Finite Group Databasebeta

← #27320406

Subgroups — #27320406

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×#16810560C211yesmaximal, characteristic
3C2×#16810514C211yesmaximal, characteristic
4C2×C4×C4⋊Dic53C211yesmaximal, characteristic, complemented
5#20272969C211yesmaximal, characteristic, complemented
6#20272968C212yesmaximal, complemented
7#20272961C211yesmaximal, characteristic
8#20272957C212yesmaximal
9#20272951C212yesmaximal
10#20272949C212yesmaximal, complemented
11C53×C4⋊#95C211yesmaximal, characteristic, complemented
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×#11719921C2211yescharacteristic
18C22×#11719921C2212yes—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2212yes—
21C22×C4×Dic53C2212yes—
22#16810563—24no—
23#16810560—24no—
24#16810514—24no—
25C4×C4⋊Dic53—24no—
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C22×C4⋊C4×C53C2212yes—
35C22×C42×C53C2211yescharacteristic
36C2×#95×C53C2212yes—
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C24×D53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2312yes—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—24no—
49C2×C4⋊Dic53—24no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53D414yes—
53C2×#11719919—22no—
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719921—22no—
57C2×#11719921—22no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921—22no—
61C2×#11719921—24no—
62C2×#11719921—22no—
63C2×#11719921—24no—
64C2×#11719921—24no—
65C2×#11719921—24no—
66C2×#11719921—42no—
67C2×#11719921—42no—
68C2×#11719921—42no—
69C2×#11719921Q814yes—
70C2×C4×Dic53—24no—
71C2×C4×Dic53—42no—
72C2×C4×Dic53—42no—
73#13133124—22no—
74#13133124—22no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—42no—
80#13133124—42no—
81C23×C4×C53C2312yes—
82C23×C4×C53C2312yes—
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—24no—
88C2×C4⋊C4×C53—42no—
89C2×C42×C53—24no—
90C2×C42×C53D414yes—
91#95×C53—22no—
92#95×C53—22no—
93#95×C53—24no—
94C24×C53C2411yescharacteristic
95C23×D53—21no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—21no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53—21no—
104C23×D53—21no—
105C23×D53C2×Q812yes—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—21no—
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—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×C53C2×D412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×Q812yes—
131C22×C4×C53#4114yes—
132C22×C4×C53#4112yes—
133C22×C4×C53#4112yes—
134C22×C4×C53#4114yes—
135C22×Dic53—21no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—22no—
158C22×Dic53—21no—
159C22×Dic53—22no—
160C22×Dic53—21no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53C2×D414yes—
164C22×Dic53C2×Q812yes—
165C22×Dic53#4114yes—
166C4⋊C4×C53—48no—
167C42×C53—28no—
168#11719921—28no—
169#11719921—48no—
170#11719921—48no—
171#11719921—44no—
172#11719921—44no—
173C4⋊Dic53—28no—
174C4⋊Dic53—28no—
175C4⋊Dic53—48no—
176#11719919—44no—
177C4×Dic53—48no—
178C23×C53C22×D411yescharacteristic
179C23×C53C22×D411yescharacteristic
180C23×C53C22×Q811yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4111yescharacteristic
183C23×C53C2×#4112yes—
184C23×C53C2×#4112yes—
185C23×C53C2×#4111yescharacteristic
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C53H(5,𝔽2)12yes—
188C23×C532−1+411yescharacteristic
189C22×D53—22no—
190C22×D53—24no—
191C22×D53—22no—
192C22×D53—24no—
193C22×D53—22no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—41no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—41no—
211C22×D53—42no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—42no—
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—42no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53#12114yes—
239C2×C4×C53C22⋊Q814yes—
240C2×C4×C53#12314yes—
241C2×Dic53—24no—
242C2×Dic53—22no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272#15103—531nomaximal, Sylow
273C22×C53C2×#12111yescharacteristic
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53C2×#12311yescharacteristic
276C22×C53#52511yescharacteristic
277C22×C53#52912yes—
278C22×C53#52912yes—
279C22×C53#53212yes—
280C22×C53#53211yescharacteristic
281C22×C53#53312yes—
282C22×C53#53511yescharacteristic
283C22×C53#53711yescharacteristic
284C22×C53#53812yes—
285C22×C53#53912yes—
286C22×C53#54612yes—
287C22×C53D4×Q811yescharacteristic
288C22×C53#55011yescharacteristic
289C22×C53#55112yes—
290C22×C53#55212yes—
291C22×C53#55312yes—
292C22×C53#55412yes—
293C22×C53#55411yescharacteristic
294C22×C53#55611yescharacteristic
295C22×C53#55912yes—
296C2×D53—44no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—44no—
301C2×D53—82no—
302C2×D53—81no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—82no—
307C4×C53—24no—
308C4×C53—24no—
309C4×C53—24no—
310C4×C53—44no—
311C4×C53—82no—
312Dic53—44no—
313Dic53—42no—
314Dic53—44no—
315Dic53—81no—
316Dic53—81no—
317C2×#400—531no—
318C2×#399—531no—
319C2×#388—531no—
320C2×C4×C4⋊C4—531no—
321#1441—531no—
322#1441—532no—
323#1439—531no—
324#1439—532no—
325#1437—532no—
326#1436—532no—
327C4⋊#95D5311yescharacteristic, complemented
328C2×C53#239611yescharacteristic
329C2×C53#248711yescharacteristic
330C2×C53#249811yescharacteristic
331C2×C53#253811yescharacteristic
332C2×C53#254411yescharacteristic
333C2×C53#255412yes—
334C2×C53#255612yes—
335C2×C53#258912yes—
336C2×C53#259612yes—
337C2×C53#263611yescharacteristic
338C2×C53#285411yescharacteristic
339D53—82no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—532no—
344C22×C4⋊C4C2×D5312yes—
345C22×C22⋊C4—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C42—532no—
351C22×C42C2×D5311yescharacteristic
352#400—1064no—
353#399—1064no—
354#388—1064no—
355C4×C4⋊C4—1064no—
356C2×#95—532no—
357C2×#95—532no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—531no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95C2×D5311yescharacteristic
365C2×#95C2×D5311yescharacteristic
366C2×#95C2×D5312yes—
367C53#1510311yescharacteristic, complemented, Sylow
368C25—531no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4C22×D5312yes—
375C23×C4C22×D5312yes—
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—42no—
382C2×C4⋊C4—534no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—2122no—
393C2×C22⋊C4—534no—
394C2×C22⋊C4—1062no—
395C2×C22⋊C4—1062no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C22⋊C4—1064no—
401C2×C22⋊C4—1062no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—2122no—
405C2×C22⋊C4—2122no—
406C2×C22⋊C4—2122no—
407C2×C42—24no—
408C2×C42—1064no—
409C2×C42—2122no—
410C2×C42—2122no—
411C2×C42D21214yes—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1064no—
419#95—1062no—
420#95—1062no—
421#95—2122no—
422#95—2122no—
423C24—532no—
424C24—1061no—
425C24—1061no—
426C24—1062no—
427C24—1062no—
428C24—1061no—
429C24—1062no—
430C24—1061no—
431C24—1061no—
432C24—1062no—
433C24—1061no—
434C24C23×D5311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—534no—
456C22×C4—532no—
457C22×C4—534no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1062no—
486C22×C4C2×D21212yes—
487C22×C4C2×D21212yes—
488C22×C4C2×D21212yes—
489C22×C4#1171994514yes—
490C22×C4D4×D5312yes—
491C22×C4#1171994712yes—
492C22×C4#1171994714yes—
493C22×C4Q8×D5312yes—
494C22×C4#1171994912yes—
495C4⋊C4—48no—
496C4⋊C4—1068no—
497C4⋊C4—1068no—
498C4⋊C4—2128no—
499C4⋊C4—2124no—
500C22⋊C4—1068no—
501C22⋊C4—2128no—
502C22⋊C4—2128no—
503C22⋊C4—2124no—
504C22⋊C4—2124no—
505C42—28no—
506C42—2128no—
507C23—1062no—
508C23—1064no—
509C23—1062no—
510C23—1062no—
511C23—1064no—
512C23—2122no—
513C23—2121no—
514C23—2122no—
515C23—2122no—
516C23—2122no—
517C23—2121no—
518C23—2122no—
519C23—2121no—
520C23—2121no—
521C23—2121no—
522C23—2122no—
523C23—2122no—
524C23—2122no—
525C23—2121no—
526C23—2121no—
527C23—2121no—
528C23—2122no—
529C23—2122no—
530C23C2×D4×D5311yescharacteristic
531C23C22×D21211yescharacteristic
532C23C2×Q8×D5311yescharacteristic
533C23#41×D5312yes—
534C23C2×#1171994711yescharacteristic
535C23C2×#1171994712yes—
536C23C2×#1171994512yes—
537C23C2×#1171994911yescharacteristic
538C23#1313326211yescharacteristic
539C23#1313326312yes—
540C23#1313326511yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—1064no—
568C2×C4—1064no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1064no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2121no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4—2121no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4#1313319214yes—
599C2×C4#1313319714yes—
600C2×C4#1313320114yes—
601C22—4241no—
602C22—4241no—
603C22—4241no—
604C22—4241no—
605C22—4242no—
606C22—4242no—
607C22—2122no—
608C22—2124no—
609C22—2122no—
610C22—2124no—
611C22—2122no—
612C22C2×#1313319211yescharacteristic
613C22C2×#1313319711yescharacteristic
614C22C2×#1313320111yescharacteristic
615C22#1681113511yescharacteristic
616C22#1681115312yes—
617C22#1681115912yes—
618C22#1681116812yes—
619C22#1681117311yescharacteristic
620C22#1681117712yes—
621C22#1681119211yescharacteristic
622C22#1681120411yescharacteristic
623C22#1681120712yes—
624C22#1681122112yes—
625C22#1681126912yes—
626C22#1681128811yescharacteristic
627C22#1681129311yescharacteristic
628C22#1681129912yes—
629C22#1681130812yes—
630C22#1681132512yes—
631C22#1681133112yes—
632C22#1681133411yescharacteristic
633C22#1681134711yescharacteristic
634C22#1681135812yes—
635C4—24no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—82no—
640C4—4241no—
641C4—4241no—
642C4—2124no—
643C4—2122no—
644C4—2124no—
645C2—4242no—
646C2#2027798711yescharacteristic
647C2#2027903311yescharacteristic
648C2#2027915911yescharacteristic
649C2#2027963811yescharacteristic
650C2#2027968611yescharacteristic
651C2#2027977212yes—
652C2#2027979612yes—
653C2#2028017012yes—
654C2#2028025512yes—
655C2#2028061411yescharacteristic
656C2#2028304711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×C4⋊#95C2
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#15103
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C2×#16810563C2semidirect
4. C2×C4×C4⋊Dic53C2semidirect
5. #20272969C2semidirect
6. #20272968C2semidirect
10. #20272949C2semidirect
11. C53×C4⋊#95C2semidirect
327. C4⋊#95D53semidirect
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×#168105602
C2×#168105142
C2×C4×C4⋊Dic532
#202729692
#202729682
#202729612
#202729572
#202729512
#202729492
C53×C4⋊#952
#1510353