Finite Group Databasebeta

← #27315383

Subgroups — #27315383

Order 13568.

2734 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133195C211yesmaximal, characteristic
2C22×C42×D53C211yesmaximal, characteristic
3C2×#16810527C212yesmaximal, complemented
4C2×#16810482C211yesmaximal, characteristic
5#20272950C211yesmaximal, characteristic, complemented
6#20272948C211yesmaximal, characteristic, complemented
7#20272947C211yesmaximal, characteristic, complemented
8#20272943C212yesmaximal, complemented
9#20272936C212yesmaximal
10#20272936C212yesmaximal
11C53×#1436C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2212yes—
14C23×C4×D53C2211yescharacteristic
15C2×#13133195—24no—
16C2×#13133195—24no—
17C2×#13133195C418yescomplemented
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×#11719921C2212yes—
22C22×#11719921C2211yescharacteristic
23C22×#11719921C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25C22×C4×Dic53C2212yes—
26C2×C42×D53—24no—
27C2×C42×D53—28no—
28#16810527—28no—
29#16810482—24no—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C2×#13133124C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C2×#95×C53C2211yescharacteristic
42C24×D53C2311yescharacteristic
43C22×Dic106—24no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106—24no—
47C22×Dic106C2×C414yescomplemented
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2312yes—
52C23×Dic53C2311yescharacteristic
53C22×C4×D53—24no—
54C22×C4×D53—24no—
55C22×C4×D53—24no—
56C22×C4×D53—24no—
57C22×C4×D53—24no—
58C22×C4×D53—24no—
59C22×C4×D53—24no—
60C22×C4×D53—24no—
61C22×C4×D53C2×C414yes—
62C22×C4×D53D414yes—
63C22×C4×D53D414yes—
64C22×C4×D53Q818yescomplemented
65#13133195—232no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×#11719919—22no—
70C2×#11719919—24no—
71C2×#11719919—24no—
72C2×#11719919—24no—
73C2×#11719919C2×C414yescomplemented
74C2×#11719919C2×C414yescomplemented
75C2×#11719921—22no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C2×#11719921—22no—
79C2×#11719921—24no—
80C2×#11719921—24no—
81C2×#11719921—24no—
82C2×#11719921C2×C414yes—
83C2×#11719921C2×C414yes—
84C2×C4×Dic53—22no—
85C2×C4×Dic53—24no—
86C2×C4×Dic53—24no—
87C2×C4×Dic53—24no—
88C2×C4×Dic53D418yes—
89C42×D53—232no—
90#13133124—24no—
91#13133124—22no—
92#13133124—24no—
93#13133124—24no—
94#13133124—24no—
95#13133124—24no—
96#13133124—24no—
97C23×C4×C53C2312yes—
98C23×C4×C53C2311yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101C23×C4×C53C2312yes—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2×C414yescomplemented
105C2×C42×C53—22no—
106C2×C42×C53—24no—
107#95×C53—24no—
108#95×C53—22no—
109#95×C53—24no—
110C24×C53C2411yescharacteristic
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—22no—
115C23×D53C22×C412yes—
116C23×D53C2×D412yes—
117C23×D53C2×D412yes—
118C23×D53C2×Q814yes—
119C23×D53#4114yes—
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×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53C22×C412yes—
142C22×C4×C53C22×C412yes—
143C22×C4×C53C22×C412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×Q814yes—
148C22×C4×C53#4114yes—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C22×C412yes—
171C22×Dic53C2×D414yes—
172C22×Dic53C2×D412yes—
173C22×Dic53C2×D412yes—
174C22×Dic53C2×D414yes—
175C22×Dic53C2×Q814yes—
176C22×Dic53#4114yes—
177C22×Dic53#4114yes—
178C2×C4×D53—216no—
179C2×C4×D53—216no—
180C2×C4×D53—216no—
181C2×C4×D53—216no—
182C2×C4×D53—216no—
183C2×C4×D53—216no—
184C2×C4×D53—216no—
185C2×C4×D53—28no—
186C2×C4×D53—216no—
187C2×C4×D53—216no—
188C2×C4×D53—28no—
189C2×C4×D53C22⋊C4116yes—
190C2×Dic106—28no—
191C2×Dic106—216no—
192C2×Dic106—44no—
193C2×Dic106—48no—
194C2×Dic106—44no—
195C4⋊C4×C53—28no—
196C42×C53—28no—
197#11719921—28no—
198#11719921—28no—
199#11719921—48no—
200#11719919—28no—
201#11719919—28no—
202#11719919—48no—
203C4×Dic53—28no—
204C4×Dic53—216no—
205C23×C53C23×C411yescharacteristic
206C23×C53C22×D411yescharacteristic
207C23×C53C22×D411yescharacteristic
208C23×C53C22×D412yes—
209C23×C53C22×Q812yes—
210C23×C53C2×#4112yes—
211C23×C53C2×#4112yes—
212C23×C53C2×#4112yes—
213C23×C53C2×#4112yes—
214C22×D53—24no—
215C22×D53—22no—
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—22no—
222C22×D53—24no—
223C22×D53—24no—
224C22×D53—24no—
225C22×D53—24no—
226C22×D53—24no—
227C22×D53—24no—
228C22×D53—24no—
229C22×D53—24no—
230C22×D53C2×C22⋊C414yes—
231C22×D53C4×Q818yes—
232C22×D53C22⋊Q818yes—
233C22×D53#12414yes—
234C22×D53C4⋊Q814yes—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—24no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53—41no—
253C2×C4×C53—42no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—42no—
259C2×C4×C53—42no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53C2×C22⋊C414yes—
263C2×C4×C53C22⋊Q818yescomplemented
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—24no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—42no—
280C2×Dic53—42no—
281C2×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287C2×Dic53C2×C22⋊C414yes—
288C2×Dic53C4×D418yes—
289C2×Dic53#12014yes—
290C2×Dic53#12118yes—
291C2×Dic53C22⋊Q818yescomplemented
292C2×Dic53#12314yes—
293C4×D53—264no—
294C4×D53—232no—
295Dic106—416no—
296#14702—531nomaximal, Sylow
297C22×C53C22×C22⋊C411yescharacteristic
298C22×C53C2×C4×D412yes—
299C22×C53C2×C4×Q812yes—
300C22×C53C4×#4111yescharacteristic
301C22×C53C4×#4111yescharacteristic
302C22×C53C2×#12011yescharacteristic
303C22×C53C2×#12112yes—
304C22×C53C2×C22⋊Q812yes—
305C22×C53C2×C22⋊Q812yes—
306C22×C53C2×C22⋊Q812yes—
307C22×C53C2×#12311yescharacteristic
308C22×C53#52512yes—
309C22×C53#52512yes—
310C22×C53#52512yes—
311C22×C53C2×#12411yescharacteristic
312C22×C53#52912yes—
313C22×C53#52912yes—
314C22×C53C2×C4⋊Q811yescharacteristic
315C22×C53#53212yes—
316C22×C53#53312yes—
317C22×C53#53312yes—
318C2×D53—28no—
319C2×D53—28no—
320C2×D53—28no—
321C2×D53—28no—
322C2×D53—24no—
323C2×D53—24no—
324C2×D53—28no—
325C2×D53—28no—
326C2×D53#39118yescomplemented
327C4×C53—28no—
328C4×C53—24no—
329C4×C53—42no—
330C4×C53—42no—
331C4×C53—82no—
332Dic53—28no—
333Dic53—42no—
334Dic53—42no—
335Dic53—82no—
336Dic53#38618yes—
337C22×C22⋊Q8—531no—
338C23×C42—531no—
339C2×#391—532no—
340C2×#380—531no—
341#1436—531no—
342#1436—531no—
343#1436—531no—
344#1436D5311yescharacteristic, complemented
345C4⋊#95—532no—
346C22⋊#95—532no—
347C22⋊#95—532no—
348C2×C53C2×#38611yescharacteristic
349C2×C53C2×#39111yescharacteristic
350C2×C53#229711yescharacteristic
351C2×C53C4×C22⋊Q812yes—
352C2×C53C4×C22⋊Q812yes—
353C2×C53#284812yes—
354C2×C53#285012yes—
355C2×C53#285512yes—
356C2×C53#285712yes—
357D53—28no—
358C23×Q8—531no—
359C24×C4—531no—
360C24×C4—532no—
361C2×C22⋊Q8—538no—
362C2×C22⋊Q8—1064no—
363C2×C22⋊Q8—1064no—
364C22×C4⋊C4—531no—
365C22×C4⋊C4—531no—
366C22×C4⋊C4—532no—
367C22×C4⋊C4C2×D5311yescharacteristic
368C22×C22⋊C4—531no—
369C22×C22⋊C4—531no—
370C22×C22⋊C4—532no—
371C22×C42—532no—
372C22×C42—531no—
373C22×C42—1068no—
374C22×C42—1064no—
375C22×C42C2×D5311yescharacteristic
376#391—1068no—
377#380—1064no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—532no—
382C2×#95—532no—
383C2×#95—532no—
384C2×#95—531no—
385C2×#95C2×D5312yes—
386C2×#95C2×D5312yes—
387C2×#95C2×D5311yescharacteristic
388C53#1470211yescharacteristic, complemented, Sylow
389C25—531no—
390C22×Q8—534no—
391C22×Q8—1062no—
392C22×Q8—1062no—
393C22×Q8—1064no—
394C22×Q8—1064no—
395C23×C4—531no—
396C23×C4—532no—
397C23×C4—534no—
398C23×C4—534no—
399C23×C4—534no—
400C23×C4—531no—
401C23×C4—531no—
402C23×C4—538no—
403C23×C4—532no—
404C23×C4—1064no—
405C23×C4—1064no—
406C23×C4—1064no—
407C23×C4—1064no—
408C23×C4—1064no—
409C23×C4—1064no—
410C23×C4—1064no—
411C23×C4—1064no—
412C23×C4C22×D5312yes—
413C23×C4C22×D5311yescharacteristic
414C23×C4C22×D5311yescharacteristic
415C23×C4C22×D5311yescharacteristic
416C23×C4C22×D5312yes—
417C22⋊Q8—10632no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—534no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—1062no—
423C2×C4⋊C4—1062no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1064no—
427C2×C4⋊C4—1064no—
428C2×C4⋊C4—1064no—
429C2×C4⋊C4C4×D5314yescomplemented
430C2×C22⋊C4—534no—
431C2×C22⋊C4—534no—
432C2×C22⋊C4—1062no—
433C2×C22⋊C4—1062no—
434C2×C22⋊C4—1062no—
435C2×C22⋊C4—1062no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—1064no—
438C2×C22⋊C4—1064no—
439C2×C42—22no—
440C2×C42—24no—
441C2×C42—538no—
442C2×C42—10632no—
443C2×C42—1064no—
444C2×C42—1064no—
445C2×C42—1064no—
446C2×C42—1062no—
447#95—24no—
448#95—22no—
449#95—24no—
450#95—1064no—
451#95—1064no—
452#95—1064no—
453#95—1064no—
454#95—1064no—
455#95—1062no—
456#95—1064no—
457C24—532no—
458C24—532no—
459C24—532no—
460C24—534no—
461C24—534no—
462C24—1062no—
463C24—1062no—
464C24—1062no—
465C24—1062no—
466C24C23×D5311yescharacteristic
467C2×Q8—1068no—
468C2×Q8—10616no—
469C2×Q8—2124no—
470C2×Q8—2128no—
471C2×Q8—2124no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—532no—
494C22×C4—534no—
495C22×C4—532no—
496C22×C4—532no—
497C22×C4—534no—
498C22×C4—534no—
499C22×C4—534no—
500C22×C4—534no—
501C22×C4—5316no—
502C22×C4—532no—
503C22×C4—532no—
504C22×C4—1068no—
505C22×C4—10616no—
506C22×C4—10616no—
507C22×C4—10616no—
508C22×C4—1068no—
509C22×C4—10616no—
510C22×C4—10616no—
511C22×C4—10616no—
512C22×C4—10616no—
513C22×C4—10616no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—10616no—
519C22×C4—1062no—
520C22×C4—1061no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4—1061no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—1062no—
532C22×C4—1062no—
533C22×C4—1062no—
534C22×C4C2×C4×D5312yes—
535C22×C4C2×C4×D5312yes—
536C22×C4C2×C4×D5312yes—
537C22×C4#1171994514yes—
538C22×C4D4×D5312yes—
539C22×C4D4×D5312yes—
540C22×C4D4×D5314yes—
541C22×C4Q8×D5314yes—
542C4⋊C4—28no—
543C4⋊C4—1068no—
544C4⋊C4—1068no—
545C4⋊C4—2128no—
546C22⋊C4—1068no—
547C22⋊C4—1068no—
548C22⋊C4—2128no—
549C42—28no—
550C42—10616no—
551C42—1068no—
552C23—534no—
553C23—538no—
554C23—538no—
555C23—534no—
556C23—534no—
557C23—1064no—
558C23—1062no—
559C23—1062no—
560C23—1062no—
561C23—1064no—
562C23—1064no—
563C23—1064no—
564C23—1062no—
565C23—1064no—
566C23—1064no—
567C23—1064no—
568C23—1064no—
569C23—1064no—
570C23—1064no—
571C23—1064no—
572C23—1064no—
573C23C22×C4×D5311yescharacteristic
574C23C2×D4×D5311yescharacteristic
575C23C2×D4×D5311yescharacteristic
576C23C2×D4×D5312yes—
577C23C2×Q8×D5312yes—
578C23#41×D5312yes—
579C23#41×D5312yes—
580C23#41×D5312yes—
581C23C2×#1171994512yes—
582Q8—21216no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—534no—
611C2×C4—538no—
612C2×C4—534no—
613C2×C4—538no—
614C2×C4—534no—
615C2×C4—538no—
616C2×C4—10632no—
617C2×C4—10664no—
618C2×C4—1064no—
619C2×C4—1064no—
620C2×C4—1064no—
621C2×C4—1064no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—1062no—
625C2×C4—1062no—
626C2×C4—1064no—
627C2×C4—1062no—
628C2×C4—1062no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2121no—
637C2×C4—2122no—
638C2×C4—2121no—
639C2×C4—2122no—
640C2×C4—2122no—
641C2×C4C22⋊C4×D5314yes—
642C2×C4#1313319518yescomplemented
643C22—538no—
644C22—1068no—
645C22—1068no—
646C22—1068no—
647C22—1068no—
648C22—1064no—
649C22—1068no—
650C22—1068no—
651C22—1064no—
652C22C2×C22⋊C4×D5311yescharacteristic
653C22C4×D4×D5312yes—
654C22C4×Q8×D5312yes—
655C22#1681109111yescharacteristic
656C22C4×#1171994511yescharacteristic
657C22#120×D5311yescharacteristic
658C22#121×D5312yes—
659C22C22⋊Q8×D5312yes—
660C22C22⋊Q8×D5312yes—
661C22C2×#1313319512yes—
662C22#123×D5311yescharacteristic
663C22#1681113112yes—
664C22#1681113212yes—
665C22#1681113212yes—
666C22#124×D5311yescharacteristic
667C22#1681115112yes—
668C22#1681115612yes—
669C22C4⋊Q8×D5311yescharacteristic
670C22#1681117012yes—
671C22#1681117712yes—
672C22#1681117912yes—
673C4—28no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—82no—
678C4—538no—
679C4—1068no—
680C4—4242no—
681C4—2122no—
682C4—2122no—
683C2—1068no—
684C2#386×D5311yescharacteristic
685C2#391×D5311yescharacteristic
686C2#2027720111yescharacteristic
687C2C4×#1313319512yes—
688C2#2027724912yes—
689C2#2028300112yes—
690C2#2028301312yes—
691C2#2028305012yes—
692C2#2028306612yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1436C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1436D53
hypocenterC53#14702
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. C2×#16810527C2semidirect
5. #20272950C2semidirect
6. #20272948C2semidirect
7. #20272947C2semidirect
8. #20272943C2semidirect
11. C53×#1436C2semidirect
17. C2×#13133195C4semidirect
47. C22×Dic106C2×C4semidirect
64. C22×C4×D53Q8semidirect
73. C2×#11719919C2×C4semidirect
74. C2×#11719919C2×C4semidirect
104. C2×C4⋊C4×C53C2×C4semidirect
263. C2×C4×C53C22⋊Q8semidirect
291. C2×Dic53C22⋊Q8semidirect
326. C2×D53#391semidirect
344. #1436D53semidirect
388. C53#14702semidirect
429. C2×C4⋊C4C4×D53semidirect
642. C2×C4#13133195semidirect

Maximal subgroups

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

subgroupindex
C22×#131331952
C22×C42×D532
C2×#168105272
C2×#168104822
#202729502
#202729482
#202729472
#202729432
#202729362
C53×#14362
#1470253