Finite Group Databasebeta

← #27399744

Subgroups — #27399744

Order 13568.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810701C211yesmaximal, characteristic, complemented
2C2×#16810624C212yesmaximal, complemented
3D4×C4⋊Dic53C211yesmaximal, characteristic, complemented
4#20275639C211yesmaximal, characteristic, complemented
5#20275518C211yesmaximal, characteristic, complemented
6#20275149C212yesmaximal, complemented
7#20275023C212yesmaximal, complemented
8#20274715C211yesmaximal, characteristic, complemented
9#20274655C212yesmaximal, complemented
10C4×#13133138C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C22×#11719917C2212yes—
13C2×D4×Dic53—21no—
14C22×C4⋊Dic53C2212yes—
15#16810701—24no—
16C2×C4⋊#88322C2211yescharacteristic
17#16810624—28no—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2212yes—
26C2×C4×#88322C2211yescharacteristic
27#16810522—21no—
28#16810498—22no—
29#16810493C2211yescharacteristic
30#16810483C2212yes—
31C4×C4⋊Dic53C2211yescharacteristic
32#16810385C2212yes—
33#16810385C2211yescharacteristic
34#16810385C2211yescharacteristic
35C22×#41×C53C2212yes—
36C53×#547—24no—
37C53×#547C418yescomplemented
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53C2212yes—
41C2×#121×C53—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53—21no—
45C23×Dic53—22no—
46C2×#11719917—24no—
47C2×#11719917—24no—
48C2×#11719917—42no—
49C22×#88322C2311yescharacteristic
50C22×#88322C2312yes—
51C22×#88322C2311yescharacteristic
52D4×Dic53—44no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—24no—
57C2×C4⋊Dic53—42no—
58C2×C4⋊Dic53C2312yes—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C22⋊Dic53—22no—
62C2×C4×Dic53—21no—
63C4⋊#88322—22no—
64#13133141—24no—
65#13133141—24no—
66#13133141—42no—
67#13133141—42no—
68#13133138—24no—
69#13133138—22no—
70#13133138D414yescomplemented
71C22⋊#88322—24no—
72C4×#88322—22no—
73#13133124—21no—
74#13133124—22no—
75C2×#41×C53—24no—
76C2×#41×C53—24no—
77C2×#41×C53—24no—
78C2×#41×C53—42no—
79C2×#41×C53—42no—
80C2×#41×C53C2×C418yescomplemented
81C22×Q8×C53C2312yes—
82C22×D4×C53—22no—
83C22×D4×C53D411yescharacteristic
84C22×D4×C53C2311yescharacteristic
85C23×C4×C53—21no—
86C23×C4×C53—22no—
87C23×C4×C53C2312yes—
88C4⋊Q8×C53—22no—
89C4⋊Q8×C53C2×C414yescomplemented
90#123×C53—216no—
91C22⋊Q8×C53—24no—
92C22⋊Q8×C53C2×C418yescomplemented
93#121×C53—28no—
94C4×D4×C53—22no—
95C4×D4×C53C2×C414yes—
96C4×D4×C53C2×C414yescomplemented
97C4×D4×C53D414yes—
98C2×C4⋊C4×C53—28no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—22no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D412yes—
107C2×C42×C53C2311yescharacteristic
108C24×C53—21no—
109C24×C53C2×D412yes—
110#41×C53—216no—
111#41×C53—48no—
112#41×C53—48no—
113#41×C53—48no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53C22×C414yes—
120C2×D4×C53—28no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—44no—
127C2×D4×C53—44no—
128C2×D4×C53C22⋊C414yes—
129C2×D4×C53C22×C412yes—
130C2×D4×C53C2×D412yes—
131C2×D4×C53#4112yes—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—24no—
139C22×C4×C53—28no—
140C22×C4×C53—44no—
141C22×C4×C53—44no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53C22×C414yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D414yes—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—44no—
154C22×Dic53—44no—
155C22×Dic53—82no—
156C4⋊C4×C53—24no—
157C4⋊C4×C53—28no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53C22⋊C414yes—
161C4⋊C4×C53C22×C412yes—
162C22⋊C4×C53—28no—
163C22⋊C4×C53—24no—
164C22⋊C4×C53C22⋊C418yes—
165C42×C53C22×C412yes—
166C42×C53C2×D412yes—
167C22⋊Dic53—44no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—22no—
173C4⋊Dic53—44no—
174C4⋊Dic53—42no—
175C4⋊Dic53—81no—
176C4⋊Dic53C2×D412yes—
177C4×Dic53—42no—
178#11719917—48no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—22no—
187C2×#88322C2×D412yes—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53C2×C22⋊C414yes—
202C23×C53C22×D411yescharacteristic
203Q8×C53—24no—
204Q8×C53—24no—
205Q8×C53—42no—
206Q8×C53—42no—
207D4×C53—22no—
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—28no—
211D4×C53—416no—
212D4×C53C4×D414yescomplemented
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—416no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—48no—
229C2×C4×C53C2×C22⋊C414yes—
230C2×C4×C53C2×C22⋊C412yes—
231C2×C4×C53C2×C22⋊C412yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53#13712yes—
234C2×C4×C53C23×C411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53H(5,𝔽2)11yescharacteristic
239C2×C4×C53H(5,𝔽2)11yescharacteristic
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—41no—
247C2×Dic53—42no—
248C2×Dic53—41no—
249C2×Dic53—88no—
250#88322—22no—
251#88322—42no—
252#88322—41no—
253#22431—531nomaximal, Sylow
254C22×C53—24no—
255C22×C53—24no—
256C22×C53—24no—
257C22×C53—24no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53C22×C22⋊C411yescharacteristic
269C22×C53D4211yescharacteristic
270C22×C53#54611yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C4×C53—24no—
276C4×C53—22no—
277C4×C53—44no—
278C4×C53—42no—
279C4×C53#41714yescomplemented
280C4×C53C2×C4×D411yescharacteristic
281C4×C53#51811yescharacteristic
282C4×C53D4211yescharacteristic
283C4×C53#54611yescharacteristic
284Dic53—42no—
285Dic53—81no—
286Dic53—82no—
287C2×#547D5311yescharacteristic
288C2×#436—531no—
289C2×#417—532no—
290D4×C4⋊C4—531no—
291#1981—531no—
292#1956—531no—
293#1894—532no—
294#1872—532no—
295#1802—531no—
296#1788—532no—
297C4×#102—531no—
298C2×C53—44no—
299C2×C53—44no—
300C2×C53D4×C22⋊C411yescharacteristic
301C2×C53C2×#41711yescharacteristic
302C2×C53#289811yescharacteristic
303C2×C53#327211yescharacteristic
304C2×C53#327611yescharacteristic
305C2×C53#333011yescharacteristic
306C2×C53#333111yescharacteristic
307C22×#41C2×D5312yes—
308C22×M16—532no—
309#547—24no—
310#547Dic5318yescomplemented
311C2×C4⋊Q8C2×D5311yescharacteristic
312C2×#123—22no—
313C2×C22⋊Q8C2×D5312yes—
314C2×#121—21no—
315C2×C4×D4—1061no—
316C2×C4×D4C2×D5311yescharacteristic
317C2×C4×D4C2×D5311yescharacteristic
318C22×C4⋊C4—21no—
319C22×C4⋊C4—532no—
320#436—1064no—
321C2×C4⋊C8—531no—
322#417—1068no—
323C2×#103—532no—
324C2×#103—532no—
325C2×#103—532no—
326C2×#103—532no—
327C2×#102—531no—
328C2×#102—532no—
329C2×#102—531no—
330C2×C22⋊C8—532no—
331C2×C4×C8—531no—
332#389—1061no—
333#385—1062no—
334#384—531no—
335#380—532no—
336C4×C4⋊C4—531no—
337#340—532no—
338#340—531no—
339#340—531no—
340C53#2243111yescharacteristic, complemented, Sylow
341C2×#41—24no—
342C2×#41—24no—
343C2×#41—24no—
344C2×#41—42no—
345C2×#41—42no—
346C2×#41C2×Dic5318yescomplemented
347C22×Q8C22×D5312yes—
348C22×D4—22no—
349C22×D4C22⋊D5311yescharacteristic
350C22×D4C22×D5311yescharacteristic
351C23×C4—21no—
352C23×C4—22no—
353C23×C4—1062no—
354C23×C4C22×D5312yes—
355C2×M16—1064no—
356C2×M16—1064no—
357C2×M16—2122no—
358C22×C8—531no—
359C22×C8—532no—
360C22×C8—531no—
361C4⋊Q8—22no—
362C4⋊Q8C2×Dic5314yescomplemented
363#123—216no—
364C22⋊Q8—24no—
365C22⋊Q8C2×Dic5318yescomplemented
366#121—28no—
367C4×D4—22no—
368C4×D4—2124no—
369C4×D4C2×Dic5314yes—
370C4×D4C2×Dic5314yescomplemented
371C4×D4C22⋊D5314yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—532no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—2122no—
384C2×C4⋊C4C22⋊D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C22⋊C4—22no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—1062no—
389C2×C22⋊C4C22⋊D5312yes—
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392C4⋊C8—1062no—
393#103—1064no—
394#103—1064no—
395#103—2122no—
396#103—2122no—
397#102—534no—
398#102—1064no—
399#102—1062no—
400C22⋊C8—1064no—
401C4×C8—1062no—
402#95—1061no—
403#95—1062no—
404C24—21no—
405C24C2×C22⋊D5312yes—
406#41—216no—
407#41—48no—
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8C22×Dic5314yes—
416C2×D4—28no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4C22⋊Dic5314yes—
425C2×D4D4×D5312yes—
426C2×D4#1171994712yes—
427C2×D4C22×Dic5312yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1062no—
443C22×C4—4242no—
444C22×C4—2124no—
445C22×C4—2124no—
446C22×C4C22×Dic5314yes—
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5312yes—
449C22×C4C2×C22⋊D5314yes—
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452M16—2128no—
453C2×C8—532no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1062no—
457C2×C8—1061no—
458C2×C8—1062no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1062no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—532no—
467C4⋊C4—1061no—
468C4⋊C4—1062no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4—4241no—
473C4⋊C4—2124no—
474C4⋊C4—2122no—
475C4⋊C4C22⋊Dic5314yes—
476C4⋊C4C22×Dic5312yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—2124no—
480C22⋊C4C22⋊Dic5318yes—
481C42—2122no—
482C42C22×Dic5312yes—
483C42C2×C22⋊D5312yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×C22⋊Dic5314yes—
498C23C22×C22⋊D5311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—42no—
502Q8—42no—
503D4—22no—
504D4—22no—
505D4—28no—
506D4—22no—
507D4—416no—
508D4D4×Dic5314yescomplemented
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—416no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—1062no—
526C2×C4—4248no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2121no—
532C2×C4—2122no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4C2×C22⋊Dic5312yes—
536C2×C4C2×C22⋊Dic5314yes—
537C2×C4C2×C22⋊Dic5312yes—
538C2×C4#1313324212yes—
539C2×C4#1313324712yes—
540C2×C4C23×Dic5311yescharacteristic
541C2×C4C2×D4×D5311yescharacteristic
542C2×C4C22×C22⋊D5311yescharacteristic
543C2×C4C2×#1171994711yescharacteristic
544C2×C4#1313326211yescharacteristic
545C2×C4#1313326211yescharacteristic
546C8—1062no—
547C8—2121no—
548C8—2122no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C22×C22⋊Dic5311yescharacteristic
564C22D4×C22⋊D5311yescharacteristic
565C22#1681126711yescharacteristic
566C22C2×#1313324211yescharacteristic
567C22C2×#1313324711yescharacteristic
568C22#1681143111yescharacteristic
569C22#1681143811yescharacteristic
570C4—24no—
571C4—22no—
572C4—44no—
573C4—42no—
574C4—4241no—
575C4—4242no—
576C4—2122no—
577C4#1681062414yescomplemented
578C4C2×D4×Dic5311yescharacteristic
579C4#1681109811yescharacteristic
580C4D4×C22⋊D5311yescharacteristic
581C4#1681126711yescharacteristic
582C2—44no—
583C2—44no—
584C2D4×C22⋊Dic5311yescharacteristic
585C2C2×#1681062411yescharacteristic
586C2#2028324811yescharacteristic
587C2#2028631511yescharacteristic
588C2#2028635711yescharacteristic
589C2#2028702011yescharacteristic
590C2#2028703111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenterC2×#547D53
hypocenterC53#22431
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810701C2semidirect
2. C2×#16810624C2semidirect
3. D4×C4⋊Dic53C2semidirect
4. #20275639C2semidirect
5. #20275518C2semidirect
6. #20275149C2semidirect
7. #20275023C2semidirect
8. #20274715C2semidirect
9. #20274655C2semidirect
10. C4×#13133138C2semidirect
37. C53×#547C4semidirect
70. #13133138D4semidirect
80. C2×#41×C53C2×C4semidirect
89. C4⋊Q8×C53C2×C4semidirect
92. C22⋊Q8×C53C2×C4semidirect
96. C4×D4×C53C2×C4semidirect
212. D4×C53C4×D4semidirect
279. C4×C53#417semidirect
310. #547Dic53semidirect
340. C53#22431semidirect
346. C2×#41C2×Dic53semidirect
362. C4⋊Q8C2×Dic53semidirect
365. C22⋊Q8C2×Dic53semidirect
370. C4×D4C2×Dic53semidirect
508. D4D4×Dic53semidirect
577. C4#16810624semidirect

Maximal subgroups

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

subgroupindex
C2×#168107012
C2×#168106242
D4×C4⋊Dic532
#202756392
#202755182
#202751492
#202750232
#202747152
#202746552
C4×#131331382
C2×C53×#5472
#2243153