Finite Group Databasebeta

← #27320642

Subgroups — #27320642

Order 13568.

2038 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133215C211yesmaximal, characteristic, complemented
2C2×#16810536C211yesmaximal, characteristic
3C2×#16810534C211yesmaximal, characteristic
4C2×#16810482C212yesmaximal
5C2×#16810470C211yesmaximal, characteristic
6#20272985C211yesmaximal, characteristic, complemented
7#20272973C212yesmaximal, complemented
8#20272959C212yesmaximal
9#20272943C212yesmaximal, complemented
10#20272943C211yesmaximal, characteristic, complemented
11C53×#1446C211yesmaximal, characteristic, complemented
12C23×Dic106C2211yescharacteristic
13C23×C4×D53C2211yescharacteristic
14C2×#13133215—28no—
15C2×#13133215—24no—
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×#11719921C2211yescharacteristic
22C22×#11719921C2211yescharacteristic
23C22×C4×Dic53C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25#16810536—24no—
26#16810534—24no—
27#16810482—28no—
28#16810470—24no—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C2×#13133124C2212yes—
36C23×Q8×C53C2211yescharacteristic
37C2×#95×C53C2211yescharacteristic
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C2×#95×C53C2212yes—
41C24×D53C2311yescharacteristic
42C22×Dic106—22no—
43C22×Dic106—24no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106—42no—
47C23×Dic53C2312yes—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C22×C4×D53—22no—
53C22×C4×D53—24no—
54C22×C4×D53—24no—
55C22×C4×D53—22no—
56C22×C4×D53Q814yescomplemented
57#13133215—232no—
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×#11719919D418yes—
67C2×#11719921—24no—
68C2×#11719921—24no—
69C2×#11719921—22no—
70C2×#11719921—24no—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921—24no—
74C2×#11719921—22no—
75C2×#11719921—42no—
76C2×#11719921—42no—
77C2×#11719921D414yes—
78C2×#11719921Q814yescomplemented
79C2×C4×Dic53—24no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53—24no—
82C2×C4×Dic53D414yes—
83#13133124—22no—
84#13133124—24no—
85#13133124—24no—
86#13133124—22no—
87#13133124—24no—
88#13133124—24no—
89#13133124—42no—
90C22×Q8×C53—24no—
91C22×Q8×C53—22no—
92C22×Q8×C53—22no—
93C22×Q8×C53—22no—
94C22×Q8×C53—42no—
95C23×C4×C53C2312yes—
96C23×C4×C53C2312yes—
97C23×C4×C53C2311yescharacteristic
98C23×C4×C53C2311yescharacteristic
99C23×C4×C53C2311yescharacteristic
100#95×C53—22no—
101#95×C53—22no—
102#95×C53—24no—
103#95×C53—42no—
104C24×C53C2411yescharacteristic
105C23×D53—21no—
106C23×D53—22no—
107C23×D53—21no—
108C23×D53—21no—
109C23×D53—21no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53C2×D412yes—
114C23×D53C2×Q812yes—
115C23×D53C2×Q812yes—
116C2×Q8×C53—28no—
117C2×Q8×C53—48no—
118C2×Q8×C53—48no—
119C2×Q8×C53—44no—
120C2×Q8×C53—44no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
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×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53C2×D414yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×Q812yes—
148C22×C4×C53C2×Q812yes—
149C22×C4×C53#4112yes—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
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×Dic53—22no—
169C22×Dic53C2×D412yes—
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D412yes—
172C22×Dic53C2×D412yes—
173C22×Dic53C2×D414yes—
174C22×Dic53C2×Q812yes—
175C22×Dic53C2×Q812yes—
176C22×Dic53#4112yes—
177C22×Dic53#4114yes—
178C22×Dic53#4112yes—
179C2×C4×D53—28no—
180C2×C4×D53—216no—
181C2×C4×D53—44no—
182C2×C4×D53—44no—
183C2×C4×D53—48no—
184C2×Dic106—28no—
185C2×Dic106—48no—
186C2×Dic106—48no—
187C2×Dic106—44no—
188C2×Dic106—44no—
189#11719921—28no—
190#11719921—28no—
191#11719921—48no—
192#11719921—48no—
193#11719919—216no—
194#11719919—216no—
195#11719919—48no—
196C4×Dic53—28no—
197C4×Dic53—44no—
198C23×C53C22×D411yescharacteristic
199C23×C53C22×D412yes—
200C23×C53C22×D411yescharacteristic
201C23×C53C22×Q811yescharacteristic
202C23×C53C22×Q811yescharacteristic
203C23×C53C2×#4111yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4111yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C23×C53H(5,𝔽2)12yes—
208C23×C532−1+412yes—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—24no—
212C22×D53—22no—
213C22×D53—22no—
214C22×D53—22no—
215C22×D53—22no—
216C22×D53—24no—
217C22×D53—24no—
218C22×D53—24no—
219C22×D53—24no—
220C22×D53—24no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—42no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—42no—
231C22×D53C4⋊Q814yescomplemented
232Q8×C53—416no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—22no—
239C2×C4×C53—24no—
240C2×C4×C53—22no—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—41no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53—42no—
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#12114yes—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—24no—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—42no—
272C2×Dic53—41no—
273C2×Dic53—42no—
274C2×Dic53—41no—
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×Dic53—42no—
284C2×Dic53—42no—
285C2×Dic53#12014yes—
286C2×Dic53C22⋊Q814yes—
287C2×Dic53C22⋊Q814yes—
288C2×Dic53C22⋊Q818yes—
289C2×Dic53#12414yes—
290C2×Dic53#12418yes—
291C4×D53—416no—
292Dic106—416no—
293#15125—531nomaximal, Sylow
294C22×C53C2×#12011yescharacteristic
295C22×C53C2×#12111yescharacteristic
296C22×C53C2×C22⋊Q811yescharacteristic
297C22×C53C2×C22⋊Q811yescharacteristic
298C22×C53C2×C22⋊Q812yes—
299C22×C53C2×#12411yescharacteristic
300C22×C53C2×#12412yes—
301C22×C53C2×C4⋊Q811yescharacteristic
302C22×C53#53211yescharacteristic
303C22×C53#53311yescharacteristic
304C22×C53#53311yescharacteristic
305C22×C53#53511yescharacteristic
306C22×C53#53611yescharacteristic
307C22×C53#54212yes—
308C22×C53#54412yes—
309C22×C53D4212yes—
310C22×C53#54612yes—
311C22×C53#54712yes—
312C22×C53#54812yes—
313C22×C53D4×Q812yes—
314C22×C53#55412yes—
315C22×C53#55512yes—
316C22×C53#55612yes—
317C2×D53—24no—
318C2×D53—28no—
319C2×D53—42no—
320C2×D53—42no—
321C2×D53—42no—
322C2×D53—42no—
323C2×D53—42no—
324C2×D53—44no—
325C2×D53—44no—
326C2×D53—44no—
327C2×D53—42no—
328C4×C53—24no—
329C4×C53—42no—
330C4×C53—42no—
331C4×C53—44no—
332C4×C53—82no—
333Dic53—24no—
334Dic53—44no—
335Dic53—42no—
336Dic53—82no—
337Dic53#39318yescomplemented
338C22×C4⋊Q8—531no—
339C2×#393—531no—
340C2×#393—531no—
341C2×#380—532no—
342C2×C4×C22⋊C4—531no—
343#1446—531no—
344#1446D5311yescharacteristic, complemented
345#1442—532no—
346#1439—532no—
347C4⋊#95—532no—
348C4⋊#95—531no—
349C2×C53C2×#39311yescharacteristic
350C2×C53#240811yescharacteristic
351C2×C53#240911yescharacteristic
352C2×C53#243912yes—
353C2×C53#244012yes—
354C2×C53#249112yes—
355C2×C53#249212yes—
356C2×C53#254711yescharacteristic
357C2×C53#256311yescharacteristic
358C2×C53#262811yescharacteristic
359C2×C53#285711yescharacteristic
360D53—44no—
361C23×Q8—531no—
362C23×Q8C2×D5311yescharacteristic
363C24×C4—531no—
364C2×C4⋊Q8—1064no—
365C2×C4⋊Q8—1068no—
366C22×C4⋊C4—532no—
367C22×C4⋊C4—532no—
368C22×C4⋊C4—532no—
369C22×C22⋊C4—531no—
370C22×C22⋊C4—531no—
371C22×C22⋊C4—532no—
372C22×C22⋊C4—532no—
373C22×C42—531no—
374C22×C42—531no—
375#393—1064no—
376#393—1064no—
377#380—1068no—
378C4×C22⋊C4—1064no—
379C2×#95—532no—
380C2×#95—532no—
381C2×#95—531no—
382C2×#95—532no—
383C2×#95—531no—
384C2×#95—532no—
385C2×#95—532no—
386C2×#95C2×D5311yescharacteristic
387C2×#95C2×D5312yes—
388C2×#95C2×D5311yescharacteristic
389C2×#95C2×D5312yes—
390C53#1512511yescharacteristic, complemented, Sylow
391C25—531no—
392C22×Q8—22no—
393C22×Q8—24no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—42no—
397C22×Q8—1064no—
398C22×Q8—1062no—
399C22×Q8—1062no—
400C22×Q8—1062no—
401C22×Q8—2122no—
402C23×C4—531no—
403C23×C4—531no—
404C23×C4—532no—
405C23×C4—531no—
406C23×C4—534no—
407C23×C4—532no—
408C23×C4—1062no—
409C23×C4—1064no—
410C23×C4—1064no—
411C23×C4—1062no—
412C23×C4C22×D5312yes—
413C23×C4C22×D5312yes—
414C23×C4C22×D5311yescharacteristic
415C23×C4C22×D5311yescharacteristic
416C23×C4C22×D5311yescharacteristic
417C4⋊Q8—10632no—
418C2×C4⋊C4—538no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1064no—
423C2×C4⋊C4—1064no—
424C2×C4⋊C4—1064no—
425C2×C4⋊C4—1064no—
426C2×C4⋊C4—1064no—
427C2×C22⋊C4—534no—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—1062no—
430C2×C22⋊C4—1062no—
431C2×C22⋊C4—1062no—
432C2×C22⋊C4—1062no—
433C2×C22⋊C4—1064no—
434C2×C22⋊C4—1064no—
435C2×C22⋊C4—1064no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—2122no—
438C2×C22⋊C4—2122no—
439C2×C42—534no—
440C2×C42—1064no—
441C2×C42—1062no—
442C2×C42—1064no—
443#95—22no—
444#95—24no—
445#95—22no—
446#95—42no—
447#95—1064no—
448#95—1062no—
449#95—1064no—
450#95—1064no—
451#95—1064no—
452#95—1062no—
453#95—2122no—
454C24—532no—
455C24—532no—
456C24—532no—
457C24—1062no—
458C24—1061no—
459C24—1061no—
460C24—1062no—
461C24—1061no—
462C24—1061no—
463C24—1062no—
464C24—1062no—
465C24C23×D5311yescharacteristic
466C2×Q8—28no—
467C2×Q8—48no—
468C2×Q8—48no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—1068no—
472C2×Q8—2128no—
473C2×Q8—2128no—
474C2×Q8—2124no—
475C2×Q8—2124no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—532no—
500C22×C4—534no—
501C22×C4—532no—
502C22×C4—534no—
503C22×C4—532no—
504C22×C4—532no—
505C22×C4—532no—
506C22×C4—532no—
507C22×C4—532no—
508C22×C4—534no—
509C22×C4—1068no—
510C22×C4—10616no—
511C22×C4—1062no—
512C22×C4—1062no—
513C22×C4—1061no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—1062no—
517C22×C4—1062no—
518C22×C4—1061no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—2124no—
531C22×C4—2124no—
532C22×C4—2128no—
533C22×C4#1171994512yes—
534C22×C4D4×D5314yes—
535C22×C4D4×D5312yes—
536C22×C4D4×D5312yes—
537C22×C4Q8×D5312yes—
538C22×C4Q8×D5312yes—
539C4⋊C4—10616no—
540C4⋊C4—10616no—
541C4⋊C4—2128no—
542C22⋊C4—1068no—
543C22⋊C4—1068no—
544C22⋊C4—2128no—
545C22⋊C4—2128no—
546C42—1068no—
547C42—2124no—
548C23—534no—
549C23—1064no—
550C23—1064no—
551C23—1064no—
552C23—1062no—
553C23—1062no—
554C23—1064no—
555C23—1062no—
556C23—1062no—
557C23—1062no—
558C23—1062no—
559C23—1064no—
560C23—1064no—
561C23—2122no—
562C23—2122no—
563C23—2122no—
564C23—2121no—
565C23—2122no—
566C23—2121no—
567C23—2121no—
568C23—2121no—
569C23—2122no—
570C23—2122no—
571C23C2×D4×D5311yescharacteristic
572C23C2×D4×D5312yes—
573C23C2×D4×D5311yescharacteristic
574C23C2×Q8×D5311yescharacteristic
575C23C2×Q8×D5311yescharacteristic
576C23#41×D5312yes—
577C23#41×D5311yescharacteristic
578C23#41×D5311yescharacteristic
579C23C2×#1171994511yescharacteristic
580C23#1313326212yes—
581C23#1313326512yes—
582Q8—416no—
583Q8—21216no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
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—42no—
611C2×C4—42no—
612C2×C4—534no—
613C2×C4—538no—
614C2×C4—534no—
615C2×C4—534no—
616C2×C4—538no—
617C2×C4—534no—
618C2×C4—1064no—
619C2×C4—1064no—
620C2×C4—1062no—
621C2×C4—1062no—
622C2×C4—1064no—
623C2×C4—1064no—
624C2×C4—1062no—
625C2×C4—1064no—
626C2×C4—1064no—
627C2×C4—21216no—
628C2×C4—2122no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4—2122no—
636C2×C4—2122no—
637C2×C4—2121no—
638C2×C4—2122no—
639C2×C4—2121no—
640C2×C4—2122no—
641C2×C4—2122no—
642C2×C4#1313319014yes—
643C22—1068no—
644C22—1064no—
645C22—2124no—
646C22—2124no—
647C22—2122no—
648C22—2122no—
649C22—2122no—
650C22—2124no—
651C22—2122no—
652C22—2122no—
653C22—2122no—
654C22#120×D5311yescharacteristic
655C22C2×#1313319011yescharacteristic
656C22C22⋊Q8×D5311yescharacteristic
657C22C22⋊Q8×D5312yes—
658C22C22⋊Q8×D5311yescharacteristic
659C22#124×D5312yes—
660C22#124×D5311yescharacteristic
661C22C4⋊Q8×D5311yescharacteristic
662C22#1681117211yescharacteristic
663C22#1681117711yescharacteristic
664C22#1681117711yescharacteristic
665C22#1681118811yescharacteristic
666C22#1681119811yescharacteristic
667C22#1681124312yes—
668C22#1681125012yes—
669C22#1681125512yes—
670C22#1681126312yes—
671C22#1681127512yes—
672C22#1681128312yes—
673C22#1681128812yes—
674C22#1681132712yes—
675C22#1681133512yes—
676C22#1681134512yes—
677C4—24no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—82no—
682C4—538no—
683C4—1064no—
684C4—4242no—
685C4—2124no—
686C4—2122no—
687C2—2124no—
688C2#393×D5311yescharacteristic
689C2#2027807011yescharacteristic
690C2#2027807511yescharacteristic
691C2#2027840312yes—
692C2#2027841012yes—
693C2#2027907312yes—
694C2#2027908812yes—
695C2#2027971211yescharacteristic
696C2#2027988311yescharacteristic
697C2#2028053811yescharacteristic
698C2#2028306411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, cosocleC24×C53C24
FittingC53×#1446C2
prime radicalC24×D53C23
hypercenter#1446D53
hypocenterC53#15125
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1446

Complements

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

NH ≅ G/Ntype
1. C22×#13133215C2semidirect
6. #20272985C2semidirect
7. #20272973C2semidirect
9. #20272943C2semidirect
10. #20272943C2semidirect
11. C53×#1446C2semidirect
56. C22×C4×D53Q8semidirect
78. C2×#11719921Q8semidirect
231. C22×D53C4⋊Q8semidirect
337. Dic53#393semidirect
344. #1446D53semidirect
390. C53#15125semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
C2×#168105362
C2×#168105342
C2×#168104822
C2×#168104702
#202729852
#202729732
#202729592
#202729432
C53×#14462
#1512553