Finite Group Databasebeta

← #27441253

Subgroups — #27441253

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×#16810929C211yesmaximal, characteristic, complemented
2C22×#13133140C211yesmaximal, characteristic
3#20278433C211yesmaximal, characteristic
4#20277547C211yesmaximal, characteristic
5#20276260C211yesmaximal, characteristic, complemented
6#20276105C211yesmaximal, characteristic, complemented
7#20276097C211yesmaximal, characteristic, complemented
8#20275865C211yesmaximal, characteristic, complemented
9#20274984C211yesmaximal, characteristic, complemented
10#20274966C211yesmaximal, characteristic, complemented
11#20274892C211yesmaximal, characteristic
12#20274738C211yesmaximal, characteristic
13#20274729C211yesmaximal, characteristic, complemented
14#20274651C211yesmaximal, characteristic
15C53×#1852C211yesmaximal, characteristic
16C23×Dic106C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133208C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133182C2211yescharacteristic
21#16810929—24no—
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133140—24no—
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×#13133140C2211yescharacteristic
34C2×#13133140C2211yescharacteristic
35#16810562—21no—
36#16810556—21no—
37#16810528C2211yescharacteristic
38#16810528C2211yescharacteristic
39#16810527—21no—
40#16810527—21no—
41#16810522C2211yescharacteristic
42#16810519C2211yescharacteristic
43#16810496—21no—
44#16810488C2211yescharacteristic
45#16810483—21no—
46#16810483C2211yescharacteristic
47#16810481—21no—
48C22⋊C4×Dic53—21no—
49C4×C22⋊Dic53C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
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
76#13133208—22no—
77#13133208—41no—
78#13133182—22no—
79#13133182—22no—
80#13133182—41no—
81#13133182—41no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53D411yescharacteristic
92C2×C22⋊Dic53D411yescharacteristic
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97#13133148—22no—
98#13133148—41no—
99#13133146—22no—
100#13133146—41no—
101#13133142—22no—
102#13133142—22no—
103#13133142—41no—
104#13133142—41no—
105#13133140—22no—
106#13133140—22no—
107#13133140—22no—
108#13133140D414yes—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124D411yescharacteristic
119#13133124D411yescharacteristic
120C23×C4×C53—21no—
121C23×C4×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53—22no—
126C2×C4⋊C4×C53—41no—
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132#107×C53—22no—
133#107×C53—41no—
134C22⋊C8×C53—22no—
135#95×C53—21no—
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×Q812yes—
150C22×C4×C53#4112yes—
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—24no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
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—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—21no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—42no—
196C4⋊C4×C53C2×D412yes—
197C22⋊C4×C53—42no—
198C22⋊Dic53—22no—
199C22⋊Dic53—22no—
200C22⋊Dic53—42no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—42no—
206C4⋊Dic53—81no—
207#11719919—42no—
208#11719919—42no—
209#11719919—42no—
210#11719919—81no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—21no—
215C4×Dic53—42no—
216C4×Dic53—42no—
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—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—84no—
251C2×C4×C53C22⋊Q812yes—
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
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—
292C8×C53—41no—
293C8×C53—41no—
294#88322—22no—
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—42no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53C2×C22⋊Q811yescharacteristic
317C4×C53#54311yescharacteristic
318C4×C53#54611yescharacteristic
319C4×C53#55111yescharacteristic
320Dic53—24no—
321Dic53—42no—
322Dic53—41no—
323Dic53—41no—
324Dic53—41no—
325Dic53—41no—
326Dic53—81no—
327Dic53—81no—
328C2×#479—531no—
329C22×#103—531no—
330#2442—531no—
331Q8×C22⋊C4—531no—
332#2097—531no—
333#2068—531no—
334#2066—531no—
335#2026—531no—
336#1865—531no—
337#1863—531no—
338#1852D5311yescharacteristic
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×#125—531no—
354C2×C4×Q8—531no—
355C2×C4×Q8—531no—
356C22×C4⋊C4C2×D5311yescharacteristic
357#479—1064no—
358C2×#107C2×D5311yescharacteristic
359C2×#107C2×D5311yescharacteristic
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⋊C8C2×D5311yescharacteristic
374#400—1061no—
375#398—1061no—
376#391—531no—
377#391—531no—
378#391—1061no—
379#391—1061no—
380#389—531no—
381#389—531no—
382#385—1061no—
383#382—531no—
384#380—531no—
385#380—1061no—
386#380—1061no—
387#380C2×D5311yescharacteristic
388C4×C22⋊C4—531no—
389C4×C22⋊C4—1061no—
390#340—531no—
391#340—531no—
392#340C2×D5311yescharacteristic
393#340C2×D5311yescharacteristic
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—1062no—
412C22×C8—1062no—
413C22×C8—2121no—
414C22×C8C22×D5311yescharacteristic
415C22×C8C22×D5311yescharacteristic
416#125—1062no—
417#125—2121no—
418C4×Q8—1062no—
419C4×Q8—1062no—
420C4×Q8—2121no—
421C4×Q8—2121no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—41no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
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—531no—
439C2×C22⋊C4—531no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445#107—22no—
446#107—41no—
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—22no—
460#95—21no—
461#95—531no—
462#95—531no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472C24D4×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—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—1061no—
508C22×C4—1064no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—2122no—
513C22×C4—2122no—
514C22×C4#1171994512yes—
515C22×C4D4×D5311yescharacteristic
516C22×C4Q8×D5312yes—
517C22×C4C23×D5311yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—532no—
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—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—4241no—
544C4⋊C4—4241no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4D4×D5312yes—
550C22⋊C4—42no—
551C22⋊C4—1062no—
552C22⋊C4—1062no—
553C22⋊C4—2122no—
554C42—1061no—
555C42—1061no—
556C42—1061no—
557C42—1061no—
558C42—2122no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1313319512yes—
565C23#1313323612yes—
566C23#1313324912yes—
567C23C2×D4×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—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1062no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—1061no—
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#1313319512yes—
622C2×C4C2×D4×D5311yescharacteristic
623C2×C4C2×Q8×D5311yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4C2×#1171994511yescharacteristic
626C2×C4#1313326211yescharacteristic
627C2×C4#1313326311yescharacteristic
628C8—41no—
629C8—41no—
630C8—1062no—
631C8—2121no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1681092914yescomplemented
639C22C2×#1313319511yescharacteristic
640C22#1681125511yescharacteristic
641C22#1681126511yescharacteristic
642C22C2×#1313323611yescharacteristic
643C22C2×#1313324911yescharacteristic
644C22#1681143311yescharacteristic
645C22#1681143911yescharacteristic
646C4—22no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—1064no—
652C4—4241no—
653C4—4241no—
654C4—2121no—
655C4—2122no—
656C4—2121no—
657C4—2121no—
658C4—2121no—
659C4C22⋊Q8×D5311yescharacteristic
660C4#1681124811yescharacteristic
661C4#1681126911yescharacteristic
662C4#1681130211yescharacteristic
663C2—24no—
664C2#2027838011yescharacteristic
665C2C2×#1681092911yescharacteristic
666C2#2028455511yescharacteristic
667C2#2028628911yescharacteristic
668C2#2028642611yescharacteristic
669C2#2028660011yescharacteristic
670C2#2028688411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1852D53
hypocenterC53#25957
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#16810929C2semidirect
5. #20276260C2semidirect
6. #20276105C2semidirect
7. #20276097C2semidirect
8. #20275865C2semidirect
9. #20274984C2semidirect
10. #20274966C2semidirect
13. #20274729C2semidirect
303. C22×C53#479semidirect
396. C53#25957semidirect
638. C22#16810929semidirect

Maximal subgroups

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

subgroupindex
C2×#168109292
C22×#131331402
#202784332
#202775472
#202762602
#202761052
#202760972
#202758652
#202749842
#202749662
#202748922
#202747382
#202747292
#202746512
C53×#18522
#2595753