Finite Group Databasebeta

← #27310469

Subgroups — #27310469

Order 13568.

2142 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133195C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic
3C2×#16810527C212yesmaximal, complemented
4C2×#16810482C211yesmaximal, characteristic
5#20272969C211yesmaximal, characteristic, complemented
6#20272968C211yesmaximal, characteristic, complemented
7#20272965C211yesmaximal, characteristic, complemented
8#20272963C212yesmaximal
9#20272963C212yesmaximal
10#20272943C212yesmaximal, complemented
11C53×#1441C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C2×#13133195—24no—
15C2×#13133195—24no—
16C2×#13133195C418yescomplemented
17C2×#13133175—24no—
18C2×#13133175—28no—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×#11719919C2211yescharacteristic
22C22×#11719919C2211yescharacteristic
23C22×#11719921C2212yes—
24C22×#11719921C2211yescharacteristic
25C22×#11719921C2211yescharacteristic
26C22×#11719921C2212yes—
27C22×C4×Dic53C2212yes—
28#16810527—28no—
29#16810482—24no—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C2×#13133124C2212yes—
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2211yescharacteristic
41C2×#95×C53C2212yes—
42C24×D53C2311yescharacteristic
43C22×Dic106—24no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106—24no—
47C22×Dic106C2×C414yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2312yes—
52C23×Dic53C2311yescharacteristic
53C22×C4×D53—24no—
54C22×C4×D53—24no—
55C22×C4×D53C2×C414yes—
56C22×C4×D53D414yes—
57C22×C4×D53D414yes—
58#13133195—232no—
59#13133175—232no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—24no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919—24no—
69C2×#11719919C2×C414yes—
70C2×#11719919C2×C414yes—
71C2×#11719921—24no—
72C2×#11719921—22no—
73C2×#11719921—22no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—24no—
77C2×#11719921—24no—
78C2×#11719921—24no—
79C2×#11719921—42no—
80C2×#11719921—42no—
81C2×#11719921C2×C414yes—
82C2×#11719921C2×C414yes—
83C2×C4×Dic53—24no—
84C2×C4×Dic53—24no—
85C2×C4×Dic53D418yes—
86#13133124—24no—
87#13133124—22no—
88#13133124—24no—
89#13133124—24no—
90#13133124—42no—
91#13133124—42no—
92#13133124—42no—
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2312yes—
97C23×C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53—24no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53C2×C414yes—
103#95×C53—24no—
104#95×C53—22no—
105#95×C53—42no—
106C24×C53C2411yescharacteristic
107C23×D53—22no—
108C23×D53—22no—
109C23×D53—21no—
110C23×D53—21no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—21no—
114C23×D53—21no—
115C23×D53C22×C412yes—
116C23×D53C2×D412yes—
117C23×D53C2×D412yes—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53C22×C412yes—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C22×C412yes—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D412yes—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—22no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53C22×C412yes—
173C22×Dic53C22×C412yes—
174C22×Dic53C22×C412yes—
175C22×Dic53C2×D414yes—
176C22×Dic53C2×D412yes—
177C22×Dic53C2×D412yes—
178C22×Dic53C2×D414yes—
179C22×Dic53#4114yes—
180C2×C4×D53—216no—
181C2×C4×D53—28no—
182C2×C4×D53—216no—
183C2×C4×D53—28no—
184C2×C4×D53C22⋊C4116yes—
185C2×Dic106—28no—
186C2×Dic106—216no—
187C2×Dic106—44no—
188C2×Dic106—48no—
189C2×Dic106—44no—
190C4⋊C4×C53—28no—
191C4⋊C4×C53—28no—
192#11719921—216no—
193#11719921—28no—
194#11719921—28no—
195#11719921—48no—
196C4⋊Dic53—28no—
197#11719919—28no—
198#11719919—28no—
199#11719919—48no—
200C4×Dic53—216no—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×D411yescharacteristic
204C23×C53C22×D412yes—
205C23×C53C2×#4112yes—
206C23×C53H(5,𝔽2)11yescharacteristic
207C23×C53H(5,𝔽2)11yescharacteristic
208C23×C532−1+412yes—
209C23×C532−1+411yescharacteristic
210C23×C532−1+412yes—
211C23×C532−1+411yescharacteristic
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—24no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—24no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—42no—
231C22×D53—42no—
232C22×D53—42no—
233C22×D53—42no—
234C22×D53C2×C22⋊C414yes—
235C2×C4×C53—24no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—24no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—24no—
245C2×C4×C53—24no—
246C2×C4×C53—24no—
247C2×C4×C53—41no—
248C2×C4×C53—42no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—41no—
254C2×C4×C53—42no—
255C2×C4×C53—41no—
256C2×C4×C53—42no—
257C2×C4×C53—41no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53C2×C22⋊C414yes—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—24no—
274C2×Dic53—41no—
275C2×Dic53—42no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53C2×C22⋊C414yes—
293C2×Dic53C4×D418yes—
294C2×Dic53#12014yes—
295C2×Dic53#12118yes—
296C2×Dic53#12314yes—
297C4×D53—232no—
298Dic106—416no—
299#14258—531nomaximal, Sylow
300C22×C53C22×C22⋊C411yescharacteristic
301C22×C53C2×C4×D412yes—
302C22×C53#51912yes—
303C22×C53#52011yescharacteristic
304C22×C53#52011yescharacteristic
305C22×C53C2×#12011yescharacteristic
306C22×C53C2×#12112yes—
307C22×C53C2×#12311yescharacteristic
308C22×C53#53511yescharacteristic
309C22×C53#53611yescharacteristic
310C22×C53#53612yes—
311C22×C53#53612yes—
312C22×C53#53611yescharacteristic
313C22×C53#53611yescharacteristic
314C22×C53#53612yes—
315C22×C53#53712yes—
316C22×C53#53711yescharacteristic
317C22×C53#53711yescharacteristic
318C22×C53#53712yes—
319C22×C53#54112yes—
320C22×C53#54112yes—
321C22×C53#54212yes—
322C22×C53#54212yes—
323C2×D53—24no—
324C2×D53—24no—
325C2×D53—24no—
326C2×D53—44no—
327C2×D53—44no—
328C2×D53—42no—
329C2×D53—42no—
330C2×D53—44no—
331C2×D53—44no—
332C2×D53—42no—
333C2×D53—42no—
334C4×C53—24no—
335C4×C53—42no—
336C4×C53—44no—
337C4×C53—42no—
338C4×C53—82no—
339Dic53—42no—
340Dic53—44no—
341Dic53—42no—
342Dic53—82no—
343Dic53#38618yes—
344C22×C22⋊Q8—531no—
345C22×#117—531no—
346C2×#391—532no—
347C2×#380—531no—
348#1441—531no—
349#1441—531no—
350#1441—531no—
351#1441D5311yescharacteristic, complemented
352#1440—532no—
353#1440—532no—
354C4⋊#95—532no—
355C2×C53C2×#38611yescharacteristic
356C2×C53#231011yescharacteristic
357C2×C53#231111yescharacteristic
358C2×C53#232312yes—
359C2×C53#232312yes—
360C2×C53#261611yescharacteristic
361C2×C53#261611yescharacteristic
362C2×C53#262012yes—
363C2×C53#262112yes—
364C2×C53#262511yescharacteristic
365C2×C53#262511yescharacteristic
366D53—44no—
367C23×Q8—531no—
368C24×C4—531no—
369C2×C22⋊Q8—538no—
370C2×C22⋊Q8—1064no—
371C2×C22⋊Q8—1064no—
372C2×#117—1068no—
373C2×#117—1064no—
374C22×C4⋊C4—531no—
375C22×C4⋊C4—531no—
376C22×C4⋊C4—532no—
377C22×C4⋊C4—531no—
378C22×C4⋊C4C2×D5311yescharacteristic
379C22×C4⋊C4C2×D5311yescharacteristic
380C22×C22⋊C4—531no—
381C22×C22⋊C4—531no—
382C22×C22⋊C4—532no—
383C22×C22⋊C4—532no—
384C22×C42—532no—
385#391—1068no—
386#380—1064no—
387C2×#95—532no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—531no—
391C2×#95—532no—
392C2×#95—532no—
393C2×#95—532no—
394C2×#95C2×D5312yes—
395C2×#95C2×D5311yescharacteristic
396C2×#95C2×D5312yes—
397C53#1425811yescharacteristic, complemented, Sylow
398C25—531no—
399C22×Q8—534no—
400C22×Q8—1062no—
401C22×Q8—1062no—
402C22×Q8—1064no—
403C22×Q8—1064no—
404C23×C4—531no—
405C23×C4—534no—
406C23×C4—534no—
407C23×C4—534no—
408C23×C4—531no—
409C23×C4—531no—
410C23×C4—532no—
411C23×C4—532no—
412C23×C4—1064no—
413C23×C4—1064no—
414C23×C4C22×D5312yes—
415C23×C4C22×D5311yescharacteristic
416C23×C4C22×D5311yescharacteristic
417C23×C4C22×D5312yes—
418C23×C4C22×D5311yescharacteristic
419C22⋊Q8—10632no—
420#117—10632no—
421C2×C4⋊C4—24no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—534no—
426C2×C4⋊C4—534no—
427C2×C4⋊C4—1064no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1062no—
433C2×C4⋊C4—1064no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1064no—
436C2×C4⋊C4C4×D5314yes—
437C2×C22⋊C4—534no—
438C2×C22⋊C4—534no—
439C2×C22⋊C4—1064no—
440C2×C22⋊C4—1064no—
441C2×C22⋊C4—1062no—
442C2×C22⋊C4—1062no—
443C2×C22⋊C4—1062no—
444C2×C22⋊C4—1062no—
445C2×C22⋊C4—1064no—
446C2×C22⋊C4—1064no—
447C2×C22⋊C4—2122no—
448C2×C22⋊C4—2122no—
449C2×C42—538no—
450C2×C42—1064no—
451C2×C42—1064no—
452#95—22no—
453#95—24no—
454#95—42no—
455#95—1064no—
456#95—1064no—
457#95—1062no—
458#95—1064no—
459#95—2122no—
460#95—2122no—
461#95—2122no—
462C24—532no—
463C24—532no—
464C24—532no—
465C24—1062no—
466C24—1062no—
467C24—1061no—
468C24—1061no—
469C24—1062no—
470C24—1061no—
471C24—1062no—
472C24—1061no—
473C24C23×D5311yescharacteristic
474C2×Q8—1068no—
475C2×Q8—10616no—
476C2×Q8—2124no—
477C2×Q8—2128no—
478C2×Q8—2124no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—532no—
505C22×C4—534no—
506C22×C4—532no—
507C22×C4—532no—
508C22×C4—534no—
509C22×C4—5316no—
510C22×C4—532no—
511C22×C4—532no—
512C22×C4—534no—
513C22×C4—1068no—
514C22×C4—10616no—
515C22×C4—10616no—
516C22×C4—1068no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1062no—
520C22×C4—1061no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4—1061no—
524C22×C4—1062no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1062no—
528C22×C4—1061no—
529C22×C4—1061no—
530C22×C4—1062no—
531C22×C4—1062no—
532C22×C4—1062no—
533C22×C4—1062no—
534C22×C4—1062no—
535C22×C4—1062no—
536C22×C4—1062no—
537C22×C4—1062no—
538C22×C4—1062no—
539C22×C4—1062no—
540C22×C4C2×C4×D5312yes—
541C22×C4C2×C4×D5312yes—
542C22×C4C2×C4×D5312yes—
543C22×C4D4×D5314yes—
544C22×C4D4×D5312yes—
545C22×C4D4×D5312yes—
546C4⋊C4—28no—
547C4⋊C4—28no—
548C4⋊C4—1068no—
549C4⋊C4—1068no—
550C4⋊C4—1068no—
551C4⋊C4—2128no—
552C22⋊C4—1068no—
553C22⋊C4—1068no—
554C22⋊C4—10616no—
555C22⋊C4—2128no—
556C42—10616no—
557C23—534no—
558C23—1064no—
559C23—1064no—
560C23—1062no—
561C23—1062no—
562C23—1062no—
563C23—1064no—
564C23—1062no—
565C23—1062no—
566C23—1062no—
567C23—1064no—
568C23—1062no—
569C23—1062no—
570C23—1062no—
571C23—1062no—
572C23—2122no—
573C23—2122no—
574C23—2122no—
575C23—2122no—
576C23—2122no—
577C23—2122no—
578C23—2122no—
579C23—2122no—
580C23C22×C4×D5311yescharacteristic
581C23C2×D4×D5311yescharacteristic
582C23C2×D4×D5311yescharacteristic
583C23C2×D4×D5312yes—
584C23#41×D5312yes—
585C23#1313326211yescharacteristic
586C23#1313326311yescharacteristic
587C23#1313326412yes—
588C23#1313326411yescharacteristic
589C23#1313326512yes—
590C23#1313326511yescharacteristic
591Q8—21216no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—534no—
623C2×C4—538no—
624C2×C4—534no—
625C2×C4—538no—
626C2×C4—534no—
627C2×C4—10632no—
628C2×C4—1062no—
629C2×C4—1062no—
630C2×C4—1064no—
631C2×C4—1062no—
632C2×C4—1062no—
633C2×C4—1064no—
634C2×C4—1064no—
635C2×C4—1064no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4—2122no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4—2122no—
643C2×C4—2121no—
644C2×C4—2122no—
645C2×C4—2121no—
646C2×C4—2121no—
647C2×C4—2121no—
648C2×C4—2122no—
649C2×C4—2122no—
650C2×C4—2121no—
651C2×C4—2121no—
652C2×C4—2122no—
653C2×C4—2122no—
654C2×C4C22⋊C4×D5314yes—
655C22—1064no—
656C22—1064no—
657C22—1064no—
658C22—2122no—
659C22—2122no—
660C22—2124no—
661C22—2124no—
662C22—2124no—
663C22—2124no—
664C22—2122no—
665C22—2122no—
666C22C2×C22⋊C4×D5311yescharacteristic
667C22C4×D4×D5312yes—
668C22#1681109912yes—
669C22#1681110311yescharacteristic
670C22#1681110411yescharacteristic
671C22#120×D5311yescharacteristic
672C22#121×D5312yes—
673C22#123×D5311yescharacteristic
674C22#1681118911yescharacteristic
675C22#1681119311yescharacteristic
676C22#1681119512yes—
677C22#1681119612yes—
678C22#1681119611yescharacteristic
679C22#1681119811yescharacteristic
680C22#1681119812yes—
681C22#1681120012yes—
682C22#1681120211yescharacteristic
683C22#1681120311yescharacteristic
684C22#1681120312yes—
685C22#1681123212yes—
686C22#1681123212yes—
687C22#1681123912yes—
688C22#1681124312yes—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—82no—
694C4—538no—
695C4—4242no—
696C4—2122no—
697C4—2124no—
698C4—2122no—
699C2—2124no—
700C2#386×D5311yescharacteristic
701C2#2027729811yescharacteristic
702C2#2027730411yescharacteristic
703C2#2027739212yes—
704C2#2027739612yes—
705C2#2028044311yescharacteristic
706C2#2028044711yescharacteristic
707C2#2028047412yes—
708C2#2028048312yes—
709C2#2028051611yescharacteristic
710C2#2028051811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1441D53
hypocenterC53#14258
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#16810527C2semidirect
5. #20272969C2semidirect
6. #20272968C2semidirect
7. #20272965C2semidirect
10. #20272943C2semidirect
11. C53×#1441C2semidirect
16. C2×#13133195C4semidirect
351. #1441D53semidirect
397. C53#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#131331952
C22×#131331752
C2×#168105272
C2×#168104822
#202729692
#202729682
#202729652
#202729632
#202729432
C53×#14412
#1425853