Finite Group Databasebeta

← #27349993

Subgroups — #27349993

Order 13568.

1350 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811096C211yesmaximal, characteristic
2#20283741C212yesmaximal, complemented
3#20273054C212yesmaximal, complemented
4#20273050C212yesmaximal, complemented
5#20273046C212yesmaximal, complemented
6#20273040C212yesmaximal, complemented
7#20273031C212yesmaximal, complemented
8C2×#16810347C211yesmaximal, characteristic, complemented
9C2×C53×#323C211yesmaximal, characteristic, complemented
10C23×C22⋊D53C411yescharacteristic
11#16811096—24no—
12#16811096—24no—
13C2×C4×C22⋊D53C411yescharacteristic
14C2×C4×C22⋊D53C411yescharacteristic
15C2×C4×C22⋊D53C2211yescharacteristic
16C2×C4×C22⋊D53C2211yescharacteristic
17C2×#13133177C411yescharacteristic
18C2×#13133177C411yescharacteristic
19C2×#13133177C2211yescharacteristic
20C2×#13133177C2211yescharacteristic
21C2×#13133173C411yescharacteristic
22C2×#13133173C2211yescharacteristic
23C22×C22⋊Dic53C411yescharacteristic
24C2×C22⋊C4×D53C411yescharacteristic
25C2×C22⋊C4×D53C2211yescharacteristic
26#16810694—24no—
27#16810691—24no—
28#16810677C2214yescomplemented
29#16810585C2214yescomplemented
30C2×C22⋊#88322C2212yescomplemented
31#16810348C2214yescomplemented
32#16810347C2212yes—
33#16810347C2212yescomplemented
34#16810346C2214yescomplemented
35C22×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C8×C53C2212yescomplemented
37C53×#323C2212yes—
38C53×#323C2212yescomplemented
39C24×D53C2×C411yescharacteristic
40C22×C22⋊D53—24no—
41C22×C22⋊D53—22no—
42C22×C22⋊D53—22no—
43C22×C22⋊D53—24no—
44C22×C22⋊D53C2×C411yescharacteristic
45C22×C22⋊D53C2×C411yescharacteristic
46C22×C22⋊D53C2×C411yescharacteristic
47C22×C22⋊D53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C23×Dic53C2×C411yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C2×#11719912—24no—
56C22×#88322—22no—
57C4×C22⋊D53—22no—
58C4×C22⋊D53—22no—
59C4×C22⋊D53—22no—
60C4×C22⋊D53—22no—
61C4×C22⋊D53—22no—
62C4×C22⋊D53—22no—
63C4×C22⋊D53D414yes—
64C4×C22⋊D53D414yes—
65#13133177—24no—
66#13133177—24no—
67#13133177—24no—
68#13133177—24no—
69#13133173—24no—
70#13133173—24no—
71C2×#11719919C2×C411yescharacteristic
72C2×#11719919C2×C411yescharacteristic
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2311yescharacteristic
75C2×C22⋊Dic53—22no—
76C2×C22⋊Dic53—22no—
77C2×C22⋊Dic53C2×C411yescharacteristic
78C2×C22⋊Dic53C2×C411yescharacteristic
79C2×C22⋊Dic53C2×C411yescharacteristic
80C2×C22⋊Dic53C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic
82C2×#11719921C2×C411yescharacteristic
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2311yescharacteristic
85C22⋊C4×D53—24no—
86C22⋊C4×D53—24no—
87C2×C4×Dic53C2×C411yescharacteristic
88C2×C4×Dic53C2×C411yescharacteristic
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133146—24no—
92C22⋊#88322—22no—
93C22⋊#88322C2314yes—
94#13133129—24no—
95#13133127—24no—
96C25×C53C2×C411yescharacteristic
97C23×C4×C53C2×C411yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53—22no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—22no—
102C2×C22⋊C4×C53C2×C411yescharacteristic
103C2×C22⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53C2311yescharacteristic
106C22⋊C8×C53—22no—
107C22⋊C8×C53C2314yes—
108C24×C53—21no—
109C24×C53—21no—
110C24×C53—22no—
111C24×C53—81no—
112C24×C53—81no—
113C24×C53M1614yescomplemented
114C24×C53C22×C411yescharacteristic
115C24×C53C22×C411yescharacteristic
116C24×C53C22×C411yescharacteristic
117C23×D53—22no—
118C23×D53—22no—
119C23×D53—22no—
120C23×D53—81no—
121C23×D53—81no—
122C23×D53C22×C411yescharacteristic
123C23×D53C22×C411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—81no—
127C22×C4×C53—81no—
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C22×C411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2411yescharacteristic
134C2×C22⋊D53—216no—
135C2×C22⋊D53—216no—
136C2×C22⋊D53—21no—
137C2×C22⋊D53—21no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C2×C22⋊D53—22no—
141C2×C22⋊D53—22no—
142C2×C22⋊D53—22no—
143C2×C22⋊D53—22no—
144C2×C22⋊D53—22no—
145C2×C22⋊D53—22no—
146C2×C22⋊D53—22no—
147C2×C22⋊D53C22⋊C412yes—
148C2×C22⋊D53C22⋊C412yes—
149C2×C22⋊D53M1614yescomplemented
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157C22×Dic53C22×C411yescharacteristic
158C22×Dic53C22×C411yescharacteristic
159C22×Dic53C22×C411yescharacteristic
160C22×Dic53C22×C411yescharacteristic
161C22×Dic53C22×C411yescharacteristic
162C22×Dic53C22×C411yescharacteristic
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—81no—
167C2×C4×D53—81no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C2×C4×D53C2×D414yes—
171C2×C8×C53—24no—
172C2×C8×C53—82no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊Dic53—21no—
178C22⋊Dic53—21no—
179C22⋊Dic53—22no—
180C22⋊Dic53—22no—
181C22⋊Dic53C22⋊C412yes—
182C22⋊Dic53C22⋊C412yes—
183#11719921—22no—
184#11719921—22no—
185#11719921—22no—
186#11719921C22⋊C414yes—
187#11719919—22no—
188#11719919—22no—
189#11719919—22no—
190#11719919C22⋊C414yes—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—22no—
194C4×Dic53C2×D414yes—
195C2×#88322—24no—
196C2×#88322—82no—
197#11719912—84no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—81no—
211C23×C53—81no—
212C23×C53—81no—
213C23×C53—81no—
214C23×C53—84no—
215C23×C53—84no—
216C23×C53C2×C22⋊C411yescharacteristic
217C23×C53C2×C22⋊C411yescharacteristic
218C23×C53C2×M1612yes—
219C23×C53C2×M1612yes—
220C23×C53C2×M1612yes—
221C23×C53C2×M1612yes—
222C23×C53C23×C411yescharacteristic
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—21no—
226C22×D53—21no—
227C22×D53—28no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—82no—
233C22×D53—82no—
234C22×D53—84no—
235C22×D53—84no—
236C22×D53C2×C22⋊C412yes—
237C22×D53C2×M1612yes—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—81no—
247C2×C4×C53—81no—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C22×D412yes—
250C22⋊D53—28no—
251C22⋊D53—28no—
252C22⋊D53—24no—
253C22⋊D53—24no—
254C22⋊D53—24no—
255C22⋊D53—24no—
256C2×Dic53—28no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—82no—
268C2×Dic53—82no—
269C2×Dic53C2×C22⋊C412yes—
270C2×Dic53C2×C22⋊C412yes—
271C2×Dic53C2×C22⋊C412yes—
272C2×Dic53C2×M1612yes—
273C4×D53—44no—
274C4×D53—82no—
275C4×D53—82no—
276C4×D53—82no—
277C8×C53—44no—
278#88322—44no—
279#17367—531nomaximal, Sylow
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—84no—
299C22×C53—84no—
300C22×C53#40712yes—
301C22×C53#40712yes—
302C22×C53C22×C22⋊C411yescharacteristic
303C22×C53C22×M1611yescharacteristic
304C22×C53C22×M1611yescharacteristic
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—22no—
308C2×D53—22no—
309C2×D53—42no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—41no—
313C2×D53—41no—
314C2×D53—42no—
315C2×D53—88no—
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—41no—
319C4×C53—41no—
320Dic53—22no—
321Dic53—22no—
322Dic53—22no—
323Dic53—22no—
324Dic53—41no—
325Dic53—41no—
326C2×#518—531no—
327#2973—532no—
328#1466—532no—
329#1465—532no—
330#1464—532no—
331#1462—532no—
332#1459—532no—
333C2×#323—531no—
334C2×#323D5311yescharacteristic, complemented
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—22no—
338C2×C53—22no—
339C2×C53—42no—
340C2×C53—42no—
341C2×C53C2×#40711yescharacteristic
342C2×C53#288311yescharacteristic
343C2×C53#288811yescharacteristic
344D53—42no—
345D53—42no—
346C23×D4—531no—
347#518—1064no—
348#518—1064no—
349C2×C4×D4—531no—
350C2×C4×D4—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—531no—
354C2×C4×D4—531no—
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×#117—531no—
358C2×#117—531no—
359C22×C22⋊C4—531no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4C2×D5311yescharacteristic
363#435—1064no—
364#435—1064no—
365#432—534no—
366#407—534no—
367C2×C22⋊C8—532no—
368C2×C22⋊C8C2×D5312yescomplemented
369#324—534no—
370#323—532no—
371#323—532no—
372#323—534no—
373#323C2×D5312yes—
374#323C2×D5312yescomplemented
375C53#1736711yescharacteristic, complemented, Sylow
376C25—531no—
377C25C4×D5311yescharacteristic
378C22×D4—531no—
379C22×D4—531no—
380C22×D4—531no—
381C22×D4—531no—
382C22×D4—1062no—
383C22×D4—1062no—
384C22×D4—1064no—
385C22×D4—1064no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4C4×D5311yescharacteristic
394C23×C4C22×D5311yescharacteristic
395C2×M16—1064no—
396C22×C8—22no—
397C22×C8—1062no—
398C4×D4—534no—
399C4×D4—534no—
400C4×D4—1064no—
401C4×D4—1064no—
402C4×D4—1064no—
403C4×D4—1064no—
404C4×D4—1062no—
405C4×D4—1062no—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—1062no—
409C4×D4—1062no—
410#117—1064no—
411#117—1064no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—531no—
419C2×C22⋊C4—531no—
420C2×C22⋊C4—531no—
421C2×C22⋊C4—531no—
422C2×C22⋊C4—531no—
423C2×C22⋊C4—531no—
424C2×C22⋊C4—531no—
425C2×C22⋊C4—531no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—1062no—
430C2×C22⋊C4C4×D5311yescharacteristic
431C2×C22⋊C4C4×D5311yescharacteristic
432C2×C22⋊C4C22×D5311yescharacteristic
433C2×C22⋊C4C22×D5311yescharacteristic
434C2×C42—531no—
435C2×C42—531no—
436C2×C42—531no—
437C2×C42—531no—
438C4⋊C8—1064no—
439C22⋊C8—22no—
440C22⋊C8—534no—
441C22⋊C8—1064no—
442C22⋊C8—1062no—
443C22⋊C8C22×D5314yes—
444#97—1064no—
445C24—21no—
446C24—21no—
447C24—22no—
448C24—81no—
449C24—81no—
450C24—531no—
451C24—531no—
452C24—1062no—
453C24—1062no—
454C24—1062no—
455C24—4241no—
456C24—4241no—
457C24#1171991214yescomplemented
458C24C2×C4×D5311yescharacteristic
459C24C2×C4×D5311yescharacteristic
460C24C2×C4×D5311yescharacteristic
461C2×D4—532no—
462C2×D4—532no—
463C2×D4—534no—
464C2×D4—1062no—
465C2×D4—1062no—
466C2×D4—1062no—
467C2×D4—1062no—
468C2×D4—10616no—
469C2×D4—10616no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1062no—
473C2×D4—1062no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—1062no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—534no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—4241no—
497C22×C4—4241no—
498C22×C4—4241no—
499C22×C4—4241no—
500C22×C4—4241no—
501C22×C4—4241no—
502C22×C4C2×C4×D5311yescharacteristic
503C22×C4C2×C4×D5311yescharacteristic
504C22×C4C2×C4×D5311yescharacteristic
505C22×C4D4×D5312yes—
506C22×C4D4×D5312yes—
507C22×C4C23×D5311yescharacteristic
508M16—4244no—
509C2×C8—24no—
510C2×C8—82no—
511C2×C8—1064no—
512C2×C8—4242no—
513C4⋊C4—534no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—1062no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—532no—
522C22⋊C4—532no—
523C22⋊C4—534no—
524C22⋊C4—1062no—
525C22⋊C4—1062no—
526C22⋊C4—1062no—
527C22⋊C4—1062no—
528C22⋊C4—1061no—
529C22⋊C4—1061no—
530C22⋊C4—1062no—
531C42—534no—
532C42—1062no—
533C42—1062no—
534C42—1062no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—24no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—81no—
548C23—81no—
549C23—81no—
550C23—81no—
551C23—84no—
552C23—84no—
553C23—532no—
554C23—532no—
555C23—1068no—
556C23—1061no—
557C23—1061no—
558C23—1061no—
559C23—1061no—
560C23—4242no—
561C23—4242no—
562C23—4244no—
563C23—4244no—
564C23—2121no—
565C23—2121no—
566C23—2121no—
567C23—2121no—
568C23C22⋊C4×D5311yescharacteristic
569C23C22⋊C4×D5311yescharacteristic
570C23M16×D5312yes—
571C23C2×#1171991212yes—
572C23C2×#1171991212yes—
573C23C2×#1171991212yes—
574C23C22×C4×D5311yescharacteristic
575D4—1068no—
576D4—1068no—
577D4—1064no—
578D4—1064no—
579D4—1064no—
580D4—1064no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—532no—
594C2×C4—532no—
595C2×C4—1068no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—4242no—
603C2×C4—4242no—
604C2×C4—4242no—
605C2×C4—4242no—
606C2×C4—4242no—
607C2×C4—2124no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4C22⋊C4×D5312yes—
613C2×C4C2×D4×D5312yes—
614C8—44no—
615C8—2124no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—84no—
635C22—84no—
636C22—1062no—
637C22—1062no—
638C22—1062no—
639C22—1062no—
640C22—4248no—
641C22—2122no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22—2121no—
646C22—2121no—
647C22#1681058512yes—
648C22#1681058512yes—
649C22C2×C22⋊C4×D5311yescharacteristic
650C22C2×M16×D5311yescharacteristic
651C22C22×#1171991211yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—1062no—
657C4—1062no—
658C4—1062no—
659C4—1062no—
660C4—2121no—
661C4—2121no—
662C2—21no—
663C2—21no—
664C2—22no—
665C2—22no—
666C2—42no—
667C2—42no—
668C2—2122no—
669C2—2122no—
670C2C2×#1681058511yescharacteristic
671C2#2028317311yescharacteristic
672C2#2028319111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#323C2
socleC22×C53C22×C22⋊C4
prime radicalC23×C22⋊D53C4
cosocleC22×C4×C53C24
hypercenterC2×#323D53
hypocenterC53#17367
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #20283741C2semidirect
3. #20273054C2semidirect
4. #20273050C2semidirect
5. #20273046C2semidirect
6. #20273040C2semidirect
7. #20273031C2semidirect
8. C2×#16810347C2semidirect
9. C2×C53×#323C2semidirect
28. #16810677C22semidirect
29. #16810585C22semidirect
30. C2×C22⋊#88322C22semidirect
31. #16810348C22semidirect
33. #16810347C22semidirect
34. #16810346C22semidirect
36. C2×C22⋊C8×C53C22semidirect
38. C53×#323C22semidirect
113. C24×C53M16semidirect
149. C2×C22⋊D53M16semidirect
334. C2×#323D53semidirect
368. C2×C22⋊C8C2×D53semidirect
374. #323C2×D53semidirect
375. C53#17367semidirect
457. C24#11719912semidirect

Maximal subgroups

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

subgroupindex
C2×#168110962
#202837412
#202730542
#202730502
#202730462
#202730402
#202730312
C2×#168103472
C2×C53×#3232
#1736753