Finite Group Databasebeta

← #27386262

Subgroups — #27386262

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810662C211yesmaximal, characteristic
2#20277196C211yesmaximal, characteristic
3C4×C4⋊C4×D53C211yesmaximal, characteristic
4#20275308C211yesmaximal, characteristic, complemented
5#20275307C211yesmaximal, characteristic, complemented
6#20275303C211yesmaximal, characteristic, complemented
7#20275302C211yesmaximal, characteristic, complemented
8#20275290C211yesmaximal, characteristic, complemented
9#20275285C211yesmaximal, characteristic, complemented
10#20275284C211yesmaximal, characteristic, complemented
11#20274433C211yesmaximal, characteristic
12#20274433C211yesmaximal, characteristic
13#20274433C211yesmaximal, characteristic
14#20274407C211yesmaximal, characteristic
15C53×#1919C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C42×D53—21no—
23C2×C42×D53C2211yescharacteristic
24#16810662C418yescomplemented
25C2×#13133151C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810497—21no—
32#16810494—21no—
33C4⋊C4×Dic53—21no—
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4×#11719919—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38#16810474—21no—
39#16810474—21no—
40#95×D53—21no—
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53D411yescharacteristic
63C22×C4×D53D411yescharacteristic
64C22×C4×D53C2311yescharacteristic
65C2×#11719912—22no—
66C2×#11719912—22no—
67C2×#11719912—22no—
68C2×#11719912—22no—
69C2×#11719912—22no—
70C2×#11719912C2×C414yes—
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133175—22no—
74#13133175—22no—
75#13133175—22no—
76#13133175C2×C414yes—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C4⋊C4×D53—22no—
87C4⋊C4×D53—22no—
88C4⋊C4×D53C2×C414yes—
89C4⋊C4×D53Q814yes—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53D411yescharacteristic
98C2×C4×Dic53D411yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100C42×D53—22no—
101C42×D53—22no—
102C42×D53—44no—
103#13133151C2×C414yescomplemented
104#13133150C2×C414yescomplemented
105#13133148C2×C414yescomplemented
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×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C42×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#106×C53C2×C414yescomplemented
122#95×C53—21no—
123#95×C53—21no—
124C23×D53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—24no—
143C2×C4×D53—24no—
144C2×C4×D53—22no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×C4×D53C22×C412yes—
152C2×C4×D53C2×Q812yes—
153C2×C4×D53#4112yes—
154C2×C4×D53#4112yes—
155C2×C4×D53#4112yes—
156C2×C4×D53#4112yes—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53C22×C412yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53C22×C412yes—
170C4⋊C4×C53C22×C412yes—
171C4⋊C4×C53C2×Q812yes—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—42no—
175#11719921—24no—
176#11719921—42no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53C22×C412yes—
184C4⋊Dic53C22×C412yes—
185C4⋊Dic53C2×Q812yes—
186#11719919—24no—
187#11719919—24no—
188#11719919—42no—
189#11719919—42no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—24no—
193C4×Dic53—22no—
194C4×Dic53—22no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C22×C412yes—
205#11719912—28no—
206#11719912—44no—
207C23×C53C22×D411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53C4×D412yes—
213C22×D53C22⋊Q812yes—
214C22×D53#12312yes—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
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×C53#13712yes—
233C2×C4×C53C23×C411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53C4×D412yes—
257C2×Dic53C22⋊Q812yes—
258C2×Dic53#12312yes—
259C2×Dic53AGL(1,ℤ/8)12yes—
260C4×D53—28no—
261C4×D53—48no—
262C4×D53—44no—
263C4×D53—48no—
264C4×D53#11714yes—
265C4×D53C4×Q814yes—
266C4×D53#12514yes—
267C4×D53#12614yes—
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272#21033—531nomaximal, Sylow
273C22×C53C2×C4×D411yescharacteristic
274C22×C53C2×C22⋊Q811yescharacteristic
275C22×C53C2×#12311yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53C2×#13711yescharacteristic
278C22×C53#57511yescharacteristic
279C22×C53#57511yescharacteristic
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53—22no—
284C2×D53—22no—
285C2×D53—22no—
286C2×D53#38714yes—
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—42no—
292C4×C53—81no—
293C4×C53C2×#11711yescharacteristic
294C4×C53C2×C4×Q811yescharacteristic
295C4×C53C2×#12511yescharacteristic
296C4×C53C2×#12611yescharacteristic
297Dic53—22no—
298Dic53—42no—
299Dic53—41no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53#38714yes—
304C2×#428—531no—
305#2296—531no—
306C2×C4×C4⋊C4—531no—
307#1921—531no—
308#1921—531no—
309#1920—531no—
310#1920—531no—
311#1919—531no—
312#1919D5311yescharacteristic, complemented
313#1918—531no—
314#1918—531no—
315#1736—531no—
316#1736—531no—
317#1736—531no—
318#1728—531no—
319C2×C53C2×#38711yescharacteristic
320C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
321C2×C53C4×#13711yescharacteristic
322C2×C53#307711yescharacteristic
323C2×C53#307811yescharacteristic
324C2×C53#309211yescharacteristic
325C2×C53#309311yescharacteristic
326D53—24no—
327C22×M16—531no—
328C22×M16—531no—
329C2×#117—531no—
330C2×#117—531no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4—531no—
333C22×C42—531no—
334C22×C42—1061no—
335#428—538no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#106—531no—
339C2×#106C2×D5311yescharacteristic
340#389—531no—
341#389—531no—
342#389—531no—
343#389C2×D5311yescharacteristic
344#385—1061no—
345#385—1061no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350C4×C4⋊C4—1061no—
351C4×C4⋊C4—1061no—
352C4×C4⋊C4—1061no—
353C4×C4⋊C4C2×D5311yescharacteristic
354C2×#95—1061no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340C2×D5311yescharacteristic
365#340C2×D5311yescharacteristic
366#340C2×D5311yescharacteristic
367#336—531no—
368#336—531no—
369#336—531no—
370#336C2×D5311yescharacteristic
371C53#2103311yescharacteristic, complemented, Sylow
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—1061no—
376C23×C4—1061no—
377C2×M16—534no—
378C2×M16—1062no—
379C2×M16—1062no—
380C2×M16—1062no—
381C2×M16—1062no—
382C2×M16—1062no—
383C22×C8—531no—
384C22×C8—531no—
385C22×C8C22×D5311yescharacteristic
386C22×C8C22×D5311yescharacteristic
387#117—534no—
388#117—1062no—
389#117—1062no—
390#117—1062no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—534no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—531no—
415C2×C42—1062no—
416C2×C42—1062no—
417C2×C42—1061no—
418C2×C42—1061no—
419C2×C42—1061no—
420C2×C42—1061no—
421C2×C42—1061no—
422C2×C42—2124no—
423C2×C42C22×D5311yescharacteristic
424#107—534no—
425#107—534no—
426#106—534no—
427#106C4×D5314yescomplemented
428#95—21no—
429#95—21no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—531no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—531no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—532no—
447C22×C4—532no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1064no—
454C22×C4—1064no—
455C22×C4—1062no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—2121no—
461C22×C4—2121no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4C23×D5311yescharacteristic
469M16—1068no—
470M16—2124no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8C2×C4×D5312yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—532no—
490C4⋊C4—532no—
491C4⋊C4—532no—
492C4⋊C4—1064no—
493C4⋊C4—1064no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4C2×C4×D5312yes—
503C4⋊C4C2×C4×D5312yes—
504C4⋊C4Q8×D5312yes—
505C22⋊C4—1064no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—1062no—
511C42—1062no—
512C42—1064no—
513C42—1061no—
514C42—1061no—
515C42—2122no—
516C42—2122no—
517C42—2122no—
518C42—2122no—
519C23—532no—
520C23—532no—
521C23—532no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1061no—
526C23C2×D4×D5311yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—534no—
548C2×C4—534no—
549C2×C4—534no—
550C2×C4—534no—
551C2×C4—532no—
552C2×C4—1068no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—2128no—
562C2×C4—2128no—
563C2×C4—2124no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313324612yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4C2×Q8×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—534no—
585C22—1062no—
586C22—1062no—
587C22—1062no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22C4×D4×D5311yescharacteristic
592C22C22⋊Q8×D5311yescharacteristic
593C22#123×D5311yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22C2×#1313324611yescharacteristic
596C22#1681142011yescharacteristic
597C22#1681142211yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—534no—
605C4—1062no—
606C4—4241no—
607C4—2122no—
608C4—2122no—
609C4—2122no—
610C4—2121no—
611C4#117×D5311yescharacteristic
612C4C4×Q8×D5311yescharacteristic
613C4#125×D5311yescharacteristic
614C4#126×D5311yescharacteristic
615C2—1064no—
616C2#387×D5311yescharacteristic
617C2#2028352011yescharacteristic
618C2C4×#1313324611yescharacteristic
619C2#2028451011yescharacteristic
620C2#2028451511yescharacteristic
621C2#2028463711yescharacteristic
622C2#2028464711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1919C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1919D53
hypocenterC53#21033
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #20275308C2semidirect
5. #20275307C2semidirect
6. #20275303C2semidirect
7. #20275302C2semidirect
8. #20275290C2semidirect
9. #20275285C2semidirect
10. #20275284C2semidirect
15. C53×#1919C2semidirect
24. #16810662C4semidirect
103. #13133151C2×C4semidirect
104. #13133150C2×C4semidirect
105. #13133148C2×C4semidirect
121. #106×C53C2×C4semidirect
312. #1919D53semidirect
371. C53#21033semidirect
427. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106622
#202771962
C4×C4⋊C4×D532
#202753082
#202753072
#202753032
#202753022
#202752902
#202752852
#202752842
#202744332
#202744072
C53×#19192
#2103353