Finite Group Databasebeta

← #27435667

Subgroups — #27435667

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811027C211yesmaximal, characteristic, complemented
2C2×#16810902C211yesmaximal, characteristic
3#20279667C211yesmaximal, characteristic
4#20277774C211yesmaximal, characteristic, complemented
5#20276189C211yesmaximal, characteristic, complemented
6#20276187C211yesmaximal, characteristic, complemented
7#20276102C211yesmaximal, characteristic
8#20276098C211yesmaximal, characteristic
9#20275665C211yesmaximal, characteristic, complemented
10#20275472C211yesmaximal, characteristic, complemented
11#20275430C211yesmaximal, characteristic, complemented
12#20275335C211yesmaximal, characteristic
13#20275333C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1947C211yesmaximal, characteristic, complemented
16C2×#13133216C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133195—21no—
19C2×#13133195—21no—
20C2×C4×Dic106—21no—
21C2×#13133179C2211yescharacteristic
22#16811027—24no—
23#16810902—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35#16810558—21no—
36#16810558—21no—
37#16810527—21no—
38#16810525C2211yescharacteristic
39#16810524C2211yescharacteristic
40#16810522C2211yescharacteristic
41#16810521C2211yescharacteristic
42#16810510—21no—
43#16810507—21no—
44#16810492C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C4×#11719919C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106—21no—
63C22×D212C2311yescharacteristic
64C22×C4×D53—21no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216—22no—
69#13133216—22no—
70#13133216D414yes—
71#13133195—44no—
72#13133195—44no—
73C4×Dic106—44no—
74#13133179—22no—
75#13133179—22no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C2×#11719919D411yescharacteristic
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133151—22no—
97#13133150—22no—
98#13133150—22no—
99#13133150—22no—
100#13133146—22no—
101#13133146—22no—
102#13133139—22no—
103#13133139—22no—
104#13133137—22no—
105#13133137Q814yescomplemented
106C8×Dic53—22no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124D411yescharacteristic
112#13133124D411yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123#107×C53—22no—
124#107×C53—22no—
125C4⋊C8×C53—22no—
126#95×C53—21no—
127C23×D53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C2×Q812yes—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—42no—
151C2×Dic106—24no—
152C2×Dic106—42no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106—42no—
156C2×Dic106—42no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53#4112yes—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—42no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53C2×D412yes—
172C4⋊C4×C53C2×Q812yes—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179#11719921—42no—
180#11719921—42no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—42no—
187C4⋊Dic53C2×D412yes—
188#11719919—24no—
189#11719919—22no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919D814yes—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—42no—
198C4×Dic53—42no—
199C4×Dic53C2×D412yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C2×Q812yes—
207C23×C53C22×D411yescharacteristic
208C22×D53—22no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53C2×D812yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53#12112yes—
256C2×Dic53#12112yes—
257C2×Dic53C2×D812yes—
258C2×Dic53C2×D812yes—
259C2×Dic53AGL(1,ℤ/8)12yes—
260D212—22no—
261D212—22no—
262D212—41no—
263D212—41no—
264C4×D53—84no—
265Dic106—48no—
266Dic106—84no—
267C8×C53—22no—
268C8×C53—41no—
269#88322—22no—
270#88322—41no—
271#25449—531nomaximal, Sylow
272C22×C53C2×#12111yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53D4×Q811yescharacteristic
275C22×C53C22×D811yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C53#57811yescharacteristic
278C22×C53#57811yescharacteristic
279C2×D53—42no—
280C2×D53—42no—
281C2×D53—42no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C2×D53—81no—
286C4×C53—22no—
287C4×C53—41no—
288C4×C53—42no—
289C4×C53—42no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#53311yescharacteristic
293C4×C53#54111yescharacteristic
294C4×C53#54811yescharacteristic
295C4×C53D4×Q811yescharacteristic
296Dic53—22no—
297Dic53—22no—
298Dic53—41no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302Dic53#45914yes—
303C2×#500—531no—
304C2×#474—531no—
305#2542—531no—
306#2365—531no—
307#2083—531no—
308#2083—531no—
309#2067—531no—
310#2067—531no—
311#1988—531no—
312#1947—531no—
313#1947D5311yescharacteristic, complemented
314#1939—531no—
315#1925—531no—
316#1925—531no—
317C4×#102—531no—
318C2×C53#242911yescharacteristic
319C2×C53C2×#45911yescharacteristic
320C2×C53#304411yescharacteristic
321C2×C53#333411yescharacteristic
322C2×C53#335311yescharacteristic
323C2×C53Q8×D811yescharacteristic
324C2×C53#338611yescharacteristic
325D53—82no—
326C2×C4⋊Q8—531no—
327C2×C4⋊Q8—531no—
328C2×C22⋊Q8—1061no—
329C2×C22⋊Q8—1061no—
330C2×C4×Q8—1061no—
331C2×C4×D4—531no—
332#500—1064no—
333#474—1064no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#107C2×D5311yescharacteristic
339C2×#107C2×D5311yescharacteristic
340C2×C4⋊C8—531no—
341C2×C4⋊C8—531no—
342C2×C4⋊C8C2×D5311yescharacteristic
343C2×#102—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×C4×C8—531no—
348#399—1061no—
349#399—1061no—
350#391—1061no—
351#390—531no—
352#390—531no—
353#389—531no—
354#389—531no—
355#387—1061no—
356#387—1061no—
357#384—531no—
358#384C2×D5311yescharacteristic
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—531no—
362C4×C4⋊C4C2×D5311yescharacteristic
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#340C2×D5311yescharacteristic
369#336—531no—
370#336—531no—
371C53#2544911yescharacteristic, complemented, Sylow
372C22×Q8—1061no—
373C22×Q8—1061no—
374C22×D4—531no—
375C23×C4—1061no—
376C22×C8—531no—
377C22×C8—531no—
378C22×C8C22×D5311yescharacteristic
379C22×C8C22×D5311yescharacteristic
380C4⋊Q8—534no—
381C4⋊Q8—1062no—
382C4⋊Q8—1062no—
383C4⋊Q8—1062no—
384C22⋊Q8—2124no—
385C22⋊Q8—2124no—
386C4×Q8—2124no—
387C4×D4—1062no—
388C4×D4—1062no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4—1061no—
411C2×C42—21no—
412C2×C42—531no—
413C2×C42—531no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42C22×D5311yescharacteristic
417#107—22no—
418#107—22no—
419#107—1062no—
420#107—1062no—
421#107—1062no—
422#107—1062no—
423C4⋊C8—22no—
424C4⋊C8—1062no—
425C4⋊C8—1062no—
426#102—534no—
427#102—1062no—
428#102—1062no—
429#102—1062no—
430C4×C8—1062no—
431#95—21no—
432#95—531no—
433#95—531no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438C24—1061no—
439C2×Q8—1064no—
440C2×Q8—2122no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C2×Q8—2122no—
444C2×Q8—2122no—
445C2×D4—532no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—1061no—
450C2×D4—1061no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—2122no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4D4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—532no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8#1171994512yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—534no—
489C4⋊C4—532no—
490C4⋊C4—1062no—
491C4⋊C4—1064no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4D4×D5312yes—
503C4⋊C4Q8×D5312yes—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C22⋊C4—2122no—
508C22⋊C4—2122no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—532no—
513C42—532no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C42—2122no—
518C23—1062no—
519C23—2121no—
520C23—2121no—
521C23—2121no—
522C23—2121no—
523C23—2121no—
524C23—2121no—
525C23C2×D4×D5311yescharacteristic
526Q8—4244no—
527Q8—2128no—
528D4—1062no—
529D4—1062no—
530D4—2121no—
531D4—2121no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—532no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1061no—
561C2×C4—1061no—
562C2×C4—4244no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4D8×D5312yes—
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C2×Q8×D5311yescharacteristic
574C2×C4#41×D5311yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C2×C4#1313326411yescharacteristic
577C2×C4#1313326511yescharacteristic
578C8—22no—
579C8—41no—
580C8—1062no—
581C8—2121no—
582C22—4241no—
583C22—4241no—
584C22—4241no—
585C22—4241no—
586C22—2122no—
587C22—2122no—
588C22—2122no—
589C22#121×D5311yescharacteristic
590C22#1681127411yescharacteristic
591C22#1681128611yescharacteristic
592C22C2×D8×D5311yescharacteristic
593C22AGL(1,ℤ/8)×D5311yescharacteristic
594C22#1681143611yescharacteristic
595C22#1681143911yescharacteristic
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—81no—
601C4—81no—
602C4—534no—
603C4—1062no—
604C4—1062no—
605C4—4241no—
606C4—4241no—
607C4—2121no—
608C4—2122no—
609C4#1681117711yescharacteristic
610C4#1681123211yescharacteristic
611C4#1681128211yescharacteristic
612C4#1681128811yescharacteristic
613C2—4242no—
614C2#2027831311yescharacteristic
615C2#459×D5311yescharacteristic
616C2#2028424011yescharacteristic
617C2#2028706011yescharacteristic
618C2#2028729511yescharacteristic
619C2#2028756511yescharacteristic
620C2#2028763911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1947C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1947D53
hypocenterC53#25449
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. C2×#16811027C2semidirect
4. #20277774C2semidirect
5. #20276189C2semidirect
6. #20276187C2semidirect
9. #20275665C2semidirect
10. #20275472C2semidirect
11. #20275430C2semidirect
15. C53×#1947C2semidirect
105. #13133137Q8semidirect
313. #1947D53semidirect
371. C53#25449semidirect

Maximal subgroups

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

subgroupindex
C2×#168110272
C2×#168109022
#202796672
#202777742
#202761892
#202761872
#202761022
#202760982
#202756652
#202754722
#202754302
#202753352
#202753332
#202745292
C53×#19472
#2544953