Finite Group Databasebeta

← #27350239

Subgroups — #27350239

Order 13568.

952 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811216C211yesmaximal, characteristic
2#20283754C211yesmaximal, characteristic, complemented
3#20283741C211yesmaximal, characteristic, complemented
4#20273081C212yesmaximal, complemented
5#20273080C211yesmaximal, characteristic, complemented
6#20273078C211yesmaximal, characteristic, complemented
7#20273069C212yesmaximal, complemented
8#20273063C211yesmaximal, characteristic, complemented
9#20273050C212yesmaximal, complemented
10#20273043C212yesmaximal, complemented
11C53×#1468C211yesmaximal, characteristic, complemented
12#16811216—22no—
13#16811216—22no—
14#16811216—22no—
15#16811216—22no—
16C2×#13133210C411yescharacteristic
17C2×#13133210C2211yescharacteristic
18C2×#13133208C411yescharacteristic
19C2×#13133207C2211yescharacteristic
20C2×#13133199C411yescharacteristic
21C2×#13133199C411yescharacteristic
22C2×#13133198C411yescharacteristic
23C2×#13133196C2211yescharacteristic
24C2×#13133195C411yescharacteristic
25C2×#13133194C411yescharacteristic
26C2×#13133188C411yescharacteristic
27C2×C4×C22⋊D53C2211yescharacteristic
28C2×#13133177C2211yescharacteristic
29C2×C4⋊C4×D53C2211yescharacteristic
30#16810699—22no—
31#16810698—22no—
32#16810694—22no—
33#16810691—22no—
34#16810677C2212yes—
35#16810673C2212yes—
36#16810587C2211yescharacteristic
37#16810585C2212yes—
38#16810585C2212yes—
39C2×C22⋊#88322C2211yescharacteristic
40#16810349C2212yes—
41#16810348C2212yes—
42#16810348C2212yes—
43#16810348C2212yes—
44#16810348C2212yes—
45#16810347C2212yes—
46C2×#123×C53C2211yescharacteristic
47C53×#407C2211yescharacteristic
48C2×C22⋊C8×C53C2211yescharacteristic
49C53×#324C2212yes—
50C53×#323C2212yes—
51C22×Dic106C2×C411yescharacteristic
52C22×C22⋊D53C2×C411yescharacteristic
53C22×C22⋊D53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2311yescharacteristic
58C2×#11719917—21no—
59C2×#11719912—22no—
60C2×#11719912—22no—
61C22×#88322—21no—
62#13133210—22no—
63#13133210—22no—
64#13133210—22no—
65#13133210—22no—
66#13133208—22no—
67#13133208—22no—
68#13133207—22no—
69#13133207—22no—
70#13133199—22no—
71#13133199—22no—
72#13133199—22no—
73#13133199—22no—
74#13133198—22no—
75#13133198—22no—
76#13133196—21no—
77#13133196—21no—
78#13133196—21no—
79#13133196—21no—
80#13133195—22no—
81#13133195—22no—
82#13133194—22no—
83#13133194—22no—
84#13133188—22no—
85#13133188—22no—
86C4×C22⋊D53D412yes—
87C4×C22⋊D53D412yes—
88C4×C22⋊D53D412yes—
89C4×C22⋊D53D412yes—
90#13133177—22no—
91#13133177—22no—
92C2×C4⋊Dic53C2×C411yescharacteristic
93C2×C4⋊Dic53C2×C411yescharacteristic
94C2×C4⋊Dic53C2×C411yescharacteristic
95C2×C4⋊Dic53C2×C411yescharacteristic
96C2×#11719919C2×C411yescharacteristic
97C2×#11719919C2×C411yescharacteristic
98C2×#11719919C2×C411yescharacteristic
99C2×#11719919C2×C411yescharacteristic
100C2×#11719919C2×C411yescharacteristic
101C2×#11719919C2×C411yescharacteristic
102C2×#11719919C2311yescharacteristic
103C2×#11719919C2311yescharacteristic
104C4⋊C4×D53—22no—
105C4⋊C4×D53—22no—
106C2×C22⋊Dic53C2×C411yescharacteristic
107C2×C22⋊Dic53C2×C411yescharacteristic
108C2×C22⋊Dic53C2×C411yescharacteristic
109C2×C22⋊Dic53C2×C411yescharacteristic
110C2×C22⋊Dic53C2×C411yescharacteristic
111C2×#11719921C2×C411yescharacteristic
112C2×#11719921C2×C411yescharacteristic
113C2×C4×Dic53C2×C411yescharacteristic
114C2×C4×Dic53C2311yescharacteristic
115#13133146—22no—
116#13133146—22no—
117C22⋊#88322—21no—
118C22⋊#88322—21no—
119C22⋊#88322C2312yes—
120C22⋊#88322C2312yes—
121#13133129—22no—
122#13133129—22no—
123#13133127—22no—
124C8×Dic53—22no—
125C22×D4×C53C2×C411yescharacteristic
126C23×C4×C53C2311yescharacteristic
127C2×M16×C53—21no—
128C22×C8×C53—21no—
129#123×C53—21no—
130#123×C53—21no—
131#123×C53—21no—
132#123×C53—21no—
133C2×C4⋊C4×C53C2×C411yescharacteristic
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C22⋊C4×C53C2×C411yescharacteristic
136C2×C22⋊C4×C53C2×C411yescharacteristic
137C2×C22⋊C4×C53C2311yescharacteristic
138C22⋊C8×C53—21no—
139C22⋊C8×C53—21no—
140C22⋊C8×C53C2312yes—
141C22⋊C8×C53C2312yes—
142C24×C53C22×C411yescharacteristic
143C24×C53C22×C411yescharacteristic
144C23×D53C22×C411yescharacteristic
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—81no—
148C22×C4×C53—81no—
149C22×C4×C53C22×C411yescharacteristic
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C2×C22⋊D53—22no—
159C2×C22⋊D53—42no—
160C2×C22⋊D53—42no—
161C2×C22⋊D53C22⋊C412yes—
162C2×C22⋊D53C22⋊C412yes—
163C22×Dic53—22no—
164C22×Dic53—81no—
165C22×Dic53C22×C411yescharacteristic
166C22×Dic53C22×C411yescharacteristic
167C22×Dic53C22×C411yescharacteristic
168C22×Dic53C22×C411yescharacteristic
169C22×Dic53C22×C411yescharacteristic
170C22×Dic53C22×C411yescharacteristic
171C22×Dic53C22×C411yescharacteristic
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—42no—
175C2×C4×D53—81no—
176C2×C4×D53—81no—
177C2×C4×D53C2×D412yes—
178C2×C4×D53C2×D412yes—
179C2×Dic106—21no—
180C2×Dic106—21no—
181C2×Dic106—81no—
182M16×C53—81no—
183C2×C8×C53—22no—
184C2×C8×C53—22no—
185C2×C8×C53—81no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C22⋊C4×C53—21no—
191C22⋊C4×C53—21no—
192C22⋊C4×C53—21no—
193C22⋊C4×C53—21no—
194C22⋊C4×C53—21no—
195C22⋊C4×C53—21no—
196C22⋊Dic53—21no—
197C22⋊Dic53—21no—
198C22⋊Dic53—21no—
199C22⋊Dic53—21no—
200C22⋊Dic53—22no—
201C22⋊Dic53—22no—
202C22⋊Dic53C22⋊C412yes—
203C22⋊Dic53C22⋊C412yes—
204#11719921—22no—
205#11719921C22⋊C412yes—
206#11719921C22⋊C412yes—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—22no—
210C4⋊Dic53—22no—
211C4⋊Dic53—22no—
212#11719919—21no—
213#11719919—21no—
214#11719919—21no—
215#11719919—21no—
216#11719919—22no—
217#11719919—22no—
218#11719919—22no—
219#11719919—22no—
220#11719919—22no—
221#11719919C22⋊C412yes—
222#11719919C22⋊C412yes—
223C4×Dic53—22no—
224C4×Dic53C2×D412yes—
225C4×Dic53C2×D412yes—
226#11719917—81no—
227C2×#88322—22no—
228C2×#88322—22no—
229C2×#88322—81no—
230#11719912—82no—
231#11719912—82no—
232C23×C53—22no—
233C23×C53—21no—
234C23×C53—22no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53—41no—
238C23×C53—41no—
239C23×C53C2×C22⋊C411yescharacteristic
240C23×C53C2×C22⋊C411yescharacteristic
241C23×C53C23×C411yescharacteristic
242C22×D53—22no—
243C22×D53—41no—
244C22×D53—41no—
245C22×D53C2×C22⋊C412yes—
246D4×C53—81no—
247D4×C53—81no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
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×C53C2×C22⋊C411yescharacteristic
261C2×C4×C53C2×C22⋊C411yescharacteristic
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C22⋊D53—42no—
265C22⋊D53—42no—
266C22⋊D53—44no—
267C2×Dic53—22no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—82no—
279C2×Dic53C2×C22⋊C412yes—
280C2×Dic53C2×C22⋊C412yes—
281C2×Dic53C2×C22⋊C412yes—
282C4×D53—42no—
283C4×D53—42no—
284C4×D53—44no—
285C4×D53—82no—
286Dic106—81no—
287Dic106—81no—
288C8×C53—42no—
289C8×C53—42no—
290#88322—42no—
291#88322—42no—
292#17406—531nomaximal, Sylow
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53C22×C22⋊C411yescharacteristic
307C22×C53#56812yes—
308C2×D53—22no—
309C2×D53—22no—
310C2×D53—22no—
311C2×D53—44no—
312C4×C53—21no—
313C4×C53—21no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53—41no—
319Dic53—22no—
320Dic53—22no—
321Dic53—22no—
322Dic53—41no—
323Dic53—41no—
324Dic53—41no—
325Dic53—41no—
326C2×#539—531no—
327#2975—531no—
328#2973—531no—
329#1473—532no—
330#1472—531no—
331#1472—531no—
332#1470—532no—
333#1468—531no—
334#1468D5311yescharacteristic, complemented
335#1465—532no—
336#1463—532no—
337C2×C53—22no—
338C2×C53—41no—
339C2×C53—41no—
340C2×C53—81no—
341C2×C53#287911yescharacteristic
342C2×C53#288311yescharacteristic
343C2×C53#288911yescharacteristic
344D53—42no—
345#539—1062no—
346#539—1062no—
347#539—1062no—
348#539—1062no—
349C2×#126—531no—
350C2×#126—531no—
351C2×#125—531no—
352C2×#125—531no—
353C2×#123—531no—
354C2×#123—531no—
355C2×#123—531no—
356C2×#123C2×D5311yescharacteristic
357C2×C22⋊Q8—531no—
358C2×C22⋊Q8—531no—
359C2×C22⋊Q8—531no—
360C2×#121—531no—
361C2×C4×D4—531no—
362C2×C4×D4—531no—
363C22×C4⋊C4—531no—
364#436—1062no—
365#436—1062no—
366#435—1062no—
367#435—1062no—
368#432—532no—
369#431—532no—
370#407—531no—
371#407—532no—
372#407—532no—
373#407C2×D5311yescharacteristic
374C2×C22⋊C8—531no—
375C2×C22⋊C8C2×D5311yescharacteristic
376#324—532no—
377#324—532no—
378#324—532no—
379#324—532no—
380#324—532no—
381#324C2×D5312yes—
382#323—532no—
383#323C2×D5312yes—
384C53#1740611yescharacteristic, complemented, Sylow
385C22×Q8—531no—
386C22×D4—531no—
387C22×D4—531no—
388C22×D4C4×D5311yescharacteristic
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4C22×D5311yescharacteristic
394C2×M16—21no—
395C2×M16—1062no—
396C2×M16—1062no—
397C2×M16—1061no—
398C22×C8—21no—
399C22×C8—1061no—
400#126—1062no—
401#126—1062no—
402#126—1062no—
403#126—1062no—
404#125—1062no—
405#125—1062no—
406#125—1062no—
407#125—1062no—
408#123—21no—
409#123—21no—
410#123—21no—
411#123—21no—
412#123—1062no—
413#123—1062no—
414#123—1062no—
415#123—1062no—
416#123—1062no—
417#123—1062no—
418C22⋊Q8—1062no—
419C22⋊Q8—1062no—
420C22⋊Q8—1062no—
421C22⋊Q8—1062no—
422C22⋊Q8—1061no—
423C22⋊Q8—1061no—
424C22⋊Q8—1061no—
425C22⋊Q8—1061no—
426#121—1062no—
427#121—1062no—
428C4×D4—532no—
429C4×D4—532no—
430C4×D4—532no—
431C4×D4—532no—
432C4×D4—1062no—
433C4×D4—1062no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—531no—
446C2×C4⋊C4—1062no—
447C2×C4⋊C4—1062no—
448C2×C4⋊C4C4×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C22⋊C4—531no—
451C2×C22⋊C4—531no—
452C2×C22⋊C4—531no—
453C2×C22⋊C4—531no—
454C2×C22⋊C4—531no—
455C2×C22⋊C4—531no—
456C2×C22⋊C4—531no—
457C2×C22⋊C4C4×D5311yescharacteristic
458C2×C22⋊C4C4×D5311yescharacteristic
459C2×C22⋊C4C22×D5311yescharacteristic
460C2×C42—531no—
461C2×C42—531no—
462C4⋊C8—1062no—
463C4⋊C8—1062no—
464C22⋊C8—21no—
465C22⋊C8—21no—
466C22⋊C8—532no—
467C22⋊C8—532no—
468C22⋊C8—1062no—
469C22⋊C8—1062no—
470C22⋊C8—1061no—
471C22⋊C8—1061no—
472C22⋊C8C22×D5312yes—
473C22⋊C8C22×D5312yes—
474#97—1062no—
475C4×C8—1062no—
476C24—531no—
477C24C2×C4×D5311yescharacteristic
478C24C2×C4×D5311yescharacteristic
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—4241no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—81no—
485C2×D4—532no—
486C2×D4—532no—
487C2×D4—1062no—
488C2×D4—2122no—
489C2×D4—2122no—
490C22×C4—81no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—532no—
495C22×C4—532no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—4241no—
506C22×C4—2122no—
507C22×C4C2×C4×D5311yescharacteristic
508C22×C4C2×C4×D5311yescharacteristic
509C22×C4C2×C4×D5311yescharacteristic
510C22×C4C2×C4×D5311yescharacteristic
511C22×C4D4×D5311yescharacteristic
512C22×C4D4×D5311yescharacteristic
513C22×C4D4×D5311yescharacteristic
514C22×C4D4×D5311yescharacteristic
515C22×C4C23×D5311yescharacteristic
516M16—81no—
517M16—4242no—
518M16—4242no—
519M16—4241no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—81no—
523C2×C8—1062no—
524C2×C8—1062no—
525C2×C8—4241no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—532no—
531C4⋊C4—532no—
532C4⋊C4—1062no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1062no—
539C4⋊C4—1062no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—532no—
553C22⋊C4—532no—
554C22⋊C4—532no—
555C22⋊C4—532no—
556C22⋊C4—1062no—
557C22⋊C4—1062no—
558C22⋊C4—1062no—
559C22⋊C4—1061no—
560C22⋊C4—1061no—
561C22⋊C4—1061no—
562C22⋊C4—1061no—
563C42—532no—
564C42—532no—
565C42—1062no—
566C23—22no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—532no—
574C23—1062no—
575C23—2121no—
576C23—2121no—
577C23C22⋊C4×D5311yescharacteristic
578C23C22⋊C4×D5311yescharacteristic
579C23C22×C4×D5311yescharacteristic
580Q8—4241no—
581Q8—4241no—
582D4—81no—
583D4—81no—
584D4—2122no—
585D4—2122no—
586D4—2124no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—532no—
600C2×C4—532no—
601C2×C4—532no—
602C2×C4—1062no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—4242no—
610C2×C4—4242no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2124no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4C22⋊C4×D5311yescharacteristic
619C2×C4C22⋊C4×D5311yescharacteristic
620C2×C4C2×D4×D5311yescharacteristic
621C2×C4C2×D4×D5311yescharacteristic
622C8—42no—
623C8—42no—
624C8—2122no—
625C8—2122no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—82no—
638C22—82no—
639C22—1062no—
640C22—1062no—
641C22—1062no—
642C22—2124no—
643C22C2×C22⋊C4×D5311yescharacteristic
644C22#1681138812yes—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—1062no—
653C4—1062no—
654C4—1062no—
655C4—2121no—
656C4—2121no—
657C4—2121no—
658C4—2121no—
659C2—22no—
660C2—41no—
661C2—41no—
662C2—81no—
663C2—2122no—
664C2#2028315711yescharacteristic
665C2#2028317311yescharacteristic
666C2#2028319611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1468C2
socleC22×C53C22×C22⋊C4
prime radicalC2×#13133188C4
cosocleC22×C4×C53C24
hypercenter#1468D53
hypocenterC53#17406
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1468

Complements

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

NH ≅ G/Ntype
2. #20283754C2semidirect
3. #20283741C2semidirect
4. #20273081C2semidirect
5. #20273080C2semidirect
6. #20273078C2semidirect
7. #20273069C2semidirect
8. #20273063C2semidirect
9. #20273050C2semidirect
10. #20273043C2semidirect
11. C53×#1468C2semidirect
334. #1468D53semidirect
384. C53#17406semidirect

Maximal subgroups

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

subgroupindex
C2×#168112162
#202837542
#202837412
#202730812
#202730802
#202730782
#202730692
#202730632
#202730502
#202730432
C53×#14682
#1740653