Finite Group Databasebeta

← #27388407

Subgroups — #27388407

Order 13568.

950 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810733C211yesmaximal, characteristic
2#20277180C211yesmaximal, characteristic, complemented
3#20277176C211yesmaximal, characteristic, complemented
4#20275588C211yesmaximal, characteristic
5#20275578C211yesmaximal, characteristic
6#20275457C211yesmaximal, characteristic, complemented
7#20275442C211yesmaximal, characteristic, complemented
8#20275413C211yesmaximal, characteristic, complemented
9#20275409C211yesmaximal, characteristic, complemented
10#20275399C211yesmaximal, characteristic, complemented
11#20275395C211yesmaximal, characteristic, complemented
12#20274613C211yesmaximal, characteristic, complemented
13#20274541C211yesmaximal, characteristic, complemented
14#97×Dic53C211yesmaximal, characteristic, complemented
15C53×#1762C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×#13133176—21no—
19C2×#13133174—21no—
20#16810733C418yescomplemented
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C8×Dic53C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810526C2211yescharacteristic
35#16810524C2211yescharacteristic
36#16810524C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810519C2211yescharacteristic
41#16810516—21no—
42#16810516—21no—
43#16810512—21no—
44#16810512—21no—
45C4×C4⋊Dic53C2211yescharacteristic
46C4×#11719921—21no—
47C4×C22⋊Dic53—21no—
48C42×Dic53C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C2×#11719922—22no—
63C2×#11719922—22no—
64C2×#11719922C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×D212—22no—
68C4×D212C2×C414yes—
69#13133176—44no—
70#13133174—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133150—22no—
95#13133148—22no—
96#13133148C2×C414yes—
97#13133139—22no—
98#13133139—22no—
99#13133138C2×C414yes—
100#13133137C2×C414yes—
101#13133128C2×C414yes—
102#13133127—22no—
103#13133127—22no—
104C8×Dic53—22no—
105C8×Dic53—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111C22×D4×C53C2311yescharacteristic
112C23×C4×C53—21no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4×D4×C53—22no—
116C4×D4×C53C2×C414yes—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#102×C53—22no—
124#102×C53—22no—
125#97×C53—22no—
126#95×C53—21no—
127C24×C53—21no—
128C23×D53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53C22×C412yes—
135C22×C4×C53—21no—
136C22×C4×C53—24no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212C22×C412yes—
157C2×C4×D53—24no—
158C2×C4×D53—42no—
159C2×C4×D53—42no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53#4112yes—
165C2×C8×C53#4112yes—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C22×C412yes—
172C22⋊C4×C53—24no—
173C42×C53—42no—
174C42×C53C22×C412yes—
175C42×C53#4112yes—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—42no—
180#11719922—28no—
181#11719921—24no—
182#11719921—42no—
183#11719921—42no—
184#11719921—42no—
185#11719921—42no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—42no—
192C4⋊Dic53—42no—
193C4⋊Dic53C22×C412yes—
194#11719919—42no—
195#11719919—42no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—22no—
201C4×Dic53—22no—
202C4×Dic53—41no—
203C4×Dic53—41no—
204C4×Dic53—42no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322C22×C412yes—
208C2×#88322C22×C412yes—
209C2×#88322#4112yes—
210C2×#88322#4112yes—
211C23×C53—22no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226D4×C53—22no—
227D4×C53—22no—
228D4×C53—41no—
229D4×C53—41no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—22no—
235C2×C4×C53—48no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C4×D412yes—
245C2×C4×C53C4×D412yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53AGL(1,ℤ/8)12yes—
272C2×Dic53AGL(1,ℤ/8)12yes—
273D212—22no—
274D212—22no—
275D212—41no—
276D212—41no—
277C4×D53—48no—
278C8×C53—22no—
279C8×C53—22no—
280#88322—22no—
281#88322#11714yescomplemented
282#21286—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×C4×D411yescharacteristic
291C22×C53#53211yescharacteristic
292C22×C53#53211yescharacteristic
293C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C53#57511yescharacteristic
296C22×C53#57511yescharacteristic
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—81no—
301C2×D53—81no—
302C2×D53—81no—
303C2×D53—81no—
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—42no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53C2×#11711yescharacteristic
310C4×C53C4×#4111yescharacteristic
311C4×C53#52911yescharacteristic
312C4×C53#52911yescharacteristic
313Dic53—22no—
314Dic53—22no—
315Dic53—41no—
316Dic53—41no—
317Dic53—42no—
318Dic53—81no—
319Dic53—81no—
320C2×#442—531no—
321#2293—531no—
322#2293—531no—
323#1972—531no—
324#1970—531no—
325#1944—531no—
326#1941—531no—
327#1936—531no—
328#1936—531no—
329#1934—531no—
330#1934—531no—
331#1776—531no—
332#1762—531no—
333#1762D5311yescharacteristic, complemented
334C4×#97—531no—
335C2×C53—82no—
336C2×C53#229311yescharacteristic
337C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
339C2×C53#318011yescharacteristic
340C2×C53#318011yescharacteristic
341C2×C53#318311yescharacteristic
342C2×C53#318411yescharacteristic
343D53—82no—
344C22×D8—531no—
345C2×C4×D4—531no—
346C2×C4×D4C2×D5311yescharacteristic
347C2×#117—1061no—
348C2×#117—1061no—
349#442—538no—
350C2×#107—531no—
351C2×#106—531no—
352C2×#106—531no—
353C2×#102—531no—
354C2×#102—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102C2×D5311yescharacteristic
358C2×#102C2×D5311yescharacteristic
359C2×#97—531no—
360C2×#97—531no—
361C2×#97—531no—
362C2×#97C2×D5311yescharacteristic
363C2×C4×C8—531no—
364C2×C4×C8—531no—
365#390—531no—
366#390—531no—
367#390—531no—
368#390—531no—
369#389—531no—
370#389—531no—
371#389—531no—
372#389—531no—
373#388—1061no—
374#388—1061no—
375#388—1061no—
376#388—1061no—
377C4×C4⋊C4—531no—
378C4×C4⋊C4C2×D5311yescharacteristic
379C4×C22⋊C4—1061no—
380C4×C22⋊C4—1061no—
381C43—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388C53#2128611yescharacteristic, complemented, Sylow
389C22×D4—531no—
390C22×D4C22×D5311yescharacteristic
391C23×C4—21no—
392C23×C4—1061no—
393C2×D8—534no—
394C2×D8—1062no—
395C2×D8—1062no—
396C22×C8—531no—
397C22×C8—531no—
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400C4×D4—22no—
401C4×D4—534no—
402C4×D4—1062no—
403C4×D4C4×D5314yes—
404#117—2124no—
405#117—2124no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4C22×D5311yescharacteristic
414C2×C4⋊C4C22×D5311yescharacteristic
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1061no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C42—21no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—531no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—531no—
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435#107—1062no—
436#106—534no—
437#106—1062no—
438#102—22no—
439#102—22no—
440#102—534no—
441#102—534no—
442#102—1062no—
443#102—1062no—
444#97—22no—
445#97—534no—
446#97—1062no—
447#97—1062no—
448C4×C8—1062no—
449C4×C8—1062no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24—21no—
457C24—1061no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—532no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C2×D4C2×C4×D5312yes—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—1061no—
481C22×C4—1064no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—2122no—
485C22×C4—2122no—
486C22×C4D4×D5311yescharacteristic
487C22×C4D4×D5311yescharacteristic
488C22×C4C23×D5311yescharacteristic
489D8—1068no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—532no—
495C2×C8—532no—
496C2×C8—532no—
497C2×C8—532no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8#1171994512yes—
501C2×C8#1171994512yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—532no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C4⋊C4C2×C4×D5312yes—
518C22⋊C4—24no—
519C22⋊C4—1064no—
520C22⋊C4—2122no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C22⋊C4—2122no—
526C22⋊C4—2122no—
527C22⋊C4—2122no—
528C42—42no—
529C42—1062no—
530C42—1062no—
531C42—1061no—
532C42—1061no—
533C42—1061no—
534C42—1061no—
535C42—2121no—
536C42—2121no—
537C42—2122no—
538C42C2×C4×D5312yes—
539C42#1171994712yes—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—1062no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23C2×D4×D5311yescharacteristic
555D4—22no—
556D4—22no—
557D4—41no—
558D4—41no—
559D4—1062no—
560D4—1062no—
561D4—2121no—
562D4—2121no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—532no—
578C2×C4—532no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1062no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—2128no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4#1313317712yes—
600C2×C4#1313317712yes—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4#41×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994511yescharacteristic
606C2×C4C2×#1171994511yescharacteristic
607C8—22no—
608C8—22no—
609C8—534no—
610C8—1062no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—4241no—
619C22—4241no—
620C22—4241no—
621C22—4241no—
622C22—2122no—
623C22—2122no—
624C22—2122no—
625C22C2×#1313317711yescharacteristic
626C22#1681117011yescharacteristic
627C22#1681117011yescharacteristic
628C22AGL(1,ℤ/8)×D5311yescharacteristic
629C22AGL(1,ℤ/8)×D5311yescharacteristic
630C22#1681141911yescharacteristic
631C22#1681142211yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—1062no—
638C4—1062no—
639C4—4241no—
640C4—4241no—
641C4—2121no—
642C4—2121no—
643C4—2122no—
644C4#117×D5311yescharacteristic
645C4C4×#1171994511yescharacteristic
646C4#1681115311yescharacteristic
647C4#1681115311yescharacteristic
648C2—82no—
649C2—4242no—
650C2#2027717811yescharacteristic
651C2#2028351911yescharacteristic
652C2#2028352811yescharacteristic
653C2#2028540911yescharacteristic
654C2#2028541111yescharacteristic
655C2#2028544011yescharacteristic
656C2#2028545011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1762C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
cosocleC22×C4×C53C24
hypercenter#1762D53
hypocenterC53#21286
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. #20277180C2semidirect
3. #20277176C2semidirect
6. #20275457C2semidirect
7. #20275442C2semidirect
8. #20275413C2semidirect
9. #20275409C2semidirect
10. #20275399C2semidirect
11. #20275395C2semidirect
12. #20274613C2semidirect
13. #20274541C2semidirect
14. #97×Dic53C2semidirect
15. C53×#1762C2semidirect
20. #16810733C4semidirect
281. #88322#117semidirect
333. #1762D53semidirect
388. C53#21286semidirect

Maximal subgroups

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

subgroupindex
C2×#168107332
#202771802
#202771762
#202755882
#202755782
#202754572
#202754422
#202754132
#202754092
#202753992
#202753952
#202746132
#202745412
#97×Dic532
C53×#17622
#2128653