Finite Group Databasebeta

← #27432614

Subgroups — #27432614

Order 13568.

864 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic
2C2×#16810867C211yesmaximal, characteristic
3#20279200C211yesmaximal, characteristic, complemented
4#20276196C211yesmaximal, characteristic, complemented
5#20276128C211yesmaximal, characteristic, complemented
6#20276093C211yesmaximal, characteristic, complemented
7#20276032C211yesmaximal, characteristic, complemented
8#20275483C211yesmaximal, characteristic, complemented
9#20275349C211yesmaximal, characteristic, complemented
10#20275049C211yesmaximal, characteristic, complemented
11#20275027C211yesmaximal, characteristic, complemented
12#20274900C211yesmaximal, characteristic, complemented
13#20274725C211yesmaximal, characteristic, complemented
14#20274436C211yesmaximal, characteristic, complemented
15C53×#2363C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810939—24no—
21#16810867—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810563—21no—
35#16810563—21no—
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810489C2211yescharacteristic
39#16810489C2211yescharacteristic
40#16810484—21no—
41#16810484—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#121×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#390—21no—
54C53×#390C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#386—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53—21no—
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63C2×#11719923—22no—
64C2×#11719923—22no—
65C2×#11719923—41no—
66C2×#11719917—22no—
67C2×#11719917—22no—
68C2×#11719917—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133196—22no—
74#13133196—22no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83#13133150—22no—
84#13133148—22no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—41no—
88#13133140—22no—
89#13133140—22no—
90#13133140—41no—
91#13133138—22no—
92#13133138—22no—
93#13133138—22no—
94C22⋊#88322—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C23×C4×C53C2311yescharacteristic
107#121×C53—22no—
108#121×C53D414yes—
109C4×D4×C53—22no—
110C4×D4×C53—41no—
111#117×C53—22no—
112#117×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—24no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—81no—
141C2×D4×C53—81no—
142C2×D4×C53C2×D412yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—42no—
151C22×C4×C53—42no—
152C22×C4×C53—81no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D412yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—81no—
179C4⋊C4×C53C2×D412yes—
180C4⋊C4×C53#4112yes—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—41no—
183C22⋊C4×C53—41no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—81no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—42no—
193C42×C53—81no—
194C22⋊Dic53—42no—
195#11719923—44no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203#11719919—42no—
204#11719917—44no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322#4112yes—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—22no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53#12112yes—
231C23×C53C22×D411yescharacteristic
232D4×C53—22no—
233D4×C53—22no—
234D4×C53—41no—
235D4×C53—41no—
236D4×C53—84no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
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—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—84no—
265C2×C4×C53#12112yes—
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×C4×C532−1+411yescharacteristic
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—84no—
285Dic106—41no—
286Dic106—41no—
287Dic106—41no—
288Dic106—41no—
289#88322—22no—
290#88322—41no—
291#88322—41no—
292#88322—41no—
293#25225—531nomaximal, Sylow
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53C2×#12111yescharacteristic
309C22×C53#54611yescharacteristic
310C22×C53#54711yescharacteristic
311C22×C53#57511yescharacteristic
312C22×C53#57511yescharacteristic
313C22×C53#57711yescharacteristic
314C22×C53#57711yescharacteristic
315C4×C53—22no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53—42no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53—42no—
322C4×C53—81no—
323C4×C53#52911yescharacteristic
324C4×C53#53911yescharacteristic
325C4×C53#54611yescharacteristic
326C4×C53#54711yescharacteristic
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330Dic53—81no—
331C2×#481—531no—
332C2×#468—531no—
333#2503—531no—
334#2363D5311yescharacteristic
335#2085—531no—
336#2073—531no—
337#2066—531no—
338#2055—531no—
339#1949—531no—
340#1927—531no—
341#1877—531no—
342#1873—531no—
343#1851—531no—
344#1804—531no—
345#1736—531no—
346C2×C53—42no—
347C2×C53—82no—
348C2×C53#242511yescharacteristic
349C2×C53#304011yescharacteristic
350C2×C53#304211yescharacteristic
351C2×C53#331711yescharacteristic
352C2×C53#332311yescharacteristic
353C2×C53#334811yescharacteristic
354C2×C53#335111yescharacteristic
355C22×SD16—531no—
356C22×M16—531no—
357C2×C22⋊Q8—531no—
358C2×#121C2×D5311yescharacteristic
359C2×C4×D4C2×D5311yescharacteristic
360C2×#117C2×D5311yescharacteristic
361C22×C4⋊C4—531no—
362#481—1064no—
363#468—1064no—
364C2×#107—531no—
365C2×#106—531no—
366C2×#106—531no—
367C2×C4⋊C8—531no—
368C2×C4⋊C8—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#102—531no—
373C2×#102—531no—
374C2×#102—531no—
375C2×C22⋊C8—531no—
376#400—1061no—
377#400—1061no—
378#391—531no—
379#391—531no—
380#390—21no—
381#390C2×D5311yescharacteristic
382#389C2×D5311yescharacteristic
383#388—21no—
384#386—21no—
385#382—531no—
386#382—531no—
387#381—1061no—
388#381—1061no—
389C4×C4⋊C4C2×D5311yescharacteristic
390C4×C22⋊C4—21no—
391C4×C22⋊C4C2×D5311yescharacteristic
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#336—531no—
399#336—531no—
400C53#2522511yescharacteristic, complemented, Sylow
401C22×Q8—531no—
402C22×D4—21no—
403C22×D4C22×D5311yescharacteristic
404C23×C4—21no—
405C23×C4—1061no—
406C23×C4C22×D5311yescharacteristic
407C2×SD16—1062no—
408C2×SD16—1062no—
409C2×SD16—2121no—
410C2×M16—1062no—
411C2×M16—1062no—
412C2×M16—2121no—
413C22×C8—531no—
414C22×C8—531no—
415C22×C8—531no—
416C22×C8—531no—
417C22⋊Q8—1062no—
418C22⋊Q8—1062no—
419#121—22no—
420#121D21214yes—
421C4×D4—22no—
422C4×D4—41no—
423#117—22no—
424#117—41no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4C22⋊D5311yescharacteristic
443C2×C22⋊C4C22⋊D5311yescharacteristic
444C2×C42—21no—
445C2×C42—21no—
446C2×C42C22×D5311yescharacteristic
447C2×C42C22×D5311yescharacteristic
448#107—1062no—
449#106—1062no—
450#106—1062no—
451C4⋊C8—1062no—
452C4⋊C8—2121no—
453#103—1062no—
454#103—1062no—
455#103—2121no—
456#102—1062no—
457#102—1062no—
458#102—1062no—
459C22⋊C8—1062no—
460#95—21no—
461#95—21no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95C22⋊D5311yescharacteristic
471#95C22⋊D5311yescharacteristic
472C24—21no—
473C24C2×C22⋊D5311yescharacteristic
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—81no—
487C2×D4—81no—
488C2×D4C2×D21212yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4C2×D21212yes—
507C22×C4#1171994512yes—
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C23×D5311yescharacteristic
514SD16—2124no—
515M16—2124no—
516C2×C8—532no—
517C2×C8—1061no—
518C2×C8—1061no—
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—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—81no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4C2×D21212yes—
542C4⋊C4#1171994712yes—
543C22⋊C4—24no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—2122no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—81no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23#1313319112yes—
571C23C22×C22⋊D5311yescharacteristic
572Q8—2121no—
573Q8—2121no—
574Q8—2121no—
575Q8—2121no—
576D4—22no—
577D4—22no—
578D4—41no—
579D4—41no—
580D4—84no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—84no—
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—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4#1313319112yes—
623C2×C4C22×D21211yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4C2×#1171994711yescharacteristic
626C2×C4C2×#1171994511yescharacteristic
627C2×C4#1313326211yescharacteristic
628C2×C4#1313326411yescharacteristic
629C8—1062no—
630C8—2121no—
631C8—2121no—
632C8—2121no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#1313319111yescharacteristic
648C22#1681126711yescharacteristic
649C22#1681127211yescharacteristic
650C22#1681142111yescharacteristic
651C22#1681142111yescharacteristic
652C22#1681143411yescharacteristic
653C22#1681143411yescharacteristic
654C4—22no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—4241no—
663C4—4241no—
664C4—4241no—
665C4—4241no—
666C4#1681115311yescharacteristic
667C4#1681121511yescharacteristic
668C4#1681126811yescharacteristic
669C4#1681127711yescharacteristic
670C2—42no—
671C2—82no—
672C2#2027826011yescharacteristic
673C2#2028420911yescharacteristic
674C2#2028422711yescharacteristic
675C2#2028684111yescharacteristic
676C2#2028693111yescharacteristic
677C2#2028723011yescharacteristic
678C2#2028726911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2363C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2363D53
hypocenterC53#25225
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
3. #20279200C2semidirect
4. #20276196C2semidirect
5. #20276128C2semidirect
6. #20276093C2semidirect
7. #20276032C2semidirect
8. #20275483C2semidirect
9. #20275349C2semidirect
10. #20275049C2semidirect
11. #20275027C2semidirect
12. #20274900C2semidirect
13. #20274725C2semidirect
14. #20274436C2semidirect
400. C53#25225semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168108672
#202792002
#202761962
#202761282
#202760932
#202760322
#202754832
#202753492
#202750492
#202750272
#202749002
#202747252
#202744362
C53×#23632
#2522553