Finite Group Databasebeta

← #27306502

Subgroups — #27306502

Order 13568.

1854 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133182C211yesmaximal, characteristic
2C2×#16810527C212yesmaximal
3C2×#16810492C211yesmaximal, characteristic
4C2×#16810474C211yesmaximal, characteristic
5#20272969C211yesmaximal, characteristic
6#20272968C211yesmaximal, characteristic
7#20272965C211yesmaximal, characteristic
8#20272943C212yesmaximal
9#20272943C212yesmaximal
10#20272943C212yesmaximal
11C53×#1441C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×#13133182—24no—
14C2×#13133182—24no—
15C2×#13133182C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2211yescharacteristic
21C22×#11719919C2212yes—
22C22×C4×Dic53C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25C22×C4×Dic53C2211yescharacteristic
26#16810527—28no—
27#16810492—24no—
28#16810474—24no—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C2×#95×C53C2212yes—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106C2×C414yes—
46C22×Dic106D414yes—
47C22×Dic106Q814yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2312yes—
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53C23×Dic53C2311yescharacteristic
54#13133182—232no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—42no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—24no—
67C2×#11719919—24no—
68C2×#11719919—24no—
69C2×#11719919C2×C414yes—
70C2×#11719919C2×C414yes—
71C2×#11719919Q818yescomplemented
72C2×C4×Dic53—22no—
73C2×C4×Dic53—22no—
74C2×C4×Dic53—22no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—24no—
77C2×C4×Dic53—24no—
78C2×C4×Dic53—24no—
79C2×C4×Dic53C2×C414yes—
80C2×C4×Dic53C2×C414yes—
81C2×C4×Dic53C2×C414yes—
82C2×C4×Dic53D414yes—
83C2×C4×Dic53D414yes—
84#13133124—24no—
85#13133124—22no—
86#13133124—24no—
87#13133124—24no—
88#13133124—42no—
89#13133124—42no—
90#13133124—42no—
91C23×C4×C53C2312yes—
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2312yes—
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99C2×C4⋊C4×C53—42no—
100C2×C4⋊C4×C53C2×C414yes—
101#95×C53—24no—
102#95×C53—22no—
103#95×C53—42no—
104C24×C53C2411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
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—22no—
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—22no—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×Q814yes—
135C22×C4×C53C2×Q812yes—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—21no—
143C22×Dic53—21no—
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×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C22×C412yes—
160C22×Dic53C22×C412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53C2×D412yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×Q814yes—
168C22×Dic53C2×Q812yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53C2×Q814yes—
171C22×Dic53#4112yes—
172C22×Dic53#4112yes—
173C22×Dic53#4114yes—
174C2×Dic106—28no—
175C2×Dic106—28no—
176C2×Dic106—28no—
177C2×Dic106—28no—
178C2×Dic106—28no—
179C2×Dic106—28no—
180C2×Dic106C4⋊C4116yescomplemented
181C4⋊C4×C53—28no—
182C4⋊C4×C53—44no—
183C4⋊Dic53—44no—
184#11719919—28no—
185#11719919—28no—
186#11719919—216no—
187#11719919—48no—
188#11719919—48no—
189C4×Dic53—28no—
190C4×Dic53—28no—
191C4×Dic53—48no—
192C4×Dic53C22⋊C4116yes—
193C23×C53C23×C411yescharacteristic
194C23×C53C22×D411yescharacteristic
195C23×C53C22×D411yescharacteristic
196C23×C53C22×D411yescharacteristic
197C23×C53C22×Q811yescharacteristic
198C23×C53C22×Q812yes—
199C23×C53C2×#4111yescharacteristic
200C23×C53C2×#4111yescharacteristic
201C23×C53C2×#4112yes—
202C23×C53H(5,𝔽2)12yes—
203C23×C532−1+412yes—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53C2×C22⋊C414yes—
234C2×C4×C53C2×C4⋊C414yes—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
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—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53C2×C22⋊C414yes—
261C2×Dic53C2×C22⋊C414yes—
262C2×Dic53C2×C4⋊C414yes—
263C2×Dic53C2×C4⋊C414yes—
264C2×Dic53#11714yes—
265C2×Dic53C4×Q818yes—
266C2×Dic53#12114yes—
267C2×Dic53#12114yes—
268C2×Dic53C22⋊Q814yes—
269C2×Dic53C22⋊Q814yes—
270C2×Dic53C22⋊Q818yes—
271C2×Dic53#12414yes—
272C2×Dic53C4⋊Q814yes—
273Dic106—232no—
274#13926—531nomaximal, Sylow
275C22×C53C22×C22⋊C411yescharacteristic
276C22×C53C22×C4⋊C411yescharacteristic
277C22×C53C2×#11711yescharacteristic
278C22×C53C2×C4×Q812yes—
279C22×C53#52012yes—
280C22×C53C2×#12111yescharacteristic
281C22×C53C2×#12111yescharacteristic
282C22×C53C2×C22⋊Q811yescharacteristic
283C22×C53C2×C22⋊Q811yescharacteristic
284C22×C53C2×C22⋊Q812yes—
285C22×C53C2×#12411yescharacteristic
286C22×C53C2×C4⋊Q811yescharacteristic
287C22×C53#53511yescharacteristic
288C22×C53#53611yescharacteristic
289C22×C53#53712yes—
290C22×C53D4×Q812yes—
291C22×C53#55012yes—
292C22×C53#55412yes—
293C22×C53#55612yes—
294C22×C53#55712yes—
295C22×C53#55712yes—
296C22×C53Q8212yes—
297C22×C53#55912yes—
298C4×C53—24no—
299C4×C53—24no—
300C4×C53—42no—
301C4×C53—44no—
302C4×C53—82no—
303Dic53—24no—
304Dic53—42no—
305Dic53—44no—
306Dic53—82no—
307Dic53#38018yes—
308Dic53#39118yescomplemented
309C22×C4×Q8—531no—
310C2×#391—532no—
311C2×#384—531no—
312C2×C4×C4⋊C4—531no—
313#1441—531no—
314#1441—531no—
315#1441—531no—
316#1441D5311yescharacteristic
317C4⋊#95—532no—
318C4⋊#95—532no—
319C4⋊#95—532no—
320C2×C53C2×#38011yescharacteristic
321C2×C53C2×#39111yescharacteristic
322C2×C53#231111yescharacteristic
323C2×C53Q8×C4⋊C412yes—
324C2×C53#235512yes—
325C2×C53#253812yes—
326C2×C53#254611yescharacteristic
327C2×C53#254811yescharacteristic
328C2×C53#256711yescharacteristic
329C2×C53#256811yescharacteristic
330C2×C53#258212yes—
331C23×Q8—531no—
332C2×C4×Q8—538no—
333C2×C4×Q8—1064no—
334C2×C4×Q8—1064no—
335C22×C4⋊C4—531no—
336C22×C4⋊C4—531no—
337C22×C4⋊C4—532no—
338C22×C4⋊C4—532no—
339C22×C4⋊C4—532no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C42—531no—
344C22×C42—531no—
345C22×C42—531no—
346C22×C42—532no—
347#391—1068no—
348#384—1064no—
349C4×C4⋊C4—1064no—
350C2×#95—532no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—531no—
354C2×#95—532no—
355C2×#95—532no—
356C2×#95—532no—
357C2×#95C2×D5312yes—
358C2×#95C2×D5311yescharacteristic
359C2×#95C2×D5312yes—
360C53#1392611yescharacteristic, complemented, Sylow
361C22×Q8—534no—
362C22×Q8—534no—
363C22×Q8—534no—
364C22×Q8—1062no—
365C22×Q8—1062no—
366C22×Q8—1062no—
367C22×Q8—1062no—
368C23×C4—531no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—532no—
373C23×C4—532no—
374C23×C4C22×D5312yes—
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5312yes—
377C23×C4C22×D5311yescharacteristic
378C23×C4C22×D5311yescharacteristic
379C4×Q8—10632no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—534no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—538no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4—1064no—
396C2×C4⋊C4—1064no—
397C2×C4⋊C4—1064no—
398C2×C4⋊C4—1064no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—2122no—
401C2×C4⋊C4C4×D5314yes—
402C2×C42—534no—
403C2×C42—534no—
404C2×C42—534no—
405C2×C42—534no—
406C2×C42—534no—
407C2×C42—1062no—
408C2×C42—1062no—
409C2×C42—1062no—
410C2×C42—1062no—
411C2×C42—1064no—
412C2×C42—1064no—
413C2×C42—1064no—
414#95—22no—
415#95—24no—
416#95—42no—
417#95—1064no—
418#95—1064no—
419#95—1062no—
420#95—1064no—
421#95—2122no—
422#95—2122no—
423#95—2122no—
424C24C23×D5311yescharacteristic
425C2×Q8—5316no—
426C2×Q8—1068no—
427C2×Q8—1068no—
428C2×Q8—1068no—
429C2×Q8—1068no—
430C2×Q8—1068no—
431C2×Q8—1068no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
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—532no—
456C22×C4—532no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—532no—
462C22×C4—534no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—534no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—534no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4C2×C4×D5312yes—
494C22×C4C2×C4×D5312yes—
495C22×C4C2×C4×D5312yes—
496C22×C4D4×D5312yes—
497C22×C4D4×D5312yes—
498C22×C4D4×D5312yes—
499C22×C4Q8×D5314yes—
500C22×C4Q8×D5312yes—
501C4⋊C4—28no—
502C4⋊C4—44no—
503C4⋊C4—1068no—
504C4⋊C4—10616no—
505C4⋊C4—1068no—
506C4⋊C4—2124no—
507C4⋊C4—2128no—
508C4⋊C4—2128no—
509C42—5316no—
510C42—1068no—
511C42—1068no—
512C42—2128no—
513C23C22×C4×D5311yescharacteristic
514C23C2×D4×D5311yescharacteristic
515C23C2×D4×D5311yescharacteristic
516C23C2×D4×D5311yescharacteristic
517C23C2×Q8×D5312yes—
518C23C2×Q8×D5311yescharacteristic
519C23#41×D5312yes—
520C23#41×D5311yescharacteristic
521C23#41×D5311yescharacteristic
522C23#1313326212yes—
523C23#1313326412yes—
524Q8—10632no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—534no—
555C2×C4—534no—
556C2×C4—534no—
557C2×C4—534no—
558C2×C4—538no—
559C2×C4—534no—
560C2×C4—534no—
561C2×C4—538no—
562C2×C4—534no—
563C2×C4—534no—
564C2×C4—534no—
565C2×C4—534no—
566C2×C4—534no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4C22⋊C4×D5314yes—
593C2×C4C4⋊C4×D5314yes—
594C22C2×C22⋊C4×D5311yescharacteristic
595C22C2×C4⋊C4×D5311yescharacteristic
596C22#117×D5311yescharacteristic
597C22C4×Q8×D5312yes—
598C22#1681110212yes—
599C22#121×D5311yescharacteristic
600C22#121×D5311yescharacteristic
601C22C22⋊Q8×D5311yescharacteristic
602C22C22⋊Q8×D5312yes—
603C22C22⋊Q8×D5311yescharacteristic
604C22#124×D5311yescharacteristic
605C22C4⋊Q8×D5311yescharacteristic
606C22#1681118811yescharacteristic
607C22#1681119511yescharacteristic
608C22#1681120012yes—
609C22#1681128612yes—
610C22#1681129112yes—
611C22#1681132712yes—
612C22#1681134012yes—
613C22#1681135012yes—
614C22#1681135012yes—
615C22#1681135312yes—
616C22#1681135712yes—
617C4—24no—
618C4—24no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—538no—
623C4—538no—
624C4—1064no—
625C4—4242no—
626C4—2122no—
627C4—2124no—
628C2#380×D5311yescharacteristic
629C2#391×D5311yescharacteristic
630C2#2027730211yescharacteristic
631C2#2027762912yes—
632C2#2027766912yes—
633C2#2027963212yes—
634C2#2027969911yescharacteristic
635C2#2027971311yescharacteristic
636C2#2027991911yescharacteristic
637C2#2027992611yescharacteristic
638C2#2028008412yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1441D53
hypocenterC53#13926
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
15. C2×#13133182C4semidirect
71. C2×#11719919Q8semidirect
180. C2×Dic106C4⋊C4semidirect
308. Dic53#391semidirect
360. C53#13926semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168105272
C2×#168104922
C2×#168104742
#202729692
#202729682
#202729652
#202729432
C53×#14412
#1392653