Finite Group Databasebeta

← #27320399

Subgroups — #27320399

Order 13568.

1462 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810563C211yesmaximal, characteristic, complemented
2C2×#16810559C211yesmaximal, characteristic
3C2×#16810517C211yesmaximal, characteristic
4C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
5#20272966C212yesmaximal, complemented
6#20272965C211yesmaximal, characteristic, complemented
7#20272958C212yesmaximal
8#20272952C212yesmaximal
9#20272946C212yesmaximal, complemented
10#20272943C211yesmaximal, characteristic, complemented
11C53×#1439C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×#11719919C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21#16810563—24no—
22#16810559—24no—
23#16810517—24no—
24C4⋊C4×Dic53—24no—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2211yescharacteristic
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2212yes—
34C22×C4⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53D414yes—
48C2×#11719919—22no—
49C2×#11719919—22no—
50C2×#11719919—22no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—42no—
56C2×#11719919—42no—
57C2×C22⋊Dic53—22no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53—24no—
62C2×C22⋊Dic53—24no—
63C2×C22⋊Dic53—42no—
64C2×C22⋊Dic53Q814yes—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—42no—
68C2×C4×Dic53—42no—
69C2×C4×Dic53D414yes—
70#13133124—22no—
71#13133124—24no—
72#13133124—22no—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—42no—
79C25×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—24no—
86C2×C22⋊C4×C53—22no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—24no—
89C2×C22⋊C4×C53—22no—
90C2×C22⋊C4×C53—42no—
91C2×C22⋊C4×C53—42no—
92#95×C53—22no—
93#95×C53—42no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53—21no—
98C24×C53—21no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—21no—
102C24×C53—22no—
103C24×C53—21no—
104C24×C53C2×Q812yes—
105C24×C53C2411yescharacteristic
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
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—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×D412yes—
128C22×C4×C53#4114yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—21no—
158C22×Dic53C2×D414yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×Q812yes—
162C22×Dic53C2×Q812yes—
163C22×Dic53#4114yes—
164C22×Dic53#4112yes—
165C22×Dic53#4114yes—
166C4⋊C4×C53—28no—
167C22⋊C4×C53—48no—
168C22⋊C4×C53—44no—
169C22⋊Dic53—28no—
170C22⋊Dic53—48no—
171C22⋊Dic53—44no—
172C4⋊Dic53—28no—
173#11719919—48no—
174#11719919—48no—
175#11719919—44no—
176C4×Dic53—28no—
177C4×Dic53—48no—
178C23×C53—22no—
179C23×C53—24no—
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—22no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—41no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—41no—
200C23×C53—42no—
201C23×C53C22×D411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C22×Q811yescharacteristic
204C23×C53C2×#4111yescharacteristic
205C23×C53C2×#4112yes—
206C23×C53C2×#4112yes—
207C23×C53C2×#4112yes—
208C23×C53C2×#4111yescharacteristic
209C23×C53H(5,𝔽2)12yes—
210C23×C53H(5,𝔽2)11yescharacteristic
211C23×C532−1+411yescharacteristic
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53#12114yes—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—22no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—42no—
259C2×Dic53—41no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—41no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53C22⋊Q814yes—
271C2×Dic53#12314yes—
272#15103—531nomaximal, Sylow
273C22×C53—44no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—44no—
278C22×C53—82no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53—82no—
284C22×C53C2×#12111yescharacteristic
285C22×C53C2×C22⋊Q811yescharacteristic
286C22×C53C2×#12311yescharacteristic
287C22×C53#52511yescharacteristic
288C22×C53#52912yes—
289C22×C53#52912yes—
290C22×C53#53212yes—
291C22×C53#53211yescharacteristic
292C22×C53#53312yes—
293C22×C53#53511yescharacteristic
294C22×C53#53711yescharacteristic
295C22×C53#53812yes—
296C22×C53#53912yes—
297C22×C53#54612yes—
298C22×C53D4×Q811yescharacteristic
299C22×C53#55011yescharacteristic
300C22×C53#55112yes—
301C22×C53#55212yes—
302C22×C53#55312yes—
303C22×C53#55412yes—
304C22×C53#55411yescharacteristic
305C22×C53#55611yescharacteristic
306C22×C53#55912yes—
307C4×C53—24no—
308C4×C53—44no—
309C4×C53—81no—
310C4×C53—82no—
311Dic53—24no—
312Dic53—24no—
313Dic53—42no—
314Dic53—44no—
315Dic53—44no—
316Dic53—81no—
317C2×#400—531no—
318C2×#399—531no—
319C2×#388—531no—
320C2×C4×C4⋊C4—531no—
321#1441—532no—
322#1441—531no—
323#1439—532no—
324#1439D5311yescharacteristic
325#1437—532no—
326#1436—532no—
327C4⋊#95—531no—
328C2×C53—82no—
329C2×C53#239611yescharacteristic
330C2×C53#248711yescharacteristic
331C2×C53#249811yescharacteristic
332C2×C53#253811yescharacteristic
333C2×C53#254411yescharacteristic
334C2×C53#255412yes—
335C2×C53#255612yes—
336C2×C53#258912yes—
337C2×C53#259612yes—
338C2×C53#263611yescharacteristic
339C2×C53#285411yescharacteristic
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—532no—
344C22×C4⋊C4C2×D5311yescharacteristic
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—532no—
348C22×C22⋊C4C2×D5311yescharacteristic
349C22×C22⋊C4C2×D5312yes—
350C22×C42—531no—
351C22×C42—532no—
352#400—1064no—
353#399—1064no—
354#388—1064no—
355C4×C4⋊C4—1064no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—532no—
359C2×#95—532no—
360C2×#95—531no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—531no—
364C2×#95—532no—
365C2×#95C2×D5311yescharacteristic
366C2×#95C2×D5312yes—
367C53#1510311yescharacteristic, complemented, Sylow
368C25C22×D5311yescharacteristic
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5312yes—
377C23×C4C22×D5312yes—
378C23×C4C22×D5311yescharacteristic
379C2×C4⋊C4—22no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—2122no—
392C2×C4⋊C4—2122no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—42no—
398C2×C22⋊C4—42no—
399C2×C22⋊C4—534no—
400C2×C22⋊C4—1062no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1062no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—2122no—
407C2×C42—534no—
408C2×C42—1064no—
409C2×C42—1064no—
410C2×C42—2122no—
411C2×C42—2122no—
412#95—22no—
413#95—42no—
414#95—1064no—
415#95—1064no—
416#95—1062no—
417#95—1062no—
418#95—1064no—
419#95—1064no—
420#95—1062no—
421#95—1064no—
422#95—2122no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—21no—
427C24—22no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24Q8×D5312yes—
434C24C23×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—534no—
460C22×C4—534no—
461C22×C4—532no—
462C22×C4—534no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1061no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1062no—
480C22×C4—1061no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4#1171994512yes—
492C22×C4D4×D5312yes—
493C22×C4D4×D5312yes—
494C22×C4#1171994714yes—
495C4⋊C4—28no—
496C4⋊C4—1068no—
497C4⋊C4—2128no—
498C4⋊C4—2128no—
499C4⋊C4—2124no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—1068no—
503C22⋊C4—2124no—
504C22⋊C4—2128no—
505C42—1068no—
506C42—2128no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23C2×D4×D5311yescharacteristic
531C23C2×D4×D5311yescharacteristic
532C23C2×Q8×D5311yescharacteristic
533C23#41×D5312yes—
534C23#41×D5311yescharacteristic
535C23#41×D5312yes—
536C23C2×#1171994712yes—
537C23C2×#1171994511yescharacteristic
538C23#1313326212yes—
539C23#1313326211yescharacteristic
540C23#1313326411yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—534no—
565C2×C4—534no—
566C2×C4—1064no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1062no—
580C2×C4—1064no—
581C2×C4—1062no—
582C2×C4—1064no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2121no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2121no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2121no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—2121no—
599C2×C4—2122no—
600C2×C4#1313319014yes—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22C2×#1313319011yescharacteristic
613C22C22⋊Q8×D5311yescharacteristic
614C22#123×D5311yescharacteristic
615C22#1681112911yescharacteristic
616C22#1681115212yes—
617C22#1681115312yes—
618C22#1681116812yes—
619C22#1681117211yescharacteristic
620C22#1681117512yes—
621C22#1681118611yescharacteristic
622C22#1681120211yescharacteristic
623C22#1681120812yes—
624C22#1681121812yes—
625C22#1681127012yes—
626C22#1681128611yescharacteristic
627C22#1681129111yescharacteristic
628C22#1681130112yes—
629C22#1681130912yes—
630C22#1681131812yes—
631C22#1681132712yes—
632C22#1681132811yescharacteristic
633C22#1681134111yescharacteristic
634C22#1681135612yes—
635C4—24no—
636C4—44no—
637C4—81no—
638C4—82no—
639C4—1064no—
640C4—1064no—
641C4—4241no—
642C4—2122no—
643C4—2124no—
644C4—2124no—
645C2—82no—
646C2#2027798211yescharacteristic
647C2#2027902511yescharacteristic
648C2#2027916211yescharacteristic
649C2#2027963911yescharacteristic
650C2#2027968911yescharacteristic
651C2#2027977012yes—
652C2#2027979212yes—
653C2#2028016312yes—
654C2#2028025612yes—
655C2#2028060911yescharacteristic
656C2#2028304311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1439C2
prime radicalC25×C53C23
hypercenter#1439D53
hypocenterC53#15103
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C2×#16810563C2semidirect
4. C2×C4⋊C4×Dic53C2semidirect
5. #20272966C2semidirect
6. #20272965C2semidirect
9. #20272946C2semidirect
10. #20272943C2semidirect
367. C53#15103semidirect

Maximal subgroups

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

subgroupindex
C2×#168105632
C2×#168105592
C2×#168105172
C2×C4⋊C4×Dic532
#202729662
#202729652
#202729582
#202729522
#202729462
#202729432
C53×#14392
#1510353