Finite Group Databasebeta

← #27305376

Subgroups — #27305376

Order 13568.

2102 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810497C212yesmaximal
2C2×C4×#11719921C211yesmaximal, characteristic
3C2×#16810470C211yesmaximal, characteristic
4C2×#95×D53C211yesmaximal, characteristic
5#20272949C212yesmaximal, complemented
6#20272946C212yesmaximal, complemented
7#20272945C211yesmaximal, characteristic, complemented
8#20272943C211yesmaximal, characteristic, complemented
9#20272941C211yesmaximal, characteristic, complemented
10#20272936C212yesmaximal
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2212yes—
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2212yes—
18C22×#11719921C2211yescharacteristic
19C22×C4×Dic53C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810497C4116yescomplemented
22C4×#11719921C418yescomplemented
23#16810470—24no—
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33#95×D53—24no—
34C22×C42×C53C2211yescharacteristic
35C2×#95×C53C2211yescharacteristic
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2212yes—
39C24×D53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—42no—
48C22×C4×D53—42no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53C2×C418yes—
54C22×C4×D53C2×C414yes—
55C22×C4×D53D414yes—
56C22×C4×D53Q814yes—
57C2×C4⋊Dic53—42no—
58C2×C4⋊Dic53—42no—
59C2×C4⋊Dic53C2×C418yescomplemented
60C2×#11719919—42no—
61C2×#11719919—42no—
62C2×#11719919C2×C418yescomplemented
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—24no—
66C2×#11719921C2×C418yes—
67C2×#11719921C2×C414yes—
68C2×#11719921C2×C418yes—
69C2×#11719921C2×C414yes—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53—42no—
73C2×C4×Dic53—42no—
74C2×C4×Dic53C2×C414yescomplemented
75#13133124—22no—
76#13133124—22no—
77#13133124—24no—
78#13133124—42no—
79#13133124—42no—
80#13133124—42no—
81#13133124C2×C418yescomplemented
82#13133124C2×C414yescomplemented
83#13133124C2×C414yescomplemented
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2312yes—
86C23×C4×C53C2312yes—
87C23×C4×C53C2311yescharacteristic
88C23×C4×C53C2311yescharacteristic
89C2×C42×C53—42no—
90C2×C42×C53C2×C414yescomplemented
91#95×C53—22no—
92#95×C53—22no—
93#95×C53—42no—
94#95×C53C2×C418yescomplemented
95C24×C53C2411yescharacteristic
96C23×D53—22no—
97C23×D53—22no—
98C23×D53—21no—
99C23×D53—21no—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53C22×C412yes—
103C23×D53C22×C414yes—
104C23×D53C2×D412yes—
105C23×D53C2×Q812yes—
106C23×D53#4114yes—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
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—22no—
120C22×C4×C53—21no—
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—21no—
128C22×C4×C53C22×C414yes—
129C22×C4×C53C22×C414yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C414yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×Q812yes—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
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—21no—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C414yes—
159C22×Dic53C22×C414yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C22×C414yes—
162C22×Dic53C22×C412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×Q812yes—
165C2×C4×D53—216no—
166C2×C4×D53—44no—
167C2×C4×D53—44no—
168C2×C4×D53—48no—
169C2×C4×D53—48no—
170C2×C4×D53—48no—
171C2×C4×D53—48no—
172C2×C4×D53—48no—
173C2×C4×D53—48no—
174C2×C4×D53—48no—
175C2×C4×D53—48no—
176C2×C4×D53C4⋊C4116yescomplemented
177C42×C53—44no—
178#11719921—28no—
179#11719921—216no—
180#11719921C42116yescomplemented
181C4⋊Dic53—48no—
182#11719919—48no—
183C4×Dic53—44no—
184C4×Dic53—48no—
185C23×C53C23×C411yescharacteristic
186C23×C53C23×C412yes—
187C23×C53C22×D411yescharacteristic
188C23×C53C22×Q811yescharacteristic
189C23×C53C2×#4112yes—
190C23×C53H(5,𝔽2)12yes—
191C23×C53H(5,𝔽2)12yes—
192C23×C53H(5,𝔽2)11yescharacteristic
193C23×C53H(5,𝔽2)11yescharacteristic
194C23×C532−1+411yescharacteristic
195C23×C532−1+411yescharacteristic
196C22×D53—24no—
197C22×D53—22no—
198C22×D53—22no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—22no—
202C22×D53—22no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53C2×C4214yes—
215C22×D53C2×C4⋊C414yes—
216C22×D53#11714yes—
217C22×D53C4×D418yes—
218C22×D53C4×Q818yes—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
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—24no—
231C2×C4×C53—24no—
232C2×C4×C53—22no—
233C2×C4×C53—24no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
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×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53C2×C4214yes—
247C2×C4×C53C2×C4⋊C414yes—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—24no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53C2×C4214yes—
276C2×Dic53C2×C4⋊C414yes—
277C4×D53—416no—
278C4×D53—432no—
279#13782—531nomaximal, Sylow
280C22×C53C22×C4211yescharacteristic
281C22×C53C22×C4⋊C411yescharacteristic
282C22×C53C2×#11711yescharacteristic
283C22×C53C2×C4×D412yes—
284C22×C53C2×C4×Q812yes—
285C22×C53#51812yes—
286C22×C53#51811yescharacteristic
287C22×C53#51811yescharacteristic
288C22×C53#51812yes—
289C22×C53#52012yes—
290C22×C53#52011yescharacteristic
291C22×C53#52011yescharacteristic
292C22×C53#52012yes—
293C22×C53#53412yes—
294C22×C53#53511yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#53812yes—
297C22×C53#53812yes—
298C22×C53#53912yes—
299C22×C53#53912yes—
300C22×C53#54312yes—
301C22×C53#54411yescharacteristic
302C22×C53#54411yescharacteristic
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—24no—
306C2×D53—24no—
307C2×D53—28no—
308C2×D53—28no—
309C2×D53—28no—
310C2×D53—28no—
311C2×D53—24no—
312C2×D53—24no—
313C2×D53C4×C4⋊C418yes—
314C4×C53—24no—
315C4×C53—24no—
316C4×C53—42no—
317C4×C53—44no—
318C4×C53—44no—
319Dic53—24no—
320Dic53—24no—
321Dic53—42no—
322Dic53—44no—
323Dic53—44no—
324C2×#385—532no—
325C2×C4×C22⋊C4—531no—
326C2×C4×C22⋊C4—531no—
327C22×#95—531no—
328#1436—532no—
329#1436—532no—
330C4⋊#95—531no—
331C4⋊#95—531no—
332#1434—531no—
333#1434D5311yescharacteristic, complemented
334C22⋊#95—532no—
335C2×C53C2×C4×C4⋊C411yescharacteristic
336C2×C53#227511yescharacteristic
337C2×C53#227511yescharacteristic
338C2×C53#231511yescharacteristic
339C2×C53#231711yescharacteristic
340C2×C53#231811yescharacteristic
341C2×C53#231811yescharacteristic
342C2×C53#232112yes—
343C2×C53#232212yes—
344C2×C53#232912yes—
345C2×C53#232912yes—
346D53—28no—
347C24×C4—531no—
348C24×C4—532no—
349C22×C4⋊C4—532no—
350C22×C4⋊C4—532no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—531no—
353C22×C22⋊C4—532no—
354C22×C42—531no—
355C22×C42—532no—
356C22×C42C2×D5311yescharacteristic
357#385—5316no—
358C4×C22⋊C4—538no—
359C4×C22⋊C4—1064no—
360C2×#95—532no—
361C2×#95—531no—
362C2×#95—531no—
363C2×#95—531no—
364C2×#95—531no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—532no—
369C2×#95—1064no—
370C2×#95C2×D5311yescharacteristic
371C2×#95C2×D5311yescharacteristic
372C2×#95C2×D5312yes—
373C2×#95C2×D5312yes—
374C53#1378211yescharacteristic, complemented, Sylow
375C25—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—534no—
379C23×C4—534no—
380C23×C4—534no—
381C23×C4—532no—
382C23×C4—531no—
383C23×C4—532no—
384C23×C4—538no—
385C23×C4—1064no—
386C23×C4—1064no—
387C23×C4—2122no—
388C23×C4—2122no—
389C23×C4—2122no—
390C23×C4—2122no—
391C23×C4—2122no—
392C23×C4—2122no—
393C23×C4C22×D5311yescharacteristic
394C23×C4C22×D5312yes—
395C23×C4C22×D5312yes—
396C23×C4C22×D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C2×C4⋊C4—538no—
399C2×C4⋊C4—538no—
400C2×C4⋊C4—2122no—
401C2×C4⋊C4—2122no—
402C2×C4⋊C4—2122no—
403C2×C4⋊C4—2122no—
404C2×C22⋊C4—534no—
405C2×C22⋊C4—538no—
406C2×C22⋊C4—534no—
407C2×C22⋊C4—538no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1064no—
411C2×C42—42no—
412C2×C42—534no—
413C2×C42—1064no—
414C2×C42—2122no—
415C2×C42—2122no—
416C2×C42—2122no—
417C2×C42C4×D5314yescomplemented
418#95—22no—
419#95—22no—
420#95—42no—
421#95—534no—
422#95—534no—
423#95—538no—
424#95—1064no—
425#95—1062no—
426#95—1062no—
427#95—2122no—
428#95—2122no—
429#95—2122no—
430#95C4×D5318yescomplemented
431C24—532no—
432C24—534no—
433C24—532no—
434C24—532no—
435C24—534no—
436C24—1062no—
437C24—1061no—
438C24—1062no—
439C24—1061no—
440C24—1061no—
441C24—1061no—
442C24C23×D5311yescharacteristic
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—534no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—5316no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—534no—
473C22×C4—10616no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1061no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1061no—
495C22×C4—2124no—
496C22×C4—2124no—
497C22×C4—2128no—
498C22×C4—2128no—
499C22×C4—2128no—
500C22×C4—2128no—
501C22×C4—2128no—
502C22×C4—2128no—
503C22×C4—2128no—
504C22×C4—2128no—
505C22×C4C2×C4×D5314yes—
506C22×C4C2×C4×D5312yes—
507C22×C4C2×C4×D5314yes—
508C22×C4C2×C4×D5312yes—
509C22×C4C2×C4×D5314yes—
510C22×C4C2×C4×D5312yes—
511C22×C4D4×D5312yes—
512C22×C4Q8×D5312yes—
513C4⋊C4—2128no—
514C4⋊C4—2128no—
515C22⋊C4—5316no—
516C22⋊C4—1068no—
517C22⋊C4—10616no—
518C42—44no—
519C42—2124no—
520C42—2128no—
521C23—534no—
522C23—534no—
523C23—534no—
524C23—538no—
525C23—538no—
526C23—1064no—
527C23—1062no—
528C23—1062no—
529C23—1064no—
530C23—1064no—
531C23—1062no—
532C23—1062no—
533C23—1064no—
534C23—1064no—
535C23—1062no—
536C23—1062no—
537C23—1064no—
538C23—1064no—
539C23—1064no—
540C23—1064no—
541C23—1064no—
542C23—1062no—
543C23—1062no—
544C23C22×C4×D5311yescharacteristic
545C23C22×C4×D5312yes—
546C23C2×D4×D5311yescharacteristic
547C23C2×Q8×D5311yescharacteristic
548C23#41×D5312yes—
549C23#1313326212yes—
550C23#1313326311yescharacteristic
551C23#1313326312yes—
552C23#1313326311yescharacteristic
553C23#1313326411yescharacteristic
554C23#1313326411yescharacteristic
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
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—1062no—
585C2×C4—1062no—
586C2×C4—1064no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1062no—
598C2×C4—1064no—
599C2×C4—1064no—
600C2×C4—1062no—
601C2×C4—21216no—
602C2×C4—21232no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4C42×D5314yes—
614C2×C4C4⋊C4×D5314yes—
615C22—538no—
616C22—1064no—
617C22—1064no—
618C22—1064no—
619C22—1064no—
620C22—1064no—
621C22—1064no—
622C22—1068no—
623C22—1068no—
624C22—1068no—
625C22—1068no—
626C22C2×C42×D5311yescharacteristic
627C22C2×C4⋊C4×D5311yescharacteristic
628C22#117×D5311yescharacteristic
629C22C4×D4×D5312yes—
630C22C4×Q8×D5312yes—
631C22#1681109512yes—
632C22#1681109611yescharacteristic
633C22#1681109711yescharacteristic
634C22#1681109712yes—
635C22#1681110212yes—
636C22#1681110311yescharacteristic
637C22#1681110311yescharacteristic
638C22#1681110312yes—
639C22#1681118112yes—
640C22#1681119111yescharacteristic
641C22#1681119311yescharacteristic
642C22#1681120912yes—
643C22#1681121212yes—
644C22#1681121412yes—
645C22#1681121612yes—
646C22#1681124812yes—
647C22#1681125311yescharacteristic
648C22#1681125311yescharacteristic
649C4—24no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—1064no—
655C4—1064no—
656C4—2122no—
657C4—2124no—
658C4—2124no—
659C2—1068no—
660C2C4×C4⋊C4×D5311yescharacteristic
661C2#2027707411yescharacteristic
662C2#2027707711yescharacteristic
663C2#2027732811yescharacteristic
664C2#2027734011yescharacteristic
665C2#2027734411yescharacteristic
666C2#2027734611yescharacteristic
667C2#2027736812yes—
668C2#2027738512yes—
669C2#2027744212yes—
670C2#2027744412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13782
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. #20272949C2semidirect
6. #20272946C2semidirect
7. #20272945C2semidirect
8. #20272943C2semidirect
9. #20272941C2semidirect
11. C53×#1434C2semidirect
21. #16810497C4semidirect
22. C4×#11719921C4semidirect
59. C2×C4⋊Dic53C2×C4semidirect
62. C2×#11719919C2×C4semidirect
74. C2×C4×Dic53C2×C4semidirect
81. #13133124C2×C4semidirect
82. #13133124C2×C4semidirect
83. #13133124C2×C4semidirect
90. C2×C42×C53C2×C4semidirect
94. #95×C53C2×C4semidirect
176. C2×C4×D53C4⋊C4semidirect
180. #11719921C42semidirect
333. #1434D53semidirect
374. C53#13782semidirect
417. C2×C42C4×D53semidirect
430. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104972
C2×C4×#117199212
C2×#168104702
C2×#95×D532
#202729492
#202729462
#202729452
#202729432
#202729412
#202729362
C53×#14342
#1378253