Finite Group Databasebeta

← #27434268

Subgroups — #27434268

Order 13568.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810927C211yesmaximal, characteristic, complemented
2C2×D4×#88322C211yesmaximal, characteristic
3#20279624C211yesmaximal, characteristic
4#20276152C211yesmaximal, characteristic, complemented
5#20276141C211yesmaximal, characteristic, complemented
6#20275918C211yesmaximal, characteristic, complemented
7#20275911C211yesmaximal, characteristic, complemented
8#20275681C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275136C211yesmaximal, characteristic, complemented
11#20275136C211yesmaximal, characteristic, complemented
12#20274742C211yesmaximal, characteristic, complemented
13#20274738C211yesmaximal, characteristic, complemented
14C4×#13133140C211yesmaximal, characteristic, complemented
15C53×#2562C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133196C2211yescharacteristic
20C2×C4×Dic106C2211yescharacteristic
21#16810927—24no—
22D4×#88322—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810554—21no—
36#16810537—21no—
37#16810511—21no—
38#16810511—21no—
39#16810489C2211yescharacteristic
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
62C22×#88322—22no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322—22no—
66C22×#88322—41no—
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⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89#13133150—22no—
90#13133150—41no—
91#13133148—22no—
92#13133148—41no—
93C4⋊#88322D414yes—
94#13133140—22no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98C22⋊#88322—22no—
99C22⋊#88322—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
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—22no—
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—42no—
191C22⋊Dic53—42no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198#11719919—42no—
199#11719919—42no—
200#11719919—42no—
201C4×Dic53—42no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—44no—
212C2×#88322—44no—
213C2×#88322C2×D412yes—
214C2×#88322C2×D412yes—
215C2×#88322#4112yes—
216C23×C53—21no—
217C23×C53—21no—
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#13512yes—
229C23×C53#13512yes—
230C23×C53C22×D411yescharacteristic
231D4×C53—28no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
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—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—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#12112yes—
263C2×C4×C53#12112yes—
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C532−1+411yescharacteristic
270C2×Dic53—22no—
271C2×Dic53—41no—
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—
282Dic106—22no—
283Dic106—22no—
284Dic106—41no—
285Dic106—41no—
286#88322—22no—
287#88322—22no—
288#88322—22no—
289#88322—41no—
290#25343—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
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×C53C2×#12111yescharacteristic
306C22×C53#54611yescharacteristic
307C22×C53#54711yescharacteristic
308C22×C53C2×#13511yescharacteristic
309C22×C53C2×#13511yescharacteristic
310C22×C53#57711yescharacteristic
311C22×C53#57811yescharacteristic
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53—81no—
319C4×C53—81no—
320C4×C53#52911yescharacteristic
321C4×C53#54211yescharacteristic
322C4×C53#54611yescharacteristic
323C4×C53#54811yescharacteristic
324Dic53—42no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328C2×#479—531no—
329C2×C8×D4—531no—
330#2562D5311yescharacteristic
331#2536—531no—
332#2077—531no—
333#2075—531no—
334#2036—531no—
335#2035—531no—
336#1991—531no—
337#1949—531no—
338#1892—531no—
339#1892—531no—
340#1808—531no—
341#1807—531no—
342C4×#103—531no—
343C2×C53—42no—
344C2×C53—42no—
345C2×C53#242711yescharacteristic
346C2×C53#305011yescharacteristic
347C2×C53#306311yescharacteristic
348C2×C53#330011yescharacteristic
349C2×C53#330111yescharacteristic
350C2×C53#333611yescharacteristic
351C2×C53#333711yescharacteristic
352C23×C8—531no—
353C23×C8—531no—
354C2×#125C2×D5311yescharacteristic
355C2×C22⋊Q8—531no—
356C2×C22⋊Q8—531no—
357C2×C4×Q8—531no—
358C2×C4×D4C2×D5311yescharacteristic
359#479—1064no—
360C8×D4—1064no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#106—531no—
364C2×#106—531no—
365C2×C4⋊C8—531no—
366C2×#103—531no—
367C2×#103—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×C22⋊C8—531no—
371C2×C22⋊C8—531no—
372C2×C4×C8—531no—
373#400—21no—
374#398—1061no—
375#394—21no—
376#394—1061no—
377#388—21no—
378#388—21no—
379#388—1061no—
380#388—1061no—
381#382—531no—
382#380—531no—
383#380—531no—
384#380C2×D5311yescharacteristic
385#380C2×D5311yescharacteristic
386#380C2×D5311yescharacteristic
387#380C2×D5311yescharacteristic
388C4×C4⋊C4C2×D5311yescharacteristic
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397C53#2534311yescharacteristic, complemented, Sylow
398C22×Q8—531no—
399C22×D4C22⋊D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—1062no—
407C22×C8—1062no—
408C22×C8—1062no—
409C22×C8—1062no—
410C22×C8—2121no—
411C22×C8—2121no—
412#125—22no—
413#125—22no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—2121no—
417C22⋊Q8—2121no—
418C4×Q8—1062no—
419C4×Q8—1062no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22⋊D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4C22⋊D5311yescharacteristic
444C2×C22⋊C4C22⋊D5311yescharacteristic
445C2×C42—21no—
446C2×C42—1061no—
447C2×C42C22×D5311yescharacteristic
448#107—1062no—
449#107—2121no—
450#106—1062no—
451#106—2121no—
452C4⋊C8—534no—
453#103—1062no—
454#103—1062no—
455#103—1062no—
456#103—1062no—
457C22⋊C8—1062no—
458C22⋊C8—1062no—
459C4×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—
469C24C2×C22⋊D5311yescharacteristic
470C24C2×C22⋊D5311yescharacteristic
471C2×Q8—532no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C22×C4—21no—
481C22×C4—21no—
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—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C23×D5311yescharacteristic
505C2×C8—532no—
506C2×C8—532no—
507C2×C8—532no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—2124no—
518C2×C8—2124no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—2122no—
546C22⋊C4—2122no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C42—81no—
550C42—2122no—
551C42D4×D5312yes—
552C42#1171994712yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1313323712yes—
566C23#1313323712yes—
567C23C22×C22⋊D5311yescharacteristic
568Q8—1062no—
569Q8—1062no—
570Q8—2121no—
571Q8—2121no—
572D4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
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—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1062no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4#1313318912yes—
616C2×C4#1313318912yes—
617C2×C4C2×D4×D5311yescharacteristic
618C2×C4#41×D5311yescharacteristic
619C2×C4#41×D5311yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4#1313326311yescharacteristic
622C2×C4#1313326411yescharacteristic
623C8—1062no—
624C8—1062no—
625C8—1062no—
626C8—2121no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1313318911yescharacteristic
642C22#1681126111yescharacteristic
643C22#1681127211yescharacteristic
644C22C2×#1313323711yescharacteristic
645C22C2×#1313323711yescharacteristic
646C22#1681143411yescharacteristic
647C22#1681143811yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4—2122no—
660C4#1681115211yescharacteristic
661C4#1681124211yescharacteristic
662C4#1681126611yescharacteristic
663C4#1681128111yescharacteristic
664C2—42no—
665C2—42no—
666C2#2027829511yescharacteristic
667C2#2028429311yescharacteristic
668C2#2028440811yescharacteristic
669C2#2028659611yescharacteristic
670C2#2028661111yescharacteristic
671C2#2028707711yescharacteristic
672C2#2028708811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2562C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#2562D53
hypocenterC53#25343
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. #20276152C2semidirect
5. #20276141C2semidirect
6. #20275918C2semidirect
7. #20275911C2semidirect
8. #20275681C2semidirect
9. #20275483C2semidirect
10. #20275136C2semidirect
11. #20275136C2semidirect
12. #20274742C2semidirect
13. #20274738C2semidirect
14. C4×#13133140C2semidirect
397. C53#25343semidirect

Maximal subgroups

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

subgroupindex
C2×#168109272
C2×D4×#883222
#202796242
#202761522
#202761412
#202759182
#202759112
#202756812
#202754832
#202751362
#202747422
#202747382
C4×#131331402
C53×#25622
#2534353