Finite Group Databasebeta

← #27433424

Subgroups — #27433424

Order 13568.

848 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810977C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic, complemented
3#20277809C211yesmaximal, characteristic, complemented
4#20276110C211yesmaximal, characteristic, complemented
5#20276089C211yesmaximal, characteristic, complemented
6#20275951C211yesmaximal, characteristic, complemented
7#20275946C211yesmaximal, characteristic
8#20275648C211yesmaximal, characteristic, complemented
9#20275648C211yesmaximal, characteristic, complemented
10#20275643C211yesmaximal, characteristic
11#20275434C211yesmaximal, characteristic, complemented
12#20275346C211yesmaximal, characteristic, complemented
13#20275332C211yesmaximal, characteristic, complemented
14C4×#13133138C211yesmaximal, characteristic, complemented
15C53×#2544C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810977—24no—
18#16810923—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35C2×#13133138C2211yescharacteristic
36C2×C4×#88322C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810518—21no—
40#16810505—21no—
41#16810504—21no—
42C4×C4⋊Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810474—21no—
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#127×C53C2211yescharacteristic
51C2×#124×C53C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C53×#394—21no—
55C53×#394—21no—
56C53×#391C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#386—21no—
59C53×#386—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209D414yescomplemented
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78#13133150—22no—
79#13133148—22no—
80C4⋊#88322—22no—
81C4⋊#88322—41no—
82C4⋊#88322—41no—
83#13133141—22no—
84#13133141—22no—
85#13133141—22no—
86#13133141—41no—
87#13133138—22no—
88#13133138—22no—
89#13133138—22no—
90#13133138—22no—
91#13133138—22no—
92#13133138—22no—
93#13133138—22no—
94#13133138—41no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×Q8×C53C2311yescharacteristic
102C22×D4×C53—21no—
103C22×D4×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C22×D4×C53C2311yescharacteristic
106C23×C4×C53—21no—
107#127×C53—22no—
108#127×C53—41no—
109#124×C53—22no—
110#124×C53—22no—
111#124×C53—22no—
112#124×C53—41no—
113C4×D4×C53—22no—
114C4×D4×C53—22no—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53—21no—
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C24×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—42no—
152C2×D4×C53—42no—
153C2×D4×C53—42no—
154C2×D4×C53—42no—
155C2×D4×C53—81no—
156C2×D4×C53—81no—
157C2×D4×C53#4112yes—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—42no—
161C22×C4×C53—81no—
162C22×C4×C53—81no—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2411yescharacteristic
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—21no—
173C22×Dic53—21no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—41no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—42no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—81no—
184C22⋊C4×C53—81no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53C2×D412yes—
190C42×C53#4112yes—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53C2×D412yes—
197C4⋊Dic53C2×D412yes—
198C4⋊Dic53#4112yes—
199#11719919—24no—
200#11719919—24no—
201#11719919—42no—
202#11719919—81no—
203C4×Dic53—42no—
204C4×Dic53—81no—
205C4×Dic53—81no—
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—21no—
217C2×#88322—21no—
218C23×C53—22no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53C22×D411yescharacteristic
233Q8×C53—41no—
234Q8×C53—41no—
235Q8×C53—41no—
236Q8×C53—41no—
237D4×C53—22no—
238D4×C53—22no—
239D4×C53—41no—
240D4×C53—41no—
241D4×C53—41no—
242D4×C53—41no—
243D4×C53—41no—
244D4×C53—41no—
245D4×C53—84no—
246D4×C53—84no—
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—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—84no—
269C2×C4×C53#12112yes—
270C2×C4×C53#12112yes—
271C2×C4×C53#13512yes—
272C2×C4×C53C22×D411yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C53H(5,𝔽2)11yescharacteristic
277C2×C4×C532−1+411yescharacteristic
278C2×Dic53—22no—
279C2×Dic53—22no—
280C2×Dic53—22no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290#88322—41no—
291#88322—41no—
292#88322—41no—
293#88322—41no—
294#25281—531nomaximal, Sylow
295C22×C53—42no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53—81no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53C2×#12111yescharacteristic
310C22×C53#54611yescharacteristic
311C22×C53#54811yescharacteristic
312C22×C53C2×#13511yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57611yescharacteristic
315C22×C53#57711yescharacteristic
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53—41no—
322C4×C53—81no—
323C4×C53—81no—
324C4×C53#52911yescharacteristic
325C4×C53#54211yescharacteristic
326C4×C53#54811yescharacteristic
327C4×C53#55011yescharacteristic
328Dic53—42no—
329Dic53—42no—
330Dic53—42no—
331Dic53—81no—
332C2×#488—531no—
333C2×#478—531no—
334#2544D5311yescharacteristic
335#2370—531no—
336#2068—531no—
337#2065—531no—
338#2042—531no—
339#2041—531no—
340#1983—531no—
341#1983—531no—
342#1982—531no—
343#1939—531no—
344#1927—531no—
345#1925—531no—
346C4×#102—531no—
347C2×C53—82no—
348C2×C53—161no—
349C2×C53#242611yescharacteristic
350C2×C53#303511yescharacteristic
351C2×C53#304511yescharacteristic
352C2×C53#329611yescharacteristic
353C2×C53#332211yescharacteristic
354C2×C53#336111yescharacteristic
355C2×C53#337111yescharacteristic
356C2×#127C2×D5311yescharacteristic
357C2×#125—531no—
358C2×#124C2×D5311yescharacteristic
359C2×#124C2×D5311yescharacteristic
360C2×C4×D4C2×D5311yescharacteristic
361#488—1064no—
362#478—1064no—
363C2×#107—531no—
364C2×#106—531no—
365C2×C4⋊C8—531no—
366C2×C4⋊C8—531no—
367C2×C4⋊C8—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#102—531no—
373C2×#102—531no—
374C2×#102—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×#102—531no—
378C2×#102—531no—
379C2×#102—531no—
380C2×C4×C8—531no—
381#394—21no—
382#394—21no—
383#391C2×D5311yescharacteristic
384#390C2×D5311yescharacteristic
385#389—531no—
386#389—531no—
387#389—1061no—
388#387—1061no—
389#387—1061no—
390#386—21no—
391#386—21no—
392C4×C4⋊C4—531no—
393C4×C4⋊C4—531no—
394C4×C4⋊C4—531no—
395C4×C4⋊C4—1061no—
396C4×C4⋊C4C2×D5311yescharacteristic
397#340—531no—
398#340—531no—
399#336—531no—
400#336—531no—
401C53#2528111yescharacteristic, complemented, Sylow
402C22×Q8C22×D5311yescharacteristic
403C22×D4—21no—
404C22×D4—21no—
405C22×D4C22×D5311yescharacteristic
406C22×D4C22×D5311yescharacteristic
407C23×C4—21no—
408C22×C8—531no—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—531no—
412#127—22no—
413#127—41no—
414#125—534no—
415#125—1062no—
416#124—22no—
417#124—22no—
418#124—22no—
419#124—41no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1061no—
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×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5311yescharacteristic
445C2×C42C22×D5311yescharacteristic
446#107—1062no—
447#106—1062no—
448C4⋊C8—1062no—
449C4⋊C8—2121no—
450C4⋊C8—2121no—
451#103—1062no—
452#103—1062no—
453#103—1062no—
454#103—2121no—
455#102—1062no—
456#102—1062no—
457#102—1062no—
458#102—1062no—
459#102—1062no—
460#102—1062no—
461#102—1062no—
462#102—2121no—
463C4×C8—1062no—
464#95—21no—
465#95—21no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95C22⋊D5311yescharacteristic
472#95C22⋊D5311yescharacteristic
473C24—21no—
474C24—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—81no—
497C2×D4—81no—
498C2×D4#1171994712yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4C2×C22⋊D5311yescharacteristic
509C22×C4C2×C22⋊D5311yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511C22×C4C2×C22⋊D5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C23×D5311yescharacteristic
515C2×C8—1061no—
516C2×C8—1061no—
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—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—41no—
531C4⋊C4—532no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—4241no—
542C4⋊C4—2122no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—81no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—4241no—
555C42—4241no—
556C42—2122no—
557C42D4×D5312yes—
558C42#1171994712yes—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C22×C22⋊D5311yescharacteristic
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—41no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—84no—
587D4—84no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
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—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
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#1313318812yes—
623C2×C4#1313318812yes—
624C2×C4#1313323712yes—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4#41×D5311yescharacteristic
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994711yescharacteristic
629C2×C4#1313326211yescharacteristic
630C2×C4#1313326411yescharacteristic
631C8—2121no—
632C8—2121no—
633C8—2121no—
634C8—2121no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×#1313318811yescharacteristic
650C22#1681126711yescharacteristic
651C22#1681128011yescharacteristic
652C22C2×#1313323711yescharacteristic
653C22#1681142111yescharacteristic
654C22#1681142611yescharacteristic
655C22#1681143411yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—81no—
663C4—81no—
664C4—4241no—
665C4—2122no—
666C4—2122no—
667C4—2122no—
668C4#1681115411yescharacteristic
669C4#1681124011yescharacteristic
670C4#1681127911yescharacteristic
671C4#1681129111yescharacteristic
672C2—82no—
673C2—161no—
674C2#2027827711yescharacteristic
675C2#2028416411yescharacteristic
676C2#2028425011yescharacteristic
677C2#2028655011yescharacteristic
678C2#2028691611yescharacteristic
679C2#2028737111yescharacteristic
680C2#2028747711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2544C2
socleC23×C53C22×D4
prime radicalC2×#127×C53C22
hypercenter#2544D53
hypocenterC53#25281
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. C2×#16810977C2semidirect
2. C2×#16810923C2semidirect
3. #20277809C2semidirect
4. #20276110C2semidirect
5. #20276089C2semidirect
6. #20275951C2semidirect
8. #20275648C2semidirect
9. #20275648C2semidirect
11. #20275434C2semidirect
12. #20275346C2semidirect
13. #20275332C2semidirect
14. C4×#13133138C2semidirect
66. #13133209D4semidirect
401. C53#25281semidirect

Maximal subgroups

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

subgroupindex
C2×#168109772
C2×#168109232
#202778092
#202761102
#202760892
#202759512
#202759462
#202756482
#202756432
#202754342
#202753462
#202753322
C4×#131331382
C53×#25442
#2528153