Finite Group Databasebeta

← #27420545

Subgroups — #27420545

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic
3#20280073C211yesmaximal, characteristic
4#20276050C211yesmaximal, characteristic, complemented
5#20276045C211yesmaximal, characteristic, complemented
6#20275858C211yesmaximal, characteristic
7#20275848C211yesmaximal, characteristic
8#20275681C211yesmaximal, characteristic, complemented
9#20275652C211yesmaximal, characteristic, complemented
10#20275070C211yesmaximal, characteristic
11#20275070C211yesmaximal, characteristic
12#20274983C211yesmaximal, characteristic, complemented
13#20274969C211yesmaximal, characteristic, complemented
14#20274730C211yesmaximal, characteristic, complemented
15C2×C53×#557C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810920—24no—
19C4×#11719924—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810558—21no—
32#16810534—21no—
33#16810525C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810494—21no—
38#16810494—21no—
39#16810489C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C53×#557—24no—
51C53×#557—24no—
52C2×C4⋊Q8×C53—21no—
53C2×C4⋊Q8×C53C2211yescharacteristic
54C2×#125×C53—21no—
55C2×#125×C53—21no—
56C2×#125×C53C2211yescharacteristic
57C2×#125×C53C2211yescharacteristic
58C2×C4×Q8×C53—21no—
59C2×C4×Q8×C53C2211yescharacteristic
60C2×C4×Q8×C53C2211yescharacteristic
61C2×C4×Q8×C53C2211yescharacteristic
62C2×C4×Q8×C53C2211yescharacteristic
63C22×D212C2311yescharacteristic
64C22×C4×D53—21no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924Q814yescomplemented
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C4×D212—22no—
73C4×D212—22no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84#13133150—22no—
85#13133148—22no—
86#13133148—22no—
87C4⋊#88322—22no—
88#13133141—22no—
89#13133141—22no—
90#13133141—22no—
91#13133137—22no—
92#13133137—22no—
93#13133137—22no—
94C4×#88322—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99C22×Q8×C53—21no—
100C22×Q8×C53D411yescharacteristic
101C22×Q8×C53C2311yescharacteristic
102C4⋊Q8×C53—28no—
103C4⋊Q8×C53—22no—
104C4⋊Q8×C53—22no—
105#125×C53—28no—
106#125×C53—28no—
107#125×C53—22no—
108#125×C53—22no—
109#125×C53—22no—
110#125×C53D414yes—
111C4×Q8×C53—28no—
112C4×Q8×C53—22no—
113C4×Q8×C53—22no—
114C4×Q8×C53—22no—
115C4×Q8×C53—22no—
116C4×Q8×C53—22no—
117C4×Q8×C53—22no—
118C4×Q8×C53—22no—
119C4×Q8×C53D414yes—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53—21no—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C4⋊C4×C53D411yescharacteristic
130C2×C4⋊C4×C53D411yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C4⋊C4×C53C2311yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C42×C53—21no—
135C2×C42×C53—21no—
136C2×C42×C53—21no—
137C2×C42×C53C2311yescharacteristic
138C2×C42×C53C2311yescharacteristic
139C2×C42×C53C2311yescharacteristic
140C23×D53—21no—
141C2×Q8×C53—24no—
142C2×Q8×C53—22no—
143C2×Q8×C53—22no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—41no—
148C2×Q8×C53—42no—
149C2×Q8×C53—42no—
150C2×Q8×C53C2×D412yes—
151C2×Q8×C53C2×Q812yes—
152C2×Q8×C53#4112yes—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2411yescharacteristic
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212—21no—
172C2×D212C2×Q812yes—
173C2×C4×D53—42no—
174C2×C4×D53—81no—
175C2×C4×D53—81no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—24no—
181C4⋊C4×C53—24no—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—24no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—22no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53C2×D412yes—
191C4⋊C4×C53C2×D412yes—
192C4⋊C4×C53C2×D412yes—
193C42×C53—24no—
194C42×C53—24no—
195C42×C53—24no—
196C42×C53—21no—
197C42×C53—21no—
198C42×C53—21no—
199C42×C53—21no—
200C42×C53C2×D412yes—
201C42×C53C2×D412yes—
202#11719924—28no—
203#11719921—42no—
204#11719921—42no—
205#11719921—42no—
206#11719921—81no—
207#11719921—81no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211C4⋊Dic53—21no—
212C4⋊Dic53—21no—
213C4⋊Dic53—21no—
214#11719919—81no—
215#11719919—81no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322C2×Q812yes—
228C23×C53C22×D411yescharacteristic
229C22×D53—22no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—41no—
235C22×D53—41no—
236Q8×C53—22no—
237Q8×C53—22no—
238Q8×C53—22no—
239Q8×C53—44no—
240Q8×C53—48no—
241Q8×C53C22⋊Q814yes—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53#12012yes—
270C2×C4×C53#12012yes—
271C2×C4×C53#13512yes—
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C22×Q811yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C532−1+411yescharacteristic
277C2×C4×C532−1+411yescharacteristic
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287D212—22no—
288D212—22no—
289D212—41no—
290D212—41no—
291C4×D53—84no—
292#88322—22no—
293#88322—41no—
294#88322—41no—
295#88322—41no—
296#24334—531nomaximal, Sylow
297C22×C53C2×#12011yescharacteristic
298C22×C53#54811yescharacteristic
299C22×C53D4×Q811yescharacteristic
300C22×C53C2×#13511yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C2×D53—81no—
311C4×C53—24no—
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—22no—
315C4×C53—41no—
316C4×C53—42no—
317C4×C53—42no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53—42no—
321C4×C53C2×C22⋊Q811yescharacteristic
322C4×C53#53611yescharacteristic
323C4×C53#54811yescharacteristic
324C4×C53D4×Q811yescharacteristic
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328C2×#557D5311yescharacteristic, complemented
329C2×#478—531no—
330C2×C4×SD16—531no—
331#2580—531no—
332#2059—531no—
333#2058—531no—
334#2025—531no—
335#2023—531no—
336#1991—531no—
337#1984—531no—
338#1880—531no—
339#1880—531no—
340#1865—531no—
341#1863—531no—
342#1805—531no—
343C2×C53#240911yescharacteristic
344C2×C53#300211yescharacteristic
345C2×C53#301011yescharacteristic
346C2×C53#336511yescharacteristic
347C2×C53#337111yescharacteristic
348C2×C53#338111yescharacteristic
349C2×C53#338811yescharacteristic
350D53—82no—
351C22×SD16—531no—
352#557—24no—
353#557—24no—
354C2×C4⋊Q8—21no—
355C2×C4⋊Q8C2×D5311yescharacteristic
356C2×#125—21no—
357C2×#125—21no—
358C2×#125C2×D5311yescharacteristic
359C2×#125C2×D5311yescharacteristic
360C2×C4×Q8—21no—
361C2×C4×Q8C2×D5311yescharacteristic
362C2×C4×Q8C2×D5311yescharacteristic
363C2×C4×Q8C2×D5311yescharacteristic
364C2×C4×Q8C2×D5311yescharacteristic
365C2×C4×D4—531no—
366#478—1064no—
367C4×SD16—1064no—
368C2×#107—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×C4⋊C8—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×#102—531no—
378C2×C4×C8—531no—
379#399—1061no—
380#393—1061no—
381#390—531no—
382#390—531no—
383#389—531no—
384#389—531no—
385#385—1061no—
386#385—1061no—
387#382—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396#336—531no—
397#336—531no—
398C53#2433411yescharacteristic, complemented, Sylow
399C22×Q8—21no—
400C22×Q8C22⋊D5311yescharacteristic
401C22×Q8C22×D5311yescharacteristic
402C22×D4—531no—
403C23×C4—1061no—
404C2×SD16—534no—
405C2×SD16—1062no—
406C2×SD16—1062no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8—531no—
410C22×C8—531no—
411C4⋊Q8—28no—
412C4⋊Q8—22no—
413C4⋊Q8—22no—
414#125—28no—
415#125—28no—
416#125—22no—
417#125—22no—
418#125—22no—
419#125C22⋊D5314yes—
420C4×Q8—28no—
421C4×Q8—22no—
422C4×Q8—22no—
423C4×Q8—22no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8C22⋊D5314yes—
429C4×D4—1062no—
430C4×D4—1062no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4C22⋊D5311yescharacteristic
445C2×C4⋊C4C22⋊D5311yescharacteristic
446C2×C4⋊C4C22⋊D5311yescharacteristic
447C2×C4⋊C4C22×D5311yescharacteristic
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4—1061no—
455C2×C42—21no—
456C2×C42—21no—
457C2×C42—21no—
458C2×C42C22×D5311yescharacteristic
459C2×C42C22×D5311yescharacteristic
460C2×C42C22×D5311yescharacteristic
461#107—1062no—
462#106—1062no—
463#106—1062no—
464C4⋊C8—1062no—
465#103—1062no—
466#103—1062no—
467#103—1062no—
468#102—1062no—
469#102—1062no—
470#102—1062no—
471C4×C8—1062no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476C24—1061no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—41no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8Q8×D5312yes—
487C2×Q8#1171994912yes—
488C2×Q8C2×C22⋊D5312yes—
489C2×D4—532no—
490C2×D4—1061no—
491C2×D4—1061no—
492C2×D4—1061no—
493C2×D4—1061no—
494C2×D4—1061no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—4241no—
503C22×C4—4241no—
504C22×C4—2122no—
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C23×D5311yescharacteristic
512SD16—1068no—
513C2×C8—532no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—4241no—
546C4⋊C4—4241no—
547C4⋊C4C2×C22⋊D5312yes—
548C4⋊C4C2×C22⋊D5312yes—
549C4⋊C4C2×C22⋊D5312yes—
550C22⋊C4—4241no—
551C22⋊C4—4241no—
552C22⋊C4—2122no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42C2×C22⋊D5312yes—
563C42C2×C22⋊D5312yes—
564C23—1062no—
565C23—2121no—
566C23—2121no—
567C23—2121no—
568C23—2121no—
569C23—2121no—
570C23—2121no—
571C23C22×C22⋊D5311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—44no—
576Q8—48no—
577Q8#1313319314yes—
578D4—1062no—
579D4—1062no—
580D4—2121no—
581D4—2121no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—4244no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4#1313318612yes—
620C2×C4#1313318612yes—
621C2×C4#1313323712yes—
622C2×C4C22×C22⋊D5311yescharacteristic
623C2×C4C22×C22⋊D5311yescharacteristic
624C2×C4C2×Q8×D5311yescharacteristic
625C2×C4C2×#1171994911yescharacteristic
626C2×C4#1313326511yescharacteristic
627C2×C4#1313326511yescharacteristic
628C8—1062no—
629C8—2121no—
630C8—2121no—
631C8—2121no—
632C22—4241no—
633C22—4241no—
634C22—4241no—
635C22—4241no—
636C22—2122no—
637C22—2122no—
638C22—2122no—
639C22C2×#1313318611yescharacteristic
640C22#1681128411yescharacteristic
641C22Q8×C22⋊D5311yescharacteristic
642C22C2×#1313323711yescharacteristic
643C22#1681142111yescharacteristic
644C22#1681143111yescharacteristic
645C22#1681143111yescharacteristic
646C4—22no—
647C4—22no—
648C4—24no—
649C4—22no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—42no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4C2×#1313319311yescharacteristic
660C4#1681119711yescharacteristic
661C4#1681128411yescharacteristic
662C4Q8×C22⋊D5311yescharacteristic
663C2—4242no—
664C2#2027807411yescharacteristic
665C2#2028393311yescharacteristic
666C2#2028400111yescharacteristic
667C2#2028741111yescharacteristic
668C2#2028748311yescharacteristic
669C2#2028758611yescharacteristic
670C2#2028765411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#557C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenterC2×#557D53
hypocenterC53#24334
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #20276050C2semidirect
5. #20276045C2semidirect
8. #20275681C2semidirect
9. #20275652C2semidirect
12. #20274983C2semidirect
13. #20274969C2semidirect
14. #20274730C2semidirect
15. C2×C53×#557C2semidirect
67. C2×#11719924Q8semidirect
328. C2×#557D53semidirect
398. C53#24334semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C2×C4×#117199242
#202800732
#202760502
#202760452
#202758582
#202758482
#202756812
#202756522
#202750702
#202749832
#202749692
#202747302
C2×C53×#5572
#2433453