Finite Group Databasebeta

← #27386802

Subgroups — #27386802

Order 13568.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810624C211yesmaximal, characteristic
2C2×M16×Dic53C211yesmaximal, characteristic, complemented
3#20277206C211yesmaximal, characteristic
4C2×#16810522C211yesmaximal, characteristic, complemented
5#20275414C212yesmaximal, complemented
6#20275407C212yesmaximal, complemented
7#20275400C212yesmaximal, complemented
8#20275393C212yesmaximal, complemented
9#20274435C211yesmaximal, characteristic, complemented
10#20274434C211yesmaximal, characteristic, complemented
11C2×C53×#417C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C22×C4⋊Dic53—21no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17#16810624—24no—
18C2×#13133141C2212yes—
19C2×#13133138C2212yes—
20M16×Dic53—24no—
21C2×#13133127C2212yes—
22C2×C8×Dic53C2212yes—
23#16810530C2211yescharacteristic
24#16810530C2211yescharacteristic
25#16810526C2211yescharacteristic
26#16810526C2211yescharacteristic
27#16810522—21no—
28#16810522—21no—
29#16810522C2212yes—
30#16810522C2212yes—
31#16810503—21no—
32#16810503—21no—
33C4×C22⋊Dic53—21no—
34C4×C22⋊Dic53—21no—
35C2×#13133124—21no—
36#16810386C2212yes—
37#16810386C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40C22×#41×C53C2211yescharacteristic
41C22×M16×C53C2211yescharacteristic
42C22×C4⋊C4×C53C2211yescharacteristic
43C53×#417C418yescomplemented
44C2×#103×C53C2212yes—
45C2×#102×C53C2212yes—
46C23×Dic53—21no—
47C23×Dic53D411yescharacteristic
48C23×Dic53D411yescharacteristic
49C2×#11719917—22no—
50C2×#11719917—22no—
51C2×#11719917D414yes—
52C22×#88322C2312yes—
53#13133176—22no—
54#13133176—22no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—22no—
60C2×C4⋊Dic53—44no—
61C2×C4⋊Dic53C2312yes—
62C2×C4⋊Dic53C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×C22⋊Dic53—21no—
69C2×C4×Dic53—22no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53—22no—
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53D411yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75C2×C4×Dic53C2311yescharacteristic
76#13133141—24no—
77#13133138—24no—
78#13133127—24no—
79C8×Dic53—24no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84#13133124—22no—
85#13133124—22no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53C2×C414yes—
92C22×Q8×C53C2311yescharacteristic
93C22×D4×C53—21no—
94C22×D4×C53C2311yescharacteristic
95C23×C4×C53—21no—
96C23×C4×C53—21no—
97C23×C4×C53C2311yescharacteristic
98C2×M16×C53—22no—
99C2×M16×C53—22no—
100C2×M16×C53C2×C414yes—
101C22×C8×C53C2312yes—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2×C414yes—
105C2×C4⋊C4×C53C2312yes—
106#103×C53C2×C418yescomplemented
107#102×C53C2×C418yescomplemented
108C24×C53—21no—
109C24×C53C2×D411yescharacteristic
110#41×C53—28no—
111#41×C53—28no—
112#41×C53—44no—
113#41×C53—44no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53C22×C412yes—
120C2×D4×C53—24no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53C22×C412yes—
129C22×C4×C53—24no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—24no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D412yes—
140C22×C4×C53#4112yes—
141C22×C4×C53#4112yes—
142C22×C4×C53#4112yes—
143C22×C4×C53#4112yes—
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C22×Dic53—41no—
153C22×Dic53—41no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53C2×D411yescharacteristic
158C22×Dic53C2×D411yescharacteristic
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161M16×C53—28no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53C22×C414yes—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C22×C414yes—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—42no—
173C4⋊Dic53—22no—
174C4⋊Dic53—22no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—44no—
180C4⋊Dic53—42no—
181C4⋊Dic53—44no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—42no—
189#11719917—28no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322C2×D414yes—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C4×D412yes—
205C23×C53#12112yes—
206C23×C53#12412yes—
207C23×C53C22×D411yescharacteristic
208Q8×C53—22no—
209Q8×C53—22no—
210Q8×C53—41no—
211Q8×C53—41no—
212D4×C53—22no—
213D4×C53—22no—
214D4×C53—48no—
215D4×C53—41no—
216D4×C53—41no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—48no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—48no—
225C2×C4×C53#11714yes—
226C2×C4×C53C4×D414yes—
227C2×C4×C53C4×D412yes—
228C2×C4×C53#12112yes—
229C2×C4×C53#12314yes—
230C2×C4×C53#12412yes—
231C2×C4×C53#12614yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—44no—
249C2×Dic53—44no—
250C2×Dic53—42no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—42no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—84no—
260C2×Dic53AGL(1,ℤ/8)12yes—
261C2×Dic53#13712yes—
262C8×C53—24no—
263#88322—24no—
264#21093—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53#38814yes—
279C22×C53C2×C4×D411yescharacteristic
280C22×C53C2×#12111yescharacteristic
281C22×C53C2×#12411yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C53C2×#13711yescharacteristic
284C22×C53#57511yescharacteristic
285C22×C53#57511yescharacteristic
286C4×C53—44no—
287C4×C53—42no—
288C4×C53#38814yes—
289C4×C53C2×#11711yescharacteristic
290C4×C53C2×C4×D411yescharacteristic
291C4×C53C2×#12311yescharacteristic
292C4×C53C2×#12611yescharacteristic
293Dic53—22no—
294Dic53—22no—
295Dic53—41no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299Dic53—82no—
300C2×#417—531no—
301C2×#417D5311yescharacteristic
302C2×C4×M16—531no—
303#2297—531no—
304C2×#389—531no—
305#1936—532no—
306#1935—532no—
307#1934—532no—
308#1933—532no—
309#1736—531no—
310#1736—531no—
311C2×C53—24no—
312C2×C53—82no—
313C2×C53C2×#38811yescharacteristic
314C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C53C4×#13711yescharacteristic
316C2×C53#305511yescharacteristic
317C2×C53#305611yescharacteristic
318C2×C53#314111yescharacteristic
319C2×C53#314211yescharacteristic
320C22×#41C2×D5311yescharacteristic
321C22×M16—531no—
322C22×M16C2×D5311yescharacteristic
323C2×#117—531no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4—1061no—
326C22×C4⋊C4C2×D5311yescharacteristic
327C22×C42—531no—
328#417—1064no—
329#417Dic5318yescomplemented
330C2×#103—532no—
331C2×#103C2×D5312yes—
332C2×#102—532no—
333C2×#102C2×D5312yes—
334C4×M16—1064no—
335C2×#97—532no—
336C2×C4×C8—532no—
337#391—531no—
338#391—531no—
339#390—531no—
340#390—531no—
341#389—532no—
342#389—532no—
343#389—1061no—
344#389—1061no—
345#386—1061no—
346#386—1061no—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C2×#95—1061no—
350#340—532no—
351#340—532no—
352#340—532no—
353#340—532no—
354C53#2109311yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41C2×Dic5314yes—
361C22×Q8C22×D5311yescharacteristic
362C22×D4—21no—
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—1061no—
369C23×C4C22×D5311yescharacteristic
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—534no—
373C2×M16—1062no—
374C2×M16—1062no—
375C2×M16C2×Dic5314yes—
376C22×C8—532no—
377C22×C8C22×D5312yes—
378#117—1062no—
379#117—1062no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—532no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—2124no—
391C2×C4⋊C4C2×Dic5314yes—
392C2×C4⋊C4C22×D5312yes—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1062no—
403C2×C42—1062no—
404C2×C42—1061no—
405#103—1064no—
406#103C2×Dic5318yescomplemented
407#102—1064no—
408#102C2×Dic5318yescomplemented
409#97—1064no—
410C4×C8—1064no—
411#95—1062no—
412#95—1062no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—21no—
418C24D4×D5311yescharacteristic
419#41—28no—
420#41—28no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8C22×Dic5312yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C22×Dic5312yes—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—2121no—
458C22×C4—2121no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4#1171994512yes—
463C22×C4#1171994512yes—
464C22×C4D4×D5311yescharacteristic
465C22×C4D4×D5312yes—
466C22×C4#1171994712yes—
467C22×C4#1171994712yes—
468C22×C4C22×Dic5312yes—
469C22×C4C23×D5311yescharacteristic
470M16—28no—
471M16—1068no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—534no—
475C2×C8—1062no—
476C2×C8—1062no—
477C2×C8C22×Dic5314yes—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—1062no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1062no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—2124no—
487C4⋊C4—2124no—
488C4⋊C4—2122no—
489C4⋊C4C22×Dic5314yes—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—1062no—
496C42—1062no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C42—1061no—
501C42—2122no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23D4×Dic5312yes—
514C23#1313319012yes—
515C23#1313320312yes—
516C23C2×D4×D5311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—4244no—
547C2×C4—2124no—
548C2×C4—2124no—
549C2×C4—2122no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2122no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2122no—
558C2×C4#1313317614yes—
559C2×C4D4×Dic5312yes—
560C2×C4D4×Dic5314yes—
561C2×C4#1313319012yes—
562C2×C4#1313319914yes—
563C2×C4#1313320312yes—
564C2×C4#1313321014yes—
565C2×C4C23×Dic5311yescharacteristic
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994511yescharacteristic
570C2×C4C2×#1171994511yescharacteristic
571C8—24no—
572C8—1064no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1681051614yes—
587C22C2×D4×Dic5311yescharacteristic
588C22C2×#1313319011yescharacteristic
589C22C2×#1313320311yescharacteristic
590C22AGL(1,ℤ/8)×D5311yescharacteristic
591C22#137×D5311yescharacteristic
592C22#1681141911yescharacteristic
593C22#1681142211yescharacteristic
594C4—44no—
595C4—42no—
596C4—1062no—
597C4—1062no—
598C4—4242no—
599C4—4241no—
600C4—4241no—
601C4—2121no—
602C4—2121no—
603C4#1681051614yes—
604C4C2×#1313317611yescharacteristic
605C4C2×D4×Dic5311yescharacteristic
606C4C2×#1313319911yescharacteristic
607C4C2×#1313321011yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1681051611yescharacteristic
611C2AGL(1,ℤ/8)×Dic5311yescharacteristic
612C2#137×Dic5311yescharacteristic
613C2#2028433711yescharacteristic
614C2#2028434811yescharacteristic
615C2#2028510611yescharacteristic
616C2#2028511311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenterC2×#417D53
hypocenterC53#21093
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic53C2semidirect
4. C2×#16810522C2semidirect
5. #20275414C2semidirect
6. #20275407C2semidirect
7. #20275400C2semidirect
8. #20275393C2semidirect
9. #20274435C2semidirect
10. #20274434C2semidirect
43. C53×#417C4semidirect
106. #103×C53C2×C4semidirect
107. #102×C53C2×C4semidirect
329. #417Dic53semidirect
354. C53#21093semidirect
406. #103C2×Dic53semidirect
408. #102C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106242
C2×M16×Dic532
#202772062
C2×#168105222
#202754142
#202754072
#202754002
#202753932
#202744352
#202744342
C2×C53×#4172
#2109353