Finite Group Databasebeta

← #27310466

Subgroups — #27310466

Order 13568.

2142 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×#13133194C211yesmaximal, characteristic
2C22×#13133173C211yesmaximal, characteristic
3C2×#16810527C212yesmaximal, complemented
4C2×#16810481C211yesmaximal, characteristic
5#20272967C212yesmaximal, complemented
6#20272966C212yesmaximal, complemented
7#20272964C211yesmaximal, characteristic
8#20272962C211yesmaximal, characteristic
9#20272962C211yesmaximal, characteristic
10#20272943C212yesmaximal, complemented
11C53×#1440C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C24×Dic53C2211yescharacteristic
14C2×#13133194—24no—
15C2×#13133194—24no—
16C2×#13133194C418yescomplemented
17C2×#13133173—24no—
18C2×#13133173—28no—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×#11719919C2212yes—
22C22×#11719919C2211yescharacteristic
23C22×#11719919C2211yescharacteristic
24C22×C22⋊Dic53C2211yescharacteristic
25C22×C22⋊Dic53C2211yescharacteristic
26C22×C22⋊Dic53C2211yescharacteristic
27C22×C4×Dic53C2212yes—
28#16810527—28no—
29#16810481—24no—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C2×#13133124C2212yes—
37C22×C22⋊C4×C53C2211yescharacteristic
38C22×C22⋊C4×C53C2211yescharacteristic
39C22×C22⋊C4×C53C2211yescharacteristic
40C2×#95×C53C2212yes—
41C2×#95×C53C2212yes—
42C22×Dic106—24no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106—24no—
46C22×Dic106C2×C414yes—
47C23×Dic53—24no—
48C23×Dic53—24no—
49C23×Dic53C2×C414yes—
50C23×Dic53D414yes—
51C23×Dic53D414yes—
52C23×Dic53C2312yes—
53C23×Dic53C2311yescharacteristic
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2312yes—
56C23×Dic53C2311yescharacteristic
57C23×Dic53C2311yescharacteristic
58#13133194—232no—
59#13133173—232no—
60C2×C4⋊Dic53—22no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53C2×C414yes—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919—22no—
69C2×#11719919—22no—
70C2×#11719919—24no—
71C2×#11719919—24no—
72C2×#11719919—24no—
73C2×#11719919C2×C414yes—
74C2×#11719919C2×C414yes—
75C2×C22⋊Dic53—22no—
76C2×C22⋊Dic53—22no—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—22no—
79C2×C22⋊Dic53—24no—
80C2×C22⋊Dic53—42no—
81C2×C22⋊Dic53C2×C414yes—
82C2×C4×Dic53—24no—
83C2×C4×Dic53—24no—
84C2×C4×Dic53D418yes—
85#13133124—24no—
86#13133124—24no—
87#13133124—24no—
88#13133124—24no—
89#13133124—42no—
90#13133124—42no—
91#13133124—42no—
92C25×C53C2311yescharacteristic
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C23×C4×C53C2312yes—
97C2×C22⋊C4×C53—22no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53—24no—
102C2×C22⋊C4×C53—42no—
103C2×C22⋊C4×C53C2×C414yes—
104#95×C53—24no—
105#95×C53—42no—
106C24×C53—22no—
107C24×C53—22no—
108C24×C53—22no—
109C24×C53—22no—
110C24×C53—22no—
111C24×C53—22no—
112C24×C53C22×C412yes—
113C24×C53C2×D412yes—
114C24×C53C2×D412yes—
115C24×C53C2411yescharacteristic
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
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×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C22×C412yes—
139C22×C4×C53C2×D414yes—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—216no—
143C22×Dic53—28no—
144C22×Dic53—216no—
145C22×Dic53—28no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
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×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53C22⋊C4116yes—
168C22×Dic53C22×C412yes—
169C22×Dic53C22×C412yes—
170C22×Dic53C22×C412yes—
171C22×Dic53C22×C412yes—
172C22×Dic53C2×D414yes—
173C22×Dic53C2×D412yes—
174C22×Dic53C2×D412yes—
175C22×Dic53C2×D412yes—
176C22×Dic53C2×D412yes—
177C22×Dic53C2×D414yes—
178C22×Dic53#4114yes—
179C2×Dic106—28no—
180C2×Dic106—216no—
181C2×Dic106—44no—
182C2×Dic106—48no—
183C2×Dic106—44no—
184C22⋊C4×C53—28no—
185C22⋊C4×C53—28no—
186C22⋊C4×C53—44no—
187C22⋊Dic53—28no—
188C22⋊Dic53—28no—
189C22⋊Dic53—44no—
190C4⋊Dic53—28no—
191#11719919—216no—
192#11719919—28no—
193#11719919—28no—
194#11719919—48no—
195C4×Dic53—216no—
196C23×C53—24no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—24no—
200C23×C53—22no—
201C23×C53—24no—
202C23×C53—22no—
203C23×C53—22no—
204C23×C53—24no—
205C23×C53—24no—
206C23×C53—22no—
207C23×C53—24no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53—42no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53C2×C22⋊C414yes—
217C23×C53C23×C411yescharacteristic
218C23×C53C22×D411yescharacteristic
219C23×C53C22×D411yescharacteristic
220C23×C53C22×D412yes—
221C23×C53C2×#4112yes—
222C23×C53H(5,𝔽2)12yes—
223C23×C532−1+412yes—
224C23×C532−1+412yes—
225C23×C532−1+412yes—
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—41no—
234C2×C4×C53—42no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—42no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
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×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—24no—
253C2×Dic53—22no—
254C2×Dic53—232no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—41no—
263C2×Dic53—42no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53—42no—
270C2×Dic53—42no—
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×Dic53C2×C22⋊C414yes—
280C2×Dic53C2×C22⋊C414yes—
281C2×Dic53C4×D418yes—
282C2×Dic53#12014yes—
283C2×Dic53#12118yes—
284C2×Dic53#12314yes—
285Dic106—416no—
286#14258—531nomaximal, Sylow
287C22×C53—24no—
288C22×C53—24no—
289C22×C53—24no—
290C22×C53—44no—
291C22×C53—44no—
292C22×C53—44no—
293C22×C53—44no—
294C22×C53—44no—
295C22×C53—44no—
296C22×C53C22×C22⋊C411yescharacteristic
297C22×C53C2×C4×D412yes—
298C22×C53#51911yescharacteristic
299C22×C53#51911yescharacteristic
300C22×C53#52012yes—
301C22×C53C2×#12011yescharacteristic
302C22×C53C2×#12112yes—
303C22×C53C2×#12311yescharacteristic
304C22×C53#53511yescharacteristic
305C22×C53#53612yes—
306C22×C53#53612yes—
307C22×C53#53612yes—
308C22×C53#53611yescharacteristic
309C22×C53#53612yes—
310C22×C53#53712yes—
311C22×C53#53712yes—
312C22×C53#53712yes—
313C22×C53#54112yes—
314C22×C53#54112yes—
315C22×C53#54212yes—
316C22×C53#54212yes—
317C4×C53—42no—
318C4×C53—44no—
319C4×C53—42no—
320C4×C53—82no—
321Dic53—24no—
322Dic53—42no—
323Dic53—44no—
324Dic53—42no—
325Dic53—82no—
326Dic53#38618yes—
327C22×C22⋊Q8—531no—
328C22×#117—531no—
329C2×#391—532no—
330C2×#380—531no—
331#1441—532no—
332#1441—532no—
333#1440—531no—
334#1440—531no—
335#1440—531no—
336#1440D5311yescharacteristic
337C4⋊#95—532no—
338C2×C53—44no—
339C2×C53C2×#38611yescharacteristic
340C2×C53#231011yescharacteristic
341C2×C53#231111yescharacteristic
342C2×C53#232312yes—
343C2×C53#232312yes—
344C2×C53#261612yes—
345C2×C53#262012yes—
346C2×C53#262112yes—
347C2×C53#262512yes—
348C23×Q8—531no—
349C24×C4—531no—
350C2×C22⋊Q8—538no—
351C2×C22⋊Q8—1064no—
352C2×C22⋊Q8—1064no—
353C2×#117—1068no—
354C2×#117—1064no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—531no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—532no—
359C22×C4⋊C4—532no—
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4—531no—
363C22×C22⋊C4C2×D5311yescharacteristic
364C22×C22⋊C4C2×D5311yescharacteristic
365C22×C22⋊C4C2×D5311yescharacteristic
366C22×C42—532no—
367#391—1068no—
368#380—1064no—
369C2×#95—532no—
370C2×#95—532no—
371C2×#95—532no—
372C2×#95—532no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—532no—
376C2×#95C2×D5312yes—
377C2×#95C2×D5312yes—
378C53#1425811yescharacteristic, complemented, Sylow
379C25C22×D5311yescharacteristic
380C22×Q8—534no—
381C22×Q8—1062no—
382C22×Q8—1062no—
383C22×Q8—1064no—
384C22×Q8—1064no—
385C23×C4—531no—
386C23×C4—534no—
387C23×C4—534no—
388C23×C4—534no—
389C23×C4—531no—
390C23×C4—531no—
391C23×C4—531no—
392C23×C4—532no—
393C23×C4—532no—
394C23×C4—1064no—
395C23×C4—1064no—
396C23×C4C22×D5312yes—
397C23×C4C22×D5311yescharacteristic
398C23×C4C22×D5311yescharacteristic
399C23×C4C22×D5312yes—
400C22⋊Q8—10632no—
401#117—10632no—
402C2×C4⋊C4—534no—
403C2×C4⋊C4—534no—
404C2×C4⋊C4—534no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1064no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—42no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1062no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—2122no—
430C2×C22⋊C4C4×D5314yes—
431C2×C42—538no—
432C2×C42—1064no—
433C2×C42—1064no—
434#95—24no—
435#95—42no—
436#95—1064no—
437#95—1064no—
438#95—1064no—
439#95—1064no—
440#95—2122no—
441#95—2122no—
442#95—2122no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24C2×C4×D5312yes—
450C24D4×D5312yes—
451C24D4×D5312yes—
452C24C23×D5311yescharacteristic
453C2×Q8—1068no—
454C2×Q8—10616no—
455C2×Q8—2124no—
456C2×Q8—2128no—
457C2×Q8—2124no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—532no—
480C22×C4—534no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—534no—
484C22×C4—5316no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—534no—
491C22×C4—1068no—
492C22×C4—10616no—
493C22×C4—10616no—
494C22×C4—1068no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1062no—
505C22×C4—1062no—
506C22×C4—1062no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1062no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4C2×C4×D5312yes—
519C22×C4C2×C4×D5312yes—
520C22×C4D4×D5314yes—
521C4⋊C4—1068no—
522C4⋊C4—1068no—
523C4⋊C4—1068no—
524C4⋊C4—10616no—
525C4⋊C4—2128no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—1068no—
530C22⋊C4—1068no—
531C22⋊C4—2124no—
532C42—10616no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C22⋊C4×D5314yes—
554C23C22×C4×D5311yescharacteristic
555C23C2×D4×D5311yescharacteristic
556C23C2×D4×D5311yescharacteristic
557C23C2×D4×D5312yes—
558C23#41×D5312yes—
559C23#1313326212yes—
560C23#1313326412yes—
561C23#1313326412yes—
562C23#1313326412yes—
563Q8—21216no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—534no—
589C2×C4—538no—
590C2×C4—534no—
591C2×C4—538no—
592C2×C4—534no—
593C2×C4—534no—
594C2×C4—10632no—
595C2×C4—1064no—
596C2×C4—1062no—
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—1064no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2122no—
609C2×C4—2122no—
610C2×C4—2122no—
611C2×C4—2122no—
612C2×C4—2122no—
613C2×C4—2121no—
614C2×C4—2122no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2122no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22C2×C22⋊C4×D5311yescharacteristic
633C22C4×D4×D5312yes—
634C22#1681110011yescharacteristic
635C22#1681110011yescharacteristic
636C22#1681110212yes—
637C22#120×D5311yescharacteristic
638C22#121×D5312yes—
639C22#123×D5311yescharacteristic
640C22#1681118811yescharacteristic
641C22#1681119312yes—
642C22#1681119512yes—
643C22#1681119512yes—
644C22#1681119511yescharacteristic
645C22#1681119512yes—
646C22#1681120012yes—
647C22#1681120012yes—
648C22#1681120212yes—
649C22#1681123012yes—
650C22#1681123012yes—
651C22#1681123712yes—
652C22#1681124112yes—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—82no—
657C4—538no—
658C4—1064no—
659C4—4242no—
660C4—2122no—
661C4—2124no—
662C4—2122no—
663C2—44no—
664C2#386×D5311yescharacteristic
665C2#2027730011yescharacteristic
666C2#2027730211yescharacteristic
667C2#2027739012yes—
668C2#2027739412yes—
669C2#2028044312yes—
670C2#2028047112yes—
671C2#2028048012yes—
672C2#2028051112yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1440C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenter#1440D53
hypocenterC53#14258
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
3. C2×#16810527C2semidirect
5. #20272967C2semidirect
6. #20272966C2semidirect
10. #20272943C2semidirect
16. C2×#13133194C4semidirect
378. C53#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#131331942
C22×#131331732
C2×#168105272
C2×#168104812
#202729672
#202729662
#202729642
#202729622
#202729432
C53×#14402
#1425853