Finite Group Databasebeta

← #27311052

Subgroups — #27311052

Order 13568.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2C2×#16810521C211yesmaximal, characteristic, complemented
3#20272984C211yesmaximal, characteristic, complemented
4#20272971C211yesmaximal, characteristic, complemented
5#20272966C212yesmaximal, complemented
6#20272960C211yesmaximal, characteristic
7#20272958C212yesmaximal
8#20272948C212yesmaximal, complemented
9#20272942C211yesmaximal, characteristic, complemented
10#20272938C212yesmaximal
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C22⋊Dic53C2212yes—
18#16810559C418yescomplemented
19#16810521—24no—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C24×C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C4×C22⋊C4×C53—24no—
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53—22no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—42no—
46C2×C4⋊Dic53C2×C414yescomplemented
47C2×#11719919—24no—
48C2×#11719919—42no—
49C2×#11719919—42no—
50C2×#11719919—42no—
51C2×#11719919—42no—
52C2×#11719919C2×C418yescomplemented
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53—42no—
57C2×C22⋊Dic53—42no—
58C2×C22⋊Dic53C2×C418yes—
59#13133124—24no—
60#13133124—24no—
61#13133124—24no—
62#13133124—22no—
63#13133124—42no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124—42no—
68#13133124C2×C414yescomplemented
69C25×C53C2311yescharacteristic
70C23×C4×C53—22no—
71C23×C4×C53—22no—
72C23×C4×C53—42no—
73C23×C4×C53—42no—
74C23×C4×C53Q814yes—
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C2×C22⋊C4×C53—22no—
82C2×C22⋊C4×C53—22no—
83C2×C22⋊C4×C53—22no—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—22no—
86C2×C22⋊C4×C53C2×C414yes—
87C2×C42×C53—22no—
88C2×C42×C53—42no—
89C2×C42×C53—42no—
90C2×C42×C53D414yes—
91#95×C53—22no—
92#95×C53—22no—
93C24×C53—21no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53C22×C412yes—
102C24×C53C2×Q812yes—
103C24×C53#4112yes—
104C24×C53C2411yescharacteristic
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—28no—
126C22×C4×C53—48no—
127C22×C4×C53—44no—
128C22×C4×C53—44no—
129C22×C4×C53—48no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×Q812yes—
136C22×C4×C53C2×Q812yes—
137C22×C4×C53#4112yes—
138C22×C4×C53#4112yes—
139C22×C4×C53#4112yes—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
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×Dic53C22×C414yes—
167C22×Dic53C22×C414yes—
168C22⋊C4×C53—28no—
169C22⋊C4×C53—28no—
170C42×C53—44no—
171C42×C53—44no—
172C22⋊Dic53—48no—
173C22⋊Dic53—48no—
174C4⋊Dic53—28no—
175C4⋊Dic53—44no—
176#11719919—48no—
177#11719919—48no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53C4×Q814yes—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×Q811yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C23×C53C2×#4111yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C23×C53C2×#4111yescharacteristic
208C23×C53H(5,𝔽2)12yes—
209C23×C53H(5,𝔽2)12yes—
210C23×C53H(5,𝔽2)12yes—
211C23×C532−1+412yes—
212C2×C4×C53—24no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—416no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53C4×D414yes—
242C2×C4×C53C22⋊Q814yes—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271#14317—531nomaximal, Sylow
272C22×C53—28no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—44no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53C2×C4×D411yescharacteristic
284C22×C53C2×C4×Q811yescharacteristic
285C22×C53C4×#4111yescharacteristic
286C22×C53#51812yes—
287C22×C53#52012yes—
288C22×C53C2×C22⋊Q811yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#52911yescharacteristic
291C22×C53#53311yescharacteristic
292C22×C53#53411yescharacteristic
293C22×C53#53511yescharacteristic
294C22×C53#53511yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#53812yes—
297C22×C53#53812yes—
298C22×C53#53912yes—
299C22×C53#53912yes—
300C22×C53#53912yes—
301C22×C53#54012yes—
302C22×C53#54012yes—
303C22×C53#54212yes—
304C22×C53#54312yes—
305C22×C53#54412yes—
306C4×C53—24no—
307C4×C53—24no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—42no—
312Dic53—44no—
313Dic53—44no—
314Dic53—82no—
315Dic53—82no—
316C2×#399—531no—
317C2×#389—531no—
318C2×C4×C22⋊C4D5311yescharacteristic
319#1445—531no—
320#1442—531no—
321#1441—532no—
322#1439—532no—
323#1439—531no—
324#1436—532no—
325#1434—531no—
326C22⋊#95—532no—
327C2×C53—44no—
328C2×C53C4×C22⋊Q811yescharacteristic
329C2×C53#232211yescharacteristic
330C2×C53#232811yescharacteristic
331C2×C53#232912yes—
332C2×C53#233512yes—
333C2×C53#262611yescharacteristic
334C2×C53#262811yescharacteristic
335C2×C53#263311yescharacteristic
336C2×C53#263511yescharacteristic
337C2×C53#264912yes—
338C2×C53#266712yes—
339C24×C4C2×D5311yescharacteristic
340C22×C4⋊C4—531no—
341C22×C4⋊C4—532no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—531no—
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4C2×D5311yescharacteristic
347C22×C22⋊C4C2×D5311yescharacteristic
348C22×C42C2×D5311yescharacteristic
349C22×C42C2×D5311yescharacteristic
350#399—538no—
351#389—1064no—
352C4×C22⋊C4—24no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95C2×D5311yescharacteristic
364C2×#95C2×D5311yescharacteristic
365C53#1431711yescharacteristic, complemented, Sylow
366C25C22×D5311yescharacteristic
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—42no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4—532no—
375C23×C4Dic10614yes—
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C23×C4C22×D5311yescharacteristic
382C2×C4⋊C4—534no—
383C2×C4⋊C4—538no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—2122no—
389C2×C4⋊C4—2122no—
390C2×C4⋊C4—2122no—
391C2×C4⋊C4—2122no—
392C2×C4⋊C4—2122no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—538no—
399C2×C22⋊C4—1064no—
400C2×C22⋊C4—2122no—
401C2×C22⋊C4—2122no—
402C2×C22⋊C4—2122no—
403C2×C22⋊C4—2122no—
404C2×C22⋊C4C4×D5314yes—
405C2×C42—22no—
406C2×C42—42no—
407C2×C42—42no—
408C2×C42C22⋊D5314yes—
409#95—22no—
410#95—22no—
411#95—534no—
412#95—1064no—
413#95—1064no—
414#95—1062no—
415#95—1064no—
416#95—2122no—
417#95—2122no—
418#95—2122no—
419#95—2122no—
420#95—2122no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×Dic10612yes—
430C24C2×C4×D5312yes—
431C24#1171994512yes—
432C24C23×D5311yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4C2×Dic10612yes—
487C22×C4C2×Dic10612yes—
488C22×C4C2×C4×D5312yes—
489C22×C4C2×C4×D5312yes—
490C22×C4#1171994512yes—
491C22×C4#1171994512yes—
492C22×C4#1171994512yes—
493C22×C4C2×C22⋊D5312yes—
494C22×C4C2×C22⋊D5312yes—
495C22×C4C2×C22⋊D5312yes—
496C4⋊C4—1068no—
497C4⋊C4—2128no—
498C4⋊C4—2128no—
499C4⋊C4—2124no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—2128no—
503C22⋊C4—2128no—
504C42—44no—
505C42—44no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23C4×Dic10614yes—
529C23C22×C4×D5311yescharacteristic
530C23C22×C22⋊D5311yescharacteristic
531C23C22×Dic10611yescharacteristic
532C23C2×#1171994511yescharacteristic
533C23C2×#1171994511yescharacteristic
534C23C2×#1171994511yescharacteristic
535C23C2×#1171994511yescharacteristic
536C23#1313326212yes—
537C23#1313326212yes—
538C23#1313326312yes—
539C23#1313326412yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4C4×C22⋊D5314yes—
598C2×C4#1313319614yes—
599C22—24no—
600C22—28no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22C2×C4×C22⋊D5311yescharacteristic
611C22C2×C4×Dic10611yescharacteristic
612C22C4×#1171994511yescharacteristic
613C22#1681109512yes—
614C22#1681110212yes—
615C22C2×#1313319611yescharacteristic
616C22#1681113411yescharacteristic
617C22#1681115711yescharacteristic
618C22#1681118011yescharacteristic
619C22#1681118211yescharacteristic
620C22#1681118711yescharacteristic
621C22#1681119011yescharacteristic
622C22#1681119411yescharacteristic
623C22#1681120612yes—
624C22#1681121112yes—
625C22#1681121512yes—
626C22#1681121512yes—
627C22#1681121512yes—
628C22#1681122412yes—
629C22#1681122412yes—
630C22#1681124412yes—
631C22#1681124912yes—
632C22#1681125112yes—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—4242no—
640C4—4242no—
641C4—2124no—
642C4—2124no—
643C2—44no—
644C2C4×#1313319611yescharacteristic
645C2#2027738011yescharacteristic
646C2#2027743711yescharacteristic
647C2#2027744312yes—
648C2#2027750712yes—
649C2#2028052011yescharacteristic
650C2#2028053611yescharacteristic
651C2#2028058311yescharacteristic
652C2#2028059911yescharacteristic
653C2#2028071212yes—
654C2#2028090912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#14317
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16810521C2semidirect
3. #20272984C2semidirect
4. #20272971C2semidirect
5. #20272966C2semidirect
8. #20272948C2semidirect
9. #20272942C2semidirect
18. #16810559C4semidirect
46. C2×C4⋊Dic53C2×C4semidirect
52. C2×#11719919C2×C4semidirect
68. #13133124C2×C4semidirect
365. C53#14317semidirect

Maximal subgroups

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

subgroupindex
C2×#168105592
C2×#168105212
#202729842
#202729712
#202729662
#202729602
#202729582
#202729482
#202729422
#202729382
C2×C4×C22⋊C4×C532
#1431753