Finite Group Databasebeta

← #27418399

Subgroups — #27418399

Order 13568.

1478 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811173C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic, complemented
4#20275897C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275667C211yesmaximal, characteristic, complemented
7#20275642C211yesmaximal, characteristic, complemented
8#20275066C212yesmaximal, complemented
9#20274961C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C4×#121×C53C211yesmaximal, characteristic, complemented
12C22×#11719945—21no—
13C22×#11719922C2212yes—
14#16811173—24no—
15#16811173—24no—
16C2×#13133217C2211yescharacteristic
17C2×#13133214C2211yescharacteristic
18C2×#13133205—21no—
19C2×#13133191—21no—
20C2×#13133191C2212yes—
21C2×C4×D212—21no—
22C2×C4×D212C2212yes—
23#16810903—24no—
24#16810818—24no—
25C2×#13133150C2212yes—
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133138C2212yes—
29C2×#13133137C2212yes—
30C2×#13133137C2212yes—
31C2×#13133137C2212yes—
32C2×C22⋊#88322C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#121×C53C2211yescharacteristic
37C2×C4×D4×C53—21no—
38C2×C4×D4×C53C2211yescharacteristic
39C22×C42×C53C2211yescharacteristic
40C53×#390C2211yescharacteristic
41C53×#389C2211yescharacteristic
42C53×#388—21no—
43C53×#386—21no—
44C53×#380C2211yescharacteristic
45C4×C4⋊C4×C53C2211yescharacteristic
46C4×C22⋊C4×C53—21no—
47C2×#11719945—28no—
48C2×#11719945—44no—
49C2×#11719945—44no—
50C22×Dic106—21no—
51C22×D212—21no—
52C22×D212—21no—
53C22×D212C2312yes—
54C22×C22⋊D53—22no—
55C22×C4×D53—22no—
56C2×#11719922—24no—
57C2×#11719922—24no—
58C2×#11719922—42no—
59C22×#88322C2312yes—
60C22×#88322C2312yes—
61#13133217—22no—
62#13133217D414yescomplemented
63#13133214—22no—
64#13133214D414yescomplemented
65#13133205—28no—
66#13133191—28no—
67#13133191—24no—
68#13133191—24no—
69C4×D212—28no—
70C4×D212—24no—
71C4×D212—24no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2312yes—
74C2×#11719921—22no—
75C2×#11719921—22no—
76#13133150—24no—
77C4⋊#88322—22no—
78C4⋊#88322—22no—
79#13133138—24no—
80#13133137—24no—
81#13133137—24no—
82#13133137—42no—
83C22⋊#88322—42no—
84C22×D4×C53—21no—
85C22×D4×C53C2311yescharacteristic
86C23×C4×C53—21no—
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90#121×C53—22no—
91#121×C53—22no—
92C4×D4×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—44no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C42×C53—22no—
105C2×C42×C53—22no—
106C2×C42×C53—21no—
107C2×C42×C53D411yescharacteristic
108C2×C42×C53D411yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114C24×C53—21no—
115C24×C53C2×D411yescharacteristic
116C23×D53—22no—
117C2×D4×C53—24no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—42no—
124C2×D4×C53—42no—
125C2×D4×C53#4112yescomplemented
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—42no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53D814yescomplemented
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C2×C22⋊D53—28no—
151C2×C22⋊D53—44no—
152C2×C22⋊D53—44no—
153C22×Dic53—22no—
154#11719945—432no—
155C2×D212—22no—
156C2×D212—22no—
157C2×D212—22no—
158C2×D212—22no—
159C2×D212—24no—
160C2×D212—24no—
161C2×D212—22no—
162C2×D212—42no—
163C2×D212—42no—
164C2×D212—42no—
165C2×D212—42no—
166C2×D212C2×D414yes—
167C2×C4×D53—28no—
168C2×C4×D53—44no—
169C2×C4×D53—44no—
170C2×Dic106—24no—
171C2×Dic106—42no—
172C2×Dic106—42no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53#4112yescomplemented
180C22⋊C4×C53—24no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—42no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—42no—
190C42×C53C2×D412yes—
191C42×C53C2×D412yes—
192#11719922—48no—
193#11719921—28no—
194#11719921—28no—
195C4⋊Dic53—24no—
196C4⋊Dic53—22no—
197C4⋊Dic53C2×D414yes—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—22no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—41no—
216C23×C53—41no—
217C23×C53C2×D812yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—24no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225C22×D53—42no—
226D4×C53—22no—
227D4×C53—22no—
228D4×C53—48no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—28no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—44no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—84no—
257C2×C4×C53#12012yes—
258C2×C4×C53#12012yes—
259C2×C4×C53C2×D812yes—
260C2×C4×C53C2×D812yes—
261C2×C4×C53AGL(1,ℤ/8)12yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C22⋊D53—416no—
269C2×Dic53—24no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272D212—24no—
273D212—24no—
274D212—48no—
275D212—42no—
276D212—42no—
277D212—48no—
278C4×D53—416no—
279Dic106—48no—
280#88322—42no—
281#88322—42no—
282#24150—531nomaximal, Sylow
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53C2×#12011yescharacteristic
298C22×C53#52511yescharacteristic
299C22×C53#53711yescharacteristic
300C22×C53C22×D811yescharacteristic
301C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57511yescharacteristic
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C2×D53—82no—
308C2×D53—82no—
309C2×D53—82no—
310C2×D53—82no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—42no—
314C4×C53—41no—
315C4×C53—41no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53#44714yes—
319C4×C53#54611yescharacteristic
320C4×C53#54711yescharacteristic
321C4×C53#54811yescharacteristic
322C4×C53#55011yescharacteristic
323Dic53—44no—
324C2×#532—531no—
325C2×#474—531no—
326C2×#459—531no—
327C4×#121D5311yescharacteristic, complemented
328#2033—532no—
329#1999—532no—
330#1988—531no—
331#1982—531no—
332#1879—532no—
333#1862—532no—
334#1797—531no—
335C2×C53—42no—
336C2×C53—82no—
337C2×C53#240711yescharacteristic
338C2×C53C2×#44711yescharacteristic
339C2×C53#300511yescharacteristic
340C2×C53#310011yescharacteristic
341C2×C53#311511yescharacteristic
342C2×C53#324511yescharacteristic
343C2×C53#325111yescharacteristic
344D53—84no—
345C22×#41—1061no—
346C22×D8—532no—
347#532—1064no—
348#532—1064no—
349C2×C4⋊Q8—531no—
350C2×#127—531no—
351C2×#124—1061no—
352C2×#121—532no—
353C2×#121—1061no—
354C2×#121C2×D5311yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4—532no—
357C2×C4×D4—1061no—
358C2×C4×D4C2×D5311yescharacteristic
359C22×C42C2×D5311yescharacteristic
360#474—1064no—
361#459—1064no—
362C2×#107—532no—
363C2×C4⋊C8—531no—
364C2×C4⋊C8—531no—
365C2×#102—532no—
366C2×#102—532no—
367C2×#102—532no—
368C2×#102—532no—
369C2×C22⋊C8—532no—
370#390C2×D5311yescharacteristic
371#389C2×D5311yescharacteristic
372#388—21no—
373#386—21no—
374#380C2×D5311yescharacteristic
375C4×C4⋊C4C2×D5311yescharacteristic
376C4×C22⋊C4—21no—
377#340—532no—
378#340—532no—
379#336—532no—
380C53#2415011yescharacteristic, complemented, Sylow
381C2×#41—1068no—
382C2×#41—2124no—
383C2×#41—2124no—
384C22×Q8—1061no—
385C22×D4—21no—
386C22×D4—532no—
387C22×D4—1062no—
388C22×D4—1061no—
389C22×D4—1061no—
390C22×D4C22×D5311yescharacteristic
391C23×C4—21no—
392C23×C4—1062no—
393C23×C4C22⋊D5311yescharacteristic
394C23×C4C22⋊D5311yescharacteristic
395C23×C4C22×D5311yescharacteristic
396C2×D8—1064no—
397C2×D8—1064no—
398C2×D8—2122no—
399C22×C8—532no—
400C22×C8—532no—
401C4⋊Q8—534no—
402C4⋊Q8—1062no—
403#127—534no—
404#127—1062no—
405#124—1068no—
406#121—22no—
407#121—22no—
408#121—1064no—
409#121—1064no—
410#121—1068no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—44no—
414C4×D4—1064no—
415C4×D4—1064no—
416C4×D4—1068no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—532no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1062no—
429C2×C22⋊C4—1062no—
430C2×C42—22no—
431C2×C42—22no—
432C2×C42—21no—
433C2×C42C22⋊D5311yescharacteristic
434C2×C42C22⋊D5311yescharacteristic
435C2×C42C22×D5311yescharacteristic
436C2×C42C22×D5311yescharacteristic
437#107—1064no—
438C4⋊C8—1062no—
439C4⋊C8—1062no—
440#102—1064no—
441#102—1064no—
442#102—1064no—
443#102—2122no—
444C22⋊C8—2122no—
445#95—21no—
446#95—21no—
447#95—21no—
448C24—21no—
449C24—1062no—
450C24C2×C22⋊D5311yescharacteristic
451#41—21232no—
452C2×Q8—1064no—
453C2×Q8—2122no—
454C2×Q8—2122no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—534no—
464C2×D4—1068no—
465C2×D4—1064no—
466C2×D4—1064no—
467C2×D4—1062no—
468C2×D4—1062no—
469C2×D4—1062no—
470C2×D4—1062no—
471C2×D4—1062no—
472C2×D4—2122no—
473C2×D4—2122no—
474C2×D4—2124no—
475C2×D4—2124no—
476C2×D4—2122no—
477C2×D4—2122no—
478C2×D4#1171994512yescomplemented
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—1068no—
497C22×C4—1062no—
498C22×C4—2124no—
499C22×C4—2124no—
500C22×C4#1171992214yescomplemented
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507D8—2128no—
508C2×C8—1062no—
509C2×C8—1062no—
510C2×C8—1062no—
511C2×C8—1062no—
512C2×C8—1062no—
513C2×C8—1062no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—534no—
521C4⋊C4—1064no—
522C4⋊C4—1062no—
523C4⋊C4#1171994512yescomplemented
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—1068no—
530C22⋊C4—1068no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42D4×D5312yes—
537C42D4×D5312yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—1064no—
552C23—2122no—
553C23—2122no—
554C23—2122no—
555C23—2122no—
556C23—2122no—
557C23—2122no—
558C23C2×#1171992212yes—
559C23C22×C22⋊D5311yescharacteristic
560Q8—2128no—
561D4—22no—
562D4—22no—
563D4—48no—
564D4—41no—
565D4—41no—
566D4—1064no—
567D4—1064no—
568D4—2128no—
569D4—21216no—
570D4—2128no—
571D4—2122no—
572D4—2122no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—1064no—
600C2×C4—21216no—
601C2×C4—2122no—
602C2×C4—2122no—
603C2×C4#1313318512yes—
604C2×C4#1313318512yes—
605C2×C4C2×#1171992212yes—
606C2×C4C2×#1171992212yes—
607C2×C4#1313324212yes—
608C2×C4C2×D4×D5311yescharacteristic
609C2×C4C2×D4×D5311yescharacteristic
610C2×C4C2×#1171994511yescharacteristic
611C2×C4C2×#1171994511yescharacteristic
612C2×C4#1313326311yescharacteristic
613C2×C4#1313326411yescharacteristic
614C8—2122no—
615C8—2122no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—4242no—
631C22—4242no—
632C22—4242no—
633C22—4242no—
634C22—2124no—
635C22—2124no—
636C22—2124no—
637C22C2×#1313318511yescharacteristic
638C22#1681113411yescharacteristic
639C22#1681120111yescharacteristic
640C22C22×#1171992211yescharacteristic
641C22C2×#1313324211yescharacteristic
642C22#1681142111yescharacteristic
643C22#1681142111yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—2124no—
652C4#1681075614yes—
653C4#1681126911yescharacteristic
654C4#1681127811yescharacteristic
655C4#1681128211yescharacteristic
656C4#1681129411yescharacteristic
657C2—42no—
658C2—82no—
659C2—4244no—
660C2#2027806011yescharacteristic
661C2C2×#1681075611yescharacteristic
662C2#2028395511yescharacteristic
663C2#2028471511yescharacteristic
664C2#2028486611yescharacteristic
665C2#2028602211yescharacteristic
666C2#2028607711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
hypercenterC4×#121D53
hypocenterC53#24150
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16811173C2semidirect
2. C2×#16810903C2semidirect
3. C2×#16810818C2semidirect
4. #20275897C2semidirect
5. #20275711C2semidirect
6. #20275667C2semidirect
7. #20275642C2semidirect
8. #20275066C2semidirect
9. #20274961C2semidirect
10. #20274695C2semidirect
11. C4×#121×C53C2semidirect
62. #13133217D4semidirect
64. #13133214D4semidirect
125. C2×D4×C53#41semidirect
143. C22×C4×C53D8semidirect
179. C4⋊C4×C53#41semidirect
327. C4×#121D53semidirect
380. C53#24150semidirect
478. C2×D4#11719945semidirect
500. C22×C4#11719922semidirect
523. C4⋊C4#11719945semidirect

Maximal subgroups

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

subgroupindex
C2×#168111732
C2×#168109032
C2×#168108182
#202758972
#202757112
#202756672
#202756422
#202750662
#202749612
#202746952
C4×#121×C532
#2415053