Finite Group Databasebeta

← #27306847

Subgroups — #27306847

Order 13568.

2150 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810497C212yesmaximal
2C2×#16810494C212yesmaximal
3C2×#16810482C211yesmaximal, characteristic
4C2×C4×#11719921C211yesmaximal, characteristic
5C2×#95×D53C211yesmaximal, characteristic
6#20272979C212yesmaximal, complemented
7#20272978C212yesmaximal, complemented
8#20272945C211yesmaximal, characteristic, complemented
9#20272943C211yesmaximal, characteristic, complemented
10#20272943C211yesmaximal, characteristic, complemented
11C53×C4⋊#95C211yesmaximal, characteristic, complemented
12C23×C4×D53C2212yes—
13C23×C4×D53C2211yescharacteristic
14C22×C4⋊Dic53C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2211yescharacteristic
20C22×#11719921C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810497—28no—
23#16810494—28no—
24#16810482—24no—
25C4×#11719921C418yescomplemented
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2211yescharacteristic
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34#95×D53—24no—
35C22×C4⋊C4×C53C2212yes—
36C22×C42×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C24×D53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46C22×C4×D53—24no—
47C22×C4×D53—24no—
48C22×C4×D53—24no—
49C22×C4×D53—24no—
50C22×C4×D53—24no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53—42no—
54C22×C4×D53—42no—
55C22×C4×D53C2×C414yes—
56C22×C4×D53D414yes—
57C22×C4×D53Q814yes—
58C2×C4⋊Dic53—24no—
59C2×C4⋊Dic53—24no—
60C2×C4⋊Dic53—42no—
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×#11719919—42no—
70C2×#11719921—24no—
71C2×#11719921—24no—
72C2×#11719921—24no—
73C2×#11719921C2×C414yes—
74C2×#11719921C2×C414yes—
75C2×#11719921D414yes—
76C2×#11719921D414yes—
77C2×#11719921Q814yes—
78C2×#11719921Q814yes—
79C2×C4×Dic53—42no—
80C2×C4×Dic53C2×C414yescomplemented
81#13133124—22no—
82#13133124—22no—
83#13133124—24no—
84#13133124—24no—
85#13133124—24no—
86#13133124—24no—
87#13133124C2×C414yescomplemented
88#13133124C2×C414yescomplemented
89C23×C4×C53C2312yes—
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—24no—
96C2×C4⋊C4×C53—42no—
97C2×C42×C53—42no—
98C2×C42×C53C2×C414yescomplemented
99#95×C53—24no—
100#95×C53—22no—
101#95×C53—22no—
102C24×C53C2411yescharacteristic
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53C22×C412yes—
108C23×D53C2×D412yes—
109C23×D53C2×D412yes—
110C23×D53C2×D412yes—
111C23×D53C2×Q812yes—
112C23×D53C2×Q812yes—
113C23×D53C2×Q812yes—
114C22×C4×C53—22no—
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×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53C22×C412yes—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×Q812yes—
140C22×C4×C53C2×Q812yes—
141C22×C4×C53C2×Q812yes—
142C22×C4×C53#4114yes—
143C22×Dic53—22no—
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×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53C22×C412yes—
163C22×Dic53C22×C412yes—
164C22×Dic53C22×C412yes—
165C22×Dic53C2×D412yes—
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D412yes—
168C22×Dic53C2×Q812yes—
169C22×Dic53C2×Q812yes—
170C22×Dic53C2×Q812yes—
171C22×Dic53#4114yes—
172C2×C4×D53—216no—
173C2×C4×D53—44no—
174C2×C4×D53—44no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53—48no—
179C2×C4×D53—48no—
180C2×C4×D53—48no—
181C2×C4×D53—48no—
182C2×C4×D53—48no—
183C2×C4×D53C4⋊C4116yescomplemented
184C4⋊C4×C53—48no—
185C42×C53—44no—
186#11719921—48no—
187#11719921C4⋊C4116yescomplemented
188#11719921C4⋊C4116yescomplemented
189C4⋊Dic53—48no—
190#11719919—48no—
191#11719919—48no—
192#11719919—48no—
193C4×Dic53—44no—
194C23×C53C23×C411yescharacteristic
195C23×C53C22×D411yescharacteristic
196C23×C53C22×D411yescharacteristic
197C23×C53C22×D411yescharacteristic
198C23×C53C22×Q811yescharacteristic
199C23×C53C22×Q811yescharacteristic
200C23×C53C22×Q811yescharacteristic
201C23×C53C2×#4112yes—
202C23×C53C2×#4112yes—
203C23×C53H(5,𝔽2)12yes—
204C23×C53H(5,𝔽2)12yes—
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—24no—
215C22×D53—24no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53C2×C4⋊C414yes—
222C22×D53C2×C4⋊C414yes—
223C22×D53C2×C4⋊C414yes—
224C22×D53#12714yes—
225C22×D53C4⋊Q814yes—
226C22×D53C4⋊Q814yes—
227C22×D53C4⋊Q814yes—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
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—24no—
239C2×C4×C53—24no—
240C2×C4×C53—24no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
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—42no—
254C2×C4×C53C2×C4⋊C414yes—
255C2×C4×C53C2×C4⋊C414yes—
256C2×C4×C53C2×C4⋊C414yes—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
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×Dic53—42no—
282C2×Dic53—42no—
283C2×Dic53C2×C4⋊C414yes—
284C2×Dic53C2×C4⋊C414yes—
285C2×Dic53C2×C4⋊C414yes—
286C4×D53—416no—
287C4×D53—432no—
288#13965—531nomaximal, Sylow
289C22×C53C22×C4⋊C411yescharacteristic
290C22×C53C22×C4⋊C411yescharacteristic
291C22×C53C22×C4⋊C411yescharacteristic
292C22×C53C4×#4111yescharacteristic
293C22×C53C4×#4111yescharacteristic
294C22×C53#51811yescharacteristic
295C22×C53#51811yescharacteristic
296C22×C53#52512yes—
297C22×C53C2×#12711yescharacteristic
298C22×C53C2×C4⋊Q811yescharacteristic
299C22×C53C2×C4⋊Q811yescharacteristic
300C22×C53C2×C4⋊Q811yescharacteristic
301C22×C53#53312yes—
302C22×C53#53412yes—
303C22×C53#54312yes—
304C22×C53#54612yes—
305C22×C53#54612yes—
306C22×C53#54612yes—
307C22×C53#54612yes—
308C22×C53#55412yes—
309C22×C53#55412yes—
310C22×C53#55412yes—
311C22×C53#55412yes—
312C2×D53—24no—
313C2×D53—28no—
314C2×D53—24no—
315C2×D53—24no—
316C2×D53—24no—
317C2×D53—24no—
318C2×D53—28no—
319C2×D53—28no—
320C2×D53—28no—
321C2×D53—24no—
322C2×D53#38418yes—
323C4×C53—24no—
324C4×C53—24no—
325C4×C53—24no—
326C4×C53—44no—
327C4×C53—82no—
328Dic53—24no—
329Dic53—24no—
330Dic53—24no—
331Dic53—44no—
332Dic53—82no—
333C2×#385—532no—
334C2×#385—532no—
335C2×#380—531no—
336C2×C4×C22⋊C4—531no—
337C22×#95—531no—
338#1444—532no—
339#1444—532no—
340C4⋊#95—531no—
341C4⋊#95—531no—
342C4⋊#95—531no—
343C4⋊#95D5311yescharacteristic, complemented
344C2×C53C2×#38411yescharacteristic
345C2×C53#229611yescharacteristic
346C2×C53#231511yescharacteristic
347C2×C53#234911yescharacteristic
348C2×C53#234911yescharacteristic
349C2×C53#234911yescharacteristic
350C2×C53#234911yescharacteristic
351C2×C53#253912yes—
352C2×C53#254912yes—
353C2×C53#254912yes—
354C2×C53#258312yes—
355D53—28no—
356C24×C4—531no—
357C24×C4—532no—
358C22×C4⋊C4—532no—
359C22×C4⋊C4—532no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4—532no—
362C22×C4⋊C4C2×D5312yes—
363C22×C22⋊C4—531no—
364C22×C22⋊C4—531no—
365C22×C22⋊C4—532no—
366C22×C42—531no—
367C22×C42C2×D5311yescharacteristic
368#385—1068no—
369#385—1068no—
370#380—1064no—
371C4×C22⋊C4—538no—
372C2×#95—531no—
373C2×#95—531no—
374C2×#95—531no—
375C2×#95—531no—
376C2×#95—532no—
377C2×#95—532no—
378C2×#95—532no—
379C2×#95—532no—
380C2×#95—1064no—
381C2×#95C2×D5311yescharacteristic
382C2×#95C2×D5311yescharacteristic
383C2×#95C2×D5312yes—
384C53#1396511yescharacteristic, complemented, Sylow
385C25—531no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—534no—
390C23×C4—534no—
391C23×C4—534no—
392C23×C4—532no—
393C23×C4—532no—
394C23×C4—1064no—
395C23×C4—1064no—
396C23×C4—1064no—
397C23×C4—1064no—
398C23×C4—1064no—
399C23×C4—2122no—
400C23×C4—2122no—
401C23×C4—2122no—
402C23×C4—2122no—
403C23×C4C22×D5312yes—
404C23×C4C22×D5312yes—
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408C2×C4⋊C4—24no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4—42no—
411C2×C4⋊C4—1064no—
412C2×C4⋊C4—1064no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1064no—
415C2×C4⋊C4—1064no—
416C2×C4⋊C4—1064no—
417C2×C4⋊C4—1064no—
418C2×C4⋊C4—2122no—
419C2×C4⋊C4—2122no—
420C2×C4⋊C4—2122no—
421C2×C4⋊C4—2122no—
422C2×C4⋊C4—2122no—
423C2×C22⋊C4—534no—
424C2×C22⋊C4—534no—
425C2×C22⋊C4—534no—
426C2×C22⋊C4—534no—
427C2×C22⋊C4—534no—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—1064no—
430C2×C22⋊C4—1064no—
431C2×C22⋊C4—1064no—
432C2×C42—42no—
433C2×C42—534no—
434C2×C42—2122no—
435C2×C42C4×D5314yescomplemented
436#95—22no—
437#95—22no—
438#95—24no—
439#95—534no—
440#95—534no—
441#95—1064no—
442#95—1064no—
443#95—1064no—
444#95—1062no—
445#95—1062no—
446#95—1064no—
447C24—532no—
448C24—532no—
449C24—532no—
450C24—532no—
451C24—532no—
452C24—532no—
453C24—532no—
454C24—1062no—
455C24—1062no—
456C24—1062no—
457C24—1062no—
458C24C23×D5311yescharacteristic
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—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—532no—
479C22×C4—532no—
480C22×C4—532no—
481C22×C4—532no—
482C22×C4—532no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—5316no—
488C22×C4—534no—
489C22×C4—10616no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
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—1062no—
509C22×C4—2124no—
510C22×C4—2124no—
511C22×C4—2128no—
512C22×C4—2128no—
513C22×C4—2128no—
514C22×C4—2128no—
515C22×C4—2128no—
516C22×C4—2128no—
517C22×C4—2128no—
518C22×C4—2128no—
519C22×C4C2×C4×D5312yes—
520C22×C4C2×C4×D5312yes—
521C22×C4C2×C4×D5312yes—
522C22×C4#1171994514yes—
523C22×C4D4×D5312yes—
524C22×C4D4×D5312yes—
525C22×C4D4×D5312yes—
526C22×C4Q8×D5312yes—
527C22×C4Q8×D5312yes—
528C22×C4Q8×D5312yes—
529C4⋊C4—48no—
530C4⋊C4—2128no—
531C4⋊C4—2128no—
532C4⋊C4—2128no—
533C4⋊C4—2128no—
534C22⋊C4—5316no—
535C22⋊C4—5316no—
536C22⋊C4—2128no—
537C42—44no—
538C42—2124no—
539C23—534no—
540C23—534no—
541C23—534no—
542C23—534no—
543C23—534no—
544C23—534no—
545C23—534no—
546C23—1064no—
547C23—1064no—
548C23—1064no—
549C23—1064no—
550C23—1064no—
551C23—1064no—
552C23—1062no—
553C23—1062no—
554C23—1062no—
555C23—1062no—
556C23—1064no—
557C23—1064no—
558C23—1064no—
559C23—1064no—
560C23—1064no—
561C23—1064no—
562C23C22×C4×D5311yescharacteristic
563C23C2×D4×D5311yescharacteristic
564C23C2×D4×D5311yescharacteristic
565C23C2×D4×D5311yescharacteristic
566C23C2×Q8×D5311yescharacteristic
567C23C2×Q8×D5311yescharacteristic
568C23C2×Q8×D5311yescharacteristic
569C23#41×D5312yes—
570C23C2×#1171994512yes—
571C23#1313326212yes—
572C23#1313326312yes—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—534no—
600C2×C4—534no—
601C2×C4—534no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1064no—
613C2×C4—1064no—
614C2×C4—1064no—
615C2×C4—1064no—
616C2×C4—1064no—
617C2×C4—21216no—
618C2×C4—21232no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2122no—
623C2×C4—2122no—
624C2×C4—2122no—
625C2×C4—2122no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4C4⋊C4×D5314yes—
631C2×C4C4⋊C4×D5314yes—
632C2×C4C4⋊C4×D5314yes—
633C22—538no—
634C22—1068no—
635C22—1064no—
636C22—1068no—
637C22—1064no—
638C22—1064no—
639C22—1064no—
640C22—1064no—
641C22—1068no—
642C22—1068no—
643C22—1064no—
644C22C2×C4⋊C4×D5311yescharacteristic
645C22C2×C4⋊C4×D5311yescharacteristic
646C22C2×C4⋊C4×D5311yescharacteristic
647C22#1681109111yescharacteristic
648C22C4×#1171994511yescharacteristic
649C22#1681109611yescharacteristic
650C22#1681109711yescharacteristic
651C22#1681113212yes—
652C22#127×D5311yescharacteristic
653C22C4⋊Q8×D5311yescharacteristic
654C22C4⋊Q8×D5311yescharacteristic
655C22C4⋊Q8×D5311yescharacteristic
656C22#1681117712yes—
657C22#1681118112yes—
658C22#1681124812yes—
659C22#1681126212yes—
660C22#1681126312yes—
661C22#1681126612yes—
662C22#1681126912yes—
663C22#1681132712yes—
664C22#1681132812yes—
665C22#1681132912yes—
666C22#1681133312yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—44no—
671C4—82no—
672C4—1064no—
673C4—1064no—
674C4—1064no—
675C4—4242no—
676C4—2124no—
677C2—1068no—
678C2#384×D5311yescharacteristic
679C2#2027719511yescharacteristic
680C2#2027732811yescharacteristic
681C2#2027761911yescharacteristic
682C2#2027762211yescharacteristic
683C2#2027762311yescharacteristic
684C2#2027762611yescharacteristic
685C2#2027964412yes—
686C2#2027972412yes—
687C2#2027972512yes—
688C2#2028009312yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13965
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
6. #20272979C2semidirect
7. #20272978C2semidirect
8. #20272945C2semidirect
9. #20272943C2semidirect
10. #20272943C2semidirect
11. C53×C4⋊#95C2semidirect
25. C4×#11719921C4semidirect
80. C2×C4×Dic53C2×C4semidirect
87. #13133124C2×C4semidirect
88. #13133124C2×C4semidirect
98. C2×C42×C53C2×C4semidirect
183. C2×C4×D53C4⋊C4semidirect
187. #11719921C4⋊C4semidirect
188. #11719921C4⋊C4semidirect
343. C4⋊#95D53semidirect
384. C53#13965semidirect
435. C2×C42C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104972
C2×#168104942
C2×#168104822
C2×C4×#117199212
C2×#95×D532
#202729792
#202729782
#202729452
#202729432
C53×C4⋊#952
#1396553