Finite Group Databasebeta

← #27310767

Subgroups — #27310767

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×#13133210C211yesmaximal, characteristic
2C2×#16810488C211yesmaximal, characteristic, complemented
3C2×#16810474C211yesmaximal, characteristic, complemented
4#20272969C211yesmaximal, characteristic, complemented
5#20272966C212yesmaximal, complemented
6#20272965C211yesmaximal, characteristic, complemented
7#20272960C211yesmaximal, characteristic
8#20272958C212yesmaximal
9#20272952C212yesmaximal
10#20272949C212yesmaximal, complemented
11C53×#1439C211yesmaximal, characteristic
12C2×#13133210—24no—
13C2×#13133210—24no—
14C2×#13133210C418yescomplemented
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×C22⋊Dic53C2211yescharacteristic
21C22×C22⋊Dic53C2211yescharacteristic
22C22×C22⋊Dic53C2212yes—
23C22×C4×Dic53C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810488—24no—
26#16810474—24no—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C22×C22⋊C4×C53C2212yes—
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46#13133210—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—42no—
51C2×C4⋊Dic53C2×C414yes—
52C2×#11719919—22no—
53C2×#11719919—22no—
54C2×#11719919—24no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919C2×C414yes—
60C2×#11719919C2×C414yes—
61C2×C22⋊Dic53—22no—
62C2×C22⋊Dic53—22no—
63C2×C22⋊Dic53—22no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—24no—
66C2×C22⋊Dic53—42no—
67C2×C22⋊Dic53—42no—
68C2×C22⋊Dic53C2×C414yes—
69C2×C22⋊Dic53C2×C414yes—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53—42no—
73C2×C4×Dic53C2×C414yes—
74C2×C4×Dic53D414yes—
75C2×C4×Dic53D414yes—
76#13133124—22no—
77#13133124—24no—
78#13133124—24no—
79#13133124—24no—
80#13133124—42no—
81#13133124—42no—
82#13133124—42no—
83#13133124—42no—
84C25×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2311yescharacteristic
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—42no—
91C2×C22⋊C4×C53—22no—
92C2×C22⋊C4×C53—22no—
93C2×C22⋊C4×C53—24no—
94C2×C22⋊C4×C53—42no—
95C2×C22⋊C4×C53—42no—
96C2×C22⋊C4×C53C2×C414yes—
97#95×C53—22no—
98#95×C53—24no—
99C24×C53—21no—
100C24×C53—21no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—21no—
104C24×C53—22no—
105C24×C53—21no—
106C24×C53—22no—
107C24×C53—21no—
108C24×C53—21no—
109C24×C53C22×C412yes—
110C24×C53C2411yescharacteristic
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
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—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53#4114yes—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
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×Dic53C22×C412yes—
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—44no—
174C22⋊C4×C53—28no—
175C22⋊C4×C53—48no—
176C22⋊Dic53—28no—
177C22⋊Dic53—28no—
178C22⋊Dic53—48no—
179C4⋊Dic53—28no—
180C4⋊Dic53—44no—
181#11719919—216no—
182#11719919—28no—
183#11719919—28no—
184C4×Dic53—48no—
185C4×Dic53C22⋊C4116yescomplemented
186C23×C53—22no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—42no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53C23×C411yescharacteristic
210C23×C53C22×D411yescharacteristic
211C23×C53C22×D411yescharacteristic
212C23×C53C2×#4112yes—
213C23×C53C2×#4112yes—
214C23×C53H(5,𝔽2)11yescharacteristic
215C23×C53H(5,𝔽2)12yes—
216C23×C53H(5,𝔽2)11yescharacteristic
217C23×C532−1+412yes—
218C23×C532−1+411yescharacteristic
219C23×C532−1+411yescharacteristic
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—42no—
243C2×C4×C53C2×C22⋊C414yes—
244C2×Dic53—24no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—41no—
270C2×Dic53—41no—
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×C53—42no—
284C22×C53—42no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53—42no—
288C22×C53—82no—
289C22×C53—82no—
290C22×C53—81no—
291C22×C53—81no—
292C22×C53—81no—
293C22×C53—81no—
294C22×C53C22×C22⋊C411yescharacteristic
295C22×C53C4×#4112yes—
296C22×C53#52012yes—
297C22×C53#52011yescharacteristic
298C22×C53#52011yescharacteristic
299C22×C53#52512yes—
300C22×C53#53211yescharacteristic
301C22×C53#53211yescharacteristic
302C22×C53#53511yescharacteristic
303C22×C53#53511yescharacteristic
304C22×C53#53611yescharacteristic
305C22×C53#53611yescharacteristic
306C22×C53#53712yes—
307C22×C53#53711yescharacteristic
308C22×C53#53711yescharacteristic
309C22×C53#53812yes—
310C22×C53#53812yes—
311C22×C53#53912yes—
312C22×C53#53912yes—
313C22×C53#54112yes—
314C22×C53#54112yes—
315C22×C53#54212yes—
316C22×C53#54212yes—
317C4×C53—24no—
318C4×C53—44no—
319C4×C53—42no—
320C4×C53—82no—
321Dic53—24no—
322Dic53—24no—
323Dic53—42no—
324Dic53—42no—
325Dic53—44no—
326Dic53—82no—
327C22×#126—531no—
328C2×#382—531no—
329C2×C4×C4⋊C4—531no—
330#1441—532no—
331#1441—531no—
332#1441—531no—
333#1439—532no—
334#1439—531no—
335#1439D5311yescharacteristic
336#1437—532no—
337#1436—532no—
338C2×C53—82no—
339C2×C53#229711yescharacteristic
340C2×C53#231111yescharacteristic
341C2×C53#231111yescharacteristic
342C2×C53#233412yes—
343C2×C53#233412yes—
344C2×C53#263612yes—
345C2×C53#263712yes—
346C2×C53#266211yescharacteristic
347C2×C53#266211yescharacteristic
348C2×C53#266411yescharacteristic
349C2×C53#266411yescharacteristic
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⋊C4—531no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C22⋊C4—531no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—532no—
362C22×C22⋊C4C2×D5311yescharacteristic
363C22×C22⋊C4C2×D5312yes—
364C22×C42—531no—
365C22×C42—532no—
366#382—1064no—
367C4×C4⋊C4—1064no—
368C2×#95—532no—
369C2×#95—532no—
370C2×#95—531no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95C2×D5312yes—
377C2×#95C2×D5311yescharacteristic
378C53#1429111yescharacteristic, complemented, Sylow
379C25C22×D5311yescharacteristic
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—532no—
385C23×C4—532no—
386C23×C4C22×D5312yes—
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5312yes—
389C23×C4C22×D5311yescharacteristic
390#126—10632no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4—534no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1064no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—2122no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4—42no—
412C2×C22⋊C4—534no—
413C2×C22⋊C4—534no—
414C2×C22⋊C4—1062no—
415C2×C22⋊C4—1062no—
416C2×C22⋊C4—1062no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4—2122no—
420C2×C22⋊C4—2122no—
421C2×C22⋊C4C4×D5314yes—
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—1064no—
431#95—1064no—
432#95—1064no—
433#95—1062no—
434#95—2122no—
435#95—2122no—
436#95—2122no—
437#95—2122no—
438C24—21no—
439C24—21no—
440C24—22no—
441C24—22no—
442C24—21no—
443C24—22no—
444C24—21no—
445C24—22no—
446C24—21no—
447C24—21no—
448C24C2×C4×D5312yes—
449C24C23×D5311yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—534no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1061no—
498C22×C4—1062no—
499C22×C4—1061no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4C2×C4×D5312yes—
508C22×C4C2×C4×D5312yes—
509C22×C4D4×D5312yes—
510C22×C4D4×D5312yes—
511C22×C4#1171994714yes—
512C4⋊C4—44no—
513C4⋊C4—1068no—
514C4⋊C4—1068no—
515C4⋊C4—10616no—
516C4⋊C4—1068no—
517C4⋊C4—2124no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—1068no—
521C22⋊C4—1068no—
522C22⋊C4—2128no—
523C42—5316no—
524C42—2128no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C22×C4×D5311yescharacteristic
549C23C2×D4×D5311yescharacteristic
550C23C2×D4×D5311yescharacteristic
551C23#41×D5312yes—
552C23C2×#1171994712yes—
553C23#1313326212yes—
554C23#1313326211yescharacteristic
555C23#1313326311yescharacteristic
556C23#1313326412yes—
557C23#1313326411yescharacteristic
558C23#1313326511yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—534no—
583C2×C4—534no—
584C2×C4—1064no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1064no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2121no—
610C2×C4—2122no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4C22⋊C4×D5314yes—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—81no—
627C22—81no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22C2×C22⋊C4×D5311yescharacteristic
633C22C4×#1171994712yes—
634C22#1681110212yes—
635C22#1681110311yescharacteristic
636C22#1681110411yescharacteristic
637C22#1681113312yes—
638C22#1681117111yescharacteristic
639C22#1681117211yescharacteristic
640C22#1681118811yescharacteristic
641C22#1681118911yescharacteristic
642C22#1681119511yescharacteristic
643C22#1681119611yescharacteristic
644C22#1681120012yes—
645C22#1681120211yescharacteristic
646C22#1681120311yescharacteristic
647C22#1681120712yes—
648C22#1681120812yes—
649C22#1681121612yes—
650C22#1681121712yes—
651C22#1681123012yes—
652C22#1681123312yes—
653C22#1681124012yes—
654C22#1681124312yes—
655C4—24no—
656C4—44no—
657C4—42no—
658C4—82no—
659C4—1064no—
660C4—1064no—
661C4—4242no—
662C4—2124no—
663C4—2122no—
664C4—2122no—
665C2—82no—
666C2#2027720211yescharacteristic
667C2#2027730211yescharacteristic
668C2#2027730411yescharacteristic
669C2#2027748512yes—
670C2#2027749712yes—
671C2#2028061712yes—
672C2#2028062812yes—
673C2#2028085711yescharacteristic
674C2#2028086111yescharacteristic
675C2#2028087911yescharacteristic
676C2#2028088311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1439C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#14291
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×#16810488C2semidirect
3. C2×#16810474C2semidirect
4. #20272969C2semidirect
5. #20272966C2semidirect
6. #20272965C2semidirect
10. #20272949C2semidirect
14. C2×#13133210C4semidirect
185. C4×Dic53C22⋊C4semidirect
378. C53#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#131332102
C2×#168104882
C2×#168104742
#202729692
#202729662
#202729652
#202729602
#202729582
#202729522
#202729492
C53×#14392
#1429153