Finite Group Databasebeta

← #27306485

Subgroups — #27306485

Order 13568.

2166 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810482C211yesmaximal, characteristic
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272967C212yesmaximal, complemented
5#20272966C212yesmaximal, complemented
6#20272950C211yesmaximal, characteristic, complemented
7#20272948C211yesmaximal, characteristic, complemented
8#20272947C211yesmaximal, characteristic, complemented
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2212yes—
19C2×C4⋊C4×D53—24no—
20C2×C4⋊C4×D53—24no—
21C2×C4⋊C4×D53C418yescomplemented
22C22×#11719921C2212yes—
23C22×#11719921C2212yes—
24C22×C4×Dic53C2211yescharacteristic
25#16810482—24no—
26C4×#11719921—24no—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2211yescharacteristic
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2312yes—
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—24no—
49C22×C4×D53—24no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—42no—
53C22×C4×D53—42no—
54C22×C4×D53—42no—
55C22×C4×D53—42no—
56C22×C4×D53C2×C414yes—
57C22×C4×D53C2×C414yes—
58C22×C4×D53C2×C414yes—
59C22×C4×D53D414yes—
60C22×C4×D53D414yes—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53C2×C414yes—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919—24no—
69C2×#11719919—42no—
70C2×#11719919—42no—
71C2×#11719919C2×C414yes—
72C2×#11719919C2×C414yes—
73C4⋊C4×D53—232no—
74C2×#11719921—24no—
75C2×#11719921—24no—
76C2×#11719921—24no—
77C2×#11719921—24no—
78C2×#11719921—24no—
79C2×#11719921Q818yes—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—42no—
82#13133124—22no—
83#13133124—24no—
84#13133124—24no—
85#13133124—24no—
86#13133124—24no—
87#13133124—42no—
88#13133124—42no—
89#13133124—42no—
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53C2×C414yes—
98C2×C42×C53—22no—
99C2×C42×C53—42no—
100#95×C53—22no—
101#95×C53—24no—
102#95×C53—42no—
103C24×C53C2411yescharacteristic
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53C22×C412yes—
109C23×D53C2×D412yes—
110C23×D53C2×D412yes—
111C23×D53C2×Q814yes—
112C23×D53#4114yes—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×Q814yes—
141C22×C4×C53#4114yes—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
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×Dic53C22×C412yes—
165C22×Dic53C22×C412yes—
166C22×Dic53C22×C412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×D412yes—
169C22×Dic53C2×Q814yes—
170C22×Dic53#4114yes—
171C2×C4×D53—216no—
172C2×C4×D53—216no—
173C2×C4×D53—28no—
174C2×C4×D53—28no—
175C2×C4×D53—28no—
176C2×C4×D53—28no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C2×C4×D53—44no—
180C2×C4×D53—44no—
181C2×C4×D53—48no—
182C2×C4×D53—48no—
183C2×C4×D53—44no—
184C2×C4×D53—44no—
185C2×C4×D53C22⋊C4116yes—
186C4⋊C4×C53—28no—
187C42×C53—44no—
188#11719921—216no—
189#11719921—48no—
190C4⋊Dic53—28no—
191#11719919—28no—
192#11719919—28no—
193#11719919—48no—
194C4×Dic53—44no—
195C23×C53C23×C411yescharacteristic
196C23×C53C22×D411yescharacteristic
197C23×C53C22×D411yescharacteristic
198C23×C53C22×Q812yes—
199C23×C53C2×#4112yes—
200C23×C53C2×#4112yes—
201C23×C53C2×#4112yes—
202C23×C53H(5,𝔽2)12yes—
203C23×C53H(5,𝔽2)12yes—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—24no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—24no—
215C22×D53—24no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53C2×C22⋊C414yes—
221C22×D53C4×Q818yes—
222C22×D53C22⋊Q818yes—
223C22×D53#12414yes—
224C22×D53C4⋊Q814yes—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—24no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—42no—
243C2×C4×C53—42no—
244C2×C4×C53—42no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53C2×C22⋊C414yes—
253C2×C4×C53#11714yes—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53C2×C22⋊C414yes—
282C2×Dic53#11714yes—
283C4×D53—232no—
284C4×D53—416no—
285C4×D53—416no—
286#13923—531nomaximal, Sylow
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C2×#11711yescharacteristic
289C22×C53C2×#11711yescharacteristic
290C22×C53C2×C4×Q812yes—
291C22×C53#51812yes—
292C22×C53C2×C22⋊Q812yes—
293C22×C53#52512yes—
294C22×C53#52512yes—
295C22×C53C2×#12411yescharacteristic
296C22×C53C2×C4⋊Q811yescharacteristic
297C22×C53#53412yes—
298C22×C53#53411yescharacteristic
299C22×C53#53411yescharacteristic
300C22×C53#55312yes—
301C22×C53#55312yes—
302C22×C53#55312yes—
303C22×C53#55312yes—
304C22×C53#55412yes—
305C22×C53#55412yes—
306C22×C53#55412yes—
307C22×C53#55412yes—
308C2×D53—24no—
309C2×D53—28no—
310C2×D53—28no—
311C2×D53—28no—
312C2×D53—28no—
313C2×D53—28no—
314C2×D53—28no—
315C2×D53—24no—
316C2×D53#39118yes—
317C4×C53—24no—
318C4×C53—24no—
319C4×C53—42no—
320C4×C53—44no—
321C4×C53—82no—
322Dic53—24no—
323Dic53—24no—
324Dic53—42no—
325Dic53—44no—
326Dic53—82no—
327C23×C4⋊C4—531no—
328C2×#380—531no—
329C2×C4×C22⋊C4—531no—
330#1441—532no—
331#1441—532no—
332#1436—531no—
333#1436—531no—
334#1436—531no—
335#1436D5311yescharacteristic, complemented
336C22⋊#95—532no—
337C22⋊#95—532no—
338C2×C53#228011yescharacteristic
339C2×C53C2×#39111yescharacteristic
340C2×C53#230811yescharacteristic
341C2×C53#234512yes—
342C2×C53#234512yes—
343C2×C53#257912yes—
344C2×C53#257912yes—
345C2×C53#258312yes—
346C2×C53#259112yes—
347D53—28no—
348C24×C4—531no—
349C24×C4—531no—
350C24×C4—531no—
351C22×C4⋊C4—531no—
352C22×C4⋊C4—531no—
353C22×C4⋊C4—531no—
354C22×C4⋊C4—538no—
355C22×C4⋊C4—532no—
356C22×C4⋊C4—1064no—
357C22×C4⋊C4—1064no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C22×C22⋊C4—532no—
360C22×C22⋊C4—532no—
361C22×C42—531no—
362C22×C42C2×D5311yescharacteristic
363#380—1064no—
364C4×C22⋊C4—1064no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95—532no—
368C2×#95—531no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95C2×D5312yes—
374C2×#95C2×D5311yescharacteristic
375C2×#95C2×D5312yes—
376C53#1392311yescharacteristic, complemented, Sylow
377C25—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—534no—
381C23×C4—534no—
382C23×C4—534no—
383C23×C4—532no—
384C23×C4—531no—
385C23×C4—534no—
386C23×C4—534no—
387C23×C4—532no—
388C23×C4—1064no—
389C23×C4—1062no—
390C23×C4—1062no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4—1064no—
394C23×C4—2122no—
395C23×C4—2122no—
396C23×C4—2122no—
397C23×C4—2122no—
398C23×C4C22×D5312yes—
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5312yes—
401C23×C4C22×D5311yescharacteristic
402C23×C4C22×D5311yescharacteristic
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—534no—
408C2×C4⋊C4—10632no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—2122no—
417C2×C4⋊C4—2122no—
418C2×C4⋊C4C4×D5314yes—
419C2×C22⋊C4—538no—
420C2×C22⋊C4—1064no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1064no—
425C2×C42—22no—
426C2×C42—42no—
427C2×C42—1062no—
428C2×C42—2122no—
429#95—24no—
430#95—22no—
431#95—42no—
432#95—1064no—
433#95—1064no—
434#95—1064no—
435#95—1064no—
436#95—1062no—
437#95—2122no—
438#95—2122no—
439#95—2122no—
440C24—532no—
441C24—532no—
442C24—532no—
443C24—534no—
444C24—534no—
445C24—1062no—
446C24—1062no—
447C24—1062no—
448C24—1062no—
449C24C23×D5311yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—532no—
473C22×C4—532no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—534no—
477C22×C4—5316no—
478C22×C4—532no—
479C22×C4—534no—
480C22×C4—1068no—
481C22×C4—1068no—
482C22×C4—10616no—
483C22×C4—10616no—
484C22×C4—1068no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1068no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—2128no—
509C22×C4—2128no—
510C22×C4—2124no—
511C22×C4—2124no—
512C22×C4—2128no—
513C22×C4—2128no—
514C22×C4—2124no—
515C22×C4—2124no—
516C22×C4C2×C4×D5312yes—
517C22×C4C2×C4×D5312yes—
518C22×C4C2×C4×D5312yes—
519C22×C4#1171994514yes—
520C22×C4D4×D5312yes—
521C22×C4D4×D5312yes—
522C22×C4Q8×D5314yes—
523C4⋊C4—28no—
524C4⋊C4—1068no—
525C4⋊C4—1068no—
526C4⋊C4—1068no—
527C4⋊C4—2128no—
528C22⋊C4—10616no—
529C22⋊C4—2128no—
530C42—44no—
531C42—2124no—
532C23—534no—
533C23—538no—
534C23—538no—
535C23—534no—
536C23—534no—
537C23—1064no—
538C23—1064no—
539C23—1064no—
540C23—1064no—
541C23—1064no—
542C23—1064no—
543C23—1064no—
544C23—1062no—
545C23—1062no—
546C23—1064no—
547C23—1064no—
548C23—1064no—
549C23—1064no—
550C23—1064no—
551C23—1062no—
552C23—1062no—
553C23C22×C4×D5311yescharacteristic
554C23C2×D4×D5311yescharacteristic
555C23C2×D4×D5311yescharacteristic
556C23C2×Q8×D5312yes—
557C23#41×D5312yes—
558C23#41×D5312yes—
559C23C2×#1171994512yes—
560C23#1313326212yes—
561C23#1313326312yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—534no—
590C2×C4—534no—
591C2×C4—10632no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1064no—
597C2×C4—1064no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—21216no—
606C2×C4—21216no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2122no—
614C2×C4—2122no—
615C2×C4—2122no—
616C2×C4—2122no—
617C2×C4—2122no—
618C2×C4—2121no—
619C2×C4—2122no—
620C2×C4—2121no—
621C2×C4C22⋊C4×D5314yes—
622C2×C4#1313317414yes—
623C22—538no—
624C22—1068no—
625C22—1068no—
626C22—1064no—
627C22—1068no—
628C22—1068no—
629C22—1064no—
630C22—1068no—
631C22—1068no—
632C22C2×C22⋊C4×D5311yescharacteristic
633C22#117×D5311yescharacteristic
634C22C2×#1313317411yescharacteristic
635C22C4×Q8×D5312yes—
636C22#1681109512yes—
637C22C22⋊Q8×D5312yes—
638C22#1681113212yes—
639C22#1681113212yes—
640C22#124×D5311yescharacteristic
641C22C4⋊Q8×D5311yescharacteristic
642C22#1681118112yes—
643C22#1681118311yescharacteristic
644C22#1681118411yescharacteristic
645C22#1681131712yes—
646C22#1681131812yes—
647C22#1681132312yes—
648C22#1681132412yes—
649C22#1681132712yes—
650C22#1681132812yes—
651C22#1681132912yes—
652C22#1681133312yes—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—44no—
657C4—82no—
658C4—1064no—
659C4—1064no—
660C4—4242no—
661C4—2124no—
662C4—2122no—
663C2—1068no—
664C2#2027709611yescharacteristic
665C2#391×D5311yescharacteristic
666C2#2027728911yescharacteristic
667C2#2027758812yes—
668C2#2027758912yes—
669C2#2028006012yes—
670C2#2028006612yes—
671C2#2028009312yes—
672C2#2028019312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1436D53
hypocenterC53#13923
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
4. #20272967C2semidirect
5. #20272966C2semidirect
6. #20272950C2semidirect
7. #20272948C2semidirect
8. #20272947C2semidirect
11. C53×#1436C2semidirect
21. C2×C4⋊C4×D53C4semidirect
335. #1436D53semidirect
376. C53#13923semidirect

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×#168104822
C2×C4×#117199212
#202729672
#202729662
#202729502
#202729482
#202729472
#202729362
C53×#14362
#1392353