Finite Group Databasebeta

← #27434402

Subgroups — #27434402

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810927C211yesmaximal, characteristic, complemented
2C2×#16810695C211yesmaximal, characteristic
3#20279637C211yesmaximal, characteristic
4#20276151C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275918C211yesmaximal, characteristic, complemented
7#20275918C211yesmaximal, characteristic, complemented
8#20275685C211yesmaximal, characteristic, complemented
9#20275473C211yesmaximal, characteristic, complemented
10#20275146C211yesmaximal, characteristic, complemented
11#20275136C211yesmaximal, characteristic, complemented
12#20274746C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic, complemented
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2562C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133196C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810927—24no—
22#16810695—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35#16810546—21no—
36#16810546—21no—
37#16810526—21no—
38#16810519—21no—
39#16810493C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#394—21no—
54C53×#388—21no—
55C53×#388—21no—
56C53×#380C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—41no—
65C22×#88322—22no—
66C22×#88322—22no—
67C22×#88322—41no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—22no—
74#13133196—41no—
75#13133196—41no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91#13133150—22no—
92#13133150—41no—
93#13133150—41no—
94#13133148—22no—
95C4⋊#88322D414yes—
96#13133140—22no—
97#13133140—22no—
98#13133140—22no—
99#13133140—22no—
100C22⋊#88322—22no—
101C22⋊#88322—22no—
102#13133128—22no—
103#13133124—21no—
104#13133124—21no—
105C22×D4×C53D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C23×C4×C53C2311yescharacteristic
108#125×C53—22no—
109#125×C53—22no—
110C4×D4×C53—22no—
111C4×D4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C22⋊C4×C53D411yescharacteristic
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132C24×C53C2×D411yescharacteristic
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—41no—
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—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106#4112yes—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—42no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53—81no—
185C42×C53—81no—
186C42×C53C2×D412yes—
187C42×C53#4112yes—
188C22⋊Dic53—42no—
189C22⋊Dic53—42no—
190C22⋊Dic53—81no—
191C22⋊Dic53—81no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203C4×Dic53—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—44no—
215C2×#88322C2×D412yes—
216C2×#88322C2×D412yes—
217C2×#88322#4112yes—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53#13712yes—
232C23×C53C22×D411yescharacteristic
233D4×C53—44no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
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—41no—
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—41no—
263C2×C4×C53—41no—
264C2×C4×C53#12112yes—
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—22no—
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—
284Dic106—22no—
285Dic106—22no—
286Dic106—41no—
287Dic106—41no—
288#88322—22no—
289#88322—22no—
290#88322—22no—
291#88322—41no—
292#25353—531nomaximal, Sylow
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53C2×#12111yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53#54711yescharacteristic
310C22×C53C2×#13711yescharacteristic
311C22×C53#57511yescharacteristic
312C22×C53#57611yescharacteristic
313C22×C53#57811yescharacteristic
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53—81no—
321C4×C53—81no—
322C4×C53#52911yescharacteristic
323C4×C53#54211yescharacteristic
324C4×C53#54611yescharacteristic
325C4×C53#54811yescharacteristic
326Dic53—42no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330C2×#479—531no—
331C2×#435—531no—
332#2562D5311yescharacteristic
333#2538—531no—
334#2077—531no—
335#2075—531no—
336#2036—531no—
337#2036—531no—
338#1992—531no—
339#1947—531no—
340#1894—531no—
341#1892—531no—
342#1809—531no—
343#1808—531no—
344#1763—531no—
345C2×C53—42no—
346C2×C53—81no—
347C2×C53#242711yescharacteristic
348C2×C53#305611yescharacteristic
349C2×C53#306111yescharacteristic
350C2×C53#330911yescharacteristic
351C2×C53#331211yescharacteristic
352C2×C53#334611yescharacteristic
353C2×C53#334911yescharacteristic
354C22×M16—531no—
355C23×C8—531no—
356C2×#125C2×D5311yescharacteristic
357C2×C22⋊Q8—531no—
358C2×C22⋊Q8—531no—
359C2×C4×Q8—531no—
360C2×C4×D4C2×D5311yescharacteristic
361#479—1064no—
362#435—1064no—
363C2×#107—531no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#106—531no—
367C2×C4⋊C8—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×C22⋊C8—531no—
373C2×C22⋊C8—531no—
374C2×#97—531no—
375#400—21no—
376#396—1061no—
377#396—1061no—
378#394—21no—
379#390—1061no—
380#389—1061no—
381#388—21no—
382#388—21no—
383#384—531no—
384#380—531no—
385#380—531no—
386#380C2×D5311yescharacteristic
387#380C2×D5311yescharacteristic
388#380C2×D5311yescharacteristic
389#380C2×D5311yescharacteristic
390C4×C4⋊C4C2×D5311yescharacteristic
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399C53#2535311yescharacteristic, complemented, Sylow
400C22×Q8—531no—
401C22×D4C22⋊D5311yescharacteristic
402C23×C4C22×D5311yescharacteristic
403C23×C4C22×D5311yescharacteristic
404C2×M16—1062no—
405C2×M16—1062no—
406C2×M16—2121no—
407C22×C8—531no—
408C22×C8—531no—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—1062no—
412C22×C8—1062no—
413C22×C8—2121no—
414#125—22no—
415#125—22no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—2121no—
419C22⋊Q8—2121no—
420C4×Q8—1062no—
421C4×Q8—1062no—
422C4×D4—22no—
423C4×D4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4C22⋊D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C4⋊C4C22×D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4C22⋊D5311yescharacteristic
448C2×C22⋊C4C22⋊D5311yescharacteristic
449C2×C42—21no—
450C2×C42—1061no—
451C2×C42C22×D5311yescharacteristic
452#107—1062no—
453#107—2121no—
454#107—2121no—
455#106—1062no—
456C4⋊C8—534no—
457#103—1062no—
458#103—1062no—
459#103—1062no—
460#103—1062no—
461C22⋊C8—1062no—
462C22⋊C8—1062no—
463#97—1062no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—1061no—
470#95—1061no—
471C24C2×C22⋊D5311yescharacteristic
472C24C2×C22⋊D5311yescharacteristic
473C2×Q8—532no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—41no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507M16—2124no—
508C2×C8—532no—
509C2×C8—532no—
510C2×C8—532no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—2124no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—4241no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—4241no—
550C22⋊C4—4241no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C42—81no—
554C42—2122no—
555C42D4×D5312yes—
556C42#1171994712yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1313324712yes—
571C23C22×C22⋊D5311yescharacteristic
572Q8—1062no—
573Q8—1062no—
574Q8—2121no—
575Q8—2121no—
576D4—44no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
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—41no—
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—1062no—
608C2×C4—2121no—
609C2×C4—2121no—
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#1313318912yes—
620C2×C4#1313318912yes—
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4#41×D5311yescharacteristic
623C2×C4#41×D5311yescharacteristic
624C2×C4C2×#1171994711yescharacteristic
625C2×C4#1313326311yescharacteristic
626C2×C4#1313326411yescharacteristic
627C8—1062no—
628C8—1062no—
629C8—1062no—
630C8—2121no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22C2×#1313318911yescharacteristic
646C22#1681126111yescharacteristic
647C22#1681127211yescharacteristic
648C22C2×#1313324711yescharacteristic
649C22#1681142111yescharacteristic
650C22#1681142611yescharacteristic
651C22#1681143811yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—4241no—
661C4—4241no—
662C4—4241no—
663C4—2122no—
664C4#1681115211yescharacteristic
665C4#1681124211yescharacteristic
666C4#1681126611yescharacteristic
667C4#1681128111yescharacteristic
668C2—42no—
669C2—81no—
670C2#2027829511yescharacteristic
671C2#2028434711yescharacteristic
672C2#2028438211yescharacteristic
673C2#2028671511yescharacteristic
674C2#2028676011yescharacteristic
675C2#2028720011yescharacteristic
676C2#2028724511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2562C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2562D53
hypocenterC53#25353
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2562

Complements

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

NH ≅ G/Ntype
1. C2×#16810927C2semidirect
4. #20276151C2semidirect
5. #20276141C2semidirect
6. #20275918C2semidirect
7. #20275918C2semidirect
8. #20275685C2semidirect
9. #20275473C2semidirect
10. #20275146C2semidirect
11. #20275136C2semidirect
12. #20274746C2semidirect
13. #20274742C2semidirect
14. #20274548C2semidirect
399. C53#25353semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C2×#168106952
#202796372
#202761512
#202761412
#202759182
#202756852
#202754732
#202751462
#202751362
#202747462
#202747422
#202745482
C53×#25622
#2535353