Finite Group Databasebeta

← #27275922

Subgroups — #27275922

Order 13568.

1040 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275033C211yesmaximal, characteristic, complemented
2#20275023C211yesmaximal, characteristic, complemented
3C2×#16810360C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic, complemented
5C2×#16810347C211yesmaximal, characteristic, complemented
6#20272960C211yesmaximal, characteristic, complemented
7C2×C53×#538C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C22×C22⋊Dic53—21no—
10C2×#13133141—21no—
11C2×#13133138—21no—
12C2×C22⋊#88322C2211yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810385—21no—
17#16810385—21no—
18#16810360—22no—
19#16810357—22no—
20#16810347—22no—
21C53×#538—22no—
22C53×#538—22no—
23C53×#538—22no—
24C53×#538C414yescomplemented
25C2×#126×C53—21no—
26C2×#124×C53—21no—
27C2×#123×C53—21no—
28C2×C22⋊Q8×C53C2211yescharacteristic
29C2×#121×C53—21no—
30C2×#121×C53C2211yescharacteristic
31C2×#120×C53—21no—
32C2×C4×D4×C53—21no—
33C22×C22⋊C4×C53C2211yescharacteristic
34C23×Dic53—21no—
35C23×Dic53—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—42no—
41C2×C22⋊Dic53—42no—
42C2×C22⋊Dic53—42no—
43C2×C22⋊Dic53—42no—
44#13133141—42no—
45#13133138—42no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—42no—
53C25×C53D411yescharacteristic
54C22×Q8×C53D411yescharacteristic
55C22×D4×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C23×C4×C53D411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62#126×C53—24no—
63#126×C53—24no—
64#124×C53—24no—
65#124×C53—24no—
66#123×C53—24no—
67#123×C53—24no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53C2×C412yes—
71C22⋊Q8×C53D412yes—
72#121×C53—24no—
73#121×C53—24no—
74#121×C53—21no—
75#121×C53—21no—
76#121×C53C2×C412yes—
77#121×C53D412yes—
78#120×C53—24no—
79#120×C53—24no—
80C4×D4×C53—24no—
81C4×D4×C53—24no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53D411yescharacteristic
85C2×C4⋊C4×C53D411yescharacteristic
86C2×C22⋊C4×C53—21no—
87C2×C22⋊C4×C53—21no—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C22⋊C4×C53—21no—
91C2×C22⋊C4×C53—21no—
92C2×C22⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53C2×C412yes—
96C2×C22⋊C4×C53D412yes—
97C2×C42×C53—21no—
98C24×C53—21no—
99C24×C53—21no—
100C24×C53—21no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—21no—
104C24×C53—21no—
105C24×C53—41no—
106C24×C53—41no—
107C24×C53—41no—
108C24×C53—41no—
109C24×C53C22⋊C412yes—
110C24×C53C2×D411yescharacteristic
111C2×Q8×C53—21no—
112C2×Q8×C53—41no—
113C2×Q8×C53—41no—
114C2×Q8×C53C22⋊C412yescomplemented
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—24no—
118C2×D4×C53—24no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53—41no—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—81no—
127C2×D4×C53—81no—
128C2×D4×C53—81no—
129C2×D4×C53—81no—
130C2×D4×C53C22⋊C412yescomplemented
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—21no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—41no—
141C22×C4×C53—81no—
142C22×C4×C53—81no—
143C22×C4×C53C22⋊C412yes—
144C22×C4×C53C22×C411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—41no—
153C22×Dic53—41no—
154C22×Dic53—41no—
155C22×Dic53—41no—
156C22×Dic53—41no—
157C22×Dic53—41no—
158C22×Dic53—41no—
159C22×Dic53—41no—
160C22×Dic53—41no—
161C22×Dic53—41no—
162C22×Dic53—41no—
163C22×Dic53—41no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53C22⋊C412yescomplemented
170C4⋊C4×C53C22⋊C412yescomplemented
171C22⋊C4×C53—24no—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C42×C53—24no—
183C22⋊Dic53—48no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—41no—
187C4⋊Dic53—41no—
188C2×#88322—41no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
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—81no—
223C23×C53—81no—
224C23×C53—81no—
225C23×C53—81no—
226C23×C53—81no—
227C23×C53—81no—
228C23×C53C2×C22⋊C411yescharacteristic
229C23×C53#12011yescharacteristic
230C23×C53#12011yescharacteristic
231C23×C53#13511yescharacteristic
232C23×C53#13511yescharacteristic
233C23×C53AGL(1,ℤ/8)11yescharacteristic
234C23×C53#13711yescharacteristic
235Q8×C53—42no—
236Q8×C53—81no—
237D4×C53—42no—
238D4×C53—42no—
239D4×C53—42no—
240D4×C53—42no—
241D4×C53—42no—
242D4×C53—81no—
243D4×C53—84no—
244D4×C53—84no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—42no—
266C2×C4×C53—81no—
267C2×C4×C53—84no—
268C2×C4×C53C2×C22⋊C411yescharacteristic
269C2×C4×C53C2×C22⋊C411yescharacteristic
270C2×C4×C53#12011yescharacteristic
271C2×C4×C53#12011yescharacteristic
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—42no—
275C2×Dic53—42no—
276C2×Dic53—42no—
277C2×Dic53—42no—
278C2×Dic53—81no—
279C2×Dic53—81no—
280C2×Dic53—81no—
281C2×Dic53—81no—
282C2×Dic53—81no—
283C2×Dic53—81no—
284C2×Dic53—81no—
285C2×Dic53—81no—
286#88322—81no—
287#88322—81no—
288#8408—531nomaximal, Sylow
289C22×C53—21no—
290C22×C53—21no—
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—21no—
299C22×C53—21no—
300C22×C53—21no—
301C22×C53—44no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53—82no—
309C22×C53—82no—
310C22×C53—82no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53—81no—
316C22×C53—81no—
317C22×C53—81no—
318C22×C53—81no—
319C22×C53#37911yescharacteristic
320C22×C53#41611yescharacteristic
321C22×C53#41711yescharacteristic
322C22×C53#44911yescharacteristic
323C22×C53#44911yescharacteristic
324C22×C53#45211yescharacteristic
325C22×C53#45211yescharacteristic
326C22×C53#45311yescharacteristic
327C22×C53#45511yescharacteristic
328C22×C53B(4,𝔽2)11yescharacteristic
329C22×C53#45811yescharacteristic
330C4×C53—21no—
331C4×C53—21no—
332C4×C53—21no—
333C4×C53—21no—
334C4×C53—42no—
335C4×C53—42no—
336C4×C53—42no—
337C4×C53—42no—
338Dic53—82no—
339Dic53—82no—
340C2×#538D5311yescharacteristic
341#1874—531no—
342#1872—531no—
343C2×#328—531no—
344C2×#327—531no—
345C2×#323—531no—
346#1439—531no—
347C2×C53—21no—
348C2×C53—21no—
349C2×C53—21no—
350C2×C53—21no—
351C2×C53—82no—
352C2×C53—82no—
353C2×C53#160011yescharacteristic
354C2×C53#160611yescharacteristic
355C2×C53#161211yescharacteristic
356C2×C53#161811yescharacteristic
357C2×C53#178811yescharacteristic
358C2×C53#179011yescharacteristic
359C2×C53#179211yescharacteristic
360#538—22no—
361#538—22no—
362#538—22no—
363#538Dic5314yescomplemented
364C2×#126—21no—
365C2×#124—21no—
366C2×#123—21no—
367C2×C22⋊Q8C2×D5311yescharacteristic
368C2×#121—21no—
369C2×#121C2×D5311yescharacteristic
370C2×#120—21no—
371C2×C4×D4—21no—
372C22×C4⋊C4—531no—
373C22×C22⋊C4—1061no—
374C22×C22⋊C4C2×D5311yescharacteristic
375C2×#103—1061no—
376C2×#102—1061no—
377C2×C22⋊C8—531no—
378C2×C22⋊C8—531no—
379C2×#95—531no—
380C2×#95—1061no—
381#340—1061no—
382#340—1061no—
383#328—1062no—
384#327—1062no—
385#323—1062no—
386C53#840811yescharacteristic, complemented, Sylow
387C25C22⋊D5311yescharacteristic
388C22×Q8C22⋊D5311yescharacteristic
389C22×D4—21no—
390C22×D4—21no—
391C22×D4—21no—
392C22×D4C22⋊D5311yescharacteristic
393C23×C4—21no—
394C23×C4—1061no—
395C23×C4—1061no—
396C23×C4C22⋊D5311yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C22×C8—1061no—
399C22×C8—1061no—
400#126—24no—
401#126—24no—
402#124—24no—
403#124—24no—
404#123—24no—
405#123—24no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8C2×Dic5312yes—
409C22⋊Q8C22⋊D5312yes—
410#121—24no—
411#121—24no—
412#121—21no—
413#121—21no—
414#121C2×Dic5312yes—
415#121C22⋊D5312yes—
416#120—24no—
417#120—24no—
418C4×D4—24no—
419C4×D4—24no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—2122no—
425C2×C4⋊C4C22⋊D5311yescharacteristic
426C2×C4⋊C4C22⋊D5311yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—2122no—
437C2×C22⋊C4—2122no—
438C2×C22⋊C4—2122no—
439C2×C22⋊C4C2×Dic5312yes—
440C2×C22⋊C4C22⋊D5312yes—
441C2×C42—21no—
442#103—2122no—
443#102—2122no—
444C22⋊C8—1061no—
445C22⋊C8—1061no—
446C22⋊C8—1061no—
447C22⋊C8—1061no—
448#95—1061no—
449#95—1061no—
450#95—2122no—
451C24—21no—
452C24—21no—
453C24—21no—
454C24—21no—
455C24—21no—
456C24—21no—
457C24—21no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24—41no—
462C24C22⋊Dic5312yes—
463C24C2×C22⋊D5311yescharacteristic
464C2×Q8—21no—
465C2×Q8—41no—
466C2×Q8—41no—
467C2×Q8C22⋊Dic5312yescomplemented
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—81no—
480C2×D4—81no—
481C2×D4—81no—
482C2×D4—81no—
483C2×D4C22⋊Dic5312yescomplemented
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—2121no—
499C22×C4—2121no—
500C22×C4—2121no—
501C22×C4—2121no—
502C22×C4—2121no—
503C22×C4—2121no—
504C22×C4—2121no—
505C22×C4—2121no—
506C22×C4—2121no—
507C22×C4—2121no—
508C22×C4—2121no—
509C22×C4—2121no—
510C22×C4C22⋊Dic5312yes—
511C22×C4C22×Dic5311yescharacteristic
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C2×C8—2121no—
518C2×C8—2121no—
519C2×C8—2121no—
520C2×C8—2121no—
521C2×C8—2121no—
522C2×C8—2121no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—2121no—
531C4⋊C4—2121no—
532C4⋊C4C22⋊Dic5312yescomplemented
533C4⋊C4C22⋊Dic5312yescomplemented
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—2128no—
546C42—24no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—81no—
576C23—81no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23C2×C22⋊Dic5311yescharacteristic
582C23#1313318611yescharacteristic
583C23#1313318611yescharacteristic
584C23#1313323711yescharacteristic
585C23#1313323711yescharacteristic
586C23#1313324211yescharacteristic
587C23#1313324711yescharacteristic
588Q8—42no—
589Q8—81no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—81no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—81no—
620C2×C4—84no—
621C2×C4—4241no—
622C2×C4—4241no—
623C2×C4—4241no—
624C2×C4—4241no—
625C2×C4—4241no—
626C2×C4—4241no—
627C2×C4—4241no—
628C2×C4—4241no—
629C2×C4—2122no—
630C2×C4—2122no—
631C2×C4—2122no—
632C2×C4—2122no—
633C2×C4—2122no—
634C2×C4—2122no—
635C2×C4C2×C22⋊Dic5311yescharacteristic
636C2×C4C2×C22⋊Dic5311yescharacteristic
637C2×C4#1313318611yescharacteristic
638C2×C4#1313318611yescharacteristic
639C8—4241no—
640C8—4241no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#1681047911yescharacteristic
672C22#1681061911yescharacteristic
673C22#1681062411yescharacteristic
674C22#1681076811yescharacteristic
675C22#1681076811yescharacteristic
676C22#1681078511yescharacteristic
677C22#1681078511yescharacteristic
678C22#1681079111yescharacteristic
679C22#1681080111yescharacteristic
680C22#1681081011yescharacteristic
681C22#1681081311yescharacteristic
682C4—21no—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—4242no—
691C4—4242no—
692C2—21no—
693C2—21no—
694C2—21no—
695C2—21no—
696C2—82no—
697C2—82no—
698C2#2027366111yescharacteristic
699C2#2027369511yescharacteristic
700C2#2027372911yescharacteristic
701C2#2027376311yescharacteristic
702C2#2027465511yescharacteristic
703C2#2027466311yescharacteristic
704C2#2027466911yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#538C2
socleC23×C53#120
cosocleC23×C4×C53C23
hypercenterC2×#538D53
hypocenterC53#8408
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. #20275033C2semidirect
2. #20275023C2semidirect
3. C2×#16810360C2semidirect
4. C2×#16810357C2semidirect
5. C2×#16810347C2semidirect
6. #20272960C2semidirect
24. C53×#538C4semidirect
114. C2×Q8×C53C22⋊C4semidirect
130. C2×D4×C53C22⋊C4semidirect
169. C4⋊C4×C53C22⋊C4semidirect
170. C4⋊C4×C53C22⋊C4semidirect
363. #538Dic53semidirect
386. C53#8408semidirect
467. C2×Q8C22⋊Dic53semidirect
483. C2×D4C22⋊Dic53semidirect
532. C4⋊C4C22⋊Dic53semidirect
533. C4⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202750332
#202750232
C2×#168103602
C2×#168103572
C2×#168103472
#202729602
C2×C53×#5382
#840853