Finite Group Databasebeta

← #27312492

Subgroups — #27312492

Order 13568.

2018 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810503C212yesmaximal
2C2×#16810498C212yesmaximal, complemented
3#20272968C212yesmaximal, complemented
4#20272960C212yesmaximal, complemented
5#20272960C212yesmaximal, complemented
6#20272945C211yesmaximal, characteristic, complemented
7#20272942C211yesmaximal, characteristic, complemented
8#20272937C212yesmaximal, complemented
9C2×C53×#390C211yesmaximal, characteristic
10C24×Dic53C2212yes—
11C22×C4⋊Dic53C2212yes—
12C22×C4⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C22⋊Dic53C2212yes—
16#16810503—28no—
17#16810498—28no—
18C2×#13133124C2212yes—
19C2×#13133124C2211yescharacteristic
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2211yescharacteristic
28C23×D4×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C22⋊C4×C53C2212yes—
31C22×C22⋊C4×C53C2212yes—
32C22×C42×C53C2211yescharacteristic
33C53×#390C418yescomplemented
34C23×Dic53—24no—
35C23×Dic53—24no—
36C23×Dic53—24no—
37C23×Dic53—42no—
38C23×Dic53—42no—
39C23×Dic53—42no—
40C23×Dic53—42no—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C2×C4⋊Dic53—24no—
46C2×C4⋊Dic53—24no—
47C2×C4⋊Dic53—24no—
48C2×C4⋊Dic53—42no—
49C2×C4⋊Dic53—42no—
50C2×C4⋊Dic53—42no—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—24no—
53C2×C22⋊Dic53—24no—
54C2×C22⋊Dic53—24no—
55C2×C22⋊Dic53—24no—
56C2×C22⋊Dic53—42no—
57C2×C22⋊Dic53—42no—
58C2×C22⋊Dic53—42no—
59C2×C22⋊Dic53—42no—
60#13133124—24no—
61#13133124—24no—
62#13133124—22no—
63#13133124—24no—
64#13133124—41no—
65#13133124—42no—
66#13133124—42no—
67#13133124—41no—
68#13133124—42no—
69#13133124D414yescomplemented
70C25×C53C2312yes—
71C22×D4×C53—42no—
72C22×D4×C53—42no—
73C22×D4×C53C2×C414yes—
74C22×D4×C53D414yes—
75C22×D4×C53Q814yes—
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2312yes—
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C2×C4⋊C4×C53—42no—
81C2×C4⋊C4×C53C2×C414yescomplemented
82C2×C22⋊C4×C53—24no—
83C2×C22⋊C4×C53—24no—
84C2×C22⋊C4×C53—42no—
85C2×C22⋊C4×C53—42no—
86C2×C22⋊C4×C53C2×C418yescomplemented
87C2×C22⋊C4×C53C2×C418yescomplemented
88C2×C42×C53—42no—
89C2×C42×C53C2×C414yescomplemented
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53C22×C414yes—
103C24×C53C2×D414yes—
104C24×C53C2×Q814yes—
105C24×C53C2411yescharacteristic
106C2×D4×C53—44no—
107C2×D4×C53—44no—
108C2×D4×C53—48no—
109C2×D4×C53—48no—
110C2×D4×C53C4⋊C4116yescomplemented
111C22×C4×C53—22no—
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—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C22×C412yes—
126C22×C4×C53C22×C414yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×Q812yes—
132C22×C4×C53#4112yes—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
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—48no—
160C22×Dic53—48no—
161C22×Dic53—48no—
162C22×Dic53—48no—
163C22×Dic53—48no—
164C22×Dic53—48no—
165C22×Dic53—48no—
166C22×Dic53C2×D414yes—
167C22×Dic53#4114yes—
168C4⋊C4×C53—44no—
169C22⋊C4×C53—216no—
170C22⋊C4×C53—48no—
171C42×C53—44no—
172C22⋊Dic53—48no—
173C22⋊Dic53—48no—
174C22⋊Dic53—48no—
175C4⋊Dic53—48no—
176C4⋊Dic53—48no—
177C23×C53—24no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—24no—
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—24no—
185C23×C53—24no—
186C23×C53—24no—
187C23×C53—24no—
188C23×C53—24no—
189C23×C53—24no—
190C23×C53—24no—
191C23×C53—24no—
192C23×C53—24no—
193C23×C53—24no—
194C23×C53—24no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53C2×C4⋊C418yes—
212C23×C53C23×C411yescharacteristic
213C23×C53C22×D411yescharacteristic
214C23×C53C22×D411yescharacteristic
215C23×C53C22×Q811yescharacteristic
216C23×C53C2×#4112yes—
217C23×C53C2×#4111yescharacteristic
218C23×C53H(5,𝔽2)12yes—
219C23×C53H(5,𝔽2)12yes—
220C23×C53H(5,𝔽2)12yes—
221C23×C53H(5,𝔽2)11yescharacteristic
222C23×C532−1+411yescharacteristic
223D4×C53—416no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—22no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
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×C53C2×C4⋊C414yes—
245C2×C4×C53C4×D414yes—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—432no—
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—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275#14438—531nomaximal, Sylow
276C22×C53—28no—
277C22×C53—28no—
278C22×C53—28no—
279C22×C53—44no—
280C22×C53—44no—
281C22×C53—44no—
282C22×C53—44no—
283C22×C53—44no—
284C22×C53—44no—
285C22×C53—44no—
286C22×C53—44no—
287C22×C53—44no—
288C22×C53—44no—
289C22×C53—44no—
290C22×C53—44no—
291C22×C53C22×C4⋊C411yescharacteristic
292C22×C53C2×C4×D411yescharacteristic
293C22×C53C4×#4111yescharacteristic
294C22×C53#51811yescharacteristic
295C22×C53#51811yescharacteristic
296C22×C53#51811yescharacteristic
297C22×C53#52011yescharacteristic
298C22×C53#53412yes—
299C22×C53#53512yes—
300C22×C53#53812yes—
301C22×C53#54012yes—
302C22×C53#54212yes—
303C22×C53#54312yes—
304C22×C53D4211yescharacteristic
305C22×C53#54612yes—
306C22×C53#54612yes—
307C22×C53#54711yescharacteristic
308C22×C53D4×Q811yescharacteristic
309C22×C53#55312yes—
310C22×C53#55411yescharacteristic
311C22×C53#55412yes—
312C22×C53#56012yes—
313C22×C53#56212yes—
314C4×C53—24no—
315C4×C53—24no—
316C4×C53—44no—
317C4×C53—42no—
318Dic53—44no—
319Dic53—44no—
320Dic53—82no—
321Dic53—82no—
322C2×#390D5311yescharacteristic
323C2×#386—532no—
324C2×#385—532no—
325#1441—532no—
326#1439—532no—
327#1439—532no—
328C4⋊#95—531no—
329#1434—531no—
330C22⋊#95—532no—
331C2×C53—48no—
332C2×C53#231511yescharacteristic
333C2×C53#232811yescharacteristic
334C2×C53#233711yescharacteristic
335C2×C53D4×C4⋊C411yescharacteristic
336C2×C53#234911yescharacteristic
337C2×C53#236411yescharacteristic
338C2×C53#237711yescharacteristic
339C2×C53#267812yes—
340C2×C53#269212yes—
341C2×C53#270112yes—
342C2×C53#278012yes—
343C23×D4C2×D5311yescharacteristic
344C24×C4—532no—
345C22×C4⋊C4—532no—
346C22×C4⋊C4—532no—
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C22⋊C4—532no—
349C22×C22⋊C4—532no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4C2×D5312yes—
352C22×C22⋊C4C2×D5312yes—
353C22×C42C2×D5311yescharacteristic
354#390Dic5318yescomplemented
355#386—1068no—
356#385—1068no—
357C2×#95—531no—
358C2×#95—532no—
359C2×#95—531no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—531no—
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—531no—
366C2×#95—532no—
367C53#1443811yescharacteristic, complemented, Sylow
368C25C22×D5312yes—
369C22×D4—42no—
370C22×D4—42no—
371C22×D4Dic10614yes—
372C22×D4D21214yes—
373C22×D4C2×Dic5314yes—
374C23×C4—532no—
375C23×C4—532no—
376C23×C4—532no—
377C23×C4—532no—
378C23×C4—1064no—
379C23×C4—1064no—
380C23×C4—1064no—
381C23×C4—2122no—
382C23×C4—2122no—
383C23×C4—2122no—
384C23×C4—2122no—
385C23×C4C22×D5312yes—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C23×C4C22×D5311yescharacteristic
389C2×C4⋊C4—42no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1064no—
393C2×C4⋊C4—2122no—
394C2×C4⋊C4—2122no—
395C2×C4⋊C4—2122no—
396C2×C4⋊C4C2×Dic5314yescomplemented
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1064no—
406C2×C22⋊C4—2122no—
407C2×C22⋊C4—2122no—
408C2×C22⋊C4—2122no—
409C2×C22⋊C4—2122no—
410C2×C22⋊C4C2×Dic5318yescomplemented
411C2×C22⋊C4C2×Dic5318yescomplemented
412C2×C42—42no—
413C2×C42C2×Dic5314yescomplemented
414#95—534no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1062no—
419#95—2122no—
420#95—2122no—
421#95—2122no—
422#95—2121no—
423#95—2121no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24C2×Dic10614yes—
437C24C2×D21214yes—
438C24C22×Dic5314yes—
439C24C23×D5311yescharacteristic
440C2×D4—48no—
441C2×D4—48no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4C4⋊Dic53116yescomplemented
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—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—534no—
460C22×C4—534no—
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—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—2128no—
488C22×C4—2128no—
489C22×C4—2128no—
490C22×C4—2128no—
491C22×C4—2128no—
492C22×C4—2128no—
493C22×C4—2128no—
494C22×C4C2×Dic10612yes—
495C22×C4C2×D21212yes—
496C22×C4D4×D5312yes—
497C22×C4#1171994712yes—
498C22×C4C22×Dic5312yes—
499C22×C4C22×Dic5314yes—
500C22×C4C22×Dic5312yes—
501C22×C4C22×Dic5312yes—
502C4⋊C4—44no—
503C4⋊C4—2128no—
504C4⋊C4—2128no—
505C22⋊C4—216no—
506C22⋊C4—48no—
507C22⋊C4—2128no—
508C22⋊C4—2128no—
509C22⋊C4—2128no—
510C42—44no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4⋊Dic5318yes—
546C23C23×Dic5311yescharacteristic
547C23C2×D4×D5311yescharacteristic
548C23C22×D21211yescharacteristic
549C23C22×Dic10611yescharacteristic
550C23#41×D5312yes—
551C23C2×#1171994711yescharacteristic
552C23#1313326212yes—
553C23#1313326212yes—
554C23#1313326212yes—
555C23#1313326311yescharacteristic
556C23#1313326411yescharacteristic
557D4—416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
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—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2122no—
598C2×C4—21232no—
599C2×C4—2122no—
600C2×C4—2122no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4—2122no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4C2×C4⋊Dic5314yes—
608C2×C4D4×Dic5314yes—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22C22×C4⋊Dic5311yescharacteristic
625C22C2×D4×Dic5311yescharacteristic
626C22#41×Dic5311yescharacteristic
627C22#1681109811yescharacteristic
628C22#1681109811yescharacteristic
629C22#1681109811yescharacteristic
630C22#1681110611yescharacteristic
631C22#1681118512yes—
632C22#1681118612yes—
633C22#1681120712yes—
634C22#1681122712yes—
635C22#1681123712yes—
636C22#1681124912yes—
637C22D4×D21211yescharacteristic
638C22#1681126212yes—
639C22#1681126812yes—
640C22#1681127711yescharacteristic
641C22D4×Dic10611yescharacteristic
642C22#1681131612yes—
643C22#1681133011yescharacteristic
644C22#1681133212yes—
645C22#1681136112yes—
646C22#1681136812yes—
647C4—24no—
648C4—24no—
649C4—42no—
650C4—44no—
651C4—4242no—
652C4—4242no—
653C4—2124no—
654C4—2124no—
655C2—48no—
656C2#2027732911yescharacteristic
657C2#2027744111yescharacteristic
658C2#2027753011yescharacteristic
659C2D4×C4⋊Dic5311yescharacteristic
660C2#2027762511yescharacteristic
661C2#2027776711yescharacteristic
662C2#2027787211yescharacteristic
663C2#2028103512yes—
664C2#2028120812yes—
665C2#2028132612yes—
666C2#2028236412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#390C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#390D53
hypocenterC53#14438
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#16810498C2semidirect
3. #20272968C2semidirect
4. #20272960C2semidirect
5. #20272960C2semidirect
6. #20272945C2semidirect
7. #20272942C2semidirect
8. #20272937C2semidirect
33. C53×#390C4semidirect
69. #13133124D4semidirect
81. C2×C4⋊C4×C53C2×C4semidirect
86. C2×C22⋊C4×C53C2×C4semidirect
87. C2×C22⋊C4×C53C2×C4semidirect
89. C2×C42×C53C2×C4semidirect
110. C2×D4×C53C4⋊C4semidirect
354. #390Dic53semidirect
367. C53#14438semidirect
396. C2×C4⋊C4C2×Dic53semidirect
410. C2×C22⋊C4C2×Dic53semidirect
411. C2×C22⋊C4C2×Dic53semidirect
413. C2×C42C2×Dic53semidirect
444. C2×D4C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105032
C2×#168104982
#202729682
#202729602
#202729452
#202729422
#202729372
C2×C53×#3902
#1443853