Finite Group Databasebeta

← #27441251

Subgroups — #27441251

Order 13568.

946 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×#16810927C211yesmaximal, characteristic, complemented
2C22×#13133140C211yesmaximal, characteristic
3C22⋊C4×Dic106C211yesmaximal, characteristic
4#20276266C211yesmaximal, characteristic, complemented
5#20276107C211yesmaximal, characteristic, complemented
6#20276093C211yesmaximal, characteristic, complemented
7#20275863C211yesmaximal, characteristic, complemented
8#20274982C211yesmaximal, characteristic, complemented
9#20274968C211yesmaximal, characteristic, complemented
10#20274903C211yesmaximal, characteristic
11#20274895C211yesmaximal, characteristic
12#20274738C211yesmaximal, characteristic
13#20274727C211yesmaximal, characteristic, complemented
14#20274649C211yesmaximal, characteristic
15C53×#2442C211yesmaximal, characteristic
16C23×Dic106C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810927—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133140—24no—
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133140C2211yescharacteristic
35C2×#13133140C2211yescharacteristic
36C2×C22⋊#88322C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810529C2211yescharacteristic
39#16810527—21no—
40#16810527—21no—
41#16810483C2211yescharacteristic
42#16810481—21no—
43C22⋊C4×Dic53—21no—
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C22×C4⋊C4×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#398—21no—
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#385—21no—
57C53×#382C2211yescharacteristic
58C53×#380—21no—
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106—22no—
62C22×Dic106—22no—
63C22×Dic106—21no—
64C22×Dic106—22no—
65C22×Dic106—41no—
66C22×Dic106—41no—
67C22×Dic106Q814yes—
68C22×Dic106C2311yescharacteristic
69C22×Dic106C2311yescharacteristic
70C23×Dic53—21no—
71C22×#88322—22no—
72C22×#88322—22no—
73C22×#88322—41no—
74C22×#88322C2311yescharacteristic
75C22×#88322C2311yescharacteristic
76C22×#88322C2311yescharacteristic
77C22×#88322C2311yescharacteristic
78C4×Dic106—22no—
79C4×Dic106—22no—
80C4×Dic106—41no—
81C4×Dic106—41no—
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89#13133150—22no—
90#13133150—41no—
91#13133148—22no—
92#13133148—41no—
93C4⋊#88322—22no—
94C4⋊#88322—41no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98#13133140—22no—
99#13133140—22no—
100#13133140—41no—
101#13133140—41no—
102#13133140D414yes—
103C22⋊#88322—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111#125×C53—22no—
112#125×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—22no—
119C2×C4⋊C4×C53—41no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C42×C53C2311yescharacteristic
128C2×C42×C53C2311yescharacteristic
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134#95×C53D411yescharacteristic
135#95×C53D411yescharacteristic
136C24×C53C2×D411yescharacteristic
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—42no—
146C22×C4×C53—42no—
147C22×C4×C53—81no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×Q812yes—
154C22×C4×C53#4112yes—
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—24no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—28no—
171C2×Dic106—24no—
172C2×Dic106—21no—
173C2×Dic106—44no—
174C2×Dic106—44no—
175C2×Dic106—44no—
176C2×Dic106—42no—
177C2×Dic106—42no—
178C2×Dic106C2×D412yes—
179C2×Dic106C2×Q812yes—
180C2×Dic106C2×Q812yes—
181C2×Dic106#4112yes—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—42no—
190C4⋊C4×C53—42no—
191C4⋊C4×C53—42no—
192C4⋊C4×C53—81no—
193C4⋊C4×C53—81no—
194C4⋊C4×C53C2×D412yes—
195C22⋊C4×C53—22no—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—42no—
198C42×C53—21no—
199C42×C53—21no—
200C42×C53—21no—
201C42×C53—21no—
202C22⋊Dic53—42no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207#11719919—42no—
208#11719919—42no—
209C4×Dic53—42no—
210C4×Dic53—42no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—44no—
223C2×#88322C2×D412yes—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53C22⋊Q812yes—
229C23×C53#13512yes—
230C23×C53#13712yes—
231C23×C53C22×D411yescharacteristic
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—84no—
263C2×C4×C53C22⋊Q812yes—
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×Q811yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—48no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283Dic106—22no—
284Dic106—22no—
285Dic106—22no—
286Dic106—22no—
287Dic106—22no—
288Dic106—48no—
289Dic106—41no—
290Dic106—41no—
291Dic106C22⋊Q814yes—
292#88322—22no—
293#88322—41no—
294#88322—41no—
295#88322—41no—
296#25957—531nomaximal, Sylow
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53#47914yescomplemented
304C22×C53C2×C22⋊Q811yescharacteristic
305C22×C53D4211yescharacteristic
306C22×C53#54611yescharacteristic
307C22×C53C2×#13511yescharacteristic
308C22×C53C2×#13711yescharacteristic
309C22×C53#57711yescharacteristic
310C22×C53#57811yescharacteristic
311C4×C53—22no—
312C4×C53—41no—
313C4×C53—41no—
314C4×C53—42no—
315C4×C53—41no—
316C4×C53—41no—
317C4×C53—81no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53C2×C22⋊Q811yescharacteristic
321C4×C53#54311yescharacteristic
322C4×C53#54611yescharacteristic
323C4×C53#55111yescharacteristic
324Dic53—24no—
325Dic53—42no—
326Dic53—81no—
327Dic53—81no—
328C2×#479—531no—
329C22×#103—531no—
330#2442D5311yescharacteristic
331Q8×C22⋊C4—531no—
332#2097—531no—
333#2068—531no—
334#2066—531no—
335#2026—531no—
336#1865—531no—
337#1863—531no—
338#1852—531no—
339#1850—531no—
340#1807—531no—
341#1805—531no—
342#1787—531no—
343C2×C53—24no—
344C2×C53#243711yescharacteristic
345C2×C53C2×#47911yescharacteristic
346C2×C53#308211yescharacteristic
347C2×C53#327011yescharacteristic
348C2×C53#328211yescharacteristic
349C2×C53#330011yescharacteristic
350C2×C53#332011yescharacteristic
351C23×Q8—531no—
352C23×C8—531no—
353C2×#125C2×D5311yescharacteristic
354C2×C4×Q8—531no—
355C2×C4×Q8—531no—
356C22×C4⋊C4C2×D5311yescharacteristic
357#479—1064no—
358C2×#107—531no—
359C2×#107—531no—
360C2×#106—531no—
361C2×#106—531no—
362C2×C4⋊C8—531no—
363C2×C4⋊C8—531no—
364C2×#103—531no—
365C2×#103—531no—
366C2×#103—531no—
367C2×#103—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—1064no—
373C2×C22⋊C8—531no—
374#400—21no—
375#398—21no—
376#391—531no—
377#391—531no—
378#391—1061no—
379#391—1061no—
380#389C2×D5311yescharacteristic
381#389C2×D5311yescharacteristic
382#385—21no—
383#382C2×D5311yescharacteristic
384#380—21no—
385#380—531no—
386#380—1061no—
387#380C2×D5311yescharacteristic
388C4×C22⋊C4—1061no—
389C4×C22⋊C4C2×D5311yescharacteristic
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396C53#2595711yescharacteristic, complemented, Sylow
397C22×Q8—534no—
398C22×Q8—531no—
399C22×Q8—531no—
400C22×Q8—1062no—
401C22×Q8—1062no—
402C22×Q8—1062no—
403C22×Q8—1061no—
404C22×Q8—2121no—
405C22×Q8—2121no—
406C23×C4—21no—
407C23×C4—1061no—
408C23×C4C22×D5311yescharacteristic
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8—531no—
413C22×C8—1062no—
414C22×C8—1062no—
415C22×C8—2121no—
416#125—22no—
417#125—41no—
418C4×Q8—1062no—
419C4×Q8—1062no—
420C4×Q8—2121no—
421C4×Q8—2121no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—41no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4C22⋊D5311yescharacteristic
440C2×C22⋊C4C22⋊D5311yescharacteristic
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5311yescharacteristic
445#107—1062no—
446#107—2121no—
447#106—1062no—
448#106—2121no—
449C4⋊C8—1062no—
450C4⋊C8—2121no—
451#103—534no—
452#103—1062no—
453#103—1062no—
454#103—1062no—
455#103—1062no—
456#103—1062no—
457#103—2121no—
458#103—2121no—
459C22⋊C8—1062no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95C22⋊D5311yescharacteristic
471#95C22⋊D5311yescharacteristic
472C24C2×C22⋊D5311yescharacteristic
473C2×Q8—532no—
474C2×Q8—532no—
475C2×Q8—532no—
476C2×Q8—532no—
477C2×Q8—1068no—
478C2×Q8—1064no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—1061no—
482C2×Q8—1061no—
483C2×Q8—1061no—
484C2×Q8—1061no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—2122no—
488C2×Q8—2122no—
489C2×Q8—2124no—
490C2×Q8—2124no—
491C2×Q8—2124no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—1061no—
504C22×C4—1064no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4Q8×D5312yes—
511C22×C4#1171994912yes—
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C22×C4C23×D5311yescharacteristic
518C2×C8—532no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—2124no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4D4×D5312yes—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—42no—
553C22⋊C4—2122no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—2122no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1313319312yes—
565C23#1313323712yes—
566C23#1313324712yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—534no—
569Q8—1062no—
570Q8—1062no—
571Q8—1062no—
572Q8—1062no—
573Q8—1062no—
574Q8—2128no—
575Q8—2121no—
576Q8—2121no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
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—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—2128no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4#1313319312yes—
622C2×C4C2×D4×D5311yescharacteristic
623C2×C4C2×Q8×D5311yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4C2×#1171994911yescharacteristic
626C2×C4#1313326211yescharacteristic
627C2×C4#1313326311yescharacteristic
628C8—1062no—
629C8—2121no—
630C8—2121no—
631C8—2121no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1681092714yescomplemented
639C22C2×#1313319311yescharacteristic
640C22D4×C22⋊D5311yescharacteristic
641C22#1681126111yescharacteristic
642C22C2×#1313323711yescharacteristic
643C22C2×#1313324711yescharacteristic
644C22#1681143411yescharacteristic
645C22#1681143811yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—1064no—
656C4—4241no—
657C4—4241no—
658C4—2122no—
659C4C22⋊Q8×D5311yescharacteristic
660C4#1681124811yescharacteristic
661C4#1681126411yescharacteristic
662C4#1681130011yescharacteristic
663C2—24no—
664C2#2027837811yescharacteristic
665C2C2×#1681092711yescharacteristic
666C2#2028455111yescharacteristic
667C2#2028628111yescharacteristic
668C2#2028641811yescharacteristic
669C2#2028659611yescharacteristic
670C2#2028688011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2442C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2442D53
hypocenterC53#25957
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#16810927C2semidirect
4. #20276266C2semidirect
5. #20276107C2semidirect
6. #20276093C2semidirect
7. #20275863C2semidirect
8. #20274982C2semidirect
9. #20274968C2semidirect
13. #20274727C2semidirect
303. C22×C53#479semidirect
396. C53#25957semidirect
638. C22#16810927semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C22×#131331402
C22⋊C4×Dic1062
#202762662
#202761072
#202760932
#202758632
#202749822
#202749682
#202749032
#202748952
#202747382
#202747272
#202746492
C53×#24422
#2595753