Finite Group Databasebeta

← #27416239

Subgroups — #27416239

Order 13568.

1690 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810756C211yesmaximal, characteristic, complemented
2C22×#13133137C211yesmaximal, characteristic, complemented
3#20278071C211yesmaximal, characteristic, complemented
4#20275848C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275144C211yesmaximal, characteristic, complemented
7#20275063C212yesmaximal, complemented
8#20275032C212yesmaximal, complemented
9#20275008C211yesmaximal, characteristic, complemented
10#20274801C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic, complemented
12C23×D212C2211yescharacteristic
13C22×#11719922C2212yes—
14C23×#88322C2211yescharacteristic
15C2×#13133191C2212yes—
16C2×#13133185—21no—
17#16810756—24no—
18C2×#13133138C2212yescomplemented
19C2×#13133137—24no—
20C2×#13133137C2212yes—
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yescomplemented
25#16810540—21no—
26#16810532—21no—
27#16810525C2212yes—
28#16810483C2211yescharacteristic
29#16810478—21no—
30#16810385C2212yescomplemented
31#16810385C2212yescomplemented
32#16810385C2212yescomplemented
33#16810378C2212yescomplemented
34C22×#41×C53—21no—
35C53×#547—24no—
36C53×#547—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic, complemented
38C2×#123×C53—21no—
39C2×#123×C53—21no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2211yescharacteristic
42C2×#121×C53C2211yescharacteristic, complemented
43C2×C4×D4×C53C2211yescharacteristic
44C2×C4×D4×C53C2211yescharacteristic, complemented
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic, complemented
47C24×D53—21no—
48C22×D212—22no—
49C22×D212—24no—
50C22×D212—22no—
51C22×D212—21no—
52C22×D212—42no—
53C22×D212C2312yes—
54C22×C22⋊D53—22no—
55C2×#11719922—24no—
56C2×#11719922—24no—
57C2×#11719922—42no—
58C22×#88322—22no—
59C22×#88322—42no—
60C22×#88322C2312yes—
61C22×#88322C2312yes—
62#13133191—24no—
63#13133191—42no—
64#13133185—44no—
65C2×C4⋊Dic53C2312yes—
66C2×C22⋊Dic53—21no—
67C2×#11719921—22no—
68C2×#11719921—22no—
69C2×#11719921—22no—
70#13133138—42no—
71#13133137—24no—
72#13133137—24no—
73#13133137—24no—
74C22⋊#88322—22no—
75C22⋊#88322—42no—
76#13133124—21no—
77C2×#41×C53—28no—
78C2×#41×C53—44no—
79C2×#41×C53—44no—
80C22×Q8×C53—21no—
81C22×D4×C53—21no—
82C22×D4×C53—21no—
83C22×D4×C53D411yescharacteristic
84C22×D4×C53C2311yescharacteristic
85C23×C4×C53—21no—
86C23×C4×C53—21no—
87C23×C4×C53—21no—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C4⋊Q8×C53—22no—
91C4⋊Q8×C53—22no—
92#123×C53—28no—
93#123×C53—28no—
94C22⋊Q8×C53—28no—
95#121×C53—22no—
96#121×C53—22no—
97#121×C53—22no—
98#121×C53D414yescomplemented
99C4×D4×C53—22no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D414yescomplemented
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C22⋊C4×C53D411yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C24×C53—21no—
121C24×C53C2×D411yescharacteristic
122C24×C53C2×D411yescharacteristic
123C23×D53—22no—
124C23×D53—44no—
125C23×D53—42no—
126C23×D53—42no—
127C23×D53—42no—
128C23×D53—82no—
129#41×C53—432no—
130C2×Q8×C53—24no—
131C2×Q8×C53—42no—
132C2×Q8×C53—42no—
133C2×D4×C53—22no—
134C2×D4×C53—22no—
135C2×D4×C53—24no—
136C2×D4×C53—24no—
137C2×D4×C53—21no—
138C2×D4×C53—22no—
139C2×D4×C53—22no—
140C2×D4×C53—42no—
141C2×D4×C53—42no—
142C2×D4×C53—42no—
143C2×D4×C53—42no—
144C2×D4×C53D814yescomplemented
145C2×D4×C53C2×D412yes—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—22no—
157C22×C4×C53—22no—
158C22×C4×C53—42no—
159C22×C4×C53—42no—
160C22×C4×C53—42no—
161C22×C4×C53—42no—
162C22×C4×C53—42no—
163C22×C4×C53—42no—
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D412yes—
168C22×C4×C53C2×D412yes—
169C22×C4×C53C2×D411yescharacteristic
170C22×C4×C53C2411yescharacteristic
171C2×C22⋊D53—44no—
172C2×C22⋊D53—44no—
173C2×C22⋊D53—82no—
174C22×Dic53—22no—
175C2×D212—28no—
176C2×D212—22no—
177C2×D212—22no—
178C2×D212—22no—
179C2×D212—22no—
180C2×D212—22no—
181C2×D212—48no—
182C2×D212—44no—
183C2×D212—44no—
184C2×D212—42no—
185C2×D212#4114yescomplemented
186C4⋊C4×C53—24no—
187C4⋊C4×C53—24no—
188C4⋊C4×C53—24no—
189C4⋊C4×C53—22no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53C2×D412yes—
194C4⋊C4×C53C2×D412yes—
195C4⋊C4×C53C2×D412yes—
196C22⋊C4×C53—24no—
197C22⋊C4×C53—24no—
198C22⋊C4×C53—22no—
199C22⋊C4×C53—24no—
200C22⋊C4×C53—22no—
201C42×C53—21no—
202C42×C53—21no—
203C22⋊Dic53—42no—
204#11719922—48no—
205#11719921—44no—
206#11719921—44no—
207#11719921—82no—
208C4⋊Dic53—22no—
209C4⋊Dic53—22no—
210C2×#88322—22no—
211C2×#88322—22no—
212C2×#88322—22no—
213C2×#88322—22no—
214C2×#88322—22no—
215C2×#88322—22no—
216C2×#88322—44no—
217C23×C53—22no—
218C23×C53—22no—
219C23×C53—21no—
220C23×C53—22no—
221C23×C53—21no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—42no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—42no—
228C23×C53#12012yes—
229C23×C53C2×D812yes—
230C23×C53C2×D812yes—
231C23×C53AGL(1,ℤ/8)12yes—
232C23×C53C22×D411yescharacteristic
233C22×D53—24no—
234C22×D53—48no—
235C22×D53—42no—
236C22×D53—42no—
237C22×D53—42no—
238C22×D53—42no—
239C22×D53—42no—
240C22×D53—42no—
241C22×D53—88no—
242C22×D53—88no—
243C22×D53—84no—
244C22×D53—84no—
245Q8×C53—48no—
246D4×C53—28no—
247D4×C53—24no—
248D4×C53—48no—
249D4×C53—48no—
250D4×C53—41no—
251D4×C53—41no—
252C2×C4×C53—21no—
253C2×C4×C53—22no—
254C2×C4×C53—21no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—24no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—22no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—22no—
264C2×C4×C53—22no—
265C2×C4×C53—48no—
266C2×C4×C53—48no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—48no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53—42no—
277C2×C4×C53#12012yes—
278C2×C4×C53C2×D812yes—
279C2×C4×C53C22×D411yescharacteristic
280C2×C4×C53C22×D411yescharacteristic
281C2×C4×C53C2×#4112yes—
282C2×C4×C53H(5,𝔽2)12yes—
283C22⋊D53—88no—
284C2×Dic53—42no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287D212—24no—
288D212—24no—
289D212—48no—
290D212—42no—
291D212—42no—
292#88322—42no—
293#88322—42no—
294#23960—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—24no—
298C22×C53—24no—
299C22×C53—22no—
300C22×C53—24no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—82no—
308C22×C53—82no—
309C22×C53#44714yes—
310C22×C53C2×#12011yescharacteristic
311C22×C53#54612yes—
312C22×C53C22×D811yescharacteristic
313C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
314C22×C53#57712yes—
315C2×D53—44no—
316C2×D53—44no—
317C2×D53—44no—
318C2×D53—816no—
319C2×D53—82no—
320C2×D53—82no—
321C2×D53—82no—
322C2×D53—82no—
323C4×C53—22no—
324C4×C53—22no—
325C4×C53—41no—
326C4×C53—42no—
327C4×C53—42no—
328C4×C53—42no—
329C4×C53—41no—
330C4×C53#52511yescharacteristic
331C4×C53#53511yescharacteristic
332C4×C53#54612yes—
333Dic53—82no—
334C2×#547D5311yescharacteristic, complemented
335C2×#447—531no—
336C22×#102—531no—
337#2408—531no—
338#2023—532no—
339#1999—532no—
340#1893—531no—
341#1879—532no—
342#1874—532no—
343#1869—531no—
344#1823—531no—
345C2×C53—24no—
346C2×C53—42no—
347C2×C53—82no—
348C2×C53#240411yescharacteristic
349C2×C53C2×#44711yescharacteristic
350C2×C53#301011yescharacteristic
351C2×C53#329412yes—
352C2×C53#330612yes—
353D53—84no—
354C22×#41—21no—
355C23×D4—531no—
356C22×D8—532no—
357C23×C8—531no—
358#547—24no—
359#547—24no—
360C2×C4⋊Q8C2×D5311yescharacteristic, complemented
361C2×#123—21no—
362C2×#123—21no—
363C2×C22⋊Q8—21no—
364C2×#121—532no—
365C2×#121C2×D5311yescharacteristic
366C2×#121C2×D5311yescharacteristic, complemented
367C2×#120—1061no—
368C2×C4×D4C2×D5311yescharacteristic
369C2×C4×D4C2×D5311yescharacteristic, complemented
370C22×C4⋊C4C2×D5311yescharacteristic
371C22×C4⋊C4C2×D5311yescharacteristic, complemented
372#447—1064no—
373C2×#102—532no—
374C2×#102—532no—
375C2×#102—532no—
376C2×#102—532no—
377C2×#102—1064no—
378C2×C22⋊C8—531no—
379C2×C22⋊C8—532no—
380#394—1061no—
381#392—1061no—
382#390—532no—
383#380—531no—
384#379—1061no—
385#340—532no—
386#340—532no—
387#340—532no—
388#336—532no—
389C53#2396011yescharacteristic, complemented, Sylow
390C25—1061no—
391C2×#41—28no—
392C2×#41—44no—
393C2×#41—44no—
394C22×Q8—21no—
395C22×D4—21no—
396C22×D4—21no—
397C22×D4—532no—
398C22×D4—1064no—
399C22×D4—1062no—
400C22×D4—1062no—
401C22×D4—1062no—
402C22×D4—1061no—
403C22×D4—2122no—
404C22×D4C22⋊D5311yescharacteristic
405C22×D4C22×D5311yescharacteristic
406C23×C4—21no—
407C23×C4—21no—
408C23×C4—21no—
409C23×C4C22×D5311yescharacteristic
410C23×C4C22×D5311yescharacteristic
411C2×D8—1064no—
412C2×D8—1064no—
413C2×D8—2122no—
414C22×C8—532no—
415C22×C8—532no—
416C22×C8—1062no—
417C22×C8—2122no—
418C4⋊Q8—22no—
419C4⋊Q8—22no—
420#123—28no—
421#123—28no—
422C22⋊Q8—28no—
423#121—22no—
424#121—22no—
425#121—22no—
426#121—1064no—
427#121—2122no—
428#121C22⋊D5314yescomplemented
429#120—2124no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—22no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—22no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4—532no—
441C2×C4⋊C4C22⋊D5311yescharacteristic
442C2×C4⋊C4C22⋊D5314yescomplemented
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1062no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1062no—
452C2×C22⋊C4—1062no—
453C2×C22⋊C4C22⋊D5311yescharacteristic
454C2×C22⋊C4C22⋊D5311yescharacteristic
455C2×C42C22×D5311yescharacteristic
456#102—1064no—
457#102—1064no—
458#102—1064no—
459#102—2122no—
460C22⋊C8—1062no—
461C22⋊C8—2122no—
462#95—1061no—
463C24—21no—
464C24—1062no—
465C24—4242no—
466C24—2124no—
467C24—2122no—
468C24—2122no—
469C24—2122no—
470C24C2×C22⋊D5311yescharacteristic
471C24C2×C22⋊D5311yescharacteristic
472#41—432no—
473C2×Q8—24no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—534no—
488C2×D4—1068no—
489C2×D4—1062no—
490C2×D4—1062no—
491C2×D4—1062no—
492C2×D4—1062no—
493C2×D4—1062no—
494C2×D4—4242no—
495C2×D4—2128no—
496C2×D4—2124no—
497C2×D4—2124no—
498C2×D4—2124no—
499C2×D4—2124no—
500C2×D4—2122no—
501C2×D4#1171992214yescomplemented
502C2×D4C2×C22⋊D5312yes—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—24no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—1062no—
522C22×C4C2×C22⋊D5311yescharacteristic
523C22×C4C2×C22⋊D5311yescharacteristic
524C22×C4C2×C22⋊D5311yescharacteristic
525C22×C4C2×C22⋊D5312yes—
526C22×C4C2×C22⋊D5312yes—
527C22×C4C2×C22⋊D5311yescharacteristic
528C22×C4C23×D5311yescharacteristic
529D8—2128no—
530C2×C8—1062no—
531C2×C8—1062no—
532C2×C8—1062no—
533C2×C8—1062no—
534C2×C8—1062no—
535C2×C8—1062no—
536C2×C8—2124no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—1062no—
545C4⋊C4—1062no—
546C4⋊C4C2×C22⋊D5312yes—
547C4⋊C4C2×C22⋊D5312yes—
548C4⋊C4C2×C22⋊D5312yes—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—22no—
552C22⋊C4—24no—
553C22⋊C4—22no—
554C22⋊C4—4242no—
555C22⋊C4—2124no—
556C22⋊C4—2122no—
557C22⋊C4—2124no—
558C42—21no—
559C42—21no—
560C23—22no—
561C23—22no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—1064no—
572C23—4244no—
573C23—4248no—
574C23—4248no—
575C23—4244no—
576C23—2128no—
577C23—2122no—
578C23—2122no—
579C23—2122no—
580C23—2122no—
581C23—2122no—
582C23—2122no—
583C23#1313318612yes—
584C23C2×#1171992212yes—
585C23C2×#1171992212yes—
586C23#1313324312yes—
587C23C22×C22⋊D5311yescharacteristic
588Q8—48no—
589D4—28no—
590D4—24no—
591D4—48no—
592D4—48no—
593D4—41no—
594D4—41no—
595D4—1064no—
596D4—1064no—
597D4—4248no—
598D4—2128no—
599D4—2122no—
600D4—2122no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—2122no—
627C2×C4—2122no—
628C2×C4—2122no—
629C2×C4#1313318612yes—
630C2×C4C2×#1171992212yes—
631C2×C4C22×C22⋊D5311yescharacteristic
632C2×C4C22×C22⋊D5311yescharacteristic
633C2×C4#41×D5312yes—
634C2×C4#1313326312yes—
635C8—2122no—
636C8—2122no—
637C22—22no—
638C22—24no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—82no—
650C22—82no—
651C22—42416no—
652C22—4242no—
653C22—4242no—
654C22—4242no—
655C22—4242no—
656C22—2124no—
657C22—2124no—
658C22—2124no—
659C22#1681075714yes—
660C22C2×#1313318611yescharacteristic
661C22#1681126112yes—
662C22C22×#1171992211yescharacteristic
663C22C2×#1313324311yescharacteristic
664C22#1681143112yes—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—4242no—
673C4#1681113011yescharacteristic
674C4#1681118711yescharacteristic
675C4#1681126112yes—
676C2—24no—
677C2—42no—
678C2—82no—
679C2—4244no—
680C2#2027803011yescharacteristic
681C2C2×#1681075711yescharacteristic
682C2#2028400111yescharacteristic
683C2#2028652212yes—
684C2#2028667012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#23960
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810756C2semidirect
2. C22×#13133137C2semidirect
3. #20278071C2semidirect
4. #20275848C2semidirect
5. #20275711C2semidirect
6. #20275144C2semidirect
7. #20275063C2semidirect
8. #20275032C2semidirect
9. #20275008C2semidirect
10. #20274801C2semidirect
11. C2×C53×#547C2semidirect
18. C2×#13133138C22semidirect
24. C2×C22⋊#88322C22semidirect
30. #16810385C22semidirect
31. #16810385C22semidirect
32. #16810385C22semidirect
33. #16810378C22semidirect
37. C2×C4⋊Q8×C53C22semidirect
42. C2×#121×C53C22semidirect
44. C2×C4×D4×C53C22semidirect
46. C22×C4⋊C4×C53C22semidirect
98. #121×C53D4semidirect
110. C2×C4⋊C4×C53D4semidirect
144. C2×D4×C53D8semidirect
185. C2×D212#41semidirect
334. C2×#547D53semidirect
360. C2×C4⋊Q8C2×D53semidirect
366. C2×#121C2×D53semidirect
369. C2×C4×D4C2×D53semidirect
371. C22×C4⋊C4C2×D53semidirect
389. C53#23960semidirect
428. #121C22⋊D53semidirect
442. C2×C4⋊C4C22⋊D53semidirect
501. C2×D4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168107562
C22×#131331372
#202780712
#202758482
#202757112
#202751442
#202750632
#202750322
#202750082
#202748012
C2×C53×#5472
#2396053