Finite Group Databasebeta

← #27391242

Subgroups — #27391242

Order 13568.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810949C211yesmaximal, characteristic
2#20277223C211yesmaximal, characteristic
3C4×#13133175C211yesmaximal, characteristic, complemented
4#20275343C211yesmaximal, characteristic, complemented
5#20275335C211yesmaximal, characteristic, complemented
6#20275304C211yesmaximal, characteristic, complemented
7#20275036C211yesmaximal, characteristic
8#20274915C211yesmaximal, characteristic, complemented
9#20274808C211yesmaximal, characteristic, complemented
10#20274607C211yesmaximal, characteristic, complemented
11#106×Dic53C211yesmaximal, characteristic, complemented
12#20274529C211yesmaximal, characteristic, complemented
13#20274529C211yesmaximal, characteristic, complemented
14#20274472C211yesmaximal, characteristic, complemented
15C53×#1920C211yesmaximal, characteristic, complemented
16C2×#13133192C2211yescharacteristic
17C2×#13133179—21no—
18C2×#13133179C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133175C2211yescharacteristic
21C2×C42×D53C2211yescharacteristic
22#16810949C418yescomplemented
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133129C2211yescharacteristic
31C2×#13133129C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34#16810525C2211yescharacteristic
35#16810513—21no—
36#16810501—21no—
37#16810480C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×#11719921—21no—
43#16810470—21no—
44#16810470—21no—
45#16810468—21no—
46C42×Dic53—21no—
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×D212—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53D411yescharacteristic
64C22×C4×D53D411yescharacteristic
65C22×C4×D53C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133192—22no—
69#13133192C2×C414yes—
70#13133179—22no—
71#13133179—22no—
72#13133179—44no—
73#13133175—22no—
74#13133175—22no—
75#13133175—22no—
76#13133175C2×C414yescomplemented
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—22no—
97C42×D53—22no—
98#13133150—22no—
99#13133150C2×C414yescomplemented
100#13133149—22no—
101#13133139—22no—
102#13133139C2×C414yescomplemented
103#13133137—22no—
104#13133137C2×C414yescomplemented
105#13133129—22no—
106#13133129C2×C414yescomplemented
107C8×Dic53—22no—
108C8×Dic53—22no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#106×C53C2×C414yescomplemented
124#95×C53—21no—
125#95×C53—21no—
126C23×D53—21no—
127C23×D53C2×D411yescharacteristic
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—24no—
146C2×D212—21no—
147C2×D212—42no—
148C2×D212—42no—
149C2×D212C22×C412yes—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—21no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—41no—
160C2×C4×D53—42no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53C22×C412yes—
164C2×C4×D53#4112yes—
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—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53C22×C412yes—
178C4⋊C4×C53C22×C412yes—
179C4⋊C4×C53#4112yes—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—42no—
183#11719921—24no—
184#11719921—24no—
185#11719921—24no—
186#11719921—42no—
187#11719921—42no—
188#11719921—42no—
189#11719921—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53C22×C412yes—
195C4⋊Dic53#4112yes—
196C4×Dic53—24no—
197C4×Dic53—24no—
198C4×Dic53—22no—
199C4×Dic53—22no—
200C4×Dic53—42no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C4×Dic53—42no—
204C4×Dic53#4112yes—
205C4×Dic53#4112yes—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322C22×C412yes—
212C23×C53C22×D411yescharacteristic
213C22×D53—21no—
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—22no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53C4×D412yes—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—41no—
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×C53#13512yes—
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—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53C4×D412yes—
265C2×Dic53#12312yes—
266C2×Dic53#12312yes—
267C2×Dic53#13512yes—
268C2×Dic53AGL(1,ℤ/8)12yes—
269D212—22no—
270D212—22no—
271D212—48no—
272D212—41no—
273D212—41no—
274C4×D53—22no—
275C4×D53—22no—
276C4×D53—22no—
277C4×D53—22no—
278C4×D53—44no—
279C4×D53—44no—
280C4×D53—84no—
281C8×C53—22no—
282C8×C53—41no—
283#88322—22no—
284#88322—41no—
285C4×#482—531nomaximal, Sylow
286C22×C53C2×C4×D411yescharacteristic
287C22×C53C2×#12311yescharacteristic
288C22×C53#52511yescharacteristic
289C22×C53C2×#13511yescharacteristic
290C22×C53C2×#13511yescharacteristic
291C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C53#57511yescharacteristic
293C2×D53—22no—
294C2×D53—22no—
295C2×D53—22no—
296C2×D53—41no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—41no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—42no—
312C4×C53—81no—
313C4×C53C4×#4111yescharacteristic
314C4×C53C4×#4111yescharacteristic
315C4×C53#52911yescharacteristic
316C4×C53#52911yescharacteristic
317Dic53—22no—
318Dic53—22no—
319Dic53—42no—
320Dic53—42no—
321Dic53—81no—
322Dic53#48214yescomplemented
323C2×#482—531no—
324C4×#121—531no—
325C4×#117—531no—
326#1926—531no—
327#1925—531no—
328#1921—531no—
329#1920D5311yescharacteristic, complemented
330#1875—531no—
331#1854—531no—
332#1825—531no—
333C4×#107—531no—
334C4×#106—531no—
335C4×#102—531no—
336C4×#102—531no—
337C4×C22⋊C8—531no—
338C2×C53C4×#12311yescharacteristic
339C2×C53C4×#13511yescharacteristic
340C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
341C2×C53C2×#48211yescharacteristic
342C2×C53#309111yescharacteristic
343C2×C53#310211yescharacteristic
344C2×C53#310611yescharacteristic
345D53—42no—
346D53—82no—
347C2×#121—531no—
348C2×C4×D4—531no—
349C2×C4×D4—1061no—
350C2×#117—531no—
351C2×#117—531no—
352C22×C42—531no—
353#482—538no—
354C2×#107—531no—
355C2×#107—531no—
356C2×#106—531no—
357C2×#106C2×D5311yescharacteristic
358C2×#102—531no—
359C2×#102—531no—
360C2×#102—531no—
361C2×#102—531no—
362C2×C22⋊C8—531no—
363C2×C22⋊C8—531no—
364C2×C4×C8—531no—
365C2×C4×C8—531no—
366#390—531no—
367#389C2×D5311yescharacteristic
368#388—1061no—
369#386—1061no—
370#380—531no—
371C4×C4⋊C4—531no—
372C4×C4⋊C4—531no—
373C4×C4⋊C4—531no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4C2×D5311yescharacteristic
376C4×C22⋊C4—1061no—
377C4×C22⋊C4—1061no—
378C4×C22⋊C4—1061no—
379#376—1061no—
380C43—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340C2×D5311yescharacteristic
387#340C2×D5311yescharacteristic
388#340C2×D5311yescharacteristic
389#336—531no—
390#336C2×D5311yescharacteristic
391C53C4×#48211yescharacteristic, complemented, Sylow
392C22×D4—531no—
393C22×D4—1061no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—1061no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402#121—534no—
403#121—1062no—
404C4×D4—1062no—
405C4×D4—1062no—
406C4×D4—2124no—
407#117—534no—
408#117—1062no—
409#117—1062no—
410#117—1062no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4C22×D5311yescharacteristic
416C2×C4⋊C4C22×D5311yescharacteristic
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
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—1062no—
431C2×C42—1062no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C22×D5311yescharacteristic
439#107—534no—
440#107—1062no—
441#106—1062no—
442#106C4×D5314yescomplemented
443#102—534no—
444#102—534no—
445#102—1062no—
446#102—1062no—
447C22⋊C8—534no—
448C22⋊C8—1062no—
449C4×C8—1062no—
450C4×C8—1062no—
451#95—21no—
452#95—21no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458C24—531no—
459C24—1061no—
460C2×D4—532no—
461C2×D4—1064no—
462C2×D4—1061no—
463C2×D4—1061no—
464C2×D4—1061no—
465C2×D4—1061no—
466C2×D4—1061no—
467C2×D4—2122no—
468C2×D4—2122no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1062no—
489C22×C4—1061no—
490C22×C4—2121no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4D4×D5311yescharacteristic
495C22×C4D4×D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—532no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8C2×C4×D5312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—532no—
516C4⋊C4—532no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4C2×C4×D5312yes—
522C4⋊C4C2×C4×D5312yes—
523C4⋊C4#1171994912yes—
524C22⋊C4—1064no—
525C22⋊C4—1064no—
526C22⋊C4—1064no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—532no—
535C42—532no—
536C42—1062no—
537C42—1062no—
538C42—1064no—
539C42—1064no—
540C42—2122no—
541C42—2122no—
542C42—2122no—
543C42—2122no—
544C23—532no—
545C23—1061no—
546C23—1061no—
547C23—1061no—
548C23—1061no—
549C23—1061no—
550C23—1061no—
551C23—1062no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23C2×D4×D5311yescharacteristic
559D4—1062no—
560D4—1062no—
561D4—2128no—
562D4—2121no—
563D4—2121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—532no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—532no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—4244no—
597C2×C4—2124no—
598C2×C4—2124no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4#1313323812yes—
607C2×C4C22×C4×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4#41×D5311yescharacteristic
610C2×C4#41×D5311yescharacteristic
611C2×C4#41×D5311yescharacteristic
612C2×C4C2×#1171994911yescharacteristic
613C8—22no—
614C8—41no—
615C8—1062no—
616C8—2121no—
617C22—1062no—
618C22—1062no—
619C22—1062no—
620C22—4241no—
621C22—4241no—
622C22—4241no—
623C22—4241no—
624C22—2122no—
625C22—2122no—
626C22—2122no—
627C22—2121no—
628C22—2121no—
629C22—2121no—
630C22—2121no—
631C22C4×D4×D5311yescharacteristic
632C22#123×D5311yescharacteristic
633C22#1681113611yescharacteristic
634C22#135×D5311yescharacteristic
635C22C2×#1313323811yescharacteristic
636C22AGL(1,ℤ/8)×D5311yescharacteristic
637C22#1681142211yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—81no—
644C4—534no—
645C4—1062no—
646C4—1062no—
647C4—4241no—
648C4—2122no—
649C4—2122no—
650C4#1681109111yescharacteristic
651C4C4×#1171994911yescharacteristic
652C4#1681115611yescharacteristic
653C4#1681115811yescharacteristic
654C2—4242no—
655C2—2122no—
656C2#2027723611yescharacteristic
657C2C4×#1313323811yescharacteristic
658C2#2028352011yescharacteristic
659C2#482×D5311yescharacteristic
660C2#2028462911yescharacteristic
661C2#2028474011yescharacteristic
662C2#2028477711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1920C2
socleC23×C53C22×D4
prime radicalC2×#13133192C22
cosocleC22×C4×C53C24
hypercenter#1920D53
hypocenterC53C4×#482
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
3. C4×#13133175C2semidirect
4. #20275343C2semidirect
5. #20275335C2semidirect
6. #20275304C2semidirect
8. #20274915C2semidirect
9. #20274808C2semidirect
10. #20274607C2semidirect
11. #106×Dic53C2semidirect
12. #20274529C2semidirect
13. #20274529C2semidirect
14. #20274472C2semidirect
15. C53×#1920C2semidirect
22. #16810949C4semidirect
76. #13133175C2×C4semidirect
99. #13133150C2×C4semidirect
102. #13133139C2×C4semidirect
104. #13133137C2×C4semidirect
106. #13133129C2×C4semidirect
123. #106×C53C2×C4semidirect
322. Dic53#482semidirect
329. #1920D53semidirect
391. C53C4×#482semidirect
442. #106C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109492
#202772232
C4×#131331752
#202753432
#202753352
#202753042
#202750362
#202749152
#202748082
#202746072
#106×Dic532
#202745292
#202744722
C53×#19202
C4×#48253