Finite Group Databasebeta

← #27387106

Subgroups — #27387106

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810725C211yesmaximal, characteristic
2#20277189C211yesmaximal, characteristic, complemented
3C4×#13133179C211yesmaximal, characteristic
4#20275355C211yesmaximal, characteristic, complemented
5#20275348C211yesmaximal, characteristic, complemented
6#20275306C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275050C211yesmaximal, characteristic
9#20275050C211yesmaximal, characteristic
10#20274973C211yesmaximal, characteristic
11#20274959C211yesmaximal, characteristic
12#20274548C211yesmaximal, characteristic, complemented
13#20274543C211yesmaximal, characteristic
14#20274463C211yesmaximal, characteristic, complemented
15C53×#1776C211yesmaximal, characteristic, complemented
16C22×#11719913C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×C42×D53—21no—
22#16810725C418yescomplemented
23C2×#13133151C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31#16810528C2211yescharacteristic
32#16810528C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810505—21no—
36#16810505—21no—
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×#11719919—21no—
40C4×#11719921—21no—
41#16810470—21no—
42#16810470—21no—
43#16810469—21no—
44C42×Dic53C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×Dic106C2311yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719913—22no—
66C2×#11719913—22no—
67C2×#11719913C2×C414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C4×Dic106—22no—
71C4×Dic106C2×C414yescomplemented
72C4×D212—22no—
73C4×D212C2×C414yes—
74#13133179—22no—
75#13133179—22no—
76#13133179—22no—
77#13133179—22no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53D411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C42×D53—44no—
96#13133151C2×C414yescomplemented
97#13133142C2×C414yescomplemented
98#13133140—22no—
99#13133140—22no—
100#13133139C2×C414yes—
101#13133137—22no—
102#13133137—22no—
103#13133128—22no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C42×C53—21no—
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53—22no—
122#106×C53—22no—
123#97×C53C2×C414yescomplemented
124#95×C53—21no—
125#95×C53—21no—
126C23×D53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C22×C412yes—
145C2×D212#4112yes—
146C2×D212#4112yes—
147C2×C4×D53—24no—
148C2×C4×D53—24no—
149C2×C4×D53—24no—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53—42no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C22×C412yes—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C22×C412yes—
167C2×C8×C53C22×C412yes—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53#4112yes—
175C4⋊C4×C53#4112yes—
176C42×C53—42no—
177C42×C53—42no—
178C42×C53C22×C412yes—
179C42×C53#4112yes—
180#11719921—24no—
181#11719921—24no—
182#11719921—24no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53C22×C412yes—
186C4⋊Dic53C22×C412yes—
187#11719919—24no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—24no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—22no—
196C4×Dic53—22no—
197C4×Dic53—41no—
198C4×Dic53—41no—
199C4×Dic53—42no—
200C4×Dic53—42no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207#11719913—28no—
208C23×C53C22×D411yescharacteristic
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53C4×D412yes—
233C2×C4×C53C4×D412yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53AGL(1,ℤ/8)12yes—
260C2×Dic53#13712yes—
261D212—22no—
262D212—22no—
263D212—22no—
264D212#11714yes—
265C4×D53—48no—
266C4×D53—48no—
267C4×D53—48no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—22no—
273C8×C53—22no—
274#88322—41no—
275#88322—41no—
276#21121—531nomaximal, Sylow
277C22×C53C2×C4×D411yescharacteristic
278C22×C53#52511yescharacteristic
279C22×C53#52511yescharacteristic
280C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C53C2×#13711yescharacteristic
282C22×C53#57511yescharacteristic
283C22×C53#57511yescharacteristic
284C2×D53—24no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—42no—
289C2×D53—42no—
290C2×D53—42no—
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—42no—
294C4×C53—42no—
295C4×C53—81no—
296C4×C53—81no—
297C4×C53C2×#11711yescharacteristic
298C4×C53C4×#4111yescharacteristic
299C4×C53#52911yescharacteristic
300C4×C53#52911yescharacteristic
301Dic53—22no—
302Dic53—22no—
303Dic53—41no—
304Dic53—41no—
305Dic53—42no—
306Dic53—42no—
307Dic53—42no—
308C2×#440—531no—
309#2295—531no—
310C42×D4—531no—
311#1928—531no—
312#1927—531no—
313#1921—531no—
314#1918—531no—
315#1877—531no—
316#1877—531no—
317#1864—531no—
318#1862—531no—
319#1776D5311yescharacteristic, complemented
320#1763—531no—
321#1762—531no—
322#1745—531no—
323C2×C53#229011yescharacteristic
324C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
325C2×C53C4×#13711yescharacteristic
326C2×C53#311011yescharacteristic
327C2×C53#311111yescharacteristic
328C2×C53#311411yescharacteristic
329C2×C53#311611yescharacteristic
330D53—44no—
331C22×SD16—531no—
332C2×C4×Q8—531no—
333C2×C4×D4—531no—
334C2×C4×D4—531no—
335C2×C4×D4—531no—
336C22×C42—1061no—
337#440—538no—
338C2×#107—531no—
339C2×#107C2×D5311yescharacteristic
340C2×#106C2×D5311yescharacteristic
341C2×#103—531no—
342C2×#103—531no—
343C2×#103—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#97—531no—
348C2×#97C2×D5311yescharacteristic
349#391—531no—
350#391—531no—
351#389—531no—
352#389—531no—
353#387—1061no—
354#387—1061no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C4⋊C4C2×D5311yescharacteristic
359C4×C4⋊C4C2×D5311yescharacteristic
360C4×C22⋊C4—1061no—
361C4×C22⋊C4—1061no—
362C4×C22⋊C4—1061no—
363#376—1061no—
364C43—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340—531no—
369#340C2×D5311yescharacteristic
370#340C2×D5311yescharacteristic
371#336—531no—
372#336—531no—
373#336—531no—
374#336—531no—
375C53#2112111yescharacteristic, complemented, Sylow
376C22×Q8—531no—
377C22×D4—531no—
378C23×C4—1061no—
379C23×C4—1061no—
380C23×C4—1061no—
381C2×SD16—534no—
382C2×SD16—1062no—
383C2×SD16—1062no—
384C22×C8—531no—
385C22×C8—531no—
386C22×C8C22×D5311yescharacteristic
387C22×C8C22×D5311yescharacteristic
388C4×Q8—534no—
389C4×Q8—1062no—
390C4×D4—534no—
391C4×D4—1062no—
392C4×D4—1062no—
393C4×D4—1062no—
394C4×D4—1062no—
395C4×D4—1062no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—2124no—
420C2×C42C22×D5311yescharacteristic
421#107—22no—
422#107—534no—
423#106—22no—
424#103—534no—
425#103—1062no—
426#103—1062no—
427#102—534no—
428#102—1062no—
429#102—1062no—
430#97—1062no—
431#97C4×D5314yescomplemented
432#95—21no—
433#95—21no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442C24—1061no—
443C2×Q8—532no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×Q8—1061no—
449C2×D4—532no—
450C2×D4—532no—
451C2×D4—532no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—1061no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1064no—
469C22×C4—2122no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4—2122no—
475C22×C4D4×D5311yescharacteristic
476C22×C4D4×D5311yescharacteristic
477C22×C4C23×D5311yescharacteristic
478SD16—1068no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—1061no—
484C2×C8—1061no—
485C2×C8—1061no—
486C2×C8—1061no—
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8C2×C4×D5312yes—
490C2×C8C2×C4×D5312yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—532no—
498C4⋊C4—532no—
499C4⋊C4—1064no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4#1171994512yes—
505C4⋊C4#1171994512yes—
506C22⋊C4—1064no—
507C22⋊C4—1064no—
508C22⋊C4—1064no—
509C42—42no—
510C42—42no—
511C42—1062no—
512C42—1062no—
513C42—1064no—
514C42—1061no—
515C42—1061no—
516C42—1061no—
517C42—1061no—
518C42—2121no—
519C42—2121no—
520C42—2122no—
521C42—2122no—
522C42C2×C4×D5312yes—
523C42#1171994912yes—
524C23—1062no—
525C23—1062no—
526C23—1062no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23C2×D4×D5311yescharacteristic
532Q8—1062no—
533Q8—1062no—
534Q8—2121no—
535Q8—2121no—
536D4—534no—
537D4—1062no—
538D4—1062no—
539D4—1062no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—1062no—
563C2×C4—1062no—
564C2×C4—1062no—
565C2×C4—1061no—
566C2×C4—1061no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—2128no—
572C2×C4—2128no—
573C2×C4—2128no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4#1313317912yes—
581C2×C4#1313317912yes—
582C2×C4C22×C4×D5311yescharacteristic
583C2×C4#41×D5311yescharacteristic
584C2×C4#41×D5311yescharacteristic
585C2×C4C2×#1171994511yescharacteristic
586C2×C4C2×#1171994511yescharacteristic
587C2×C4C2×#1171994911yescharacteristic
588C8—22no—
589C8—22no—
590C8—2121no—
591C8—2121no—
592C22—1064no—
593C22—2122no—
594C22—2122no—
595C22—2122no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22C2×#1313317911yescharacteristic
600C22#1681113211yescharacteristic
601C22#1681113211yescharacteristic
602C22AGL(1,ℤ/8)×D5311yescharacteristic
603C22#137×D5311yescharacteristic
604C22#1681141911yescharacteristic
605C22#1681142211yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—1062no—
613C4—1062no—
614C4—2122no—
615C4—2122no—
616C4—2121no—
617C4—2122no—
618C4—2121no—
619C4#117×D5311yescharacteristic
620C4C4×#1171994511yescharacteristic
621C4#1681116011yescharacteristic
622C4#1681116011yescharacteristic
623C2—2124no—
624C2#2027715611yescharacteristic
625C2#2028352211yescharacteristic
626C2#2028353311yescharacteristic
627C2#2028481211yescharacteristic
628C2#2028482211yescharacteristic
629C2#2028485111yescharacteristic
630C2#2028487511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1776C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1776D53
hypocenterC53#21121
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. #20277189C2semidirect
4. #20275355C2semidirect
5. #20275348C2semidirect
6. #20275306C2semidirect
7. #20275285C2semidirect
12. #20274548C2semidirect
14. #20274463C2semidirect
15. C53×#1776C2semidirect
22. #16810725C4semidirect
71. C4×Dic106C2×C4semidirect
96. #13133151C2×C4semidirect
97. #13133142C2×C4semidirect
123. #97×C53C2×C4semidirect
319. #1776D53semidirect
375. C53#21121semidirect
431. #97C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107252
#202771892
C4×#131331792
#202753552
#202753482
#202753062
#202752852
#202750502
#202749732
#202749592
#202745482
#202745432
#202744632
C53×#17762
#2112153