Finite Group Databasebeta

← #27310765

Subgroups — #27310765

Order 13568.

1610 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133212C211yesmaximal, characteristic
2C2×#16810487C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
4#20272968C211yesmaximal, characteristic, complemented
5#20272966C212yesmaximal, complemented
6#20272961C211yesmaximal, characteristic
7#20272959C212yesmaximal
8#20272957C211yesmaximal, characteristic
9#20272951C212yesmaximal
10#20272949C212yesmaximal, complemented
11C53×#1441C211yesmaximal, characteristic, complemented
12C2×#13133212—24no—
13C2×#13133212—24no—
14C2×#13133212C418yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2212yes—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2211yescharacteristic
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2212yes—
23C22×#11719921C2212yes—
24C22×C4×Dic53C2212yes—
25C22×C4×Dic53C2211yescharacteristic
26#16810487—24no—
27C4⋊C4×Dic53—24no—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46#13133212—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53C2×C414yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919—24no—
55C2×#11719919—42no—
56C2×#11719919—42no—
57C2×#11719919C2×C414yes—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921—24no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921—22no—
65C2×#11719921—24no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69C2×#11719921—42no—
70C2×#11719921C2×C414yes—
71C2×#11719921C2×C414yes—
72C2×#11719921C2×C414yes—
73C2×C4×Dic53—24no—
74C2×C4×Dic53—42no—
75C2×C4×Dic53—42no—
76C2×C4×Dic53C2×C414yes—
77C2×C4×Dic53D414yes—
78C2×C4×Dic53D414yes—
79#13133124—22no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124—42no—
84#13133124—42no—
85#13133124—42no—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53C2×C414yes—
96#95×C53—22no—
97#95×C53—24no—
98#95×C53—42no—
99C24×C53C2411yescharacteristic
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—21no—
106C23×D53—21no—
107C23×D53—21no—
108C23×D53—21no—
109C23×D53—22no—
110C23×D53C22×C412yes—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
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×C53C22×C412yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53#4114yes—
142C22×Dic53—22no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
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×Dic53C22×C412yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C22×C412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D412yes—
171C22×Dic53C2×D412yes—
172C22×Dic53#4114yes—
173C4⋊C4×C53—28no—
174C4⋊C4×C53—28no—
175#11719921—28no—
176#11719921—28no—
177#11719921—28no—
178#11719921—48no—
179#11719921—48no—
180C4⋊Dic53—28no—
181C4⋊Dic53—28no—
182#11719919—28no—
183#11719919—48no—
184C4×Dic53—48no—
185C4×Dic53C22⋊C4116yescomplemented
186C23×C53C23×C411yescharacteristic
187C23×C53C22×D411yescharacteristic
188C23×C53C22×D411yescharacteristic
189C23×C53C2×#4112yes—
190C23×C53C2×#4112yes—
191C23×C53H(5,𝔽2)11yescharacteristic
192C23×C53H(5,𝔽2)12yes—
193C23×C53H(5,𝔽2)11yescharacteristic
194C23×C532−1+411yescharacteristic
195C23×C532−1+411yescharacteristic
196C23×C532−1+412yes—
197C22×D53—22no—
198C22×D53—24no—
199C22×D53—22no—
200C22×D53—22no—
201C22×D53—24no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—42no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—42no—
212C22×D53—42no—
213C22×D53—42no—
214C22×D53—42no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—42no—
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×C53C2×C22⋊C414yes—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53C2×C22⋊C414yes—
281C2×Dic53C2×C22⋊C414yes—
282#14291—531nomaximal, Sylow
283C22×C53C22×C22⋊C411yescharacteristic
284C22×C53C4×#4112yes—
285C22×C53#52011yescharacteristic
286C22×C53#52011yescharacteristic
287C22×C53#52012yes—
288C22×C53#52511yescharacteristic
289C22×C53#52511yescharacteristic
290C22×C53#53212yes—
291C22×C53#53511yescharacteristic
292C22×C53#53511yescharacteristic
293C22×C53#53611yescharacteristic
294C22×C53#53611yescharacteristic
295C22×C53#53711yescharacteristic
296C22×C53#53711yescharacteristic
297C22×C53#53712yes—
298C22×C53#53812yes—
299C22×C53#53812yes—
300C22×C53#53912yes—
301C22×C53#53912yes—
302C22×C53#54112yes—
303C22×C53#54112yes—
304C22×C53#54212yes—
305C22×C53#54212yes—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—44no—
309C2×D53—44no—
310C2×D53—42no—
311C2×D53—81no—
312C2×D53—81no—
313C2×D53—81no—
314C2×D53—81no—
315C2×D53—82no—
316C2×D53—82no—
317C4×C53—24no—
318C4×C53—42no—
319C4×C53—44no—
320C4×C53—42no—
321C4×C53—82no—
322Dic53—24no—
323Dic53—24no—
324Dic53—42no—
325Dic53—44no—
326Dic53—82no—
327C22×#126—531no—
328C2×#382—531no—
329C2×C4×C4⋊C4—531no—
330#1441—532no—
331#1441—531no—
332#1441D5311yescharacteristic, complemented
333#1439—532no—
334#1439—531no—
335#1439—531no—
336#1437—532no—
337#1436—532no—
338C2×C53#229711yescharacteristic
339C2×C53#231111yescharacteristic
340C2×C53#231111yescharacteristic
341C2×C53#233412yes—
342C2×C53#233412yes—
343C2×C53#263611yescharacteristic
344C2×C53#263611yescharacteristic
345C2×C53#263711yescharacteristic
346C2×C53#263711yescharacteristic
347C2×C53#266212yes—
348C2×C53#266412yes—
349D53—82no—
350C2×#126—538no—
351C2×#126—1064no—
352C2×#126—1064no—
353C22×C4⋊C4—531no—
354C22×C4⋊C4—531no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—532no—
357C22×C4⋊C4C2×D5311yescharacteristic
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C22⋊C4—531no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4—532no—
363C22×C22⋊C4—532no—
364C22×C42—531no—
365C22×C42—532no—
366#382—1064no—
367C4×C4⋊C4—1064no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—531no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95C2×D5312yes—
376C2×#95C2×D5311yescharacteristic
377C2×#95C2×D5312yes—
378C53#1429111yescharacteristic, complemented, Sylow
379C25—531no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—532no—
384C23×C4—532no—
385C23×C4C22×D5312yes—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390#126—10632no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—2122no—
405C2×C4⋊C4—2122no—
406C2×C4⋊C4C4×D5314yes—
407C2×C22⋊C4—534no—
408C2×C22⋊C4—534no—
409C2×C22⋊C4—534no—
410C2×C22⋊C4—1062no—
411C2×C22⋊C4—1062no—
412C2×C22⋊C4—1062no—
413C2×C22⋊C4—1062no—
414C2×C22⋊C4—1062no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4—1064no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—2122no—
419C2×C22⋊C4—2122no—
420C2×C22⋊C4—2122no—
421C2×C22⋊C4—2122no—
422C2×C42—534no—
423C2×C42—534no—
424C2×C42—534no—
425C2×C42—1064no—
426C2×C42—2122no—
427C2×C42—2122no—
428#95—22no—
429#95—24no—
430#95—42no—
431#95—1064no—
432#95—1064no—
433#95—1064no—
434#95—1062no—
435#95—2122no—
436#95—2122no—
437#95—2122no—
438C24—532no—
439C24—1061no—
440C24—1061no—
441C24—1062no—
442C24—1062no—
443C24—1061no—
444C24—1061no—
445C24—1062no—
446C24—1061no—
447C24—1061no—
448C24—1062no—
449C24C23×D5311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—534no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4C2×C4×D5312yes—
507C22×C4C2×C4×D5312yes—
508C22×C4C2×C4×D5312yes—
509C22×C4D4×D5312yes—
510C22×C4D4×D5312yes—
511C22×C4#1171994714yes—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—1068no—
515C4⋊C4—1068no—
516C4⋊C4—1068no—
517C4⋊C4—2128no—
518C22⋊C4—1068no—
519C22⋊C4—1068no—
520C22⋊C4—1068no—
521C22⋊C4—2128no—
522C22⋊C4—2128no—
523C42—5316no—
524C42—2128no—
525C23—1062no—
526C23—1064no—
527C23—1062no—
528C23—1062no—
529C23—1064no—
530C23—2121no—
531C23—2121no—
532C23—2122no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23—2122no—
539C23—2121no—
540C23—2122no—
541C23—2122no—
542C23—2122no—
543C23—2122no—
544C23—2122no—
545C23—2122no—
546C23—2122no—
547C23—2122no—
548C23C22×C4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C2×D4×D5311yescharacteristic
551C23#41×D5312yes—
552C23C2×#1171994712yes—
553C23#1313326211yescharacteristic
554C23#1313326311yescharacteristic
555C23#1313326312yes—
556C23#1313326411yescharacteristic
557C23#1313326511yescharacteristic
558C23#1313326512yes—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—534no—
590C2×C4—534no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1064no—
601C2×C4—1062no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4C22⋊C4×D5314yes—
621C22—4242no—
622C22—4241no—
623C22—4241no—
624C22—4242no—
625C22—4241no—
626C22—4241no—
627C22—2122no—
628C22—2122no—
629C22—2122no—
630C22—2124no—
631C22—2124no—
632C22C2×C22⋊C4×D5311yescharacteristic
633C22C4×#1171994712yes—
634C22#1681110311yescharacteristic
635C22#1681110411yescharacteristic
636C22#1681110512yes—
637C22#1681112911yescharacteristic
638C22#1681113111yescharacteristic
639C22#1681116812yes—
640C22#1681118911yescharacteristic
641C22#1681119111yescharacteristic
642C22#1681119611yescharacteristic
643C22#1681119811yescharacteristic
644C22#1681120211yescharacteristic
645C22#1681120312yes—
646C22#1681120311yescharacteristic
647C22#1681121012yes—
648C22#1681121212yes—
649C22#1681121712yes—
650C22#1681122112yes—
651C22#1681123212yes—
652C22#1681123412yes—
653C22#1681124312yes—
654C22#1681124612yes—
655C4—24no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—1064no—
661C4—1064no—
662C4—4242no—
663C4—2124no—
664C4—2122no—
665C2—4242no—
666C2#2027720011yescharacteristic
667C2#2027730411yescharacteristic
668C2#2027730611yescharacteristic
669C2#2027748312yes—
670C2#2027749512yes—
671C2#2028061111yescharacteristic
672C2#2028061511yescharacteristic
673C2#2028062211yescharacteristic
674C2#2028062611yescharacteristic
675C2#2028085312yes—
676C2#2028087312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1441D53
hypocenterC53#14291
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×#16810487C2semidirect
3. C2×C4⋊C4×Dic53C2semidirect
4. #20272968C2semidirect
5. #20272966C2semidirect
10. #20272949C2semidirect
11. C53×#1441C2semidirect
14. C2×#13133212C4semidirect
185. C4×Dic53C22⋊C4semidirect
332. #1441D53semidirect
378. C53#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#131332122
C2×#168104872
C2×C4⋊C4×Dic532
#202729682
#202729662
#202729612
#202729592
#202729572
#202729512
#202729492
C53×#14412
#1429153