133202—22no—
66#13133200—22no—
67#13133200—22no—
68#13133199—22no—
69#13133198—22no—
70#13133196—21no—
71#13133196—21no—
72#13133196C2×C412yes—
73#13133196C2×C412yes—
74#13133196C2312yes—
75#13133195—22no—
76#13133191—21no—
77#13133191—21no—
78#13133191C2×C412yes—
79#13133191C2×C412yes—
80#13133191C2312yes—
81#13133190—22no—
82#13133189—22no—
83#13133189—22no—
84#13133186—22no—
85#13133185—22no—
86C4×Dic106C2×C412yes—
87C4×D212C2×C412yes—
88C4×C22⋊D53—22no—
89C4×C22⋊D53—22no—
90C4×C22⋊D53—22no—
91#13133177—22no—
92#13133177—22no—
93#13133176C2×C412yes—
94#13133174—22no—
95#13133173—22no—
96C2×C4⋊Dic53C2×C411yescharacteristic
97C2×C4⋊Dic53C2×C411yescharacteristic
98C2×#11719919—21no—
99C2×C22⋊Dic53—21no—
100C2×#11719921—21no—
101C22⋊C4×D53—22no—
102C4⋊#88322—22no—
103C22⋊#88322—22no—
104C22⋊#88322—21no—
105C22⋊#88322—21no—
106C22⋊#88322C2312yes—
107C22⋊#88322C2312yes—
108#13133128—22no—
109C22×D4×C53D411yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C23×C4×C53C2311yescharacteristic
112C4×D4×C53C2×C412yes—
113C4×D4×C53C2×C412yes—
114C4×D4×C53D412yes—
115C4×D4×C53D412yes—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C22⋊C4×C53D411yescharacteristic
125C2×C42×C53—21no—
126C2×C42×C53—21no—
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53C2311yescharacteristic
133#95×C53C2311yescharacteristic
134#95×C53C2311yescharacteristic
135#95×C53C2311yescharacteristic
136C24×C53C2×D411yescharacteristic
137C24×C53C2×D411yescharacteristic
138C23×D53—21no—
139C2×D4×C53—41no—
140C2×D4×C53—41no—
141C2×D4×C53C22⋊C412yes—
142C2×D4×C53C22⋊C412yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C412yes—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D412yes—
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—22no—
161C2×C22⋊D53—22no—
162C2×C22⋊D53—22no—
163C2×C22⋊D53—22no—
164C2×C22⋊D53—42no—
165C2×C22⋊D53—42no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169#11719945—82no—
170C2×D212—21no—
171C2×D212—81no—
172C2×D212C22×C412yes—
173C2×C4×D53—22no—
174C2×C4×D53—22no—
175C2×Dic106—21no—
176C2×Dic106—81no—
177C2×Dic106C22×C412yes—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53C22⋊C412yes—
181C4⋊C4×C53C22⋊C412yes—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—81no—
185C22⋊C4×C53—81no—
186C22⋊C4×C53C22⋊C412yes—
187C22⋊C4×C53C22⋊C412yes—
188C22⋊C4×C53C22⋊C412yes—
189C22⋊C4×C53C22⋊C412yes—
190C42×C53—42no—
191C42×C53—81no—
192C42×C53—81no—
193C42×C53C22×C412yes—
194C42×C53C2×D412yes—
195C22⋊Dic53—22no—
196C22⋊Dic53—22no—
197C22⋊Dic53—22no—
198C22⋊Dic53—22no—
199#11719921—22no—
200#11719921—22no—
201#11719921—22no—
202#11719921—22no—
203#11719921—22no—
204#11719921—22no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53C22×C412yes—
208C4⋊Dic53C22×C412yes—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919—22no—
213#11719919—22no—
214#11719919—22no—
215C4×Dic53—22no—
216C4×Dic53—22no—
217#11719917—81no—
218#11719917—81no—
219C2×#88322—22no—
220C2×#88322—22no—
221C2×#88322—82no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C2×C22⋊C411yescharacteristic
231C23×C53C2×C22⋊C411yescharacteristic
232C23×C53C2×C22⋊C412yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—22no—
235C22×D53—42no—
236C22×D53—81no—
237C22×D53—81no—
238D4×C53—42no—
239D4×C53—42no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53C2×C22⋊C411yescharacteristic
265C2×C4×C53C2×C22⋊C412yes—
266C2×C4×C53C2×C22⋊C412yes—
267C2×C4×C53C2×C22⋊C411yescharacteristic
268C2×C4×C53C2×C22⋊C412yes—
269C2×C4×C53C23×C411yescharacteristic
270C2×C4×C53C22×D411yescharacteristic
271C22⋊D53—44no—
272C22⋊D53—42no—
273C22⋊D53—42no—
274C22⋊D53—82no—
275C22⋊D53—82no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—22no—
279C2×Dic53—42no—
280C2×Dic53—81no—
281C2×Dic53—81no—
282D212—42no—
283D212—81no—
284C4×D53—82no—
285C4×D53—82no—
286Dic106—42no—
287Dic106—81no—
288#88322—42no—
289#88322—42no—
290#17625—531nomaximal, Sylow
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—22no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—22no—
299C22×C53—21no—
300C22×C53—22no—
301C22×C53—21no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53C22×C22⋊C411yescharacteristic
307C22×C53#57511yescharacteristic
308C22×C53#57511yescharacteristic
309C2×D53—42no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—162no—
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—22no—
317C4×C53—21no—
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—42no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53—81no—
324Dic53—42no—
325Dic53—42no—
326Dic53—42no—
327#3505—531no—
328#2973—531no—
329#2357D5311yescharacteristic, complemented
330#1516—532no—
331#1512—532no—
332#1511—531no—
333#1510—531no—
334#1509—532no—
335#1508—531no—
336#1507—531no—
337#1501—532no—
338C2×C53—21no—
339C2×C53—21no—
340C2×C53—41no—
341C2×C53—41no—
342C2×C53—42no—
343C2×C53#288311yescharacteristic
344C2×C53#289611yescharacteristic
345C2×C53#290111yescharacteristic
346D53—82no—
347#553—532no—
348#553—1062no—
349#552—532no—
350#551—1062no—
351#548—532no—
352#546—532no—
353#539—1061no—
354#538—1061no—
355#534—1061no—
356#529—532no—
357#525—532no—
358C2×C22⋊Q8—531no—
359C2×#121—531no—
360#518—1061no—
361C2×C4×D4C2×D5311yescharacteristic
362#435—1062no—
363#435—1062no—
364#432—532no—
365#407—531no—
366#407—531no—
367C2×C22⋊C8—532no—
368#390—21no—
369#388—21no—
370#388—21no—
371#382C2×D5311yescharacteristic, complemented
372#381C2×D5311yescharacteristic
373#381C2×D5311yescharacteristic, complemented
374C4×C4⋊C4—21no—
375C4×C4⋊C4C2×D5311yescharacteristic, complemented
376C4×C22⋊C4C2×D5311yescharacteristic
377C4×C22⋊C4C2×D5311yescharacteristic, complemented
378#328—532no—
379#328—532no—
380#328—532no—
381#328—532no—
382#327—532no—
383#327—532no—
384#327—532no—
385#327—532no—
386C53#1762511yescharacteristic, complemented, Sylow
387C2×#41—532no—
388C22×Q8—531no—
389C22×D4—531no—
390C22×D4—1061no—
391C22×D4C22⋊D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C23×C4C22×D5311yescharacteristic
394C2×M16—1061no—
395C2×M16—1061no—
396C22×C8—1062no—
397#126—1062no—
398#126—1062no—
399#126—1062no—
400#125—532no—
401#125—1062no—
402#124—532no—
403#124—1062no—
404#123—1062no—
405#123—1062no—
406#123—1062no—
407#123—1062no—
408#123—1062no—
409C22⋊Q8—532no—
410C22⋊Q8—532no—
411C22⋊Q8—532no—
412C22⋊Q8—1061no—
413C22⋊Q8—1061no—
414C22⋊Q8—1062no—
415#121—532no—
416#121—532no—
417#121—532no—
418#121—1062no—
419#121—1062no—
420#121—1061no—
421#121—1061no—
422#121—1062no—
423#120—1062no—
424#120—1062no—
425C4×Q8—532no—
426C4×D4—532no—
427C4×D4—1062no—
428C4×D4—1062no—
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4C4×D5312yes—
433C4×D4C4×D5312yes—
434C4×D4D21212yes—
435C4×D4D21212yes—
436#117—532no—
437#117—1062no—
438#117—1062no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4C22⋊D5311yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1062no—
452C2×C22⋊C4C22⋊D5311yescharacteristic
453C2×C22⋊C4C22⋊D5311yescharacteristic
454C2×C42—21no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42C22×D5311yescharacteristic
458C4⋊C8—1062no—
459C22⋊C8—532no—
460C22⋊C8—532no—
461C22⋊C8—1062no—
462C22⋊C8—1061no—
463C22⋊C8—1061no—
464#97—1062no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95C22×D5311yescharacteristic
469#95C22×D5311yescharacteristic
470#95C22×D5311yescharacteristic
471#95C22×D5311yescharacteristic
472C24—1061no—
473C24C2×C22⋊D5311yescharacteristic
474C24C2×C22⋊D5311yescharacteristic
475#41—4242no—
476C2×Q8—532no—
477C2×Q8—1061no—
478C2×Q8—4241no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—532no—
482C2×D4—1062no—
483C2×D4—1062no—
484C2×D4—1062no—
485C2×D4—1062no—
486C2×D4—1061no—
487C2×D4—4241no—
488C2×D4—2122no—
489C2×D4—2122no—
490C2×D4#1171992112yes—
491C2×D4#1171992112yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4C2×C4×D5311yescharacteristic
504C22×C4C2×C4×D5311yescharacteristic
505C22×C4C2×C4×D5312yes—
506C22×C4C2×D21211yescharacteristic
507C22×C4C2×D21211yescharacteristic
508C22×C4C2×D21212yes—
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C23×D5311yescharacteristic
514M16—4241no—
515M16—4241no—
516C2×C8—1062no—
517C2×C8—1062no—
518C2×C8—4242no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—532no—
522C4⋊C4—532no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4#1171992112yes—
532C4⋊C4#1171992112yes—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C22⋊C4—1062no—
538C22⋊C4—1062no—
539C22⋊C4—1062no—
540C22⋊C4—1062no—
541C22⋊C4—1062no—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C22⋊C4—1062no—
545C22⋊C4—1062no—
546C22⋊C4—1062no—
547C22⋊C4#1171992112yes—
548C22⋊C4#1171992112yes—
549C22⋊C4#1171992112yes—
550C22⋊C4#1171992112yes—
551C42—42no—
552C42—81no—
553C42—81no—
554C42—1062no—
555C42—1062no—
556C42C2×C4×D5312yes—
557C42C2×D21212yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—1062no—
567C23—4241no—
568C23—4241no—
569C23—2122no—
570C23C2×#1171992111yescharacteristic
571C23C2×#1171992112yes—
572C23C2×#1171992111yescharacteristic
573C23C22×C22⋊D5311yescharacteristic
574Q8—4241no—
575Q8—2122no—
576D4—42no—
577D4—42no—
578D4—4241no—
579D4—4242no—
580D4—4242no—
581D4—2122no—
582D4—2124no—
583D4—2122no—
584D4—2122no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—4242no—
613C2×C4—4242no—
614C2×C4—4241no—
615C2×C4—4241no—
616C2×C4—2122no—
617C2×C4C2×#1171992112yes—
618C2×C4C2×#1171992112yes—
619C2×C4C2×#1171992112yes—
620C2×C4C2×#1171992111yescharacteristic
621C2×C4C2×#1171992111yescharacteristic
622C2×C4C22×C4×D5311yescharacteristic
623C2×C4C22×D21211yescharacteristic
624C8—2122no—
625C8—2122no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—8482no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22C22×#1171992111yescharacteristic
646C22#1681142111yescharacteristic
647C22#1681142111yescharacteristic
648C4—21no—
649C4—21no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—2122no—
660C4—2122no—
661C4—2122no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—4242no—
668C2#2028317511yescharacteristic
669C2#2028323211yescharacteristic
670C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2357C2
socleC22×C53C22×C22⋊C4
prime radical#20288554C2
cosocleC22×C4×C53C24
hypercenter#2357D53
hypocenterC53#17625
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #20283744C2semidirect
3. #20273269C2semidirect
4. #20273251C2semidirect
5. #20273247C2semidirect
6. #20273243C2semidirect
7. #20273238C2semidirect
8. #20273234C2semidirect
9. #20273230C2semidirect
10. #20273204C2semidirect
11. C53×#2357C2semidirect
13. #16811319C4semidirect
14. #16811313C4semidirect
16. #16811282C4semidirect
17. #16811269C4semidirect
28. #16810680C22semidirect
31. C2×C22⋊#88322C22semidirect
32. #16810359C22semidirect
33. #16810359C22semidirect
34. #16810359C22semidirect
35. #16810359C22semidirect
44. C53×#382C22semidirect
46. C53×#381C22semidirect
48. C4×C4⋊C4×C53C22semidirect
50. C4×C22⋊C4×C53C22semidirect
329. #2357D53semidirect
371. #382C2×D53semidirect
373. #381C2×D53semidirect
375. C4×C4⋊C4C2×D53semidirect
377. C4×C22⋊C4C2×D53semidirect
386. C53#17625semidirect

Maximal subgroups

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

subgroupindex
#202885542
#202837442
#202732692
#202732512
#202732472
#202732432
#202732382
#202732342
#202732302
#202732042
C53×#23572
#1762553