Finite Group Databasebeta

← #27432431

Subgroups — #27432431

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic
2#20279201C211yesmaximal, characteristic, complemented
3#20277735C211yesmaximal, characteristic
4#20276200C211yesmaximal, characteristic, complemented
5#20276131C211yesmaximal, characteristic, complemented
6#20276106C211yesmaximal, characteristic, complemented
7#20276028C211yesmaximal, characteristic, complemented
8#20275484C211yesmaximal, characteristic, complemented
9#20275359C211yesmaximal, characteristic, complemented
10#20275051C211yesmaximal, characteristic, complemented
11#20275023C211yesmaximal, characteristic, complemented
12#20274893C211yesmaximal, characteristic, complemented
13#20274723C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C2×C53×#465C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×D4×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20#16810934—24no—
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810562—21no—
31#16810562—21no—
32#16810530C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810526C2211yescharacteristic
35#16810522C2211yescharacteristic
36#16810516—21no—
37#16810503—21no—
38#16810498—21no—
39#16810488C2211yescharacteristic
40#16810488C2211yescharacteristic
41#16810484—21no—
42#16810484—21no—
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×SD16×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×C22⋊Q8×C53C2211yescharacteristic
56C2×#121×C53C2211yescharacteristic
57C53×#465—24no—
58C2×#106×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C23×Dic53—21no—
62C23×Dic53—21no—
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66D4×Dic53—22no—
67D4×Dic53—41no—
68D4×Dic53—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D411yescharacteristic
84C2×C22⋊Dic53D411yescharacteristic
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133150—22no—
89#13133150—22no—
90#13133146—22no—
91#13133146—41no—
92#13133141—22no—
93#13133141—41no—
94#13133138—22no—
95#13133138—22no—
96C22⋊#88322—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53C2311yescharacteristic
110C22×D4×C53—21no—
111C22×D4×C53C2311yescharacteristic
112C23×C4×C53C2311yescharacteristic
113C2×SD16×C53—22no—
114C2×SD16×C53—22no—
115C2×SD16×C53—41no—
116C22×C8×C53—22no—
117C22×C8×C53—22no—
118C22×C8×C53—41no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22⋊Q8×C53—22no—
122C22⋊Q8×C53—22no—
123#121×C53—22no—
124#121×C53D414yes—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C22⋊C4×C53—21no—
129C2×C22⋊C4×C53—21no—
130#106×C53—22no—
131#103×C53—22no—
132#102×C53—22no—
133C24×C53—21no—
134C24×C53C2×D411yescharacteristic
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×D4×C53—24no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—42no—
148C2×D4×C53—42no—
149C2×D4×C53C2×D412yes—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D412yes—
159C22×C4×C53#4112yes—
160C22×C4×C53C2411yescharacteristic
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—21no—
165C22×Dic53—42no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—42no—
170C22×Dic53—42no—
171C22×Dic53—42no—
172C22×Dic53—81no—
173C22×Dic53—81no—
174C22×Dic53C2×D411yescharacteristic
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178SD16×C53—44no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—44no—
185C2×C8×C53#4112yes—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—42no—
190C4⋊C4×C53C2×D412yes—
191C22⋊C4×C53—24no—
192C22⋊C4×C53—42no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53—42no—
196C22⋊Dic53—42no—
197C22⋊Dic53—81no—
198C4⋊Dic53—21no—
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#4112yes—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—42no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—22no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53#12112yes—
232C23×C53#13512yes—
233C23×C53AGL(1,ℤ/8)12yes—
234C23×C53C22×D411yescharacteristic
235Q8×C53—41no—
236Q8×C53—41no—
237Q8×C53—41no—
238Q8×C53—41no—
239D4×C53—22no—
240D4×C53—22no—
241D4×C53—48no—
242D4×C53—41no—
243D4×C53—41no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
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#12112yes—
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53H(5,𝔽2)11yescharacteristic
265C2×C4×C532−1+411yescharacteristic
266C2×Dic53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—41no—
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—41no—
290C2×Dic53—84no—
291C2×Dic53—84no—
292C2×Dic53—84no—
293C8×C53—22no—
294C8×C53—41no—
295#88322—41no—
296#88322—41no—
297#25212—531nomaximal, Sylow
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—81no—
308C22×C53—81no—
309C22×C53—81no—
310C22×C53—81no—
311C22×C53#46314yes—
312C22×C53C2×#12111yescharacteristic
313C22×C53#54611yescharacteristic
314C22×C53#54711yescharacteristic
315C22×C53C2×#13511yescharacteristic
316C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
317C22×C53#57711yescharacteristic
318C22×C53#57811yescharacteristic
319C4×C53—22no—
320C4×C53—42no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53#52911yescharacteristic
324C4×C53#53911yescharacteristic
325C4×C53#54611yescharacteristic
326C4×C53#54711yescharacteristic
327Dic53—41no—
328Dic53—41no—
329Dic53—41no—
330Dic53—41no—
331Dic53—42no—
332Dic53—81no—
333Dic53—81no—
334Dic53—81no—
335C2×#480—531no—
336C2×#465D5311yescharacteristic
337#2503—531no—
338#2361—531no—
339#2085—531no—
340#2073—531no—
341#2068—531no—
342#2055—531no—
343#1949—531no—
344#1928—531no—
345#1877—531no—
346#1872—531no—
347#1850—531no—
348#1804—531no—
349C2×#340—531no—
350C2×C53—24no—
351C2×C53—82no—
352C2×C53#242511yescharacteristic
353C2×C53C2×#46311yescharacteristic
354C2×C53#304611yescharacteristic
355C2×C53#330011yescharacteristic
356C2×C53#331011yescharacteristic
357C2×C53#333711yescharacteristic
358C2×C53#334311yescharacteristic
359C22×SD16C2×D5311yescharacteristic
360C23×C8C2×D5311yescharacteristic
361C2×C22⋊Q8C2×D5311yescharacteristic
362C2×#121C2×D5311yescharacteristic
363C2×C4×D4—531no—
364C2×C4×D4—1061no—
365C22×C4⋊C4—531no—
366C22×C4⋊C4—531no—
367#480—1064no—
368#465—24no—
369C2×#107—531no—
370C2×#107—531no—
371C2×#106C2×D5311yescharacteristic
372C2×C4⋊C8—531no—
373C2×C4⋊C8—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103C2×D5311yescharacteristic
377C2×#102—531no—
378C2×#102—531no—
379C2×#102C2×D5311yescharacteristic
380C2×C22⋊C8—531no—
381#400—1061no—
382#400—1061no—
383#391—531no—
384#391—531no—
385#390—531no—
386#389—531no—
387#388—1061no—
388#386—1061no—
389#385—1061no—
390#382—531no—
391#382—531no—
392#381—1061no—
393#381—1061no—
394C4×C4⋊C4—531no—
395C4×C22⋊C4—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#340—531no—
401#340—531no—
402#336—531no—
403#336—531no—
404C53#2521211yescharacteristic, complemented, Sylow
405C22×Q8C22×D5311yescharacteristic
406C22×D4—21no—
407C22×D4C22×D5311yescharacteristic
408C23×C4—1061no—
409C23×C4—1061no—
410C23×C4—1061no—
411C23×C4C22×D5311yescharacteristic
412C2×SD16—22no—
413C2×SD16—22no—
414C2×SD16—41no—
415C22×C8—22no—
416C22×C8—22no—
417C22×C8—41no—
418C22×C8—531no—
419C22×C8—531no—
420C22×C8C22×D5311yescharacteristic
421C22×C8C22×D5311yescharacteristic
422C22⋊Q8—22no—
423C22⋊Q8—22no—
424#121—22no—
425#121C22⋊D5314yes—
426C4×D4—1062no—
427C4×D4—2121no—
428C4×D4—2124no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—2121no—
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—531no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1061no—
453#107—1062no—
454#107—1062no—
455#106—22no—
456C4⋊C8—1062no—
457C4⋊C8—2121no—
458#103—22no—
459#103—1062no—
460#103—2121no—
461#102—22no—
462#102—1062no—
463#102—1062no—
464C22⋊C8—1062no—
465#95—531no—
466#95—531no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477C24—21no—
478C24D4×D5311yescharacteristic
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×D4—24no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4C2×C22⋊D5312yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—531no—
502C22×C4—531no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—4241no—
510C22×C4—4241no—
511C22×C4—2122no—
512C22×C4—2122no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4—2122no—
518C22×C4D4×D5311yescharacteristic
519C22×C4#1171994712yes—
520C22×C4C2×C22⋊D5312yes—
521C22×C4C23×D5311yescharacteristic
522SD16—44no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—44no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8#1171994712yes—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—42no—
540C4⋊C4—532no—
541C4⋊C4—1061no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—2122no—
549C4⋊C4—2122no—
550C4⋊C4—2122no—
551C4⋊C4C2×C22⋊D5312yes—
552C22⋊C4—24no—
553C22⋊C4—42no—
554C22⋊C4—1062no—
555C22⋊C4—1062no—
556C22⋊C4—4241no—
557C22⋊C4—2122no—
558C22⋊C4—2122no—
559C42—1061no—
560C42—1061no—
561C42—1061no—
562C42—1061no—
563C42—2122no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23#1313318912yes—
576C23#1313323812yes—
577C23#1313324412yes—
578C23C2×D4×D5311yescharacteristic
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1061no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—1061no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1062no—
616C2×C4—4244no—
617C2×C4—4244no—
618C2×C4—4244no—
619C2×C4—2121no—
620C2×C4—2121no—
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#1313318912yes—
631C2×C4C22×C22⋊D5311yescharacteristic
632C2×C4#41×D5311yescharacteristic
633C2×C4C2×#1171994711yescharacteristic
634C2×C4C2×#1171994711yescharacteristic
635C2×C4#1313326211yescharacteristic
636C2×C4#1313326411yescharacteristic
637C8—22no—
638C8—41no—
639C8—2121no—
640C8—2121no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#1681084014yes—
655C22C2×#1313318911yescharacteristic
656C22#1681126211yescharacteristic
657C22#1681127311yescharacteristic
658C22C2×#1313323811yescharacteristic
659C22C2×#1313324411yescharacteristic
660C22#1681143011yescharacteristic
661C22#1681143711yescharacteristic
662C4—22no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—4241no—
667C4—4241no—
668C4—4241no—
669C4—2121no—
670C4—2122no—
671C4—2121no—
672C4—2121no—
673C4—2121no—
674C4#1681115411yescharacteristic
675C4#1681121811yescharacteristic
676C4#1681126711yescharacteristic
677C4#1681127211yescharacteristic
678C2—24no—
679C2—82no—
680C2#2027827211yescharacteristic
681C2C2×#1681084011yescharacteristic
682C2#2028426311yescharacteristic
683C2#2028660811yescharacteristic
684C2#2028674211yescharacteristic
685C2#2028710011yescharacteristic
686C2#2028716111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#465C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#465D53
hypocenterC53#25212
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
2. #20279201C2semidirect
4. #20276200C2semidirect
5. #20276131C2semidirect
6. #20276106C2semidirect
7. #20276028C2semidirect
8. #20275484C2semidirect
9. #20275359C2semidirect
10. #20275051C2semidirect
11. #20275023C2semidirect
12. #20274893C2semidirect
13. #20274723C2semidirect
14. C2×#16810386C2semidirect
404. C53#25212semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
#202792012
#202777352
#202762002
#202761312
#202761062
#202760282
#202754842
#202753592
#202750512
#202750232
#202748932
#202747232
C2×#168103862
C2×C53×#4652
#2521253