Finite Group Databasebeta

← #27447250

Subgroups — #27447250

Order 13568.

932 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C211yesmaximal, characteristic, complemented
2C2×C4×#11719922C211yesmaximal, characteristic
3C4⋊C4×D212C211yesmaximal, characteristic, complemented
4#20276248C211yesmaximal, characteristic, complemented
5#20276186C211yesmaximal, characteristic, complemented
6#20276089C211yesmaximal, characteristic, complemented
7#20276050C211yesmaximal, characteristic, complemented
8#20275440C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275066C211yesmaximal, characteristic
11#20275066C211yesmaximal, characteristic
12#20274975C211yesmaximal, characteristic, complemented
13#20274975C211yesmaximal, characteristic, complemented
14#20274962C211yesmaximal, characteristic, complemented
15C53×#2454C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×C4⋊C4×D53—21no—
21#16810920—24no—
22C4×#11719922—24no—
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
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×#125×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#396—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#385—21no—
59C53×#380—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C2×#11719922—22no—
66C2×#11719922—22no—
67C2×#11719922Q814yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C4×D212—22no—
73C4×D212—22no—
74C4×D212—22no—
75C4×D212—22no—
76C4×D212—41no—
77C4×D212—41no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—44no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89#13133150—22no—
90#13133150—22no—
91#13133148—22no—
92#13133148—22no—
93#13133148—41no—
94C4⋊#88322—22no—
95#13133138—22no—
96#13133138—22no—
97#13133138—41no—
98#13133137—22no—
99#13133137—22no—
100#13133137D414yes—
101C4×#88322—22no—
102#13133124—21no—
103#13133124—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106#125×C53—22no—
107#125×C53—41no—
108C4×D4×C53—22no—
109C4×D4×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53—21no—
132C23×D53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53C2×Q812yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—81no—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212C2×D412yes—
159C2×D212C2×Q812yes—
160C2×D212#4112yes—
161C2×C4×D53—24no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—42no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—81no—
180C4⋊C4×C53—81no—
181C4⋊C4×C53C2×D412yes—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—81no—
185C22⋊C4×C53—81no—
186C22⋊C4×C53—81no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—21no—
193#11719922—28no—
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×#88322C2×D412yes—
218C2×#88322C2×Q812yes—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C22×D411yescharacteristic
227C22×D53—22no—
228C22×D53—22no—
229C22×D53—22no—
230C22×D53—41no—
231C22×D53—41no—
232C22×D53—41no—
233C22×D53—41no—
234D4×C53—22no—
235D4×C53—22no—
236D4×C53—41no—
237D4×C53—41no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
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—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
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—84no—
264C2×C4×C53#13512yes—
265C2×C4×C53AGL(1,ℤ/8)12yes—
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C22×Q811yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×C4×C532−1+411yescharacteristic
272C2×Dic53—22no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281D212—22no—
282D212—22no—
283D212—22no—
284D212C22⋊Q814yes—
285C4×D53—48no—
286C4×D53—84no—
287C4×D53—84no—
288#88322—22no—
289#88322—22no—
290#88322—41no—
291#88322—41no—
292#26382—531nomaximal, Sylow
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—42no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53—81no—
300C22×C53#52511yescharacteristic
301C22×C53D4211yescharacteristic
302C22×C53D4×Q811yescharacteristic
303C22×C53C2×#13511yescharacteristic
304C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C53#57611yescharacteristic
306C22×C53#57711yescharacteristic
307C2×D53—24no—
308C2×D53—42no—
309C2×D53—42no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—42no—
313C2×D53—42no—
314C4×C53—22no—
315C4×C53—22no—
316C4×C53—41no—
317C4×C53—41no—
318C4×C53—42no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53C2×C22⋊Q811yescharacteristic
324C4×C53#53911yescharacteristic
325C4×C53#54711yescharacteristic
326C4×C53#55411yescharacteristic
327Dic53—42no—
328Dic53—81no—
329Dic53—81no—
330C2×#478—531no—
331C2×C4×D8—531no—
332#2454D5311yescharacteristic, complemented
333D4×C4⋊C4—531no—
334#2094—531no—
335#2083—531no—
336#2065—531no—
337#2059—531no—
338#1941—531no—
339#1916—531no—
340#1879—531no—
341#1879—531no—
342#1864—531no—
343#1864—531no—
344#1862—531no—
345C2×C53—82no—
346C2×C53#244911yescharacteristic
347C2×C53#310511yescharacteristic
348C2×C53#311911yescharacteristic
349C2×C53#326411yescharacteristic
350C2×C53#328411yescharacteristic
351C2×C53#338011yescharacteristic
352C2×C53#338411yescharacteristic
353D53—44no—
354C22×D8—531no—
355C2×#125C2×D5311yescharacteristic
356C2×C4×D4—531no—
357C2×C4×D4—531no—
358C2×C4×D4—531no—
359C2×C4×D4C2×D5311yescharacteristic
360C22×C4⋊C4—1061no—
361#478—1064no—
362C4×D8—1064no—
363C2×#107—531no—
364C2×#107—531no—
365C2×#106—531no—
366C2×#106—531no—
367C2×#106—531no—
368C2×C4⋊C8—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×#102—531no—
372C2×#102—531no—
373C2×#102—531no—
374C2×#102—531no—
375C2×C4×C8—531no—
376#399—21no—
377#396—21no—
378#390C2×D5311yescharacteristic
379#390C2×D5311yescharacteristic
380#389—531no—
381#389—531no—
382#389C2×D5311yescharacteristic
383#389C2×D5311yescharacteristic
384#385—21no—
385#385—1061no—
386#385—1061no—
387#384—531no—
388#380—21no—
389#380—1061no—
390C4×C4⋊C4C2×D5311yescharacteristic
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#336—531no—
396#336—531no—
397#336—531no—
398#336—531no—
399C53#2638211yescharacteristic, complemented, Sylow
400C22×D4—531no—
401C22×D4C22×D5311yescharacteristic
402C23×C4—21no—
403C23×C4—1061no—
404C23×C4—1061no—
405C23×C4—1061no—
406C2×D8—534no—
407C2×D8—1062no—
408C2×D8—1062no—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8—531no—
412C22×C8—531no—
413#125—22no—
414#125—41no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—1062no—
418C4×D4—1062no—
419C4×D4—1062no—
420C4×D4—1062no—
421C4×D4—2121no—
422C4×D4—2121no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—2124no—
435C2×C4⋊C4C22⋊D5311yescharacteristic
436C2×C4⋊C4C22⋊D5311yescharacteristic
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C42C22×D5311yescharacteristic
449C2×C42C22×D5311yescharacteristic
450C2×C42C22×D5311yescharacteristic
451#107—1062no—
452#107—1062no—
453#106—1062no—
454#106—1062no—
455#106—2121no—
456C4⋊C8—1062no—
457#102—534no—
458#102—1062no—
459#102—1062no—
460#102—1062no—
461#102—1062no—
462#102—2121no—
463C4×C8—1062no—
464#95—21no—
465#95—21no—
466#95—1061no—
467#95—1061no—
468#95C22⋊D5311yescharacteristic
469#95C22⋊D5311yescharacteristic
470C24—21no—
471C24—1061no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—532no—
478C2×D4—532no—
479C2×D4—532no—
480C2×D4—1061no—
481C2×D4—1061no—
482C2×D4—1061no—
483C2×D4Q8×D5312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—1064no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—4241no—
495C22×C4—4241no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4—2122no—
501C22×C4—2122no—
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C2×C22⋊D5311yescharacteristic
504C22×C4C2×C22⋊D5311yescharacteristic
505C22×C4C2×C22⋊D5311yescharacteristic
506C22×C4C2×C22⋊D5311yescharacteristic
507C22×C4C2×C22⋊D5311yescharacteristic
508C22×C4C23×D5311yescharacteristic
509D8—1068no—
510C2×C8—532no—
511C2×C8—532no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8—1061no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C4⋊C4—532no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—4241no—
540C4⋊C4—2122no—
541C4⋊C4—2122no—
542C4⋊C4—2122no—
543C4⋊C4D4×D5312yes—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—81no—
547C22⋊C4—81no—
548C22⋊C4—81no—
549C22⋊C4—1064no—
550C22⋊C4—2122no—
551C22⋊C4—2122no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—1062no—
566C23—1062no—
567C23—1062no—
568C23—2121no—
569C23—2121no—
570C23—2121no—
571C23—2121no—
572C23C22×C22⋊D5311yescharacteristic
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—534no—
578D4—1062no—
579D4—1062no—
580D4—1062no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—1062no—
608C2×C4—4244no—
609C2×C4—4244no—
610C2×C4—2128no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4#1313323712yes—
620C2×C4#1313324212yes—
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4C2×Q8×D5311yescharacteristic
623C2×C4#41×D5311yescharacteristic
624C2×C4#41×D5311yescharacteristic
625C2×C4#1313326211yescharacteristic
626C2×C4#1313326511yescharacteristic
627C8—1062no—
628C8—1062no—
629C8—2121no—
630C8—2121no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—1064no—
639C22—2122no—
640C22—2122no—
641C22—2122no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22#1681113011yescharacteristic
646C22D4×C22⋊D5311yescharacteristic
647C22Q8×C22⋊D5311yescharacteristic
648C22C2×#1313323711yescharacteristic
649C22C2×#1313324211yescharacteristic
650C22#1681142611yescharacteristic
651C22#1681143411yescharacteristic
652C4—22no—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—4241no—
662C4—4241no—
663C4—2122no—
664C4C22⋊Q8×D5311yescharacteristic
665C4#1681121911yescharacteristic
666C4#1681127511yescharacteristic
667C4#1681132711yescharacteristic
668C2—82no—
669C2—2124no—
670C2#2027852111yescharacteristic
671C2#2028475811yescharacteristic
672C2#2028490811yescharacteristic
673C2#2028619611yescharacteristic
674C2#2028644511yescharacteristic
675C2#2028757811yescharacteristic
676C2#2028761411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2454C2
socleC23×C53C22×D4
prime radicalC22×#11719922C22
hypercenter#2454D53
hypocenterC53#26382
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#16810920C2semidirect
3. C4⋊C4×D212C2semidirect
4. #20276248C2semidirect
5. #20276186C2semidirect
6. #20276089C2semidirect
7. #20276050C2semidirect
8. #20275440C2semidirect
9. #20275274C2semidirect
12. #20274975C2semidirect
13. #20274975C2semidirect
14. #20274962C2semidirect
15. C53×#2454C2semidirect
332. #2454D53semidirect
399. C53#26382semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C2×C4×#117199222
C4⋊C4×D2122
#202762482
#202761862
#202760892
#202760502
#202754402
#202752742
#202750662
#202749752
#202749622
C53×#24542
#2638253