Finite Group Databasebeta

← #27314852

Subgroups — #27314852

Order 13568.

1586 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810516C212yesmaximal, complemented
2#20272972C212yesmaximal, complemented
3#20272968C211yesmaximal, characteristic, complemented
4#20272960C212yesmaximal, complemented
5#20272960C212yesmaximal, complemented
6#20272960C212yesmaximal, complemented
7#20272956C212yesmaximal
8#20272944C211yesmaximal, characteristic, complemented
9C2×C53×#394C211yesmaximal, characteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2212yes—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2212yes—
16C22×C22⋊Dic53C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18#16810516—28no—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C23×D4×C53C2211yescharacteristic
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2212yes—
30C22×C22⋊C4×C53C2212yes—
31C53×#394C418yescomplemented
32C2×#95×C53C2211yescharacteristic
33C23×Dic53C2312yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—42no—
40C2×C4⋊Dic53—42no—
41C2×C4⋊Dic53—42no—
42C2×C4⋊Dic53—41no—
43C2×C4⋊Dic53—41no—
44C2×C4⋊Dic53—41no—
45C2×C22⋊Dic53—24no—
46C2×C22⋊Dic53—24no—
47C2×C22⋊Dic53—42no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53—42no—
51C2×C22⋊Dic53—42no—
52C2×C22⋊Dic53—42no—
53C2×C22⋊Dic53—42no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53—42no—
57C2×C4×Dic53—41no—
58C2×C4×Dic53—41no—
59C2×C4×Dic53D414yescomplemented
60#13133124—24no—
61#13133124—24no—
62#13133124—42no—
63#13133124—42no—
64#13133124—42no—
65#13133124—41no—
66#13133124—42no—
67#13133124—42no—
68C25×C53C2312yes—
69C22×D4×C53—41no—
70C22×D4×C53—41no—
71C22×D4×C53—42no—
72C22×D4×C53—42no—
73C22×D4×C53C2×C414yes—
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2312yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—41no—
79C2×C4⋊C4×C53—41no—
80C2×C4⋊C4×C53C2×C414yes—
81C2×C22⋊C4×C53—24no—
82C2×C22⋊C4×C53—24no—
83C2×C22⋊C4×C53—42no—
84C2×C22⋊C4×C53—42no—
85C2×C22⋊C4×C53C2×C418yes—
86C2×C22⋊C4×C53C2×C418yes—
87#95×C53C2×C414yes—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53C22×C414yes—
103C24×C53C2411yescharacteristic
104C2×D4×C53—48no—
105C2×D4×C53—44no—
106C2×D4×C53—44no—
107C2×D4×C53—48no—
108C2×D4×C53—44no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53C22×C414yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53#4112yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
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×Dic53C2×D414yes—
161C4⋊C4×C53—44no—
162C22⋊C4×C53—216no—
163C22⋊C4×C53—48no—
164C22⋊Dic53—48no—
165C22⋊Dic53—48no—
166C22⋊Dic53—48no—
167C22⋊Dic53—48no—
168C4⋊Dic53—48no—
169C4⋊Dic53—44no—
170C4⋊Dic53—44no—
171C4×Dic53—44no—
172C23×C53—24no—
173C23×C53—24no—
174C23×C53—24no—
175C23×C53—24no—
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53C23×C411yescharacteristic
208C23×C53C22×D411yescharacteristic
209C23×C53C2×#4111yescharacteristic
210C23×C53H(5,𝔽2)11yescharacteristic
211C23×C53H(5,𝔽2)12yes—
212C23×C53H(5,𝔽2)11yescharacteristic
213C23×C53H(5,𝔽2)12yes—
214C23×C53H(5,𝔽2)11yescharacteristic
215C23×C53H(5,𝔽2)12yes—
216C23×C532−1+411yescharacteristic
217C23×C532−1+412yes—
218D4×C53—416no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—22no—
231C2×C4×C53—24no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53C4×D414yes—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
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#14632—531nomaximal, Sylow
272C22×C53—44no—
273C22×C53—44no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—82no—
280C22×C53—82no—
281C22×C53—82no—
282C22×C53—82no—
283C22×C53—82no—
284C22×C53—82no—
285C22×C53—82no—
286C22×C53—82no—
287C22×C53C2×C4×D411yescharacteristic
288C22×C53#51812yes—
289C22×C53#51811yescharacteristic
290C22×C53#52012yes—
291C22×C53#52011yescharacteristic
292C22×C53#53411yescharacteristic
293C22×C53#53512yes—
294C22×C53#53711yescharacteristic
295C22×C53#53712yes—
296C22×C53#53811yescharacteristic
297C22×C53#53911yescharacteristic
298C22×C53#54012yes—
299C22×C53#54212yes—
300C22×C53#56011yescharacteristic
301C22×C53#56012yes—
302C22×C53#56012yes—
303C22×C53H(3,𝔽4)11yescharacteristic
304C22×C53H(3,𝔽4)11yescharacteristic
305C22×C53#56212yes—
306C22×C53#56212yes—
307C22×C53#56212yes—
308C22×C53#56311yescharacteristic
309C22×C53#56312yes—
310C4×C53—24no—
311C4×C53—44no—
312C4×C53—42no—
313C4×C53—42no—
314Dic53—44no—
315Dic53—82no—
316Dic53—82no—
317Dic53—82no—
318C2×#394D5311yescharacteristic
319C2×#388—532no—
320#1442—532no—
321#1441—531no—
322#1439—532no—
323#1439—532no—
324#1439—532no—
325#1438—532no—
326C4⋊#95—531no—
327C2×C53—84no—
328C2×C53#232412yes—
329C2×C53#232511yescharacteristic
330C2×C53#237511yescharacteristic
331C2×C53#237612yes—
332C2×C53#237911yescharacteristic
333C2×C53#281312yes—
334C2×C53#281911yescharacteristic
335C2×C53#282311yescharacteristic
336C2×C53#282811yescharacteristic
337C2×C53#283211yescharacteristic
338C2×C53#283712yes—
339C23×D4C2×D5311yescharacteristic
340C22×C4⋊C4—531no—
341C22×C4⋊C4—532no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4C2×D5311yescharacteristic
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4—532no—
347C22×C22⋊C4—532no—
348C22×C22⋊C4C2×D5312yes—
349C22×C22⋊C4C2×D5312yes—
350C22×C42—531no—
351#394Dic5318yescomplemented
352#388—1068no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95C2×D5311yescharacteristic
362C53#1463211yescharacteristic, complemented, Sylow
363C25C22×D5312yes—
364C22×D4—42no—
365C22×D4—42no—
366C22×D4—41no—
367C22×D4—41no—
368C22×D4C2×Dic5314yes—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4C22×D5311yescharacteristic
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C2×C4⋊C4—41no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—2121no—
382C2×C4⋊C4—2121no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4—2122no—
385C2×C4⋊C4—2122no—
386C2×C4⋊C4—2121no—
387C2×C4⋊C4C2×Dic5314yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—42no—
391C2×C22⋊C4—42no—
392C2×C22⋊C4—1064no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—2122no—
395C2×C22⋊C4—2122no—
396C2×C22⋊C4—2122no—
397C2×C22⋊C4—2122no—
398C2×C22⋊C4—2122no—
399C2×C22⋊C4—2122no—
400C2×C22⋊C4—2122no—
401C2×C22⋊C4—2122no—
402C2×C22⋊C4—2122no—
403C2×C22⋊C4—2122no—
404C2×C22⋊C4C2×Dic5318yes—
405C2×C22⋊C4C2×Dic5318yes—
406C2×C42—534no—
407C2×C42—2121no—
408C2×C42—2121no—
409#95—1064no—
410#95—1064no—
411#95—2122no—
412#95—2122no—
413#95—2121no—
414#95—2122no—
415#95—2122no—
416#95—2122no—
417#95C2×Dic5314yes—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×Dic5314yes—
433C24C23×D5311yescharacteristic
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—44no—
438C2×D4—44no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—534no—
458C22×C4—1062no—
459C22×C4—1062no—
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×C4D4×D5312yes—
486C22×C4#1171994712yes—
487C22×C4C22×Dic5314yes—
488C22×C4C22×Dic5312yes—
489C22×C4C22×Dic5312yes—
490C22×C4C22×Dic5312yes—
491C4⋊C4—44no—
492C4⋊C4—2124no—
493C4⋊C4—2128no—
494C4⋊C4—2124no—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—2128no—
498C22⋊C4—2128no—
499C22⋊C4—2128no—
500C22⋊C4—2128no—
501C42—2124no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×Dic5311yescharacteristic
538C23C2×D4×D5311yescharacteristic
539C23C2×#1171994711yescharacteristic
540C23#1313326212yes—
541C23#1313326211yescharacteristic
542C23#1313326211yescharacteristic
543C23#1313326212yes—
544C23#1313326311yescharacteristic
545C23#1313326312yes—
546C23#1313326412yes—
547C23#1313326411yescharacteristic
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
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×C4—2122no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4D4×Dic5314yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×D4×Dic5311yescharacteristic
617C22#1681109812yes—
618C22#1681109811yescharacteristic
619C22#1681110611yescharacteristic
620C22#1681110612yes—
621C22#1681118111yescharacteristic
622C22#1681118912yes—
623C22#1681120011yescharacteristic
624C22#1681120212yes—
625C22#1681121011yescharacteristic
626C22#1681121811yescharacteristic
627C22#1681122612yes—
628C22#1681124012yes—
629C22#1681136212yes—
630C22#1681136211yescharacteristic
631C22#1681136312yes—
632C22#1681136511yescharacteristic
633C22#1681136611yescharacteristic
634C22#1681136712yes—
635C22#1681137012yes—
636C22#1681137012yes—
637C22#1681137311yescharacteristic
638C22#1681137612yes—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—4242no—
644C4—4242no—
645C4—4242no—
646C4—2124no—
647C2—84no—
648C2#2027741012yes—
649C2#2027742011yescharacteristic
650C2#2027785611yescharacteristic
651C2#2027786312yes—
652C2#2027788911yescharacteristic
653C2#2028272212yes—
654C2#2028275911yescharacteristic
655C2#2028278611yescharacteristic
656C2#2028283811yescharacteristic
657C2#2028288611yescharacteristic
658C2#2028293112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#394C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#394D53
hypocenterC53#14632
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#16810516C2semidirect
2. #20272972C2semidirect
3. #20272968C2semidirect
4. #20272960C2semidirect
5. #20272960C2semidirect
6. #20272960C2semidirect
8. #20272944C2semidirect
31. C53×#394C4semidirect
59. C2×C4×Dic53D4semidirect
351. #394Dic53semidirect
362. C53#14632semidirect

Maximal subgroups

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

subgroupindex
C2×#168105162
#202729722
#202729682
#202729602
#202729562
#202729442
C2×C53×#3942
#1463253