Finite Group Databasebeta

← #27320741

Subgroups — #27320741

Order 13568.

1814 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810560C211yesmaximal, characteristic
2C2×#16810558C211yesmaximal, characteristic
3C2×#16810520C212yesmaximal, complemented
4C2×#16810492C211yesmaximal, characteristic, complemented
5C2×#16810482C212yesmaximal
6C2×#16810470C211yesmaximal, characteristic
7#20272984C211yesmaximal, characteristic, complemented
8#20272979C212yesmaximal, complemented
9#20272959C212yesmaximal
10#20272943C211yesmaximal, characteristic, complemented
11C53×#1445C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25#16810560—24no—
26#16810558—24no—
27#16810520—28no—
28#16810492—24no—
29#16810482—28no—
30#16810470—24no—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C24×D53C2311yescharacteristic
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2312yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C22×C4×D53—22no—
48C22×C4×D53—24no—
49C22×C4×D53—24no—
50C22×C4×D53—22no—
51C22×C4×D53Q814yes—
52C2×C4⋊Dic53—24no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—24no—
57C2×C4⋊Dic53—42no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919—24no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—42no—
67C2×#11719919—42no—
68C2×#11719919—42no—
69C2×#11719919D418yes—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921—22no—
73C2×#11719921—24no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—24no—
77C2×#11719921—22no—
78C2×#11719921—42no—
79C2×#11719921—42no—
80C2×#11719921D414yes—
81C2×#11719921Q814yes—
82C2×C4×Dic53—24no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—24no—
85C2×C4×Dic53D414yes—
86#13133124—22no—
87#13133124—24no—
88#13133124—22no—
89#13133124—24no—
90#13133124—24no—
91C23×C4×C53C2312yes—
92C23×C4×C53C2312yes—
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C23×C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53—22no—
99#95×C53—24no—
100#95×C53—22no—
101#95×C53—22no—
102#95×C53—24no—
103C24×C53C2411yescharacteristic
104C23×D53—21no—
105C23×D53—22no—
106C23×D53—21no—
107C23×D53—21no—
108C23×D53—21no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53C2×D412yes—
113C23×D53C2×Q812yes—
114C23×D53C2×Q812yes—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—21no—
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×C53—22no—
138C22×C4×C53C2×D414yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×Q812yes—
142C22×C4×C53C2×Q812yes—
143C22×C4×C53#4112yes—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—21no—
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×Dic53C2×D414yes—
164C22×Dic53C2×D412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D414yes—
168C22×Dic53C2×Q812yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53#4112yes—
171C22×Dic53#4114yes—
172C22×Dic53#4112yes—
173C2×C4×D53—28no—
174C2×C4×D53—216no—
175C2×C4×D53—44no—
176C2×C4×D53—48no—
177C2×C4×D53—44no—
178C4⋊C4×C53—44no—
179#11719921—28no—
180#11719921—28no—
181#11719921—48no—
182#11719921—48no—
183C4⋊Dic53—48no—
184C4⋊Dic53—44no—
185#11719919—216no—
186#11719919—48no—
187#11719919—48no—
188#11719919—48no—
189C4×Dic53—28no—
190C4×Dic53—44no—
191C23×C53C22×D411yescharacteristic
192C23×C53C22×D411yescharacteristic
193C23×C53C22×D412yes—
194C23×C53C22×Q811yescharacteristic
195C23×C53C22×Q811yescharacteristic
196C23×C53C2×#4111yescharacteristic
197C23×C53C2×#4112yes—
198C23×C53C2×#4111yescharacteristic
199C23×C53C2×#4111yescharacteristic
200C23×C53H(5,𝔽2)12yes—
201C23×C532−1+412yes—
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—24no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—42no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53C4⋊Q814yes—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—42no—
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×C53—42no—
253C2×C4×C53#12114yes—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—41no—
264C2×Dic53—42no—
265C2×Dic53—41no—
266C2×Dic53—42no—
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×Dic53C22⋊Q814yes—
278C2×Dic53C22⋊Q814yes—
279C2×Dic53C22⋊Q818yes—
280C2×Dic53#12314yes—
281C2×Dic53#12318yes—
282C2×Dic53#12714yes—
283C4×D53—416no—
284#15135—531nomaximal, Sylow
285C22×C53C2×#12111yescharacteristic
286C22×C53C2×C22⋊Q811yescharacteristic
287C22×C53C2×C22⋊Q811yescharacteristic
288C22×C53C2×C22⋊Q812yes—
289C22×C53C2×#12312yes—
290C22×C53C2×#12311yescharacteristic
291C22×C53C2×#12711yescharacteristic
292C22×C53C2×C4⋊Q811yescharacteristic
293C22×C53#53211yescharacteristic
294C22×C53#53311yescharacteristic
295C22×C53#53311yescharacteristic
296C22×C53#53511yescharacteristic
297C22×C53#53611yescharacteristic
298C22×C53#54212yes—
299C22×C53#54412yes—
300C22×C53D4212yes—
301C22×C53#54712yes—
302C22×C53#54712yes—
303C22×C53D4×Q812yes—
304C22×C53#55012yes—
305C22×C53#55212yes—
306C22×C53#55312yes—
307C22×C53#55412yes—
308C2×D53—24no—
309C2×D53—28no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—44no—
313C2×D53—44no—
314C2×D53—44no—
315C2×D53—42no—
316C2×D53—42no—
317C2×D53—42no—
318C2×D53—42no—
319C4×C53—24no—
320C4×C53—42no—
321C4×C53—44no—
322C4×C53—42no—
323C4×C53—82no—
324Dic53—24no—
325Dic53—44no—
326Dic53—42no—
327Dic53—82no—
328Dic53#39918yescomplemented
329C2×#399—531no—
330C2×#399—531no—
331C2×#389—532no—
332C2×#384—531no—
333C2×#380—532no—
334C2×C4×C22⋊C4—531no—
335#1445—531no—
336#1445D5311yescharacteristic, complemented
337#1444—532no—
338#1439—532no—
339C4⋊#95—531no—
340C2×C53C2×#39911yescharacteristic
341C2×C53#242212yes—
342C2×C53#242312yes—
343C2×C53#244012yes—
344C2×C53#244112yes—
345C2×C53#248611yescharacteristic
346C2×C53#248711yescharacteristic
347C2×C53#254111yescharacteristic
348C2×C53#254211yescharacteristic
349C2×C53#262811yescharacteristic
350C2×C53#285711yescharacteristic
351D53—44no—
352C24×C4—531no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—532no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—532no—
357C22×C4⋊C4—532no—
358C22×C4⋊C4—532no—
359C22×C4⋊C4C2×D5311yescharacteristic
360C22×C22⋊C4—531no—
361C22×C22⋊C4—531no—
362C22×C22⋊C4—532no—
363C22×C22⋊C4—532no—
364C22×C42—531no—
365C22×C42—531no—
366#399—1064no—
367#399—1064no—
368#389—1068no—
369#384—1064no—
370#380—1068no—
371C4×C22⋊C4—1064no—
372C2×#95—531no—
373C2×#95—532no—
374C2×#95—532no—
375C2×#95—531no—
376C2×#95—532no—
377C2×#95C2×D5311yescharacteristic
378C2×#95C2×D5312yes—
379C2×#95C2×D5312yes—
380C2×#95C2×D5311yescharacteristic
381C53#1513511yescharacteristic, complemented, Sylow
382C25—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—532no—
387C23×C4—534no—
388C23×C4—532no—
389C23×C4—1062no—
390C23×C4—1064no—
391C23×C4—1064no—
392C23×C4—1062no—
393C23×C4C22×D5312yes—
394C23×C4C22×D5312yes—
395C23×C4C22×D5311yescharacteristic
396C23×C4C22×D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C2×C4⋊C4—22no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—538no—
402C2×C4⋊C4—1064no—
403C2×C4⋊C4—1064no—
404C2×C4⋊C4—1064no—
405C2×C4⋊C4—1064no—
406C2×C4⋊C4—1064no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1064no—
411C2×C4⋊C4—1062no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—2122no—
416C2×C4⋊C4—2122no—
417C2×C4⋊C4—2122no—
418C2×C4⋊C4—2122no—
419C2×C22⋊C4—534no—
420C2×C22⋊C4—534no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4—1062no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1064no—
426C2×C22⋊C4—1064no—
427C2×C22⋊C4—1064no—
428C2×C22⋊C4—1064no—
429C2×C22⋊C4—2122no—
430C2×C22⋊C4—2122no—
431C2×C42—534no—
432C2×C42—1064no—
433C2×C42—1062no—
434C2×C42—1064no—
435#95—22no—
436#95—24no—
437#95—24no—
438#95—22no—
439#95—1064no—
440#95—1062no—
441#95—1064no—
442#95—1064no—
443#95—1062no—
444C24—532no—
445C24—532no—
446C24—532no—
447C24—1062no—
448C24—1061no—
449C24—1061no—
450C24—1062no—
451C24—1061no—
452C24—1061no—
453C24—1062no—
454C24—1062no—
455C24C23×D5311yescharacteristic
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
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—534no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—534no—
488C22×C4—532no—
489C22×C4—1068no—
490C22×C4—10616no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1061no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1061no—
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—1062no—
509C22×C4—1062no—
510C22×C4—2124no—
511C22×C4—2128no—
512C22×C4—2124no—
513C22×C4#1171994512yes—
514C22×C4D4×D5314yes—
515C22×C4D4×D5312yes—
516C22×C4D4×D5312yes—
517C22×C4Q8×D5312yes—
518C22×C4Q8×D5312yes—
519C4⋊C4—44no—
520C4⋊C4—10616no—
521C4⋊C4—2128no—
522C4⋊C4—2128no—
523C4⋊C4—2128no—
524C4⋊C4—2128no—
525C4⋊C4—2124no—
526C22⋊C4—1068no—
527C22⋊C4—1068no—
528C22⋊C4—2128no—
529C22⋊C4—2128no—
530C42—1068no—
531C42—2124no—
532C23—534no—
533C23—1064no—
534C23—1064no—
535C23—1064no—
536C23—1062no—
537C23—1062no—
538C23—1064no—
539C23—1062no—
540C23—1062no—
541C23—1062no—
542C23—1062no—
543C23—1064no—
544C23—1064no—
545C23—2122no—
546C23—2122no—
547C23—2122no—
548C23—2122no—
549C23—2122no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2122no—
555C23C2×D4×D5311yescharacteristic
556C23C2×D4×D5312yes—
557C23C2×D4×D5311yescharacteristic
558C23C2×Q8×D5311yescharacteristic
559C23C2×Q8×D5311yescharacteristic
560C23#41×D5311yescharacteristic
561C23#41×D5312yes—
562C23#41×D5311yescharacteristic
563C23C2×#1171994511yescharacteristic
564C23#1313326312yes—
565C23#1313326412yes—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—538no—
595C2×C4—534no—
596C2×C4—534no—
597C2×C4—538no—
598C2×C4—534no—
599C2×C4—534no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—1064no—
605C2×C4—1062no—
606C2×C4—1064no—
607C2×C4—1064no—
608C2×C4—1064no—
609C2×C4—21216no—
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—2122no—
619C2×C4—2122no—
620C2×C4—2121no—
621C2×C4—2122no—
622C2×C4—2121no—
623C2×C4—2122no—
624C2×C4#1313319014yes—
625C22—1068no—
626C22—1064no—
627C22—2122no—
628C22—2122no—
629C22—2124no—
630C22—2122no—
631C22—2124no—
632C22—2124no—
633C22—2122no—
634C22—2122no—
635C22—2122no—
636C22C2×#1313319011yescharacteristic
637C22C22⋊Q8×D5311yescharacteristic
638C22C22⋊Q8×D5312yes—
639C22C22⋊Q8×D5311yescharacteristic
640C22#123×D5312yes—
641C22#123×D5311yescharacteristic
642C22#127×D5311yescharacteristic
643C22C4⋊Q8×D5311yescharacteristic
644C22#1681117211yescharacteristic
645C22#1681117711yescharacteristic
646C22#1681117711yescharacteristic
647C22#1681119111yescharacteristic
648C22#1681119511yescharacteristic
649C22#1681123912yes—
650C22#1681125312yes—
651C22#1681125712yes—
652C22#1681127112yes—
653C22#1681127112yes—
654C22#1681128612yes—
655C22#1681129412yes—
656C22#1681130612yes—
657C22#1681132412yes—
658C22#1681132912yes—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—82no—
664C4—538no—
665C4—1064no—
666C4—4242no—
667C4—2122no—
668C4—2124no—
669C2—2124no—
670C2#399×D5311yescharacteristic
671C2#2027822412yes—
672C2#2027823912yes—
673C2#2027841212yes—
674C2#2027842712yes—
675C2#2027902211yescharacteristic
676C2#2027902711yescharacteristic
677C2#2027966011yescharacteristic
678C2#2027966311yescharacteristic
679C2#2028054311yescharacteristic
680C2#2028306411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1445C2
prime radicalC24×D53C23
hypercenter#1445D53
hypocenterC53#15135
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1445

Complements

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

NH ≅ G/Ntype
3. C2×#16810520C2semidirect
4. C2×#16810492C2semidirect
7. #20272984C2semidirect
8. #20272979C2semidirect
10. #20272943C2semidirect
11. C53×#1445C2semidirect
328. Dic53#399semidirect
336. #1445D53semidirect
381. C53#15135semidirect

Maximal subgroups

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

subgroupindex
C2×#168105602
C2×#168105582
C2×#168105202
C2×#168104922
C2×#168104822
C2×#168104702
#202729842
#202729792
#202729592
#202729432
C53×#14452
#1513553