Finite Group Databasebeta

← #27444186

Subgroups — #27444186

Order 13568.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic
2C2×C4×#11719924C211yesmaximal, characteristic
3C4⋊C4×D212C211yesmaximal, characteristic
4#20276258C211yesmaximal, characteristic, complemented
5#20276217C211yesmaximal, characteristic, complemented
6#20276123C211yesmaximal, characteristic, complemented
7#20275958C211yesmaximal, characteristic, complemented
8#20275394C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275052C211yesmaximal, characteristic
11#20275052C211yesmaximal, characteristic
12#20275052C211yesmaximal, characteristic
13#20274983C211yesmaximal, characteristic, complemented
14#20274975C211yesmaximal, characteristic
15C53×#2446C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719924C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×C4×D212C2211yescharacteristic
21C2×C4⋊C4×D53—21no—
22#16810825—24no—
23C4×#11719924—24no—
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810497—21no—
39#16810497—21no—
40#16810493C2211yescharacteristic
41#16810480—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#400—21no—
53C53×#398—21no—
54C53×#391C2211yescharacteristic
55C53×#391C2211yescharacteristic
56C53×#391C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#387—21no—
59C53×#387—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719924—22no—
66C2×#11719924—22no—
67C2×#11719924—22no—
68C2×#11719924—22no—
69C2×#11719924—22no—
70C2×#11719924D414yes—
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75C4×D212—22no—
76C4×D212—22no—
77C4×D212—22no—
78C4×D212—22no—
79C4×D212—41no—
80C4×D212—41no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C4⋊C4×D53—44no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92#13133148—22no—
93#13133148—41no—
94C4⋊#88322—22no—
95#13133141—22no—
96#13133141—22no—
97#13133141—22no—
98#13133141—22no—
99#13133141—22no—
100#13133141—41no—
101#13133137—22no—
102#13133137—22no—
103C4×#88322—22no—
104#13133124—21no—
105#13133124—21no—
106C22×Q8×C53C2311yescharacteristic
107C22×Q8×C53C2311yescharacteristic
108C4⋊Q8×C53—22no—
109C4⋊Q8×C53—41no—
110C4×Q8×C53—22no—
111C4×Q8×C53—41no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53D411yescharacteristic
117C2×C4⋊C4×C53D411yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C42×C53—21no—
121C2×C42×C53C2311yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53D411yescharacteristic
132#95×C53D411yescharacteristic
133C23×D53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53C2×D412yes—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2411yescharacteristic
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212C2×D412yes—
164C2×D212#4112yes—
165C2×D212#4112yes—
166C2×C4×D53—24no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C4×D53—81no—
174C2×C4×D53—81no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—41no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—42no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53SD1614yescomplemented
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—42no—
192#11719924—28no—
193#11719924—44no—
194#11719921—24no—
195#11719921—42no—
196#11719921—42no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—42no—
203C4⋊Dic53—81no—
204C4⋊Dic53#4112yes—
205#11719919—42no—
206#11719919—42no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322C2×D412yes—
219C23×C53C22×D411yescharacteristic
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—41no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227Q8×C53—22no—
228Q8×C53—22no—
229Q8×C53—41no—
230Q8×C53—41no—
231Q8×C53—41no—
232Q8×C53—41no—
233Q8×C53—41no—
234Q8×C53—41no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
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×C53C2×SD1612yes—
263C2×C4×C53C2×SD1612yes—
264C2×C4×C53C2×SD1612yes—
265C2×C4×C53C22×D411yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×C4×C532−1+411yescharacteristic
271C2×Dic53—22no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280D212—22no—
281D212—22no—
282D212—22no—
283D212#12314yes—
284C4×D53—48no—
285C4×D53—84no—
286C4×D53—84no—
287#88322—22no—
288#88322—41no—
289#88322—41no—
290#88322—41no—
291#26166—531nomaximal, Sylow
292C22×C53#52511yescharacteristic
293C22×C53D4211yescharacteristic
294C22×C53#54811yescharacteristic
295C22×C53C22×SD1611yescharacteristic
296C22×C53#57511yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57811yescharacteristic
299C2×D53—24no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—41no—
310C4×C53—42no—
311C4×C53—41no—
312C4×C53—41no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53C2×#12311yescharacteristic
317C4×C53#53911yescharacteristic
318C4×C53#54711yescharacteristic
319C4×C53#55311yescharacteristic
320Dic53—42no—
321Dic53—81no—
322Dic53—81no—
323C2×#461—531no—
324C2×C4×SD16—531no—
325#2446D5311yescharacteristic, complemented
326D4×C4⋊C4—531no—
327#2096—531no—
328#2088—531no—
329#2072—531no—
330#2043—531no—
331#1933—531no—
332#1916—531no—
333#1877—531no—
334#1877—531no—
335#1877—531no—
336#1865—531no—
337#1864—531no—
338C2×C53#244311yescharacteristic
339C2×C53#310111yescharacteristic
340C2×C53#312011yescharacteristic
341C2×C53#326911yescharacteristic
342C2×C53D4×SD1611yescharacteristic
343C2×C53#335911yescharacteristic
344C2×C53#337611yescharacteristic
345D53—44no—
346C22×SD16—531no—
347C22×SD16—531no—
348C2×C4⋊Q8C2×D5311yescharacteristic
349C2×C4×Q8C2×D5311yescharacteristic
350C2×C4×D4—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C22×C4⋊C4—1061no—
354#461—1064no—
355C4×SD16—1064no—
356C2×#106—531no—
357C2×#106—531no—
358C2×C4⋊C8—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×C4×C8—531no—
368#400—21no—
369#398—21no—
370#391C2×D5311yescharacteristic
371#391C2×D5311yescharacteristic
372#391C2×D5311yescharacteristic
373#389—531no—
374#389—531no—
375#389C2×D5311yescharacteristic
376#387—21no—
377#387—21no—
378#385—1061no—
379#385—1061no—
380#384—531no—
381#380—1061no—
382C4×C4⋊C4C2×D5311yescharacteristic
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#336—531no—
388#336—531no—
389#336—531no—
390#336—531no—
391C53#2616611yescharacteristic, complemented, Sylow
392C22×Q8C22×D5311yescharacteristic
393C22×Q8C22×D5311yescharacteristic
394C22×D4—531no—
395C23×C4—1061no—
396C23×C4—1061no—
397C23×C4—1061no—
398C2×SD16—534no—
399C2×SD16—1062no—
400C2×SD16—1062no—
401C2×SD16—1062no—
402C2×SD16—1062no—
403C2×SD16—1062no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8—531no—
408C4⋊Q8—22no—
409C4⋊Q8—41no—
410C4×Q8—22no—
411C4×Q8—41no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—1062no—
415C4×D4—1062no—
416C4×D4—2121no—
417C4×D4—2121no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—2124no—
430C2×C4⋊C4C22⋊D5311yescharacteristic
431C2×C4⋊C4C22⋊D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—21no—
438C2×C42C22×D5311yescharacteristic
439C2×C42C22×D5311yescharacteristic
440C2×C42C22×D5311yescharacteristic
441#106—1062no—
442#106—2121no—
443C4⋊C8—1062no—
444#103—1062no—
445#103—1062no—
446#103—1062no—
447#103—1062no—
448#103—1062no—
449#103—2121no—
450#102—1062no—
451#102—1062no—
452C4×C8—1062no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—1061no—
461#95—1061no—
462#95C22⋊D5311yescharacteristic
463#95C22⋊D5311yescharacteristic
464C24—1061no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8D4×D5312yes—
477C2×D4—532no—
478C2×D4—532no—
479C2×D4—532no—
480C2×D4—1061no—
481C2×D4—1061no—
482C2×D4—1061no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—1064no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4—2122no—
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C23×D5311yescharacteristic
506SD16—1068no—
507SD16—2124no—
508C2×C8—532no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—532no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—4241no—
536C4⋊C4—2122no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4#1171992414yescomplemented
540C22⋊C4—1064no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C23—1062no—
551C23—1062no—
552C23—1062no—
553C23—2121no—
554C23—2121no—
555C23—2121no—
556C23—2121no—
557C23C22×C22⋊D5311yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566D4—534no—
567D4—1062no—
568D4—1062no—
569D4—1062no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
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—1062no—
598C2×C4—4244no—
599C2×C4—4244no—
600C2×C4—2128no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4C2×#1171992412yes—
610C2×C4C2×#1171992412yes—
611C2×C4C2×#1171992412yes—
612C2×C4C2×D4×D5311yescharacteristic
613C2×C4#41×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4#41×D5311yescharacteristic
616C2×C4#1313326211yescharacteristic
617C2×C4#1313326411yescharacteristic
618C8—1062no—
619C8—2121no—
620C8—2121no—
621C8—2121no—
622C22—1064no—
623C22—2122no—
624C22—2122no—
625C22—2122no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22#1681113011yescharacteristic
630C22D4×C22⋊D5311yescharacteristic
631C22#1681128011yescharacteristic
632C22C22×#1171992411yescharacteristic
633C22#1681142111yescharacteristic
634C22#1681143111yescharacteristic
635C22#1681143811yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—4241no—
647C4—4241no—
648C4—2122no—
649C4#123×D5311yescharacteristic
650C4#1681121611yescharacteristic
651C4#1681127111yescharacteristic
652C4#1681131811yescharacteristic
653C2—2124no—
654C2#2027844711yescharacteristic
655C2#2028471911yescharacteristic
656C2#2028492311yescharacteristic
657C2#2028627111yescharacteristic
658C2D4×#1171992411yescharacteristic
659C2#2028735311yescharacteristic
660C2#2028753611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2446C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2446D53
hypocenterC53#26166
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2446

Complements

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

NH ≅ G/Ntype
4. #20276258C2semidirect
5. #20276217C2semidirect
6. #20276123C2semidirect
7. #20275958C2semidirect
8. #20275394C2semidirect
9. #20275274C2semidirect
13. #20274983C2semidirect
15. C53×#2446C2semidirect
184. C4⋊C4×C53SD16semidirect
325. #2446D53semidirect
391. C53#26166semidirect
539. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
C2×C4×#117199242
C4⋊C4×D2122
#202762582
#202762172
#202761232
#202759582
#202753942
#202752742
#202750522
#202749832
#202749752
C53×#24462
#2616653