Finite Group Databasebeta

← #27311270

Subgroups — #27311270

Order 13568.

1926 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133173C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic, complemented
3C2×#16810487C211yesmaximal, characteristic, complemented
4C2×#16810481C212yesmaximal
5#20272969C211yesmaximal, characteristic, complemented
6#20272966C212yesmaximal, complemented
7#20272965C211yesmaximal, characteristic, complemented
8#20272960C211yesmaximal, characteristic
9#20272958C212yesmaximal
10#20272943C212yesmaximal, complemented
11C53×#1439C211yesmaximal, characteristic
12C24×Dic53C2211yescharacteristic
13C2×#13133173—24no—
14C2×#13133173—24no—
15C2×#13133173C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×#11719919C2212yes—
22C22×C22⋊Dic53C2211yescharacteristic
23C22×C22⋊Dic53C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25C22×C4×Dic53C2211yescharacteristic
26#16810492—24no—
27#16810487—24no—
28#16810481—28no—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53C2211yescharacteristic
38C22×C22⋊C4×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C2×#95×C53C2212yes—
41C23×Dic53—22no—
42C23×Dic53—22no—
43C23×Dic53—22no—
44C23×Dic53—22no—
45C23×Dic53C2×C414yes—
46C23×Dic53D414yes—
47C23×Dic53Q814yes—
48C23×Dic53C2312yes—
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2311yescharacteristic
54#13133173—232no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—42no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919—42no—
69C2×#11719919—42no—
70C2×#11719919C2×C414yes—
71C2×#11719919C2×C414yes—
72C2×C22⋊Dic53—24no—
73C2×C22⋊Dic53—22no—
74C2×C22⋊Dic53—22no—
75C2×C22⋊Dic53—42no—
76C2×C22⋊Dic53—42no—
77C2×C22⋊Dic53C2×C414yes—
78C2×C4×Dic53—22no—
79C2×C4×Dic53—22no—
80C2×C4×Dic53C2×C414yes—
81C2×C4×Dic53C2×C414yes—
82C2×C4×Dic53D414yes—
83C2×C4×Dic53D414yes—
84#13133124—22no—
85#13133124—24no—
86#13133124—24no—
87#13133124—24no—
88#13133124—42no—
89#13133124—42no—
90#13133124—42no—
91C25×C53C2311yescharacteristic
92C23×C4×C53C2312yes—
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—42no—
98C2×C22⋊C4×C53—24no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—42no—
102C2×C22⋊C4×C53—42no—
103C2×C22⋊C4×C53C2×C414yes—
104#95×C53—22no—
105#95×C53—42no—
106C24×C53—21no—
107C24×C53—21no—
108C24×C53—21no—
109C24×C53—21no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53C22×C412yes—
115C24×C53C2×D412yes—
116C24×C53C2×Q812yes—
117C24×C53C2411yescharacteristic
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—28no—
149C22×Dic53—28no—
150C22×Dic53—28no—
151C22×Dic53—28no—
152C22×Dic53—28no—
153C22×Dic53—28no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
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×Dic53C4⋊C4116yescomplemented
173C22×Dic53C22×C412yes—
174C22×Dic53C22×C412yes—
175C22×Dic53C22×C412yes—
176C22×Dic53C22×C412yes—
177C22×Dic53C2×D412yes—
178C22×Dic53C2×D412yes—
179C22×Dic53C2×D412yes—
180C22×Dic53C2×D412yes—
181C22×Dic53C2×D412yes—
182C22×Dic53C2×D412yes—
183C22×Dic53C2×Q812yes—
184C22×Dic53C2×Q812yes—
185C22×Dic53#4112yes—
186C22×Dic53#4112yes—
187C4⋊C4×C53—44no—
188C22⋊C4×C53—28no—
189C22⋊C4×C53—48no—
190C22⋊Dic53—28no—
191C22⋊Dic53—48no—
192C4⋊Dic53—44no—
193#11719919—28no—
194#11719919—28no—
195#11719919—48no—
196#11719919—48no—
197#11719919—48no—
198C4×Dic53—28no—
199C4×Dic53C22⋊C4116yescomplemented
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—24no—
203C23×C53—24no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—24no—
211C23×C53—24no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53—42no—
220C23×C53—42no—
221C23×C53—42no—
222C23×C53C2×C4⋊C414yes—
223C23×C53C23×C411yescharacteristic
224C23×C53C22×D411yescharacteristic
225C23×C53C22×D411yescharacteristic
226C23×C53C22×D411yescharacteristic
227C23×C53C22×Q811yescharacteristic
228C23×C53C2×#4111yescharacteristic
229C23×C53C2×#4111yescharacteristic
230C23×C53H(5,𝔽2)12yes—
231C23×C53H(5,𝔽2)12yes—
232C23×C532−1+412yes—
233C23×C532−1+412yes—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53C2×C22⋊C414yes—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—24no—
268C2×Dic53—232no—
269C2×Dic53—22no—
270C2×Dic53—24no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53C2×C22⋊C414yes—
289C2×Dic53C2×C22⋊C414yes—
290C2×Dic53C2×C4⋊C414yes—
291C2×Dic53C2×C4⋊C414yes—
292C2×Dic53#11714yes—
293C2×Dic53#12114yes—
294C2×Dic53#12114yes—
295C2×Dic53C22⋊Q814yes—
296C2×Dic53C22⋊Q814yes—
297#14340—531nomaximal, Sylow
298C22×C53—28no—
299C22×C53—24no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—44no—
303C22×C53—42no—
304C22×C53—44no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53—44no—
309C22×C53C22×C22⋊C411yescharacteristic
310C22×C53C22×C4⋊C411yescharacteristic
311C22×C53C2×#11711yescharacteristic
312C22×C53#52012yes—
313C22×C53#52012yes—
314C22×C53C2×#12111yescharacteristic
315C22×C53C2×#12111yescharacteristic
316C22×C53C2×C22⋊Q811yescharacteristic
317C22×C53C2×C22⋊Q811yescharacteristic
318C22×C53#53511yescharacteristic
319C22×C53#53511yescharacteristic
320C22×C53#53512yes—
321C22×C53#53611yescharacteristic
322C22×C53#53611yescharacteristic
323C22×C53#53612yes—
324C22×C53#53712yes—
325C22×C53#53712yes—
326C22×C53#54012yes—
327C22×C53#54112yes—
328C22×C53#54212yes—
329C22×C53#54212yes—
330C22×C53#54412yes—
331C22×C53#54412yes—
332C4×C53—24no—
333C4×C53—42no—
334C4×C53—82no—
335C4×C53—82no—
336Dic53—24no—
337Dic53—24no—
338Dic53—42no—
339Dic53—82no—
340Dic53—82no—
341Dic53#38018yes—
342C22×#117—531no—
343C2×#384—531no—
344C2×#382—531no—
345C2×#380—532no—
346#1441—532no—
347#1441—531no—
348#1441—531no—
349#1439—532no—
350#1439—531no—
351#1439D5311yescharacteristic
352C4⋊#95—532no—
353C2×C53—44no—
354C2×C53C2×#38011yescharacteristic
355C2×C53#231111yescharacteristic
356C2×C53#231111yescharacteristic
357C2×C53#231412yes—
358C2×C53#231712yes—
359C2×C53#261911yescharacteristic
360C2×C53#262012yes—
361C2×C53#262111yescharacteristic
362C2×C53#262811yescharacteristic
363C2×C53#262811yescharacteristic
364C2×C53#262912yes—
365C24×C4—531no—
366C2×#117—538no—
367C2×#117—1064no—
368C2×#117—1064no—
369C22×C4⋊C4—531no—
370C22×C4⋊C4—531no—
371C22×C4⋊C4—532no—
372C22×C4⋊C4—532no—
373C22×C4⋊C4—532no—
374C22×C4⋊C4—531no—
375C22×C4⋊C4C2×D5311yescharacteristic
376C22×C22⋊C4—531no—
377C22×C22⋊C4—532no—
378C22×C22⋊C4C2×D5311yescharacteristic
379C22×C22⋊C4C2×D5312yes—
380C22×C42—531no—
381C22×C42—531no—
382#384—1064no—
383#382—1064no—
384#380—1068no—
385C2×#95—532no—
386C2×#95—531no—
387C2×#95—532no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—532no—
391C2×#95—532no—
392C2×#95C2×D5311yescharacteristic
393C2×#95C2×D5312yes—
394C53#1434011yescharacteristic, complemented, Sylow
395C25C22×D5311yescharacteristic
396C23×C4—531no—
397C23×C4—534no—
398C23×C4—534no—
399C23×C4—534no—
400C23×C4—531no—
401C23×C4—531no—
402C23×C4—531no—
403C23×C4—532no—
404C23×C4—532no—
405C23×C4—1062no—
406C23×C4—1062no—
407C23×C4—1062no—
408C23×C4—1062no—
409C23×C4C22×D5312yes—
410C23×C4C22×D5312yes—
411C23×C4C22×D5311yescharacteristic
412C23×C4C22×D5311yescharacteristic
413#117—10632no—
414C2×C4⋊C4—22no—
415C2×C4⋊C4—42no—
416C2×C4⋊C4—534no—
417C2×C4⋊C4—534no—
418C2×C4⋊C4—1062no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—1062no—
421C2×C4⋊C4—1062no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—1064no—
427C2×C4⋊C4—1064no—
428C2×C4⋊C4—1064no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—2122no—
431C2×C4⋊C4—2122no—
432C2×C4⋊C4—2122no—
433C2×C22⋊C4—22no—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—42no—
437C2×C22⋊C4—42no—
438C2×C22⋊C4—534no—
439C2×C22⋊C4—1062no—
440C2×C22⋊C4—1062no—
441C2×C22⋊C4—1064no—
442C2×C22⋊C4—2122no—
443C2×C22⋊C4—2122no—
444C2×C22⋊C4C4×D5314yes—
445C2×C42—534no—
446C2×C42—534no—
447C2×C42—534no—
448C2×C42—534no—
449C2×C42—1062no—
450C2×C42—1062no—
451#95—22no—
452#95—42no—
453#95—1064no—
454#95—1064no—
455#95—1064no—
456#95—1062no—
457#95—2122no—
458#95—2122no—
459#95—2122no—
460C24—21no—
461C24—21no—
462C24—21no—
463C24—21no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24C2×C4×D5312yes—
469C24D4×D5312yes—
470C24Q8×D5312yes—
471C24C23×D5311yescharacteristic
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—532no—
495C22×C4—532no—
496C22×C4—532no—
497C22×C4—532no—
498C22×C4—532no—
499C22×C4—5316no—
500C22×C4—532no—
501C22×C4—532no—
502C22×C4—532no—
503C22×C4—532no—
504C22×C4—532no—
505C22×C4—532no—
506C22×C4—532no—
507C22×C4—1068no—
508C22×C4—1068no—
509C22×C4—1068no—
510C22×C4—1068no—
511C22×C4—1068no—
512C22×C4—1068no—
513C22×C4—1061no—
514C22×C4—1061no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1062no—
518C22×C4—1062no—
519C22×C4—1061no—
520C22×C4—1061no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—1062no—
532C22×C4—1062no—
533C22×C4—1062no—
534C22×C4—1062no—
535C22×C4—1062no—
536C22×C4—1062no—
537C22×C4C2×C4×D5312yes—
538C22×C4C2×C4×D5312yes—
539C22×C4D4×D5312yes—
540C22×C4D4×D5312yes—
541C4⋊C4—44no—
542C4⋊C4—1068no—
543C4⋊C4—1068no—
544C4⋊C4—2124no—
545C4⋊C4—2128no—
546C4⋊C4—2128no—
547C4⋊C4—2128no—
548C22⋊C4—28no—
549C22⋊C4—48no—
550C22⋊C4—1068no—
551C22⋊C4—2128no—
552C42—5316no—
553C42—1068no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23C4⋊C4×D5314yes—
577C23C22×C4×D5311yescharacteristic
578C23C2×D4×D5311yescharacteristic
579C23C2×D4×D5311yescharacteristic
580C23C2×D4×D5311yescharacteristic
581C23C2×Q8×D5311yescharacteristic
582C23#41×D5311yescharacteristic
583C23#41×D5311yescharacteristic
584C23#1313326212yes—
585C23#1313326212yes—
586C23#1313326412yes—
587C23#1313326412yes—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—534no—
612C2×C4—534no—
613C2×C4—534no—
614C2×C4—534no—
615C2×C4—534no—
616C2×C4—534no—
617C2×C4—534no—
618C2×C4—534no—
619C2×C4—534no—
620C2×C4—10632no—
621C2×C4—1062no—
622C2×C4—1062no—
623C2×C4—1064no—
624C2×C4—1062no—
625C2×C4—1062no—
626C2×C4—1062no—
627C2×C4—1062no—
628C2×C4—1062no—
629C2×C4—1062no—
630C2×C4—1064no—
631C2×C4—1062no—
632C2×C4—1064no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2122no—
638C2×C4—2122no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4—2122no—
643C2×C4—2122no—
644C2×C4—2122no—
645C2×C4—2122no—
646C2×C4—2122no—
647C2×C4—2121no—
648C2×C4—2121no—
649C2×C4—2122no—
650C2×C4C22⋊C4×D5314yes—
651C22—28no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22C2×C22⋊C4×D5311yescharacteristic
663C22C2×C4⋊C4×D5311yescharacteristic
664C22#117×D5311yescharacteristic
665C22#1681110212yes—
666C22#1681110212yes—
667C22#121×D5311yescharacteristic
668C22#121×D5311yescharacteristic
669C22C22⋊Q8×D5311yescharacteristic
670C22C22⋊Q8×D5311yescharacteristic
671C22#1681118611yescharacteristic
672C22#1681118811yescharacteristic
673C22#1681118812yes—
674C22#1681119311yescharacteristic
675C22#1681119511yescharacteristic
676C22#1681119512yes—
677C22#1681120012yes—
678C22#1681120212yes—
679C22#1681122312yes—
680C22#1681123012yes—
681C22#1681123712yes—
682C22#1681124112yes—
683C22#1681125212yes—
684C22#1681125212yes—
685C4—24no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—538no—
690C4—1064no—
691C4—1064no—
692C4—4242no—
693C4—4242no—
694C4—2122no—
695C2—44no—
696C2#380×D5311yescharacteristic
697C2#2027730211yescharacteristic
698C2#2027730511yescharacteristic
699C2#2027731912yes—
700C2#2027733412yes—
701C2#2028046411yescharacteristic
702C2#2028047112yes—
703C2#2028048011yescharacteristic
704C2#2028053511yescharacteristic
705C2#2028054111yescharacteristic
706C2#2028054412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14340
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#16810492C2semidirect
3. C2×#16810487C2semidirect
5. #20272969C2semidirect
6. #20272966C2semidirect
7. #20272965C2semidirect
10. #20272943C2semidirect
15. C2×#13133173C4semidirect
172. C22×Dic53C4⋊C4semidirect
199. C4×Dic53C22⋊C4semidirect
394. C53#14340semidirect

Maximal subgroups

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

subgroupindex
C22×#131331732
C2×#168104922
C2×#168104872
C2×#168104812
#202729692
#202729662
#202729652
#202729602
#202729582
#202729432
C53×#14392
#1434053