Finite Group Databasebeta

← #27433544

Subgroups — #27433544

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810991C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic
3#20277809C211yesmaximal, characteristic, complemented
4#20276110C211yesmaximal, characteristic, complemented
5#20276096C211yesmaximal, characteristic, complemented
6#20275951C211yesmaximal, characteristic
7#20275951C211yesmaximal, characteristic
8#20275648C211yesmaximal, characteristic
9#20275648C211yesmaximal, characteristic
10#20275639C211yesmaximal, characteristic, complemented
11#20275439C211yesmaximal, characteristic, complemented
12#20275353C211yesmaximal, characteristic, complemented
13#20275346C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic
15C53×#2547C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810991—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×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35C2×#13133138C2211yescharacteristic
36C2×#13133128C2211yescharacteristic
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×C4⋊Q8×C53C2211yescharacteristic
51C2×#124×C53C2211yescharacteristic
52C2×#124×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C53×#397—21no—
55C53×#397—21no—
56C53×#391C2211yescharacteristic
57C53×#391C2211yescharacteristic
58C53×#380—21no—
59C53×#380—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209D414yes—
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—22no—
87#13133141—22no—
88#13133141—22no—
89#13133141—41no—
90#13133141—41no—
91#13133138—22no—
92#13133138—22no—
93#13133138—22no—
94#13133138—22no—
95#13133128—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C22×Q8×C53C2311yescharacteristic
102C22×Q8×C53C2311yescharacteristic
103C22×D4×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C4⋊Q8×C53—22no—
106C4⋊Q8×C53—22no—
107#124×C53—22no—
108#124×C53—22no—
109#124×C53—41no—
110#124×C53—41no—
111C4×D4×C53—22no—
112C4×D4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
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×C42×C53C2311yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C2×Q8×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×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×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#4112yes—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—42no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2411yescharacteristic
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—42no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53C2×D412yes—
184C42×C53#4112yes—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C2×D412yes—
191C4⋊Dic53C2×D412yes—
192C4⋊Dic53#4112yes—
193#11719919—24no—
194#11719919—24no—
195#11719919—42no—
196#11719919—81no—
197C4×Dic53—42no—
198C4×Dic53—81no—
199C4×Dic53—81no—
200C2×#88322—21no—
201C2×#88322—21no—
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—21no—
212C23×C53—22no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53C22×D411yescharacteristic
220Q8×C53—41no—
221Q8×C53—41no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53—41no—
225Q8×C53—41no—
226Q8×C53—41no—
227Q8×C53—41no—
228D4×C53—22no—
229D4×C53—22no—
230D4×C53—41no—
231D4×C53—41no—
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—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
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—84no—
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53H(5,𝔽2)11yescharacteristic
265C2×C4×C532−1+411yescharacteristic
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278#88322—41no—
279#88322—41no—
280#88322—41no—
281#88322—41no—
282#25289—531nomaximal, Sylow
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53C2×#12111yescharacteristic
291C22×C53#54611yescharacteristic
292C22×C53#54811yescharacteristic
293C22×C53#57511yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57811yescharacteristic
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53#52911yescharacteristic
307C4×C53#54211yescharacteristic
308C4×C53#54811yescharacteristic
309C4×C53#55011yescharacteristic
310Dic53—42no—
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314C2×#491—531no—
315C2×#478—531no—
316#2547D5311yescharacteristic
317#2370—531no—
318#2068—531no—
319#2066—531no—
320#2042—531no—
321#2042—531no—
322#1983—531no—
323#1983—531no—
324#1981—531no—
325#1940—531no—
326#1928—531no—
327#1927—531no—
328#1762—531no—
329C2×C53—82no—
330C2×C53#242611yescharacteristic
331C2×C53#304011yescharacteristic
332C2×C53#304611yescharacteristic
333C2×C53#330711yescharacteristic
334C2×C53#331511yescharacteristic
335C2×C53#337611yescharacteristic
336C2×C53#337711yescharacteristic
337C2×C4⋊Q8C2×D5311yescharacteristic
338C2×#125—531no—
339C2×#124C2×D5311yescharacteristic
340C2×#124C2×D5311yescharacteristic
341C2×C4×D4C2×D5311yescharacteristic
342#491—1064no—
343#478—1064no—
344C2×#107—531no—
345C2×#106—531no—
346C2×C4⋊C8—531no—
347C2×C4⋊C8—531no—
348C2×C4⋊C8—531no—
349C2×#103—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×#103—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#97—531no—
362#397—21no—
363#397—21no—
364#391C2×D5311yescharacteristic
365#391C2×D5311yescharacteristic
366#389—531no—
367#389—531no—
368#389—1061no—
369#387—1061no—
370#387—1061no—
371#380—21no—
372#380—21no—
373C4×C4⋊C4—531no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C4⋊C4—1061no—
377C4×C4⋊C4C2×D5311yescharacteristic
378#340—531no—
379#340—531no—
380#336—531no—
381#336—531no—
382C53#2528911yescharacteristic, complemented, Sylow
383C22×Q8C22×D5311yescharacteristic
384C22×Q8C22×D5311yescharacteristic
385C22×D4C22×D5311yescharacteristic
386C23×C4—21no—
387C22×C8—531no—
388C22×C8—531no—
389C22×C8—531no—
390C22×C8—531no—
391C4⋊Q8—22no—
392C4⋊Q8—22no—
393#125—534no—
394#125—1062no—
395#124—22no—
396#124—22no—
397#124—41no—
398#124—41no—
399C4×D4—22no—
400C4×D4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4C22⋊D5311yescharacteristic
412C2×C4⋊C4C22⋊D5311yescharacteristic
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—21no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42C22×D5311yescharacteristic
423C2×C42C22×D5311yescharacteristic
424C2×C42C22×D5311yescharacteristic
425#107—1062no—
426#106—1062no—
427C4⋊C8—1062no—
428C4⋊C8—2121no—
429C4⋊C8—2121no—
430#103—1062no—
431#103—1062no—
432#103—1062no—
433#103—1062no—
434#103—1062no—
435#103—1062no—
436#103—2121no—
437#103—2121no—
438#102—1062no—
439#102—1062no—
440#102—1062no—
441#102—1062no—
442#97—1062no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95C22⋊D5311yescharacteristic
453#95C22⋊D5311yescharacteristic
454C24—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1171994712yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C2×C22⋊D5311yescharacteristic
487C22×C4C2×C22⋊D5311yescharacteristic
488C22×C4C2×C22⋊D5311yescharacteristic
489C22×C4C23×D5311yescharacteristic
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—532no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—1064no—
512C4⋊C4—1064no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—2122no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—4241no—
530C42—4241no—
531C42—2122no—
532C42D4×D5312yes—
533C42#1171994712yes—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D5311yescharacteristic
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4#1313318812yes—
593C2×C4#1313318812yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4#41×D5311yescharacteristic
596C2×C4C2×#1171994711yescharacteristic
597C2×C4C2×#1171994711yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—2121no—
601C8—2121no—
602C8—2121no—
603C8—2121no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#1313318811yescharacteristic
612C22#1681126711yescharacteristic
613C22#1681128011yescharacteristic
614C22#1681142111yescharacteristic
615C22#1681142111yescharacteristic
616C22#1681143111yescharacteristic
617C22#1681143811yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—4241no—
628C4—2122no—
629C4—2122no—
630C4—2122no—
631C4#1681115411yescharacteristic
632C4#1681124011yescharacteristic
633C4#1681127911yescharacteristic
634C4#1681129111yescharacteristic
635C2—82no—
636C2#2027827711yescharacteristic
637C2#2028420511yescharacteristic
638C2#2028426111yescharacteristic
639C2#2028669111yescharacteristic
640C2#2028681111yescharacteristic
641C2#2028753611yescharacteristic
642C2#2028755111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2547C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2547D53
hypocenterC53#25289
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C2×#16810991C2semidirect
3. #20277809C2semidirect
4. #20276110C2semidirect
5. #20276096C2semidirect
10. #20275639C2semidirect
11. #20275439C2semidirect
12. #20275353C2semidirect
13. #20275346C2semidirect
382. C53#25289semidirect

Maximal subgroups

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

subgroupindex
C2×#168109912
C2×#168109232
#202778092
#202761102
#202760962
#202759512
#202756482
#202756392
#202754392
#202753532
#202753462
#202745402
C53×#25472
#2528953