Finite Group Databasebeta

← #27306203

Subgroups — #27306203

Order 13568.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810491C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic53C211yesmaximal, characteristic
3#20272964C212yesmaximal
4#20272960C212yesmaximal
5#20272949C212yesmaximal, complemented
6#20272945C211yesmaximal, characteristic, complemented
7#20272942C211yesmaximal, characteristic, complemented
8#20272942C211yesmaximal, characteristic, complemented
9#20272940C212yesmaximal, complemented
10#20272938C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
12C22×C4⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2212yes—
16#16810491—24no—
17C4×C22⋊Dic53—24no—
18C2×#13133124C2212yes—
19C2×#13133124C2211yescharacteristic
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C24×C4×C53C2211yescharacteristic
31C22×C22⋊C4×C53C2211yescharacteristic
32C22×C22⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C22×C42×C53C2211yescharacteristic
35C4×C22⋊C4×C53C418yescomplemented
36C2×#95×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C23×Dic53C2312yes—
39C23×Dic53C2312yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C2×C4⋊Dic53—24no—
43C2×C4⋊Dic53—24no—
44C2×C4⋊Dic53—42no—
45C2×C22⋊Dic53—24no—
46C2×C22⋊Dic53—24no—
47C2×C22⋊Dic53—24no—
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53—42no—
51C2×C4×Dic53—24no—
52C2×C4×Dic53—24no—
53C2×C4×Dic53—42no—
54#13133124—22no—
55#13133124—22no—
56#13133124—24no—
57#13133124—24no—
58#13133124—22no—
59#13133124—24no—
60#13133124—24no—
61#13133124—24no—
62#13133124—42no—
63#13133124—42no—
64#13133124—42no—
65#13133124D414yes—
66C25×C53C2311yescharacteristic
67C23×C4×C53—22no—
68C23×C4×C53—24no—
69C23×C4×C53—22no—
70C23×C4×C53—42no—
71C23×C4×C53C2×C414yes—
72C23×C4×C53C2311yescharacteristic
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C23×C4×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C2×C22⋊C4×C53—24no—
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53C2×C414yes—
82C2×C22⋊C4×C53C2×C414yes—
83C2×C42×C53—22no—
84C2×C42×C53—22no—
85C2×C42×C53—42no—
86C2×C42×C53C2×C414yes—
87C2×C42×C53C2×C414yes—
88#95×C53C2×C414yes—
89#95×C53C2×C414yes—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—21no—
93C24×C53—22no—
94C24×C53—21no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53C22×C412yes—
100C24×C53#4114yes—
101C24×C53C2411yescharacteristic
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
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—21no—
116C22×C4×C53—21no—
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—28no—
124C22×C4×C53—28no—
125C22×C4×C53—44no—
126C22×C4×C53—48no—
127C22×C4×C53—48no—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53#4114yes—
138C22×C4×C53#4112yes—
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×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53C2×D414yes—
166C22×Dic53#4114yes—
167C22⋊C4×C53—28no—
168C22⋊C4×C53—28no—
169C42×C53—28no—
170C42×C53—44no—
171C22⋊Dic53—216no—
172C22⋊Dic53—48no—
173C4⋊Dic53—48no—
174C4×Dic53—48no—
175C23×C53—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—42no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—41no—
195C23×C53—42no—
196C23×C53—41no—
197C23×C53#11714yes—
198C23×C53C23×C411yescharacteristic
199C23×C53C22×D411yescharacteristic
200C23×C53C2×#4112yes—
201C23×C53C2×#4111yescharacteristic
202C23×C53C2×#4112yes—
203C23×C53C2×#4112yes—
204C23×C53C2×#4111yescharacteristic
205C23×C53C2×#4111yescharacteristic
206C23×C53H(5,𝔽2)12yes—
207C23×C53H(5,𝔽2)11yescharacteristic
208C23×C532−1+411yescharacteristic
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—416no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53#11714yes—
240C2×C4×C53#11714yes—
241C2×C4×C53C4×D414yes—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
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—
270#13892—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—44no—
282C22×C53C2×#11711yescharacteristic
283C22×C53C2×#11711yescharacteristic
284C22×C53C2×#11711yescharacteristic
285C22×C53C2×C4×D411yescharacteristic
286C22×C53C4×#4111yescharacteristic
287C22×C53#51811yescharacteristic
288C22×C53#52011yescharacteristic
289C22×C53#52512yes—
290C22×C53#52912yes—
291C22×C53#52912yes—
292C22×C53#52912yes—
293C22×C53#53812yes—
294C22×C53#53912yes—
295C22×C53#54612yes—
296C22×C53#54711yescharacteristic
297C22×C53#55011yescharacteristic
298C22×C53#55112yes—
299C22×C53#55112yes—
300C22×C53#55312yes—
301C22×C53#55312yes—
302C22×C53#55511yescharacteristic
303C22×C53#55612yes—
304C22×C53#55611yescharacteristic
305C4×C53—24no—
306C4×C53—24no—
307C4×C53—24no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—42no—
311Dic53—44no—
312Dic53—44no—
313Dic53—82no—
314Dic53—82no—
315C2×#383—531no—
316C2×C4×C22⋊C4—531no—
317C2×C4×C22⋊C4D5311yescharacteristic
318#1440—532no—
319#1439—532no—
320#1436—532no—
321C4⋊#95—531no—
322#1434—531no—
323#1434—531no—
324#1434—532no—
325C22⋊#95—531no—
326C2×C53—44no—
327C2×C53#228311yescharacteristic
328C2×C53#229011yescharacteristic
329C2×C53#231811yescharacteristic
330C2×C53#234211yescharacteristic
331C2×C53#234711yescharacteristic
332C2×C53#235311yescharacteristic
333C2×C53#235411yescharacteristic
334C2×C53#246612yes—
335C2×C53#246712yes—
336C2×C53#251812yes—
337C2×C53#252312yes—
338C24×C4C2×D5311yescharacteristic
339C22×C4⋊C4—532no—
340C22×C22⋊C4—532no—
341C22×C22⋊C4—532no—
342C22×C22⋊C4C2×D5311yescharacteristic
343C22×C22⋊C4C2×D5311yescharacteristic
344C22×C42—532no—
345C22×C42C2×D5311yescharacteristic
346C22×C42C2×D5311yescharacteristic
347#383—1064no—
348C4×C22⋊C4—1064no—
349C4×C22⋊C4Dic5318yescomplemented
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—531no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95C2×D5311yescharacteristic
363C2×#95C2×D5311yescharacteristic
364C53#1389211yescharacteristic, complemented, Sylow
365C25C22×D5311yescharacteristic
366C23×C4—24no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4C2×Dic5314yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C23×C4C22×D5311yescharacteristic
380C23×C4C22×D5311yescharacteristic
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—2122no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1064no—
389C2×C22⋊C4—1064no—
390C2×C22⋊C4—1064no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4—2122no—
393C2×C22⋊C4C2×Dic5314yes—
394C2×C22⋊C4C2×Dic5314yes—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—42no—
398C2×C42—1064no—
399C2×C42—1064no—
400C2×C42—2122no—
401C2×C42C2×Dic5314yes—
402C2×C42C2×Dic5314yes—
403#95—534no—
404#95—1064no—
405#95—1064no—
406#95—1064no—
407#95—1064no—
408#95—1064no—
409#95—1062no—
410#95—1062no—
411#95—1062no—
412#95—2122no—
413#95—2122no—
414#95—2122no—
415#95C2×Dic5314yes—
416#95C2×Dic5314yes—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—21no—
421C24—21no—
422C24—22no—
423C24—22no—
424C24—21no—
425C24—21no—
426C24#1171994514yes—
427C24C22×Dic5312yes—
428C24C23×D5311yescharacteristic
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—534no—
456C22×C4—534no—
457C22×C4—1062no—
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#1171994512yes—
484C22×C4#1171994514yes—
485C22×C4#1171994512yes—
486C22×C4D4×D5312yes—
487C22×C4#1171994712yes—
488C22×C4C22×Dic5312yes—
489C22×C4C22×Dic5312yes—
490C22×C4C22×Dic5312yes—
491C22×C4C22×Dic5312yes—
492C22×C4C22×Dic5312yes—
493C22×C4C22×Dic5312yes—
494C4⋊C4—2128no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—10616no—
498C22⋊C4—2128no—
499C42—28no—
500C42—44no—
501C42—2128no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23#1313317614yes—
525C23C23×Dic5311yescharacteristic
526C23C2×D4×D5311yescharacteristic
527C23#41×D5312yes—
528C23C2×#1171994711yescharacteristic
529C23C2×#1171994511yescharacteristic
530C23C2×#1171994511yescharacteristic
531C23C2×#1171994512yes—
532C23C2×#1171994512yes—
533C23#1313326212yes—
534C23#1313326311yescharacteristic
535C23#1313326411yescharacteristic
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
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#1313317614yes—
595C2×C4#1313317614yes—
596C2×C4D4×Dic5314yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—44no—
608C22C2×#1313317611yescharacteristic
609C22C2×#1313317611yescharacteristic
610C22C2×#1313317611yescharacteristic
611C22C2×D4×Dic5311yescharacteristic
612C22#41×Dic5311yescharacteristic
613C22#1681109811yescharacteristic
614C22#1681110611yescharacteristic
615C22#1681113212yes—
616C22#1681115312yes—
617C22#1681115712yes—
618C22#1681115712yes—
619C22#1681120612yes—
620C22#1681121512yes—
621C22#1681126312yes—
622C22#1681127811yescharacteristic
623C22#1681129411yescharacteristic
624C22#1681129912yes—
625C22#1681130212yes—
626C22#1681131712yes—
627C22#1681132212yes—
628C22#1681133811yescharacteristic
629C22#1681134112yes—
630C22#1681134311yescharacteristic
631C4—24no—
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—4242no—
638C4—4242no—
639C4—2124no—
640C4—2124no—
641C2—44no—
642C2#2027711311yescharacteristic
643C2#2027716011yescharacteristic
644C2#2027734811yescharacteristic
645C2#2027756411yescharacteristic
646C2#2027760811yescharacteristic
647C2#2027765511yescharacteristic
648C2#2027766411yescharacteristic
649C2#2027875112yes—
650C2#2027876612yes—
651C2#2027936712yes—
652C2#2027944212yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C4×C22⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#13892
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16810491C2semidirect
5. #20272949C2semidirect
6. #20272945C2semidirect
7. #20272942C2semidirect
8. #20272942C2semidirect
9. #20272940C2semidirect
35. C4×C22⋊C4×C53C4semidirect
349. C4×C22⋊C4Dic53semidirect
364. C53#13892semidirect

Maximal subgroups

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

subgroupindex
C2×#168104912
C2×C4×C22⋊Dic532
#202729642
#202729602
#202729492
#202729452
#202729422
#202729402
#202729382
C2×C4×C22⋊C4×C532
#1389253