Finite Group Databasebeta

← #27440663

Subgroups — #27440663

Order 13568.

1462 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20279707C211yesmaximal, characteristic, complemented
4#20275908C212yesmaximal, complemented
5#20275901C212yesmaximal, complemented
6#20275648C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274655C212yesmaximal, complemented
9#20274646C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#546C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic, complemented
13C2×#13133204—21no—
14#16810982—24no—
15#16810913—24no—
16C2×#13133148C2212yescomplemented
17C2×C4⋊#88322C2211yescharacteristic, complemented
18C2×#13133141C2212yescomplemented
19C2×#13133141C2212yescomplemented
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yescomplemented
26C2×#13133128C2211yescharacteristic, complemented
27#16810548—21no—
28#16810548—21no—
29#16810528—21no—
30#16810483C2212yes—
31#16810483C2212yescomplemented
32C4×C4⋊Dic53C2211yescharacteristic, complemented
33#16810385C2212yescomplemented
34C22×#41×C53C2211yescharacteristic
35C23×D4×C53C2211yescharacteristic
36C53×#546—24no—
37C53×#546—24no—
38C2×#124×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53C2211yescharacteristic
41C2×#121×C53—21no—
42C2×#121×C53C2211yescharacteristic
43C2×#120×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C22×C22⋊C4×C53—21no—
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133204—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C4×Dic53—21no—
59#13133148—24no—
60C4⋊#88322—22no—
61#13133141—24no—
62#13133141—42no—
63#13133138—24no—
64#13133138—24no—
65#13133138—24no—
66#13133138—42no—
67C22⋊#88322—42no—
68C22⋊#88322—42no—
69#13133128—22no—
70#13133124—22no—
71#13133124—22no—
72#13133124—21no—
73C25×C53—21no—
74C2×#41×C53—22no—
75C2×#41×C53—22no—
76C2×#41×C53—24no—
77C2×#41×C53—42no—
78C22×Q8×C53C2311yescharacteristic
79C22×D4×C53—21no—
80C22×D4×C53—21no—
81C22×D4×C53—22no—
82C22×D4×C53—22no—
83C22×D4×C53—24no—
84C22×D4×C53—21no—
85C22×D4×C53—42no—
86C22×D4×C53D411yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C22×D4×C53C2311yescharacteristic
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93#124×C53—22no—
94#124×C53D414yes—
95#123×C53—28no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98#121×C53—28no—
99#121×C53—22no—
100#121×C53—22no—
101#120×C53—28no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C4×D4×C53—22no—
105C4×D4×C53D414yes—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—28no—
110C2×C22⋊C4×C53—21no—
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C24×C53—21no—
119C24×C53—24no—
120C24×C53—21no—
121C24×C53—44no—
122C24×C53—42no—
123C24×C53—42no—
124C24×C53—82no—
125C24×C53C2×D411yescharacteristic
126C24×C53C2×D411yescharacteristic
127#41×C53—48no—
128#41×C53—44no—
129#41×C53—44no—
130C2×Q8×C53—21no—
131C2×Q8×C53—22no—
132C2×Q8×C53—22no—
133C2×Q8×C53C2×D412yes—
134C2×D4×C53—24no—
135C2×D4×C53—24no—
136C2×D4×C53—22no—
137C2×D4×C53—24no—
138C2×D4×C53—22no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—22no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—41no—
145C2×D4×C53—44no—
146C2×D4×C53—48no—
147C2×D4×C53—44no—
148C2×D4×C53—44no—
149C2×D4×C53—44no—
150C2×D4×C53—82no—
151C2×D4×C53C2×D412yes—
152C2×D4×C53C2×D412yes—
153C2×D4×C53#4114yes—
154C22×C4×C53—24no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—22no—
162C22×C4×C53—22no—
163C22×C4×C53—22no—
164C22×C4×C53—22no—
165C22×C4×C53—24no—
166C22×C4×C53—44no—
167C22×C4×C53—82no—
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2×D411yescharacteristic
170C22×C4×C53C2×D411yescharacteristic
171C22×C4×C53C2×D411yescharacteristic
172C22×C4×C53C2411yescharacteristic
173C22×Dic53—22no—
174C22×Dic53—22no—
175C2×Dic106—42no—
176C2×Dic106—42no—
177C2×Dic106—42no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53C2×D412yes—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—24no—
186C22⋊C4×C53—24no—
187C22⋊C4×C53—24no—
188C22⋊C4×C53—22no—
189C42×C53C2×D412yes—
190C42×C53C2×D412yes—
191C22⋊Dic53—44no—
192C22⋊Dic53—44no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C23×C53—22no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—24no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—21no—
212C23×C53—22no—
213C23×C53—22no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—22no—
217C23×C53—48no—
218C23×C53—48no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—42no—
222C23×C53—42no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—42no—
226C23×C53—88no—
227C23×C53—84no—
228C23×C53—88no—
229C23×C53AGL(1,ℤ/8)12yes—
230C23×C53C22×D411yescharacteristic
231Q8×C53—24no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—22no—
235D4×C53—24no—
236D4×C53—24no—
237D4×C53—44no—
238D4×C53—48no—
239D4×C53—48no—
240D4×C53—41no—
241D4×C53—41no—
242D4×C53—84no—
243D4×C53#12414yescomplemented
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—48no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—42no—
261C2×C4×C53—42no—
262C2×C4×C53—84no—
263C2×C4×C53#12012yes—
264C2×C4×C53#12012yes—
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C2×#4112yes—
268C2×C4×C53H(5,𝔽2)12yes—
269C2×Dic53—42no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275Dic106—84no—
276#88322—42no—
277#88322—42no—
278#25898—531nomaximal, Sylow
279C22×C53—24no—
280C22×C53—22no—
281C22×C53—24no—
282C22×C53—41no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—41no—
286C22×C53—44no—
287C22×C53—42no—
288C22×C53—44no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—41no—
292C22×C53—42no—
293C22×C53—41no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—816no—
298C22×C53—82no—
299C22×C53—82no—
300C22×C53C2×#12011yescharacteristic
301C22×C53#54612yes—
302C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C53#57511yescharacteristic
304C22×C53#57712yes—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—41no—
308C4×C53—41no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53C2×#12411yescharacteristic
312C4×C53#53511yescharacteristic
313C4×C53#54612yes—
314Dic53—82no—
315Dic53—82no—
316C2×#546D5311yescharacteristic
317C2×#489—531no—
318C2×#476—531no—
319#2547—531no—
320#2035—532no—
321#2034—532no—
322#1983—531no—
323#1955—531no—
324#1788—532no—
325#1786—532no—
326#1762—531no—
327C2×C53—42no—
328C2×C53—44no—
329C2×C53—81no—
330C2×C53—82no—
331C2×C53#243511yescharacteristic
332C2×C53#300611yescharacteristic
333C2×C53#300911yescharacteristic
334C2×C53#330612yes—
335C2×C53#331112yes—
336C22×#41C2×D5311yescharacteristic
337C23×D4C2×D5311yescharacteristic
338#546—24no—
339#546—24no—
340C2×C4⋊Q8—531no—
341C2×#124—1061no—
342C2×#124C2×D5311yescharacteristic
343C2×#123—21no—
344C2×C22⋊Q8C2×D5311yescharacteristic
345C2×#121—21no—
346C2×#121C2×D5311yescharacteristic
347C2×#120—21no—
348C2×C4×D4C2×D5311yescharacteristic
349C2×C4×D4C2×D5311yescharacteristic
350C22×C22⋊C4—21no—
351#489—1064no—
352#476—1064no—
353C2×#106—532no—
354C2×C4⋊C8—531no—
355C2×#103—532no—
356C2×#103—532no—
357C2×#102—532no—
358C2×#102—532no—
359C2×#102—532no—
360C2×#102—532no—
361C2×C22⋊C8—532no—
362C2×C22⋊C8—532no—
363C2×#97—531no—
364#397—1061no—
365#397—1061no—
366#391—1061no—
367#380—532no—
368#380—532no—
369C4×C4⋊C4—531no—
370#340—532no—
371C53#2589811yescharacteristic, complemented, Sylow
372C25—21no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—24no—
376C2×#41—42no—
377C22×Q8—1061no—
378C22×Q8C22×D5311yescharacteristic
379C22×D4—21no—
380C22×D4—21no—
381C22×D4—22no—
382C22×D4—22no—
383C22×D4—24no—
384C22×D4—21no—
385C22×D4—42no—
386C22×D4C22⋊D5311yescharacteristic
387C22×D4C22×D5311yescharacteristic
388C22×D4C22×D5311yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C22×C8—532no—
394C22×C8—532no—
395C4⋊Q8—1062no—
396C4⋊Q8—1062no—
397#124—22no—
398#124—2124no—
399#124C22⋊D5314yes—
400#123—28no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403#121—28no—
404#121—22no—
405#121—22no—
406#120—28no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4C22⋊D5314yes—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—532no—
413C2×C4⋊C4—532no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C22⋊D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C22⋊C4—28no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1062no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C22⋊C4C22⋊D5311yescharacteristic
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429#106—1064no—
430C4⋊C8—1062no—
431#103—1064no—
432#103—2122no—
433#102—1064no—
434#102—1064no—
435#102—1064no—
436#102—2122no—
437C22⋊C8—2122no—
438C22⋊C8—2122no—
439#97—1062no—
440#95—1061no—
441#95—1062no—
442#95—1062no—
443C24—21no—
444C24—24no—
445C24—21no—
446C24—44no—
447C24—42no—
448C24—42no—
449C24—82no—
450C24C2×C22⋊D5311yescharacteristic
451C24C2×C22⋊D5311yescharacteristic
452#41—48no—
453#41—44no—
454#41—44no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—2122no—
459C2×Q8—2122no—
460C2×Q8—2122no—
461C2×Q8C2×C22⋊D5312yes—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C2×D4—44no—
474C2×D4—48no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—82no—
479C2×D4#1171994714yes—
480C2×D4C2×C22⋊D5312yes—
481C2×D4C2×C22⋊D5312yes—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—44no—
495C22×C4—82no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C23×D5311yescharacteristic
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C2×C8—1062no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C4⋊C4—2122no—
517C4⋊C4C2×C22⋊D5312yes—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—22no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—2124no—
526C22⋊C4—2124no—
527C42—2122no—
528C42C2×C22⋊D5312yes—
529C42C2×C22⋊D5312yes—
530C23—22no—
531C23—24no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—48no—
543C23—48no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—88no—
552C23—84no—
553C23—88no—
554C23#1313324212yes—
555C23C22×C22⋊D5311yescharacteristic
556Q8—24no—
557Q8—41no—
558Q8—41no—
559Q8—4244no—
560D4—24no—
561D4—24no—
562D4—22no—
563D4—44no—
564D4—48no—
565D4—48no—
566D4—41no—
567D4—41no—
568D4—84no—
569D4#1313320414yescomplemented
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4#1313318612yes—
596C2×C4#1313318612yes—
597C2×C4C22×C22⋊D5311yescharacteristic
598C2×C4C22×C22⋊D5311yescharacteristic
599C2×C4C2×#1171994712yes—
600C2×C4#1313326212yes—
601C8—2122no—
602C8—2122no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—44no—
613C22—42no—
614C22—41no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—816no—
622C22—82no—
623C22—82no—
624C22C2×#1313318611yescharacteristic
625C22#1681126712yes—
626C22C2×#1313324211yescharacteristic
627C22#1681142111yescharacteristic
628C22#1681143412yes—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—4242no—
636C4—4242no—
637C4C2×#1313320411yescharacteristic
638C4#1681119011yescharacteristic
639C4#1681126712yes—
640C2—42no—
641C2—44no—
642C2—81no—
643C2—82no—
644C2#2027836211yescharacteristic
645C2#2028396711yescharacteristic
646C2#2028399211yescharacteristic
647C2#2028667612yes—
648C2#2028675112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#546C2
socleC23×C53C22×D4
hypercenterC2×#546D53
hypocenterC53#25898
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#546

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. C2×#16810913C2semidirect
3. #20279707C2semidirect
4. #20275908C2semidirect
5. #20275901C2semidirect
6. #20275648C2semidirect
7. #20275514C2semidirect
8. #20274655C2semidirect
9. #20274646C2semidirect
10. #20274540C2semidirect
12. C2×#13133217C22semidirect
16. C2×#13133148C22semidirect
17. C2×C4⋊#88322C22semidirect
18. C2×#13133141C22semidirect
19. C2×#13133141C22semidirect
25. C2×C22⋊#88322C22semidirect
26. C2×#13133128C22semidirect
31. #16810483C22semidirect
32. C4×C4⋊Dic53C22semidirect
33. #16810385C22semidirect
243. D4×C53#124semidirect
371. C53#25898semidirect
569. D4#13133204semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168109132
#202797072
#202759082
#202759012
#202756482
#202755142
#202746552
#202746462
#202745402
C2×C53×#5462
#2589853