Finite Group Databasebeta

← #27432494

Subgroups — #27432494

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic, complemented
2C2×#16810871C211yesmaximal, characteristic
3#20279186C211yesmaximal, characteristic
4#20276186C211yesmaximal, characteristic, complemented
5#20276123C211yesmaximal, characteristic, complemented
6#20276100C211yesmaximal, characteristic, complemented
7#20276030C211yesmaximal, characteristic, complemented
8#20275473C211yesmaximal, characteristic, complemented
9#20275335C211yesmaximal, characteristic, complemented
10#20275070C211yesmaximal, characteristic, complemented
11#20275037C211yesmaximal, characteristic
12#20274900C211yesmaximal, characteristic, complemented
13#20274720C211yesmaximal, characteristic, complemented
14#20274436C211yesmaximal, characteristic, complemented
15C53×#2362C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133201—21no—
19C2×#13133201—21no—
20C2×#13133191C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810956—24no—
23#16810871—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×C22⋊#88322C2211yescharacteristic
36#16810560—21no—
37#16810560—21no—
38#16810525C2211yescharacteristic
39#16810525C2211yescharacteristic
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#388—21no—
56C53×#388—21no—
57C53×#387—21no—
58C4×C4⋊C4×C53—21no—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C23×Dic53—21no—
64C2×#11719924—22no—
65C2×#11719924—22no—
66C2×#11719924—41no—
67C2×#11719917—22no—
68C2×#11719917—22no—
69C2×#11719917—41no—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133201—44no—
75#13133201—44no—
76#13133191—22no—
77#13133191—22no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—22no—
84C2×C4⋊Dic53—22no—
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93#13133150—22no—
94#13133150—22no—
95#13133150—22no—
96C4⋊#88322—22no—
97C4⋊#88322—41no—
98#13133141—22no—
99#13133141—22no—
100#13133141—22no—
101#13133137—22no—
102#13133137—22no—
103#13133137—41no—
104C22⋊#88322—22no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53D414yes—
109C4×Q8×C53—22no—
110C4×Q8×C53—41no—
111#117×C53—22no—
112#117×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
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×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53—21no—
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53D411yescharacteristic
133#95×C53D411yescharacteristic
134C24×C53C2×D411yescharacteristic
135C23×D53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53C2×D412yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
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×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53#4112yes—
157C22×C4×C53C2411yescharacteristic
158C2×C22⋊D53—42no—
159C2×C22⋊D53—42no—
160C2×C22⋊D53—42no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—42no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C2×D212—21no—
168C2×D212—21no—
169C2×D212—21no—
170C2×D212—21no—
171C2×D212—21no—
172C2×D212—21no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53—42no—
180C4⋊C4×C53C2×D412yes—
181C4⋊C4×C53#4112yes—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—42no—
186C42×C53—21no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—21no—
190C42×C53—42no—
191C42×C53—42no—
192C42×C53—42no—
193#11719924—44no—
194#11719921—42no—
195#11719921—42no—
196#11719921—42no—
197#11719921—42no—
198#11719921—42no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—42no—
208C4⋊Dic53—42no—
209C4⋊Dic53—42no—
210#11719917—44no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322#4112yes—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53#12112yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—41no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—41no—
235C22×D53—41no—
236C22×D53—41no—
237C22×D53—41no—
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—41no—
241Q8×C53—41no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—21no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53#12112yes—
271C2×C4×C53AGL(1,ℤ/8)12yes—
272C2×C4×C53AGL(1,ℤ/8)12yes—
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C53C2×#4111yescharacteristic
277C2×C4×C53H(5,𝔽2)11yescharacteristic
278C2×C4×C532−1+411yescharacteristic
279C22⋊D53—84no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—84no—
290D212—41no—
291D212—41no—
292D212—41no—
293D212—41no—
294#88322—22no—
295#88322—41no—
296#88322—41no—
297#88322—41no—
298#25217—531nomaximal, Sylow
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53C2×#12111yescharacteristic
307C22×C53#54611yescharacteristic
308C22×C53#54711yescharacteristic
309C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
310C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
311C22×C53#57811yescharacteristic
312C22×C53#57811yescharacteristic
313C2×D53—81no—
314C2×D53—81no—
315C2×D53—81no—
316C2×D53—81no—
317C2×D53—81no—
318C2×D53—81no—
319C2×D53—81no—
320C4×C53—22no—
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—41no—
324C4×C53—42no—
325C4×C53—41no—
326C4×C53—42no—
327C4×C53—42no—
328C4×C53—81no—
329C4×C53#52911yescharacteristic
330C4×C53#53911yescharacteristic
331C4×C53#54611yescharacteristic
332C4×C53#54711yescharacteristic
333Dic53—81no—
334Dic53—81no—
335Dic53—81no—
336C2×#484—531no—
337C2×#468—531no—
338#2501—531no—
339#2362D5311yescharacteristic, complemented
340#2083—531no—
341#2072—531no—
342#2067—531no—
343#2055—531no—
344#1947—531no—
345#1925—531no—
346#1880—531no—
347#1875—531no—
348#1851—531no—
349#1803—531no—
350#1736—531no—
351C2×C53—42no—
352C2×C53#242511yescharacteristic
353C2×C53#303811yescharacteristic
354C2×C53#304411yescharacteristic
355C2×C53#330811yescharacteristic
356C2×C53#333011yescharacteristic
357C2×C53#334311yescharacteristic
358C2×C53#335311yescharacteristic
359D53—161no—
360C22×SD16—531no—
361C22×M16—531no—
362C2×#123—1061no—
363C2×#123—1061no—
364C2×C22⋊Q8C2×D5311yescharacteristic
365C2×#121—531no—
366C2×C4×Q8C2×D5311yescharacteristic
367C2×#117C2×D5311yescharacteristic
368C22×C4⋊C4—531no—
369#484—1064no—
370#468—1064no—
371C2×#107—531no—
372C2×#107—531no—
373C2×#107—531no—
374C2×C4⋊C8—531no—
375C2×C4⋊C8—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#102—531no—
380C2×#102—531no—
381C2×#102—531no—
382C2×C22⋊C8—531no—
383#399—1061no—
384#399—1061no—
385#391C2×D5311yescharacteristic
386#390—531no—
387#390—531no—
388#389C2×D5311yescharacteristic
389#388—21no—
390#388—21no—
391#387—21no—
392#384—531no—
393#384—531no—
394C4×C4⋊C4—21no—
395C4×C4⋊C4C2×D5311yescharacteristic
396C4×C22⋊C4C2×D5311yescharacteristic
397#340—531no—
398#340—531no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#336—531no—
404#336—531no—
405C53#2521711yescharacteristic, complemented, Sylow
406C22×Q8C22×D5311yescharacteristic
407C22×D4—531no—
408C22×D4—1061no—
409C23×C4—1061no—
410C23×C4C22×D5311yescharacteristic
411C2×SD16—1062no—
412C2×SD16—1062no—
413C2×SD16—2121no—
414C2×M16—1062no—
415C2×M16—1062no—
416C2×M16—2121no—
417C22×C8—531no—
418C22×C8—531no—
419C22×C8—531no—
420C22×C8—531no—
421#123—2124no—
422#123—2124no—
423C22⋊Q8—22no—
424C22⋊Q8D21214yes—
425#121—1062no—
426#121—1062no—
427C4×Q8—22no—
428C4×Q8—41no—
429#117—22no—
430#117—41no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—1062no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4C22⋊D5311yescharacteristic
455C2×C22⋊C4C22⋊D5311yescharacteristic
456C2×C42—21no—
457C2×C42—21no—
458C2×C42—21no—
459C2×C42C22×D5311yescharacteristic
460C2×C42C22×D5311yescharacteristic
461#107—1062no—
462#107—1062no—
463#107—1062no—
464C4⋊C8—1062no—
465C4⋊C8—2121no—
466#103—1062no—
467#103—1062no—
468#103—1062no—
469#102—1062no—
470#102—1062no—
471#102—2121no—
472C22⋊C8—1062no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—21no—
477#95C22⋊D5311yescharacteristic
478#95C22⋊D5311yescharacteristic
479C24—1061no—
480C24C2×C22⋊D5311yescharacteristic
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C2×D21212yes—
487C2×D4—1061no—
488C2×D4—1061no—
489C2×D4—1061no—
490C2×D4—1061no—
491C2×D4—1061no—
492C2×D4—1061no—
493C2×D4—2122no—
494C2×D4—2122no—
495C2×D4—2122no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1061no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4C2×D21212yes—
511C22×C4#1171994512yes—
512C22×C4C2×C22⋊D5311yescharacteristic
513C22×C4C2×C22⋊D5311yescharacteristic
514C22×C4C2×C22⋊D5311yescharacteristic
515C22×C4C2×C22⋊D5311yescharacteristic
516C22×C4C2×C22⋊D5311yescharacteristic
517C22×C4C23×D5311yescharacteristic
518SD16—2124no—
519M16—2124no—
520C2×C8—532no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C4⋊C4—24no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4—2122no—
550C4⋊C4C2×D21212yes—
551C4⋊C4#1171994712yes—
552C22⋊C4—24no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—42no—
556C22⋊C4—2122no—
557C22⋊C4—2122no—
558C22⋊C4—2122no—
559C22⋊C4—2122no—
560C22⋊C4—2122no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—42no—
567C42—42no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—2121no—
575C23—2121no—
576C23—2121no—
577C23—2121no—
578C23—2121no—
579C23—2121no—
580C23—2121no—
581C23#1313319112yes—
582C23C22×C22⋊D5311yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587D4—4244no—
588D4—2121no—
589D4—2121no—
590D4—2121no—
591D4—2121no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—4244no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4#1313319112yes—
631C2×C4#1313324212yes—
632C2×C4#1313324212yes—
633C2×C4C22×D21211yescharacteristic
634C2×C4#41×D5311yescharacteristic
635C2×C4C2×#1171994711yescharacteristic
636C2×C4C2×#1171994511yescharacteristic
637C2×C4#1313326211yescharacteristic
638C2×C4#1313326411yescharacteristic
639C8—1062no—
640C8—2121no—
641C8—2121no—
642C8—2121no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—4241no—
651C22—4241no—
652C22—4241no—
653C22—4241no—
654C22—4241no—
655C22—4241no—
656C22—4241no—
657C22C2×#1313319111yescharacteristic
658C22#1681126711yescharacteristic
659C22#1681127211yescharacteristic
660C22C2×#1313324211yescharacteristic
661C22C2×#1313324211yescharacteristic
662C22#1681143811yescharacteristic
663C22#1681143811yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—42no—
672C4—81no—
673C4—4241no—
674C4—4241no—
675C4—4241no—
676C4#1681115311yescharacteristic
677C4#1681121511yescharacteristic
678C4#1681126811yescharacteristic
679C4#1681127711yescharacteristic
680C2—42no—
681C2—8481no—
682C2#2027826011yescharacteristic
683C2#2028419511yescharacteristic
684C2#2028424511yescharacteristic
685C2#2028670611yescharacteristic
686C2#2028702011yescharacteristic
687C2#2028715511yescharacteristic
688C2#2028729311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2362C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2362D53
hypocenterC53#25217
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#16810956C2semidirect
4. #20276186C2semidirect
5. #20276123C2semidirect
6. #20276100C2semidirect
7. #20276030C2semidirect
8. #20275473C2semidirect
9. #20275335C2semidirect
10. #20275070C2semidirect
12. #20274900C2semidirect
13. #20274720C2semidirect
14. #20274436C2semidirect
15. C53×#2362C2semidirect
339. #2362D53semidirect
405. C53#25217semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168108712
#202791862
#202761862
#202761232
#202761002
#202760302
#202754732
#202753352
#202750702
#202750372
#202749002
#202747202
#202744362
C53×#23622
#2521753