Finite Group Databasebeta

← #27394128

Subgroups — #27394128

Order 13568.

1094 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719921C211yesmaximal, characteristic
2#20275329C212yesmaximal, complemented
3#20275327C212yesmaximal, complemented
4#20275322C212yesmaximal, complemented
5#20275320C212yesmaximal, complemented
6#20274695C211yesmaximal, characteristic, complemented
7#20274624C211yesmaximal, characteristic
8#20274624C211yesmaximal, characteristic
9#20274479C211yesmaximal, characteristic, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1797C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719921C411yescharacteristic
14C22×#11719921C411yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C2×C4⋊#88322C2212yescomplemented
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×#13133129C2212yes—
20C2×#13133129C2212yes—
21C2×#13133129C2212yes—
22C2×#13133129C2212yes—
23C2×#13133127C2212yescomplemented
24C2×C8×Dic53C2212yescomplemented
25C4×#11719921C411yescharacteristic
26C4×#11719921C411yescharacteristic
27C4×#11719921C411yescharacteristic
28C4×#11719921C411yescharacteristic
29C4×#11719921C2211yescharacteristic
30C4×#11719921C2211yescharacteristic
31C4×#11719921C2211yescharacteristic
32C4×#11719921C2211yescharacteristic
33C2×#13133124C411yescharacteristic
34C2×#13133124C411yescharacteristic
35#16810378C2212yescomplemented
36#16810377C2212yescomplemented
37#16810377C2212yescomplemented
38C22×C42×C53C2211yescharacteristic
39C2×C4⋊C8×C53C2212yescomplemented
40C2×C22⋊C8×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#336C2212yescomplemented
43C24×D53C2×C411yescharacteristic
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C23×Dic53C2×C411yescharacteristic
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C22×#88322C2312yes—
56C22×#88322C2312yes—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921C2×C411yescharacteristic
60C2×#11719921C2×C411yescharacteristic
61C2×#11719921C2×C411yescharacteristic
62C2×#11719921C2×C411yescharacteristic
63C2×#11719921C2×C411yescharacteristic
64C2×#11719921C2×C411yescharacteristic
65C2×#11719921C2×C411yescharacteristic
66C2×#11719921C2×C411yescharacteristic
67C2×C4×Dic53—42no—
68C2×C4×Dic53C2×C411yescharacteristic
69C2×C4×Dic53C2×C411yescharacteristic
70C2×C4×Dic53C2311yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C4⋊#88322—42no—
73C22⋊#88322—22no—
74C22⋊#88322—22no—
75#13133129—42no—
76#13133129—42no—
77#13133129—42no—
78#13133129—42no—
79#13133127—42no—
80C8×Dic53—42no—
81#13133124C2×C411yescharacteristic
82#13133124C2×C411yescharacteristic
83#13133124C2×C411yescharacteristic
84#13133124C2×C411yescharacteristic
85#13133124C2×C411yescharacteristic
86#13133124C2×C411yescharacteristic
87#13133124C2×C411yescharacteristic
88#13133124C2×C411yescharacteristic
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C22×C8×C53C2312yes—
93C22×C8×C53C2312yes—
94C2×C42×C53—42no—
95C2×C42×C53C2×C411yescharacteristic
96C2×C42×C53C2×C411yescharacteristic
97C2×C42×C53C2311yescharacteristic
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—42no—
100C22⋊C8×C53—22no—
101C22⋊C8×C53—22no—
102C24×C53C22×C411yescharacteristic
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53—41no—
109C23×D53—41no—
110C23×D53C22×C411yescharacteristic
111C23×D53C22×C411yescharacteristic
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—41no—
119C22×C4×C53—41no—
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—41no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—41no—
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—21no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53—41no—
156C2×C4×D53—41no—
157C2×C4×D53—48no—
158C2×C4×D53—48no—
159C2×C8×C53—22no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C42×C53—22no—
166C42×C53—22no—
167C42×C53—41no—
168C42×C53—41no—
169#11719921—22no—
170#11719921—22no—
171#11719921—22no—
172#11719921—22no—
173#11719921—22no—
174#11719921—22no—
175#11719921M1614yes—
176#11719921M1614yes—
177C4×Dic53—22no—
178C4×Dic53—22no—
179C4×Dic53—41no—
180C4×Dic53—41no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53#11712yes—
192C23×C53#13112yes—
193C23×C53#13112yes—
194C23×C53C23×C411yescharacteristic
195C22×D53—28no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53—21no—
208C22×D53—44no—
209C22×D53—44no—
210C22×D53—44no—
211C22×D53—44no—
212C22×D53—44no—
213C22×D53—44no—
214C22×D53C2×M1612yes—
215C22×D53C2×M1612yes—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—44no—
231C2×C4×C53—44no—
232C2×C4×C53#11712yes—
233C2×C4×C53C2×M1612yes—
234C2×C4×C53C2×M1612yes—
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)12yes—
238C2×C4×C53H(5,𝔽2)12yes—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—44no—
256C2×Dic53—44no—
257C2×Dic53—44no—
258C2×Dic53C2×M1612yes—
259C2×Dic53C2×M1612yes—
260C4×D53—44no—
261C4×D53—44no—
262C4×D53—42no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266C8×C53—42no—
267C8×C53—42no—
268#88322—42no—
269#88322—42no—
270#21904—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53#43314yes—
278C22×C53C2×#11711yescharacteristic
279C22×C53#51811yescharacteristic
280C22×C53#51811yescharacteristic
281C22×C53C22×M1611yescharacteristic
282C22×C53C22×M1611yescharacteristic
283C22×C53C2×#13111yescharacteristic
284C22×C53C2×#13111yescharacteristic
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—22no—
290C2×D53—22no—
291C2×D53—416no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—41no—
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—41no—
300C4×C53—22no—
301C4×C53—22no—
302C4×C53—22no—
303C4×C53—41no—
304C4×C53—41no—
305C4×C53#53812yes—
306C4×C53#53812yes—
307Dic53—22no—
308Dic53—22no—
309Dic53—41no—
310Dic53—41no—
311Dic53—41no—
312Dic53—41no—
313C2×C4×C22⋊C4—531no—
314#1924—532no—
315#1924—532no—
316#1923—532no—
317#1923—532no—
318#1797—531no—
319#1797D5311yescharacteristic, complemented
320#1779—531no—
321#1779—531no—
322#1749—531no—
323C4×C22⋊C8—531no—
324C2×C53—24no—
325C2×C53#231211yescharacteristic
326C2×C53C2×#43311yescharacteristic
327C2×C53#292011yescharacteristic
328C2×C53#297411yescharacteristic
329C2×C53#297411yescharacteristic
330C2×C53#297411yescharacteristic
331C2×C53#297411yescharacteristic
332D53—42no—
333D53—42no—
334C24×C4—531no—
335C22×C22⋊C4—531no—
336C22×C22⋊C4—531no—
337C22×C42—531no—
338C22×C42C2×D5311yescharacteristic
339C2×C4⋊C8—532no—
340C2×C4⋊C8C2×D5312yescomplemented
341C2×C22⋊C8—531no—
342C2×C22⋊C8—531no—
343C2×C22⋊C8—532no—
344C2×C22⋊C8—532no—
345C2×C22⋊C8—532no—
346C2×C22⋊C8—532no—
347C2×C22⋊C8C2×D5311yescharacteristic
348C2×C22⋊C8C2×D5311yescharacteristic
349C2×#97—532no—
350C2×C4×C8—532no—
351C4×C22⋊C4—531no—
352C4×C22⋊C4—531no—
353C4×C22⋊C4—531no—
354C4×C22⋊C4—531no—
355C4×C22⋊C4—531no—
356C4×C22⋊C4—531no—
357C4×C22⋊C4—531no—
358C4×C22⋊C4—531no—
359C2×#95—531no—
360C2×#95—531no—
361#336—532no—
362#336—532no—
363#336—532no—
364#336C2×D5312yescomplemented
365C53#2190411yescharacteristic, complemented, Sylow
366C25—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—1062no—
375C23×C4—1062no—
376C23×C4—2122no—
377C23×C4—2122no—
378C23×C4C4×D5311yescharacteristic
379C23×C4C4×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C22×C8—532no—
382C22×C8—532no—
383C22×C8C22×D5312yes—
384C22×C8C22×D5312yes—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—531no—
387C2×C22⋊C4—531no—
388C2×C22⋊C4—531no—
389C2×C22⋊C4—531no—
390C2×C22⋊C4—531no—
391C2×C22⋊C4—531no—
392C2×C22⋊C4—531no—
393C2×C22⋊C4—1064no—
394C2×C22⋊C4—1064no—
395C2×C42—42no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—2122no—
401C2×C42C4×D5311yescharacteristic
402C2×C42C4×D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405C4⋊C8—42no—
406C4⋊C8—2122no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—1062no—
410C22⋊C8—1062no—
411C22⋊C8—2122no—
412C22⋊C8—2122no—
413C22⋊C8—2122no—
414C22⋊C8—2122no—
415#97—2122no—
416C4×C8—2122no—
417#95—531no—
418#95—531no—
419#95—531no—
420#95—531no—
421#95—531no—
422#95—531no—
423#95—531no—
424#95—531no—
425C24—531no—
426C24—531no—
427C24—1062no—
428C24—1062no—
429C24—1062no—
430C24—1062no—
431C24—1062no—
432C24—2121no—
433C24—2121no—
434C24C2×C4×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—2121no—
465C22×C4—2121no—
466C22×C4—2121no—
467C22×C4—2121no—
468C22×C4—2128no—
469C22×C4—2128no—
470C22×C4—2121no—
471C22×C4—2121no—
472C22×C4—2121no—
473C22×C4—2121no—
474C22×C4C2×C4×D5311yescharacteristic
475C22×C4C2×C4×D5311yescharacteristic
476C22×C4C2×C4×D5311yescharacteristic
477C22×C4C2×C4×D5311yescharacteristic
478C22×C4C2×C4×D5311yescharacteristic
479C22×C4#1171994712yes—
480C22×C4#1171994912yes—
481C22×C4C23×D5311yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—1062no—
489C2×C8—1062no—
490C2×C8—1062no—
491C2×C8—1062no—
492C2×C8—1062no—
493C2×C8—1062no—
494C22⋊C4—534no—
495C22⋊C4—534no—
496C22⋊C4—1062no—
497C22⋊C4—1062no—
498C22⋊C4—1062no—
499C22⋊C4—1062no—
500C22⋊C4—1062no—
501C22⋊C4—1062no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—1062no—
507C42—1062no—
508C42—2121no—
509C42—2121no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—532no—
515C23—532no—
516C23—1068no—
517C23—1061no—
518C23—1061no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1061no—
526C23—1061no—
527C23—1061no—
528C23—1061no—
529C23—2124no—
530C23—2124no—
531C23—2124no—
532C23—2124no—
533C23—2124no—
534C23—2124no—
535C23#1313317512yes—
536C23#1313322312yes—
537C23#1313322512yes—
538C23C22×C4×D5311yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—532no—
556C2×C4—532no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1061no—
563C2×C4—1061no—
564C2×C4—1061no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1061no—
572C2×C4—1061no—
573C2×C4—2124no—
574C2×C4—2124no—
575C2×C4—2124no—
576C2×C4—2124no—
577C2×C4—2124no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4#1313317512yes—
583C2×C4M16×D5312yes—
584C2×C4M16×D5312yes—
585C2×C4C2×#1171994711yescharacteristic
586C2×C4C2×#1171994911yescharacteristic
587C2×C4#1313326212yes—
588C2×C4#1313326312yes—
589C8—42no—
590C8—42no—
591C8—2122no—
592C8—2122no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—1062no—
600C22—1062no—
601C22—1062no—
602C22—1062no—
603C22—1062no—
604C22—1062no—
605C22—21216no—
606C22—2121no—
607C22—2121no—
608C22—2121no—
609C22—2121no—
610C22—2121no—
611C22—2121no—
612C22—2121no—
613C22—2121no—
614C22#1681068314yes—
615C22C2×#1313317511yescharacteristic
616C22#1681109611yescharacteristic
617C22#1681109711yescharacteristic
618C22C2×M16×D5311yescharacteristic
619C22C2×M16×D5311yescharacteristic
620C22C2×#1313322311yescharacteristic
621C22C2×#1313322511yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—1062no—
628C4—1062no—
629C4—2121no—
630C4—2121no—
631C4—2121no—
632C4—2121no—
633C4#1681121012yes—
634C4#1681121312yes—
635C2—24no—
636C2—2122no—
637C2—2122no—
638C2#2027731211yescharacteristic
639C2C2×#1681068311yescharacteristic
640C2#2028334711yescharacteristic
641C2#2028374611yescharacteristic
642C2#2028374911yescharacteristic
643C2#2028375111yescharacteristic
644C2#2028375211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1797C2
prime radicalC24×D53C2×C4
cosocleC22×C4×C53C24
hypercenter#1797D53
hypocenterC53#21904
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #20275329C2semidirect
3. #20275327C2semidirect
4. #20275322C2semidirect
5. #20275320C2semidirect
6. #20274695C2semidirect
9. #20274479C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1797C2semidirect
16. C2×C4⋊#88322C22semidirect
23. C2×#13133127C22semidirect
24. C2×C8×Dic53C22semidirect
35. #16810378C22semidirect
36. #16810377C22semidirect
37. #16810377C22semidirect
39. C2×C4⋊C8×C53C22semidirect
42. C53×#336C22semidirect
319. #1797D53semidirect
340. C2×C4⋊C8C2×D53semidirect
364. #336C2×D53semidirect
365. C53#21904semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199212
#202753292
#202753272
#202753222
#202753202
#202746952
#202746242
#202744792
C22⋊C8×Dic532
C53×#17972
#2190453