Finite Group Databasebeta

← #27387280

Subgroups — #27387280

Order 13568.

938 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×D8×Dic53C211yesmaximal, characteristic
2#20277352C211yesmaximal, characteristic, complemented
3#20277076C211yesmaximal, characteristic, complemented
4#20275345C211yesmaximal, characteristic, complemented
5#20275339C211yesmaximal, characteristic, complemented
6#20275287C211yesmaximal, characteristic, complemented
7#20275285C211yesmaximal, characteristic, complemented
8#20275065C211yesmaximal, characteristic
9#20274976C211yesmaximal, characteristic, complemented
10#20274974C211yesmaximal, characteristic, complemented
11#20274603C211yesmaximal, characteristic, complemented
12#20274542C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic, complemented
14#20274465C211yesmaximal, characteristic, complemented
15C53×#1879C211yesmaximal, characteristic
16C2×D4×Dic53C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19D8×Dic53C418yescomplemented
20C2×#13133151C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810526C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810517—21no—
33#16810511—21no—
34#16810484—21no—
35#16810481—21no—
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810469C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×D8×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#390C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65D4×Dic53—22no—
66D4×Dic53—22no—
67D4×Dic53—22no—
68D4×Dic53—22no—
69D4×Dic53C2×C414yes—
70D4×Dic53C2×C414yescomplemented
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151C2×C414yescomplemented
89#13133148—22no—
90#13133148—22no—
91#13133138—22no—
92#13133138—22no—
93#13133138C2×C414yes—
94#13133138C2×C414yescomplemented
95#13133127—22no—
96C8×Dic53C2×C414yescomplemented
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106#13133124D411yescharacteristic
107C22×D4×C53C2311yescharacteristic
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53—21no—
110C2×D8×C53—22no—
111C2×D8×C53—22no—
112C2×D8×C53C2×C414yes—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4×D4×C53—22no—
116C4×D4×C53—22no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#102×C53—22no—
124#102×C53—22no—
125C24×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53C22×C412yes—
136C2×D4×C53C22×C412yes—
137C2×D4×C53#4112yes—
138C2×D4×C53#4112yes—
139C22×C4×C53—21no—
140C22×C4×C53—24no—
141C22×C4×C53—21no—
142C22×C4×C53—81no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—24no—
151C22×Dic53—24no—
152C22×Dic53—24no—
153C22×Dic53—21no—
154C22×Dic53—81no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157C22×Dic53—81no—
158C22×Dic53—81no—
159C22×Dic53—81no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164D8×C53—28no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53C22×C412yes—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53#4112yes—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—81no—
177C22⋊C4×C53—81no—
178C42×C53—21no—
179C42×C53—21no—
180C22⋊Dic53—24no—
181C22⋊Dic53—24no—
182C22⋊Dic53—24no—
183C22⋊Dic53—81no—
184C22⋊Dic53—81no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53C22×C412yes—
191C4⋊Dic53C22×C412yes—
192C4⋊Dic53#4112yes—
193#11719919—81no—
194#11719919—81no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—81no—
198C4×Dic53—81no—
199C4×Dic53—81no—
200C4×Dic53—81no—
201C4×Dic53C22×C412yes—
202C4×Dic53#4112yes—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322C22×C412yes—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53C22×D411yescharacteristic
224D4×C53—22no—
225D4×C53—22no—
226D4×C53—22no—
227D4×C53—22no—
228D4×C53—22no—
229D4×C53—41no—
230D4×C53—41no—
231D4×C53#11714yes—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—84no—
245C2×C4×C53C23×C411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—21no—
258C2×Dic53—22no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—84no—
274C2×Dic53—84no—
275C2×Dic53—84no—
276C2×Dic53C4×D412yes—
277C2×Dic53C4×D412yes—
278C8×C53—22no—
279C8×C53—41no—
280#88322—22no—
281#88322—41no—
282#21134—531nomaximal, Sylow
283C22×C53—24no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53C2×C4×D411yescharacteristic
298C22×C53#52511yescharacteristic
299C22×C53#52511yescharacteristic
300C22×C53#57611yescharacteristic
301C22×C53#57611yescharacteristic
302C22×C53#57711yescharacteristic
303C22×C53#57711yescharacteristic
304C4×C53—41no—
305C4×C53—42no—
306C4×C53—41no—
307C4×C53—81no—
308C4×C53C2×#11711yescharacteristic
309C4×C53C4×#4111yescharacteristic
310C4×C53#52911yescharacteristic
311C4×C53#52911yescharacteristic
312Dic53—22no—
313Dic53—22no—
314Dic53—41no—
315Dic53—42no—
316Dic53—41no—
317Dic53—42no—
318Dic53—42no—
319Dic53—81no—
320C2×C4×D8—531no—
321#2319—531no—
322#2275—531no—
323#1926—531no—
324#1926—531no—
325#1918—531no—
326#1918—531no—
327#1879—531no—
328#1879D5311yescharacteristic
329#1864—531no—
330#1864—531no—
331C4×#106—531no—
332#1762—531no—
333#1762—531no—
334#1746—531no—
335C2×C53—44no—
336C2×C53—82no—
337C2×C53#229011yescharacteristic
338C2×C53#294811yescharacteristic
339C2×C53#295211yescharacteristic
340C2×C53#311811yescharacteristic
341C2×C53#311811yescharacteristic
342C2×C53#311911yescharacteristic
343C2×C53#311911yescharacteristic
344C22×D8C2×D5311yescharacteristic
345C2×C4×D4—531no—
346C2×C4×D4—531no—
347C2×C4×D4—531no—
348C2×C4×D4C2×D5311yescharacteristic
349C4×D8—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×C4×C8—531no—
361#390—531no—
362#390C2×D5311yescharacteristic
363#389—531no—
364#389—531no—
365#388—1061no—
366#388—1061no—
367#381—1061no—
368#380—1061no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—531no—
372C4×C4⋊C4—531no—
373C4×C22⋊C4—1061no—
374C4×C22⋊C4—1061no—
375C4×C22⋊C4—1061no—
376#376—531no—
377C2×#95—1061no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340C2×D5311yescharacteristic
384#336—531no—
385#336—531no—
386#336—531no—
387#336C2×D5311yescharacteristic
388C53#2113411yescharacteristic, complemented, Sylow
389C22×D4C22×D5311yescharacteristic
390C22×D4C22×D5311yescharacteristic
391C23×C4—21no—
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4—1061no—
395C2×D8—22no—
396C2×D8—22no—
397C2×D8C4×D5314yes—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C4×D4—22no—
403C4×D4—22no—
404C4×D4—534no—
405C4×D4—534no—
406C4×D4—1062no—
407C4×D4—1062no—
408C4×D4—1062no—
409C4×D4—1062no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42C22×D5311yescharacteristic
433#107—534no—
434#106—1062no—
435#106—1062no—
436#102—22no—
437#102—22no—
438#102—534no—
439#102—534no—
440#102—1062no—
441#102—1062no—
442#97—1062no—
443C4×C8—534no—
444#95—531no—
445#95—531no—
446#95—531no—
447#95—531no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454C24—21no—
455C24—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4C2×C4×D5312yes—
465C2×D4C2×C4×D5312yes—
466C2×D4#1171994712yes—
467C2×D4#1171994712yes—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—1064no—
478C22×C4—1064no—
479C22×C4—1064no—
480C22×C4—1061no—
481C22×C4—1061no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—4241no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4—4241no—
488C22×C4—4241no—
489C22×C4—4241no—
490C22×C4D4×D5311yescharacteristic
491C22×C4D4×D5311yescharacteristic
492C22×C4C23×D5311yescharacteristic
493D8—28no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—532no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8C2×C4×D5312yes—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—532no—
510C4⋊C4—532no—
511C4⋊C4—532no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—4241no—
518C4⋊C4—4241no—
519C4⋊C4#1171994712yes—
520C22⋊C4—24no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—1064no—
524C22⋊C4—1064no—
525C22⋊C4—1064no—
526C22⋊C4—4241no—
527C22⋊C4—4241no—
528C42—21no—
529C42—21no—
530C42—532no—
531C42—532no—
532C42—1061no—
533C42—1061no—
534C42—4241no—
535C42—4241no—
536C42—4241no—
537C42—4241no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23C2×D4×D5311yescharacteristic
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4#1313317314yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—84no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1062no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—4244no—
590C2×C4—4244no—
591C2×C4—4244no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4C22×C4×D5311yescharacteristic
602C2×C4#41×D5311yescharacteristic
603C2×C4#41×D5311yescharacteristic
604C2×C4C2×#1171994711yescharacteristic
605C2×C4C2×#1171994711yescharacteristic
606C2×C4C2×#1171994711yescharacteristic
607C8—22no—
608C8—41no—
609C8—1062no—
610C8—2121no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C4×D4×D5311yescharacteristic
626C22#1681112911yescharacteristic
627C22#1681113311yescharacteristic
628C22#1681142411yescharacteristic
629C22#1681142511yescharacteristic
630C22#1681143311yescharacteristic
631C22#1681143311yescharacteristic
632C4—41no—
633C4—42no—
634C4—41no—
635C4—81no—
636C4—1062no—
637C4—1062no—
638C4—4241no—
639C4—2122no—
640C4—2122no—
641C4—2121no—
642C4—2122no—
643C4—2121no—
644C4C2×#1313317311yescharacteristic
645C4C4×#1171994711yescharacteristic
646C4#1681115011yescharacteristic
647C4#1681115411yescharacteristic
648C2—44no—
649C2—82no—
650C2#2027715511yescharacteristic
651C2#2028356211yescharacteristic
652C2#2028359711yescharacteristic
653C2#2028489211yescharacteristic
654C2#2028489611yescharacteristic
655C2#2028490711yescharacteristic
656C2#2028491111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1879C2
socleC23×C53C22×D4
prime radicalC22×D8×C53C22
cosocleC22×C4×C53C24
hypercenter#1879D53
hypocenterC53#21134
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #20277352C2semidirect
3. #20277076C2semidirect
4. #20275345C2semidirect
5. #20275339C2semidirect
6. #20275287C2semidirect
7. #20275285C2semidirect
9. #20274976C2semidirect
10. #20274974C2semidirect
11. #20274603C2semidirect
12. #20274542C2semidirect
13. #20274542C2semidirect
14. #20274465C2semidirect
19. D8×Dic53C4semidirect
70. D4×Dic53C2×C4semidirect
88. #13133151C2×C4semidirect
94. #13133138C2×C4semidirect
96. C8×Dic53C2×C4semidirect
388. C53#21134semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic532
#202773522
#202770762
#202753452
#202753392
#202752872
#202752852
#202750652
#202749762
#202749742
#202746032
#202745422
#202744652
C53×#18792
#2113453