Finite Group Databasebeta

← #27420151

Subgroups — #27420151

Order 13568.

1378 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D53C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3C4×#13133188C211yesmaximal, characteristic
4#20276041C212yesmaximal, complemented
5#20275849C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275666C211yesmaximal, characteristic, complemented
8#20275029C212yesmaximal
9#20274901C212yesmaximal, complemented
10#20274692C211yesmaximal, characteristic, complemented
11C2×C53×#474C211yesmaximal, characteristic, complemented
12C22×Q8×D53—21no—
13C2×#13133217C2211yescharacteristic
14C2×#13133216—21no—
15C2×#13133216C2212yes—
16C2×#13133215—21no—
17C4⋊Q8×D53—24no—
18C4⋊Q8×D53—24no—
19C2×#13133188C2211yescharacteristic
20C2×C4×C22⋊D53—21no—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53C2212yes—
23C2×C42×D53C2211yescharacteristic
24#16810903—24no—
25C2×#13133150C2212yes—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133138C2212yes—
28C2×#13133129C2212yes—
29#16810526C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810511—21no—
32#16810499—21no—
33#16810480C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C22⋊Dic53—21no—
36#16810386C2212yes—
37#16810386C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810377C2212yes—
41C2×C4⋊Q8×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#474—24no—
44C2×#107×C53C2212yes—
45C2×C4⋊C8×C53C2211yescharacteristic
46C2×#102×C53C2212yes—
47C22×Dic106—21no—
48C22×Dic106—22no—
49C2×Q8×D53—28no—
50C2×Q8×D53—44no—
51C2×Q8×D53—44no—
52C22×C22⋊D53—21no—
53C22×C4×D53—22no—
54C22×C4×D53D411yescharacteristic
55C22×C4×D53D411yescharacteristic
56C22×C4×D53C2311yescharacteristic
57C22×#88322C2312yes—
58#13133217—22no—
59#13133217D414yescomplemented
60#13133216—28no—
61#13133216—24no—
62#13133216—24no—
63#13133215—28no—
64#13133188—22no—
65#13133188Q814yescomplemented
66C4×C22⋊D53—44no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—22no—
72C2×#11719919—22no—
73C2×#11719919—21no—
74C4⋊C4×D53—28no—
75C4⋊C4×D53—24no—
76C4⋊C4×D53—24no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×#11719921—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C42×D53—22no—
86C42×D53—22no—
87#13133150—24no—
88C4⋊#88322—22no—
89#13133138—24no—
90#13133129—24no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C22×Q8×C53—21no—
95C22×D4×C53C2311yescharacteristic
96C23×C4×C53—21no—
97C22×C8×C53C2312yes—
98C4⋊Q8×C53—22no—
99C4⋊Q8×C53D414yescomplemented
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107#107×C53—24no—
108C4⋊C8×C53—22no—
109#102×C53—24no—
110C24×C53—21no—
111C23×D53C2×D411yescharacteristic
112C2×Q8×C53—24no—
113C2×Q8×C53—42no—
114C2×Q8×C53—42no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53C2×Q812yescomplemented
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C2×C22⋊D53—24no—
130C2×C22⋊D53—42no—
131C2×C22⋊D53—42no—
132C22×Dic53—21no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138Q8×D53—432no—
139C2×C4×D53—22no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—21no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—22no—
148C2×C4×D53—28no—
149C2×C4×D53—44no—
150C2×C4×D53—44no—
151C2×C4×D53D814yes—
152C2×C4×D53C2×Q812yes—
153C2×C4×D53#4112yes—
154C2×Dic106—28no—
155C2×Dic106—24no—
156C2×Dic106—44no—
157C2×Dic106—42no—
158C2×Dic106—42no—
159C2×Dic106—44no—
160C2×C8×C53—22no—
161C2×C8×C53—22no—
162C2×C8×C53—22no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C2×D414yes—
168C22⋊C4×C53—42no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C22⋊Dic53—24no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174#11719921—42no—
175C4⋊Dic53—24no—
176C4⋊Dic53—22no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53C2×D414yes—
181C4⋊Dic53C2×Q812yescomplemented
182#11719919—28no—
183#11719919—28no—
184#11719919—42no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53C22×D411yescharacteristic
201C22×D53—21no—
202C22×D53—21no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53C22⋊Q812yes—
206C22×D53C2×D812yes—
207C22×D53#13712yes—
208Q8×C53—48no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—24no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—84no—
222C2×C4×C53#12012yes—
223C2×C4×C53#12012yes—
224C2×C4×C53C2×D812yes—
225C2×C4×C53AGL(1,ℤ/8)12yes—
226C2×C4×C53C22×D411yescharacteristic
227C2×C4×C53C22×D411yescharacteristic
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C22⋊D53—48no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—24no—
240C2×Dic53—22no—
241C2×Dic53—42no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53C22⋊Q812yes—
249C2×Dic53C2×D812yes—
250C2×Dic53#13712yes—
251C4×D53—28no—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—28no—
257C4×D53—416no—
258Dic106—416no—
259Dic106—48no—
260C8×C53—42no—
261#88322—42no—
262#24298—531nomaximal, Sylow
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53C2×#12011yescharacteristic
271C22×C53C2×C22⋊Q811yescharacteristic
272C22×C53#53611yescharacteristic
273C22×C53C22×D811yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—22no—
282C2×D53—22no—
283C2×D53#47414yescomplemented
284C4×C53—22no—
285C4×C53—44no—
286C4×C53—81no—
287C4×C53#44714yes—
288C4×C53#54811yescharacteristic
289C4×C53#54811yescharacteristic
290C4×C53D4×Q811yescharacteristic
291C4×C53D4×Q811yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—22no—
295Dic53—44no—
296Dic53—42no—
297Dic53—81no—
298C22×C4⋊Q8—531no—
299C2×#474—531no—
300C2×#474D5311yescharacteristic, complemented
301C4×#121—531no—
302#2057—532no—
303#2023—532no—
304#1988—531no—
305#1988—531no—
306#1874—532no—
307#1852—532no—
308#1797—531no—
309C2×C53—82no—
310C2×C53#240911yescharacteristic
311C2×C53C2×#44711yescharacteristic
312C2×C53#300411yescharacteristic
313C2×C53C2×#47411yescharacteristic
314C2×C53#307911yescharacteristic
315C2×C53#322611yescharacteristic
316C2×C53#323911yescharacteristic
317D53—24no—
318C23×Q8—1061no—
319C2×C4⋊Q8—532no—
320C2×C4⋊Q8—531no—
321C2×C4⋊Q8—1061no—
322C2×C4⋊Q8—1061no—
323C2×C4⋊Q8—1064no—
324C2×C4⋊Q8—1064no—
325C2×C4⋊Q8C2×D5311yescharacteristic
326C2×#121—531no—
327C2×C4×D4—1061no—
328C2×C4×D4C2×D5311yescharacteristic
329C22×C4⋊C4—532no—
330C22×C4⋊C4—1061no—
331C22×C42—531no—
332#474—24no—
333#474—1064no—
334C2×#107—532no—
335C2×#107C2×D5312yes—
336C2×C4⋊C8—531no—
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×#102—532no—
339C2×#102C2×D5312yes—
340C2×C22⋊C8—532no—
341#390—531no—
342#389—531no—
343#388—1061no—
344#386—1061no—
345#380—531no—
346C4×C4⋊C4—531no—
347C4×C22⋊C4—1061no—
348#340—532no—
349#340—532no—
350#340—532no—
351#340—532no—
352#336—532no—
353C53#2429811yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×Q8—1062no—
356C22×Q8—1068no—
357C22×Q8—1061no—
358C22×Q8—2124no—
359C22×Q8—2124no—
360C22×D4—1061no—
361C22×D4C22×D5311yescharacteristic
362C23×C4—21no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—1062no—
367C22×C8—532no—
368C22×C8C22×D5312yes—
369C4⋊Q8—22no—
370C4⋊Q8—534no—
371C4⋊Q8—1064no—
372C4⋊Q8—1068no—
373C4⋊Q8—1068no—
374C4⋊Q8—1064no—
375C4⋊Q8—1062no—
376C4⋊Q8C22⋊D5314yescomplemented
377#121—534no—
378#121—1062no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—2124no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—532no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1068no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—1062no—
404C2×C42—1062no—
405C2×C42—1061no—
406C2×C42C22×D5311yescharacteristic
407#107—24no—
408#107—1064no—
409C4⋊C8—22no—
410C4⋊C8—1062no—
411#102—24no—
412#102—1064no—
413C22⋊C8—1064no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417C24—21no—
418C24—531no—
419C2×Q8—24no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—1068no—
423C2×Q8—1064no—
424C2×Q8—2124no—
425C2×Q8—2124no—
426C2×Q8—21232no—
427C2×Q8—2122no—
428C2×Q8—2122no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—1064no—
435C2×D4—2122no—
436C2×D4—2122no—
437C2×D4Q8×D5312yescomplemented
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—534no—
447C22×C4—532no—
448C22×C4—532no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1068no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1061no—
462C22×C4—2124no—
463C22×C4—2124no—
464C22×C4D4×D5311yescharacteristic
465C22×C4D4×D5311yescharacteristic
466C22×C4C23×D5311yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—1062no—
471C2×C8—1062no—
472C2×C8—1062no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—534no—
478C4⋊C4—532no—
479C4⋊C4—1068no—
480C4⋊C4—1068no—
481C4⋊C4—1064no—
482C4⋊C4—1062no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—2122no—
487C4⋊C4C2×C22⋊D5314yes—
488C22⋊C4—42no—
489C22⋊C4—1064no—
490C22⋊C4—2122no—
491C22⋊C4—2122no—
492C22⋊C4—2122no—
493C42—1062no—
494C42—1062no—
495C42—1061no—
496C42—1061no—
497C42—2122no—
498C42D4×D5312yes—
499C42C2×C22⋊D5312yes—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—532no—
508C23—532no—
509C23—532no—
510C23—1061no—
511C23—1061no—
512C23—1061no—
513C23—1061no—
514C23C2×D4×D5311yescharacteristic
515Q8—48no—
516Q8—21216no—
517Q8—2128no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522D4—2128no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—1068no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1064no—
548C2×C4—1062no—
549C2×C4—21216no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4#1313318512yes—
558C2×C4#1313318512yes—
559C2×C4D8×D5312yes—
560C2×C4#1313324112yes—
561C2×C4C2×D4×D5311yescharacteristic
562C2×C4C22×C22⋊D5311yescharacteristic
563C2×C4C2×Q8×D5311yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#1313326411yescharacteristic
566C2×C4#1313326511yescharacteristic
567C8—42no—
568C8—2122no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—534no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22—1062no—
582C22—1062no—
583C22C2×#1313318511yescharacteristic
584C22C22⋊Q8×D5311yescharacteristic
585C22#1681119611yescharacteristic
586C22C2×D8×D5311yescharacteristic
587C22C2×#1313324111yescharacteristic
588C22#137×D5311yescharacteristic
589C22#137×D5311yescharacteristic
590C4—22no—
591C4—44no—
592C4—81no—
593C4—1062no—
594C4—1062no—
595C4—1062no—
596C4—4241no—
597C4—2122no—
598C4—2124no—
599C4#1681075514yes—
600C4#1681128011yescharacteristic
601C4#1681128311yescharacteristic
602C4#1681128611yescharacteristic
603C4Q8×C22⋊D5311yescharacteristic
604C2—82no—
605C2—1064no—
606C2#2027807611yescharacteristic
607C2C2×#1681075511yescharacteristic
608C2#2028394711yescharacteristic
609C2#474×D5311yescharacteristic
610C2#2028452311yescharacteristic
611C2#2028583511yescharacteristic
612C2#2028594411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#474C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
hypercenterC2×#474D53
hypocenterC53#24298
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D53C2semidirect
2. C2×#16810903C2semidirect
4. #20276041C2semidirect
5. #20275849C2semidirect
6. #20275668C2semidirect
7. #20275666C2semidirect
9. #20274901C2semidirect
10. #20274692C2semidirect
11. C2×C53×#474C2semidirect
59. #13133217D4semidirect
65. #13133188Q8semidirect
99. C4⋊Q8×C53D4semidirect
120. C2×D4×C53C2×Q8semidirect
181. C4⋊Dic53C2×Q8semidirect
283. C2×D53#474semidirect
300. C2×#474D53semidirect
353. C53#24298semidirect
376. C4⋊Q8C22⋊D53semidirect
437. C2×D4Q8×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D532
C2×#168109032
C4×#131331882
#202760412
#202758492
#202756682
#202756662
#202750292
#202749012
#202746922
C2×C53×#4742
#2429853