Finite Group Databasebeta

← #27418393

Subgroups — #27418393

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×#16810900C211yesmaximal, characteristic, complemented
2C2×#16810815C211yesmaximal, characteristic, complemented
3C4×#13133191C211yesmaximal, characteristic, complemented
4#20275894C212yesmaximal, complemented
5#20275711C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275643C211yesmaximal, characteristic, complemented
8#20275063C212yesmaximal, complemented
9#20274962C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#532C211yesmaximal, characteristic, complemented
12C22×#11719922C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15C2×C4×C22⋊D53—21no—
16#16810900—24no—
17#16810815—24no—
18C2×#13133150C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×#13133137C2212yes—
25C2×C22⋊#88322C2212yes—
26#16810525C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810514—21no—
29#16810502—21no—
30#16810483C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×#11719921—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C22×#41×C53—21no—
37C53×#532—24no—
38C53×#532—24no—
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C2×#124×C53—21no—
42C2×#121×C53—21no—
43C2×#121×C53C2212yes—
44C2×C4×D4×C53—21no—
45C2×C4×D4×C53C2212yes—
46C22×C42×C53C2211yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C2×#11719922—24no—
51C2×#11719922—24no—
52C2×#11719922—42no—
53C22×#88322C2312yes—
54C22×#88322C2312yes—
55#13133191—22no—
56#13133191—22no—
57C4×D212—22no—
58C4×D212—22no—
59C4×C22⋊D53—44no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2311yescharacteristic
63C2×#11719919—21no—
64C2×C22⋊Dic53—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—21no—
70#13133150—24no—
71C4⋊#88322—22no—
72C4⋊#88322—22no—
73#13133138—24no—
74#13133138—24no—
75#13133138—42no—
76#13133137—24no—
77C22⋊#88322—42no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C2×#41×C53—28no—
82C2×#41×C53—44no—
83C2×#41×C53—44no—
84C22×Q8×C53—21no—
85C22×D4×C53—22no—
86C22×D4×C53—21no—
87C22×D4×C53—21no—
88C22×D4×C53C2312yes—
89C23×C4×C53—22no—
90C23×C4×C53D411yescharacteristic
91C23×C4×C53D411yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C4⋊Q8×C53—22no—
94C4⋊Q8×C53D414yescomplemented
95#127×C53—22no—
96#127×C53D414yescomplemented
97#124×C53—28no—
98#121×C53—28no—
99#121×C53—24no—
100#121×C53—24no—
101C4×D4×C53—28no—
102C4×D4×C53—24no—
103C4×D4×C53—24no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—22no—
107C2×C22⋊C4×C53—22no—
108C2×C42×C53—22no—
109C2×C42×C53—22no—
110C2×C42×C53D411yescharacteristic
111C2×C42×C53D411yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C24×C53—22no—
115C24×C53C2×D411yescharacteristic
116C23×D53—21no—
117#41×C53—432no—
118C2×Q8×C53—24no—
119C2×Q8×C53—42no—
120C2×Q8×C53—42no—
121C2×D4×C53—28no—
122C2×D4×C53—24no—
123C2×D4×C53—24no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53—44no—
132C2×D4×C53—44no—
133C2×D4×C53—42no—
134C2×D4×C53—42no—
135C2×D4×C53C2×D414yes—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—28no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—41no—
148C22×C4×C53—41no—
149C22×C4×C53—44no—
150C22×C4×C53—44no—
151C22×C4×C53D814yescomplemented
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C2×C22⋊D53—24no—
159C2×C22⋊D53—42no—
160C2×C22⋊D53—42no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×D212—21no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212#4112yescomplemented
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53C2×D414yes—
176C22⋊C4×C53—28no—
177C22⋊C4×C53—28no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53C2×D412yes—
183C42×C53C2×D412yes—
184C22⋊Dic53—42no—
185#11719922—48no—
186#11719921—24no—
187#11719921—42no—
188#11719921—42no—
189#11719921—42no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53#4112yescomplemented
196#11719919—42no—
197C4×Dic53—42no—
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—24no—
211C23×C53—42no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53C2×D812yes—
218C23×C53C22×D411yescharacteristic
219C22×D53—22no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226Q8×C53—48no—
227D4×C53—24no—
228D4×C53—24no—
229D4×C53—48no—
230D4×C53—416no—
231D4×C53—42no—
232D4×C53—42no—
233D4×C53—48no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—28no—
239C2×C4×C53—24no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—44no—
247C2×C4×C53—416no—
248C2×C4×C53—42no—
249C2×C4×C53—42no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53#12012yes—
255C2×C4×C53#12012yes—
256C2×C4×C53C2×D812yes—
257C2×C4×C53C2×D812yes—
258C2×C4×C53AGL(1,ℤ/8)12yes—
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C2×C4×C532−1+411yescharacteristic
265C22⋊D53—48no—
266C2×Dic53—22no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275D212—22no—
276D212—22no—
277D212—41no—
278D212—41no—
279C4×D53—84no—
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—44no—
291C22×C53—44no—
292C22×C53—44no—
293C22×C53—82no—
294C22×C53—82no—
295C22×C53—82no—
296C22×C53—82no—
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—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C2×D53—81no—
311C4×C53—22no—
312C4×C53—22no—
313C4×C53—41no—
314C4×C53—44no—
315C4×C53—41no—
316C4×C53#44714yes—
317C4×C53#54611yescharacteristic
318C4×C53#54711yescharacteristic
319C4×C53#54811yescharacteristic
320C4×C53#55011yescharacteristic
321Dic53—42no—
322Dic53—81no—
323Dic53—81no—
324C2×#532D5311yescharacteristic, complemented
325C2×#474—531no—
326C2×#459—531no—
327C4×#121—531no—
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—84no—
337C2×C53#240711yescharacteristic
338C2×C53C2×#44711yescharacteristic
339C2×C53#300511yescharacteristic
340C2×C53#310011yescharacteristic
341C2×C53#311511yescharacteristic
342C2×C53#324511yescharacteristic
343C2×C53#325111yescharacteristic
344D53—82no—
345C22×#41—21no—
346C22×D8—532no—
347#532—24no—
348#532—24no—
349C2×C4⋊Q8C2×D5311yescharacteristic
350C2×#127C2×D5311yescharacteristic
351C2×#124—21no—
352C2×#121—21no—
353C2×#121—531no—
354C2×#121C2×D5312yes—
355C2×C4×D4—21no—
356C2×C4×D4—531no—
357C2×C4×D4—1061no—
358C2×C4×D4C2×D5312yes—
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#390—531no—
371#389—531no—
372#388—1061no—
373#386—1061no—
374#380—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377#340—532no—
378#340—532no—
379#336—532no—
380C53#2415011yescharacteristic, complemented, Sylow
381C2×#41—28no—
382C2×#41—44no—
383C2×#41—44no—
384C22×Q8—21no—
385C22×D4—22no—
386C22×D4—21no—
387C22×D4—21no—
388C22×D4—531no—
389C22×D4—1061no—
390C22×D4C22×D5312yes—
391C23×C4—22no—
392C23×C4—1061no—
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—22no—
402C4⋊Q8C22⋊D5314yescomplemented
403#127—22no—
404#127C22⋊D5314yescomplemented
405#124—28no—
406#121—28no—
407#121—24no—
408#121—24no—
409#121—1062no—
410#121—1062no—
411C4×D4—28no—
412C4×D4—24no—
413C4×D4—24no—
414C4×D4—1062no—
415C4×D4—1062no—
416C4×D4—2124no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4C22×D5312yes—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C42—22no—
431C2×C42—22no—
432C2×C42—1061no—
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—1061no—
446#95—1061no—
447#95—1061no—
448C24—22no—
449C24—1061no—
450C24C2×C22⋊D5311yescharacteristic
451#41—432no—
452C2×Q8—24no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×D4—28no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—532no—
470C2×D4—1064no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—1061no—
476C2×D4—2122no—
477C2×D4—2122no—
478C2×D4C2×C22⋊D5314yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—28no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—2122no—
498C22×C4—2122no—
499C22×C4—2122no—
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—24no—
515C4⋊C4—22no—
516C4⋊C4—532no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—2122no—
523C4⋊C4C2×C22⋊D5314yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—1064no—
527C22⋊C4—2122no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—2122no—
536C42C2×C22⋊D5312yes—
537C42C2×C22⋊D5312yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—1062no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23C2×#1171992212yes—
559C23C22×C22⋊D5311yescharacteristic
560Q8—48no—
561D4—24no—
562D4—24no—
563D4—48no—
564D4—416no—
565D4—48no—
566D4—42no—
567D4—42no—
568D4—1062no—
569D4—1062no—
570D4—2128no—
571D4—2121no—
572D4—2121no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—44no—
586C2×C4—416no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—1062no—
594C2×C4—4244no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4#1313318612yes—
604C2×C4#1313318612yes—
605C2×C4C2×#1171992212yes—
606C2×C4C2×#1171992212yes—
607C2×C4#1313324312yes—
608C2×C4C22×C22⋊D5311yescharacteristic
609C2×C4C22×C22⋊D5311yescharacteristic
610C2×C4#41×D5311yescharacteristic
611C2×C4#41×D5311yescharacteristic
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—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—2122no—
635C22—2122no—
636C22—2122no—
637C22C2×#1313318611yescharacteristic
638C22#1681113011yescharacteristic
639C22#1681120111yescharacteristic
640C22C22×#1171992211yescharacteristic
641C22C2×#1313324311yescharacteristic
642C22#1681142111yescharacteristic
643C22#1681142111yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—44no—
649C4—4241no—
650C4—4241no—
651C4—2122no—
652C4#1681075714yes—
653C4#1681126111yescharacteristic
654C4#1681127211yescharacteristic
655C4#1681128011yescharacteristic
656C4#1681129211yescharacteristic
657C2—42no—
658C2—84no—
659C2—4242no—
660C2#2027805411yescharacteristic
661C2C2×#1681075711yescharacteristic
662C2#2028395711yescharacteristic
663C2#2028471111yescharacteristic
664C2#2028486011yescharacteristic
665C2#2028602211yescharacteristic
666C2#2028607111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#532C2
socleC23×C53C22×D4
hypercenterC2×#532D53
hypocenterC53#24150
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
2. C2×#16810815C2semidirect
3. C4×#13133191C2semidirect
4. #20275894C2semidirect
5. #20275711C2semidirect
6. #20275668C2semidirect
7. #20275643C2semidirect
8. #20275063C2semidirect
9. #20274962C2semidirect
10. #20274695C2semidirect
11. C2×C53×#532C2semidirect
94. C4⋊Q8×C53D4semidirect
96. #127×C53D4semidirect
151. C22×C4×C53D8semidirect
169. C2×D212#41semidirect
195. C4⋊Dic53#41semidirect
324. C2×#532D53semidirect
380. C53#24150semidirect
402. C4⋊Q8C22⋊D53semidirect
404. #127C22⋊D53semidirect
500. C22×C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
C2×#168108152
C4×#131331912
#202758942
#202757112
#202756682
#202756432
#202750632
#202749622
#202746952
C2×C53×#5322
#2415053