Finite Group Databasebeta

← #27306076

Subgroups — #27306076

Order 13568.

2510 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810497C214yesmaximal
3C2×#16810493C211yesmaximal, characteristic, complemented
4C2×#16810492C211yesmaximal, characteristic, complemented
5C2×#16810480C211yesmaximal, characteristic
6C2×C4×#11719921C211yesmaximal, characteristic
7#20272979C214yesmaximal, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C53×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D53C2212yes—
11C23×C4×D53C2211yescharacteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2214yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C2212yes—
16C22×#11719919C2214yes—
17C2×C4⋊C4×D53—24no—
18C2×C4⋊C4×D53—24no—
19C2×C4⋊C4×D53—24no—
20C22×#11719921C2212yes—
21C22×#11719921C2212yes—
22C22×C4×Dic53C2211yescharacteristic
23#16810497—216no—
24#16810493—24no—
25#16810492—24no—
26#16810480—24no—
27C4×#11719921C418yescomplemented
28C2×#13133124C2214yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C42×C53C2211yescharacteristic
34C2×#95×C53C2214yes—
35C24×D53C2311yescharacteristic
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—28no—
48C22×C4×D53—44no—
49C22×C4×D53C2×C414yes—
50C22×C4×D53D414yes—
51C22×C4×D53Q814yes—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—28no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—28no—
59C2×C4⋊Dic53—44no—
60C2×C4⋊Dic53D414yes—
61C2×#11719919—24no—
62C2×#11719919—28no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—28no—
66C2×#11719919—44no—
67C4⋊C4×D53—232no—
68C2×#11719921—28no—
69C2×#11719921—24no—
70C2×#11719921C2×C418yes—
71C2×#11719921D418yes—
72C2×#11719921Q818yes—
73C2×C4×Dic53—22no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53C2×C414yescomplemented
76#13133124—28no—
77#13133124—24no—
78#13133124C2×C418yescomplemented
79C23×C4×C53C2312yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—22no—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53D414yes—
89C2×C42×C53—22no—
90C2×C42×C53—22no—
91C2×C42×C53C2×C414yescomplemented
92#95×C53—28no—
93C24×C53C2411yescharacteristic
94C23×D53—21no—
95C23×D53—21no—
96C23×D53—21no—
97C23×D53—21no—
98C23×D53—22no—
99C23×D53—22no—
100C23×D53C22×C412yes—
101C23×D53C2×D414yes—
102C23×D53C2×D412yes—
103C23×D53C2×Q814yes—
104C23×D53C2×Q812yes—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—24no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—24no—
113C22×C4×C53—24no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—24no—
118C22×C4×C53C22×C414yes—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C2×D414yes—
121C22×C4×C53C2×D414yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53C2×Q814yes—
126C22×C4×C53C2×Q812yes—
127C22×C4×C53#4112yes—
128C22×C4×C53#4112yes—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—22no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—24no—
141C22×Dic53—22no—
142C22×Dic53—24no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—24no—
146C22×Dic53C22×C412yes—
147C22×Dic53C22×C414yes—
148C22×Dic53C2×D414yes—
149C22×Dic53C2×D412yes—
150C22×Dic53C2×D414yes—
151C22×Dic53C2×Q812yes—
152C22×Dic53C2×Q814yes—
153C2×C4×D53—28no—
154C2×C4×D53—28no—
155C2×C4×D53—28no—
156C2×C4×D53—28no—
157C2×C4×D53—28no—
158C2×C4×D53—28no—
159C2×C4×D53—216no—
160C2×C4×D53—416no—
161C2×C4×D53—416no—
162C2×C4×D53—48no—
163C2×C4×D53—48no—
164C2×C4×D53C4⋊C4116yescomplemented
165C4⋊C4×C53—28no—
166C4⋊C4×C53—44no—
167C42×C53—28no—
168#11719921—48no—
169#11719921C4⋊C4132yescomplemented
170C4⋊Dic53—28no—
171C4⋊Dic53—416no—
172C4⋊Dic53—44no—
173#11719919—216no—
174#11719919—416no—
175C4×Dic53—28no—
176C23×C53C23×C411yescharacteristic
177C23×C53C22×D411yescharacteristic
178C23×C53C22×D412yes—
179C23×C53C22×D411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C22×Q811yescharacteristic
182C23×C53C22×Q812yes—
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53H(5,𝔽2)12yes—
186C23×C532−1+412yes—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—24no—
190C22×D53—22no—
191C22×D53—22no—
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—24no—
197C22×D53—24no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53—24no—
201C22×D53—24no—
202C22×D53—24no—
203C22×D53—24no—
204C22×D53—24no—
205C22×D53C2×C4⋊C414yes—
206C22×D53C2×C4⋊C418yes—
207C22×D53#12714yes—
208C22×D53C4⋊Q814yes—
209C22×D53C4⋊Q818yes—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—28no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—44no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—44no—
225C2×C4×C53—44no—
226C2×C4×C53C2×C4⋊C414yes—
227C2×C4×C53C2×C4⋊C418yescomplemented
228C2×C4×C53C4×D414yes—
229C2×C4×C53C4×D414yes—
230C2×C4×C53#12014yes—
231C2×C4×C53C22⋊Q814yes—
232C2×C4×C53C22⋊Q814yes—
233C2×C4×C53#12314yes—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—28no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—44no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—44no—
255C2×Dic53—44no—
256C2×Dic53C2×C4⋊C414yes—
257C2×Dic53C2×C4⋊C418yescomplemented
258C4×D53—232no—
259C4×D53—432no—
260#13880—531nomaximal, Sylow
261C22×C53C22×C4⋊C411yescharacteristic
262C22×C53C22×C4⋊C412yes—
263C22×C53C2×C4×D411yescharacteristic
264C22×C53C2×C4×D411yescharacteristic
265C22×C53#52012yes—
266C22×C53C2×#12011yescharacteristic
267C22×C53C2×C22⋊Q811yescharacteristic
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53C2×#12311yescharacteristic
270C22×C53C2×#12711yescharacteristic
271C22×C53C2×C4⋊Q811yescharacteristic
272C22×C53C2×C4⋊Q812yes—
273C22×C53#53511yescharacteristic
274C22×C53#53611yescharacteristic
275C22×C53#54412yes—
276C22×C53D4212yes—
277C22×C53D4212yes—
278C22×C53#54712yes—
279C22×C53#54712yes—
280C22×C53D4×Q812yes—
281C22×C53D4×Q812yes—
282C22×C53#55412yes—
283C22×C53#55412yes—
284C2×D53—24no—
285C2×D53—28no—
286C2×D53—28no—
287C2×D53—24no—
288C2×D53—24no—
289C2×D53—28no—
290C2×D53—28no—
291C2×D53—24no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53#38418yescomplemented
295C4×C53—28no—
296C4×C53—44no—
297C4×C53—82no—
298C4×C53#38518yescomplemented
299Dic53—28no—
300Dic53—24no—
301Dic53—44no—
302Dic53—82no—
303C23×C4⋊C4—531no—
304C2×#385—534no—
305C2×#384—531no—
306C2×#384—531no—
307C2×#380—531no—
308C2×C4×C22⋊C4—531no—
309#1444—534no—
310C4⋊#95—531no—
311C4⋊#95D5311yescharacteristic, complemented
312C2×C53C2×#38411yescharacteristic
313C2×C53C2×#38511yescharacteristic
314C2×C53#231711yescharacteristic
315C2×C53D4×C4⋊C412yes—
316C2×C53D4×C4⋊C412yes—
317C2×C53#243311yescharacteristic
318C2×C53#243411yescharacteristic
319C2×C53#244012yes—
320C2×C53#244012yes—
321C2×C53#250111yescharacteristic
322C2×C53#250211yescharacteristic
323D53—28no—
324C24×C4—531no—
325C24×C4—532no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—532no—
328C22×C4⋊C4—531no—
329C22×C4⋊C4—534no—
330C22×C4⋊C4—534no—
331C22×C4⋊C4—1064no—
332C22×C4⋊C4—1064no—
333C22×C4⋊C4—1064no—
334C22×C4⋊C4C2×D5311yescharacteristic
335C22×C4⋊C4C2×D5311yescharacteristic
336C22×C22⋊C4—532no—
337C22×C22⋊C4—532no—
338C22×C42—531no—
339C22×C42C2×D5311yescharacteristic
340#385—10616no—
341#384—1064no—
342#384—1064no—
343#380—1064no—
344C4×C22⋊C4—538no—
345C2×#95—532no—
346C2×#95—534no—
347C2×#95—532no—
348C2×#95C2×D5314yes—
349C53#1388011yescharacteristic, complemented, Sylow
350C25—531no—
351C23×C4—532no—
352C23×C4—531no—
353C23×C4—534no—
354C23×C4—534no—
355C23×C4—534no—
356C23×C4—532no—
357C23×C4—532no—
358C23×C4—1064no—
359C23×C4—1064no—
360C23×C4—1068no—
361C23×C4—1062no—
362C23×C4—1062no—
363C23×C4—1062no—
364C23×C4—1062no—
365C23×C4—1064no—
366C23×C4—2124no—
367C23×C4C22×D5312yes—
368C23×C4C22×D5312yes—
369C23×C4C22×D5312yes—
370C23×C4C22×D5311yescharacteristic
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—534no—
377C2×C4⋊C4—10632no—
378C2×C4⋊C4—1064no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1068no—
384C2×C4⋊C4—1068no—
385C2×C4⋊C4—1068no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1068no—
390C2×C4⋊C4—2124no—
391C2×C4⋊C4—2124no—
392C2×C4⋊C4D21214yes—
393C2×C22⋊C4—538no—
394C2×C22⋊C4—538no—
395C2×C22⋊C4—538no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1068no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—534no—
401C2×C42—1062no—
402C2×C42—1062no—
403C2×C42C4×D5314yescomplemented
404#95—28no—
405#95—538no—
406#95—1064no—
407#95—1068no—
408C24—532no—
409C24—534no—
410C24—532no—
411C24—534no—
412C24—532no—
413C24—1061no—
414C24—1061no—
415C24—1061no—
416C24—1061no—
417C24—1062no—
418C24—1062no—
419C24C23×D5311yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—532no—
434C22×C4—534no—
435C22×C4—534no—
436C22×C4—532no—
437C22×C4—534no—
438C22×C4—532no—
439C22×C4—5316no—
440C22×C4—534no—
441C22×C4—10616no—
442C22×C4—1061no—
443C22×C4—1062no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1062no—
447C22×C4—1061no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1068no—
451C22×C4—1068no—
452C22×C4—1068no—
453C22×C4—1068no—
454C22×C4—1068no—
455C22×C4—1068no—
456C22×C4—1064no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1064no—
460C22×C4—1062no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1064no—
465C22×C4—21216no—
466C22×C4—21216no—
467C22×C4—2128no—
468C22×C4—2128no—
469C22×C4C2×C4×D5314yes—
470C22×C4C2×C4×D5312yes—
471C22×C4C2×D21214yes—
472C22×C4C2×D21212yes—
473C22×C4#1171994512yes—
474C22×C4D4×D5314yes—
475C22×C4D4×D5312yes—
476C22×C4D4×D5312yes—
477C22×C4#1171994712yes—
478C22×C4Q8×D5314yes—
479C22×C4Q8×D5312yes—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C4⋊C4—10616no—
483C4⋊C4—1068no—
484C4⋊C4—21216no—
485C4⋊C4—21216no—
486C4⋊C4—2124no—
487C22⋊C4—5332no—
488C22⋊C4—2128no—
489C42—28no—
490C42—1068no—
491C23—538no—
492C23—534no—
493C23—534no—
494C23—534no—
495C23—538no—
496C23—1064no—
497C23—1064no—
498C23—1064no—
499C23—1064no—
500C23—1064no—
501C23—1064no—
502C23—1062no—
503C23—1062no—
504C23—1064no—
505C23—1062no—
506C23—1062no—
507C23—1062no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—1064no—
512C23—1064no—
513C23—1064no—
514C23C22×C4×D5311yescharacteristic
515C23C2×D4×D5312yes—
516C23C2×D4×D5311yescharacteristic
517C23C2×D4×D5311yescharacteristic
518C23C22×D21211yescharacteristic
519C23C2×Q8×D5312yes—
520C23C2×Q8×D5311yescharacteristic
521C23C2×#1171994711yescharacteristic
522C23C2×#1171994511yescharacteristic
523C23#1313326312yes—
524C23#1313326412yes—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—534no—
542C2×C4—538no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—1062no—
547C2×C4—1064no—
548C2×C4—1062no—
549C2×C4—1064no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1064no—
555C2×C4—10632no—
556C2×C4—1064no—
557C2×C4—1068no—
558C2×C4—21232no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2124no—
563C2×C4—2124no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2124no—
567C2×C4C4⋊C4×D5314yes—
568C2×C4C4⋊C4×D5318yescomplemented
569C2×C4#1313317714yes—
570C2×C4C4×D21214yes—
571C2×C4#1313318714yes—
572C2×C4#1313319514yes—
573C2×C4#1313319714yes—
574C2×C4#1313319914yes—
575C22—538no—
576C22—1064no—
577C22—1064no—
578C22—1064no—
579C22—1068no—
580C22—1068no—
581C22—1064no—
582C22—1068no—
583C22—1064no—
584C22—1068no—
585C22—1064no—
586C22C2×C4⋊C4×D5311yescharacteristic
587C22C2×C4⋊C4×D5312yes—
588C22C2×#1313317711yescharacteristic
589C22C2×C4×D21211yescharacteristic
590C22#1681110312yes—
591C22C2×#1313318711yescharacteristic
592C22C2×#1313319511yescharacteristic
593C22C2×#1313319711yescharacteristic
594C22C2×#1313319911yescharacteristic
595C22#127×D5311yescharacteristic
596C22C4⋊Q8×D5311yescharacteristic
597C22C4⋊Q8×D5312yes—
598C22#1681119111yescharacteristic
599C22#1681119311yescharacteristic
600C22#1681125312yes—
601C22#1681125712yes—
602C22D4×D21212yes—
603C22#1681127312yes—
604C22#1681127812yes—
605C22#1681128612yes—
606C22Q8×D21212yes—
607C22#1681133112yes—
608C22#1681133312yes—
609C4—28no—
610C4—44no—
611C4—82no—
612C4—1068no—
613C4—1064no—
614C4—4242no—
615C4—2124no—
616C4#1681049718yescomplemented
617C2—1068no—
618C2#384×D5311yescharacteristic
619C2C2×#1681049711yescharacteristic
620C2#2027734011yescharacteristic
621C2#2027761112yes—
622C2C4⋊C4×D21212yes—
623C2#2027835111yescharacteristic
624C2#2027835811yescharacteristic
625C2#2027841512yes—
626C2#2027841912yes—
627C2#2027918511yescharacteristic
628C2#2027919211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13880
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16810493C2semidirect
4. C2×#16810492C2semidirect
7. #20272979C2semidirect
8. #20272944C2semidirect
9. C53×C4⋊#95C2semidirect
27. C4×#11719921C4semidirect
75. C2×C4×Dic53C2×C4semidirect
78. #13133124C2×C4semidirect
91. C2×C42×C53C2×C4semidirect
164. C2×C4×D53C4⋊C4semidirect
169. #11719921C4⋊C4semidirect
227. C2×C4×C53C2×C4⋊C4semidirect
257. C2×Dic53C2×C4⋊C4semidirect
294. C2×D53#384semidirect
298. C4×C53#385semidirect
311. C4⋊#95D53semidirect
349. C53#13880semidirect
403. C2×C42C4×D53semidirect
568. C2×C4C4⋊C4×D53semidirect
616. C4#16810497semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168104972
C2×#168104932
C2×#168104922
C2×#168104802
C2×C4×#117199212
#202729792
#202729442
C53×C4⋊#952
#1388053