Finite Group Databasebeta

← #27435812

Subgroups — #27435812

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic
3#20277773C211yesmaximal, characteristic, complemented
4#20276197C211yesmaximal, characteristic, complemented
5#20276192C211yesmaximal, characteristic, complemented
6#20276103C211yesmaximal, characteristic
7#20276089C211yesmaximal, characteristic
8#20275673C211yesmaximal, characteristic, complemented
9#20275481C211yesmaximal, characteristic, complemented
10#20275476C211yesmaximal, characteristic, complemented
11#20275439C211yesmaximal, characteristic, complemented
12#20275339C211yesmaximal, characteristic
13#20275332C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2566C211yesmaximal, characteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133182—21no—
18#16811033—24no—
19#16810913—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133128C2211yescharacteristic
34#16810528—21no—
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810508—21no—
38#16810508—21no—
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C53×#396—21no—
53C53×#396—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#385—21no—
57C53×#385—21no—
58C53×#384C2211yescharacteristic
59C53×#382C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133217—22no—
67#13133217D414yes—
68#13133182—44no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—22no—
80#13133150—22no—
81#13133148—22no—
82#13133148—22no—
83#13133148—22no—
84#13133148—22no—
85C4⋊#88322—22no—
86C4⋊#88322—22no—
87C4⋊#88322—22no—
88#13133138—22no—
89#13133138—22no—
90#13133138—22no—
91#13133138Q814yescomplemented
92#13133128—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100#125×C53—22no—
101#125×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121#95×C53—21no—
122#95×C53—21no—
123#95×C53D411yescharacteristic
124#95×C53D411yescharacteristic
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—42no—
136C22×C4×C53—81no—
137C22×C4×C53—81no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C2×Dic106—24no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—81no—
163C4⋊C4×C53—81no—
164C22⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—81no—
168C22⋊C4×C53—81no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C42×C53C2×D412yes—
174C42×C53#4112yes—
175C4⋊Dic53—24no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53C2×D412yes—
182C4⋊Dic53C2×D412yes—
183C4⋊Dic53C2×Q812yes—
184#11719919—42no—
185#11719919—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×Q812yes—
200C2×#88322#4112yes—
201C23×C53—22no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53C22×D411yescharacteristic
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—84no—
238C2×C4×C53#12112yes—
239C2×C4×C53#12112yes—
240C2×C4×C53AGL(1,ℤ/8)12yes—
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×Q811yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C532−1+411yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259Dic106—48no—
260#88322—22no—
261#88322—22no—
262#88322—41no—
263#88322—41no—
264#25461—531nomaximal, Sylow
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53C2×#12111yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53D4×Q811yescharacteristic
275C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57711yescharacteristic
278C22×C53#57711yescharacteristic
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#53311yescharacteristic
289C4×C53#54111yescharacteristic
290C4×C53#54811yescharacteristic
291C4×C53D4×Q811yescharacteristic
292Dic53—42no—
293Dic53—42no—
294Dic53—42no—
295Dic53—81no—
296C2×#501—531no—
297C2×#476—531no—
298#2566D5311yescharacteristic
299#2365—531no—
300#2085—531no—
301#2084—531no—
302#2067—531no—
303#2065—531no—
304#1989—531no—
305#1949—531no—
306#1948—531no—
307#1940—531no—
308#1926—531no—
309#1925—531no—
310#1762—531no—
311C2×C53—82no—
312C2×C53#242911yescharacteristic
313C2×C53#303611yescharacteristic
314C2×C53#304311yescharacteristic
315C2×C53#334511yescharacteristic
316C2×C53#335011yescharacteristic
317C2×C53#338411yescharacteristic
318C2×C53#338811yescharacteristic
319C2×C4⋊Q8—531no—
320C2×#125C2×D5311yescharacteristic
321C2×C4×Q8—1061no—
322C2×C4×D4C2×D5311yescharacteristic
323#501—1064no—
324#476—1064no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#106—531no—
328C2×#106—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×C4⋊C8—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#97—531no—
339#396—21no—
340#396—21no—
341#391—1061no—
342#390C2×D5311yescharacteristic
343#390C2×D5311yescharacteristic
344#389—531no—
345#389—531no—
346#387—1061no—
347#387—1061no—
348#385—21no—
349#385—21no—
350#384C2×D5311yescharacteristic
351#382C2×D5311yescharacteristic
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4C2×D5311yescharacteristic
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#336—531no—
363#336—531no—
364C53#2546111yescharacteristic, complemented, Sylow
365C22×Q8—1061no—
366C22×D4C22×D5311yescharacteristic
367C23×C4—21no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371C22×C8—531no—
372C4⋊Q8—534no—
373C4⋊Q8—1062no—
374#125—22no—
375#125—22no—
376C4×Q8—2124no—
377C4×D4—22no—
378C4×D4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4C22⋊D5311yescharacteristic
391C2×C4⋊C4C22⋊D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C42—1061no—
401C2×C42—1061no—
402C2×C42—1061no—
403C2×C42C22×D5311yescharacteristic
404C2×C42C22×D5311yescharacteristic
405C2×C42C22×D5311yescharacteristic
406#107—1062no—
407#107—1062no—
408#106—1062no—
409#106—1062no—
410#106—1062no—
411#106—1062no—
412C4⋊C8—1062no—
413C4⋊C8—1062no—
414C4⋊C8—1062no—
415#102—534no—
416#102—1062no—
417#102—1062no—
418#102—1062no—
419#97—1062no—
420#95—21no—
421#95—21no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95C22⋊D5311yescharacteristic
428#95C22⋊D5311yescharacteristic
429C24—21no—
430C2×Q8—1064no—
431C2×Q8—2122no—
432C2×Q8—2122no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4Q8×D5312yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C23×D5311yescharacteristic
456C2×C8—532no—
457C2×C8—532no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—532no—
481C4⋊C4—532no—
482C4⋊C4—532no—
483C4⋊C4—1064no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—2122no—
501C42—2122no—
502C42—2122no—
503C42D4×D5312yes—
504C42#1171994712yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C22×C22⋊D5311yescharacteristic
513Q8—2128no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4#1313318812yes—
556C2×C4#1313318812yes—
557C2×C4#1313324212yes—
558C2×C4C2×D4×D5311yescharacteristic
559C2×C4C2×Q8×D5311yescharacteristic
560C2×C4#41×D5311yescharacteristic
561C2×C4C2×#1171994711yescharacteristic
562C2×C4#1313326411yescharacteristic
563C2×C4#1313326511yescharacteristic
564C8—1062no—
565C8—1062no—
566C8—2121no—
567C8—2121no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C2×#1313318811yescharacteristic
576C22#1681127211yescharacteristic
577C22Q8×C22⋊D5311yescharacteristic
578C22C2×#1313324211yescharacteristic
579C22#1681142111yescharacteristic
580C22#1681143411yescharacteristic
581C22#1681143411yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4241no—
592C4—2122no—
593C4—2122no—
594C4—2122no—
595C4#1681117511yescharacteristic
596C4#1681123311yescharacteristic
597C4#1681128311yescharacteristic
598C4#1681128611yescharacteristic
599C2—82no—
600C2#2027831011yescharacteristic
601C2#2028417511yescharacteristic
602C2#2028423211yescharacteristic
603C2#2028717911yescharacteristic
604C2#2028725411yescharacteristic
605C2#2028761411yescharacteristic
606C2#2028766011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2566C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2566D53
hypocenterC53#25461
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
3. #20277773C2semidirect
4. #20276197C2semidirect
5. #20276192C2semidirect
8. #20275673C2semidirect
9. #20275481C2semidirect
10. #20275476C2semidirect
11. #20275439C2semidirect
91. #13133138Q8semidirect
364. C53#25461semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168109132
#202777732
#202761972
#202761922
#202761032
#202760892
#202756732
#202754812
#202754762
#202754392
#202753392
#202753322
#202745402
C53×#25662
#2546153