Finite Group Databasebeta

← #27399435

Subgroups — #27399435

Order 13568.

1474 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810597C212yesmaximal, complemented
2#20277584C212yesmaximal, complemented
3#20275552C212yesmaximal, complemented
4#20275174C214yesmaximal, complemented
5#20274703C211yesmaximal, characteristic, complemented
6#20274633C212yesmaximal, complemented
7C4×#13133132C211yesmaximal, characteristic, complemented
8C2×C53×#547C211yesmaximal, characteristic
9C2×#13133176C2214yes—
10#16810597—28no—
11C2×#13133132C2212yes—
12C2×#13133132C2214yes—
13C2×#13133132C2211yescharacteristic
14C2×#13133132C2214yes—
15C2×#13133132C2211yescharacteristic
16#16810522—22no—
17#16810509—24no—
18#16810486C2214yes—
19#16810483C2212yes—
20C4⋊C4×Dic53—22no—
21C4×C22⋊Dic53C2212yes—
22#16810390C2212yes—
23#16810390C2212yes—
24C22×#41×C53C2212yes—
25C53×#547—24no—
26C53×#547—22no—
27C53×#547—22no—
28C2×C4⋊Q8×C53C411yescharacteristic, complemented
29C2×#123×C53C412yescomplemented
30C2×#123×C53C2212yes—
31C2×C22⋊Q8×C53C412yescomplemented
32C2×#121×C53C411yescharacteristic, complemented
33C2×#121×C53C2211yescharacteristic
34C2×C4×D4×C53C411yescharacteristic, complemented
35C2×C4×D4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C411yescharacteristic, complemented
37C22×C4⋊C4×C53C2211yescharacteristic
38#13133176—28no—
39#13133176—44no—
40C2×C4⋊Dic53—24no—
41C2×C4⋊Dic53—22no—
42C2×C22⋊Dic53C2314yes—
43C2×C22⋊Dic53C2312yes—
44C2×C22⋊Dic53C2312yes—
45C2×C4×Dic53—24no—
46C2×C4×Dic53—22no—
47#13133132—28no—
48#13133132—24no—
49#13133132—22no—
50#13133132—44no—
51#13133132D414yescomplemented
52#13133124—22no—
53#13133124—22no—
54#13133124—24no—
55#13133124—24no—
56C2×#41×C53—24no—
57C2×#41×C53—24no—
58C2×#41×C53—24no—
59C2×#41×C53—24no—
60C2×#41×C53—44no—
61C22×Q8×C53C2×C412yes—
62C22×D4×C53C2×C412yes—
63C22×D4×C53C2×C411yescharacteristic
64C22×D4×C53C2312yes—
65C22×D4×C53C2311yescharacteristic
66C23×C4×C53C2×C412yes—
67C23×C4×C53C2×C412yes—
68C23×C4×C53C2×C411yescharacteristic
69C23×C4×C53C2312yes—
70C23×C4×C53C2311yescharacteristic
71C4⋊Q8×C53—22no—
72C4⋊Q8×C53—22no—
73#123×C53—24no—
74#123×C53—24no—
75#123×C53—24no—
76#123×C53—24no—
77C22⋊Q8×C53—24no—
78C22⋊Q8×C53—24no—
79#121×C53—22no—
80#121×C53—22no—
81#121×C53—22no—
82#121×C53—22no—
83C4×D4×C53—22no—
84C4×D4×C53—21no—
85C4×D4×C53—21no—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53C2×C411yescharacteristic
92C2×C4⋊C4×C53C2×C412yes—
93C2×C4⋊C4×C53C2×C411yescharacteristic
94C2×C4⋊C4×C53C2×C412yes—
95C2×C4⋊C4×C53C2×C411yescharacteristic
96C2×C4⋊C4×C53C2×C411yescharacteristic
97C2×C4⋊C4×C53C2×C412yes—
98C2×C4⋊C4×C53D414yes—
99C2×C4⋊C4×C53D412yes—
100C2×C4⋊C4×C53D412yes—
101C2×C22⋊C4×C53C2×C412yes—
102C2×C22⋊C4×C53C2×C411yescharacteristic
103C2×C22⋊C4×C53C2×C412yes—
104C2×C22⋊C4×C53C2×C412yes—
105C2×C22⋊C4×C53C2311yescharacteristic
106C2×C42×C53C2×C411yescharacteristic
107C24×C53C22×C412yes—
108C24×C53C22×C411yescharacteristic
109C24×C53C2411yescharacteristic
110#41×C53—48no—
111#41×C53—48no—
112#41×C53—48no—
113#41×C53—48no—
114C2×Q8×C53—24no—
115C2×Q8×C53—24no—
116C2×Q8×C53—42no—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—24no—
120C2×D4×C53—24no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—24no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53C2×D412yes—
130C2×D4×C53#4112yes—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—24no—
142C22×C4×C53—42no—
143C22×C4×C53—42no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53C22×C412yes—
147C22×C4×C53C22×C411yescharacteristic
148C22×C4×C53C22×C412yes—
149C22×C4×C53C22×C412yes—
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C22×C411yescharacteristic
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D414yes—
156C22×C4×C53C2×D414yes—
157C22×C4×C53C2×D412yes—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—24no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53C22⋊C412yes—
168C4⋊C4×C53C22⋊C411yescharacteristic
169C4⋊C4×C53C22⋊C411yescharacteristic
170C4⋊C4×C53C22⋊C412yes—
171C4⋊C4×C53C22⋊C412yes—
172C4⋊C4×C53C22⋊C414yes—
173C4⋊C4×C53C22⋊C414yes—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—24no—
180C42×C53—21no—
181C42×C53—21no—
182C22⋊Dic53—22no—
183C22⋊Dic53—24no—
184C22⋊Dic53—22no—
185C22⋊Dic53—22no—
186C22⋊Dic53—24no—
187C22⋊Dic53C2×D414yes—
188C4⋊Dic53—48no—
189C4⋊Dic53—44no—
190C4×Dic53—48no—
191C4×Dic53—44no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53C4×D412yes—
204C23×C53C23×C411yescharacteristic
205C23×C53C22×D411yescharacteristic
206C23×C53C22×D412yes—
207C23×C53C2×#4111yescharacteristic
208C23×C53H(5,𝔽2)12yes—
209Q8×C53—48no—
210D4×C53—24no—
211D4×C53—24no—
212D4×C53—48no—
213D4×C53—42no—
214D4×C53—44no—
215D4×C53—44no—
216D4×C53—42no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—24no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—22no—
228C2×C4×C53—24no—
229C2×C4×C53—22no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—41no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—44no—
239C2×C4×C53—44no—
240C2×C4×C53—42no—
241C2×C4×C53—42no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53C2×C22⋊C411yescharacteristic
247C2×C4×C53C2×C22⋊C414yes—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C2×C22⋊C411yescharacteristic
250C2×C4×C53C4×D412yes—
251C2×Dic53—24no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—44no—
256C2×Dic53—42no—
257C2×Dic53—44no—
258C2×Dic53—44no—
259C2×Dic53—42no—
260#22406—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—21no—
263C22×C53—22no—
264C22×C53—21no—
265C22×C53—22no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53C22×C22⋊C411yescharacteristic
279C22×C53C2×C4×D411yescharacteristic
280C22×C53#51811yescharacteristic
281C22×C53D4212yes—
282C22×C53#54612yes—
283C4×C53—22no—
284C4×C53—22no—
285C4×C53—24no—
286C4×C53—22no—
287C4×C53—21no—
288C4×C53—22no—
289C4×C53—42no—
290C4×C53—41no—
291C4×C53—41no—
292C4×C53#41012yescomplemented
293Dic53—44no—
294Dic53—84no—
295Dic53—82no—
296C2×#547D5311yescharacteristic
297C2×#410—532no—
298#2344—532no—
299#1964—532no—
300#1899—534no—
301#1799—531no—
302#1782—532no—
303C4×#99—531no—
304C2×C53—22no—
305C2×C53—22no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—42no—
309C2×C53—82no—
310C2×C53D4×C22⋊C411yescharacteristic
311C2×C53C2×#41011yescharacteristic
312C2×C53#288411yescharacteristic
313C22×#41C2×D5312yes—
314#547—22no—
315#547—24no—
316#547—22no—
317C2×C4⋊Q8Dic5311yescharacteristic, complemented
318C2×#123Dic5312yescomplemented
319C2×#123C2×D5312yes—
320C2×C22⋊Q8Dic5312yescomplemented
321C2×#121Dic5311yescharacteristic, complemented
322C2×#121C2×D5311yescharacteristic
323C2×C4×D4Dic5311yescharacteristic, complemented
324C2×C4×D4C2×D5311yescharacteristic
325C2×#117—534no—
326C22×C4⋊C4Dic5311yescharacteristic, complemented
327C22×C4⋊C4C2×D5311yescharacteristic
328#410—1068no—
329C2×#99—531no—
330C2×#99—532no—
331C2×#99—534no—
332C2×#99—531no—
333C2×#99—534no—
334#389—1062no—
335#387—1064no—
336#381—534no—
337#380—532no—
338C4×C4⋊C4—1062no—
339C4×C22⋊C4—532no—
340#342—532no—
341#342—532no—
342C53#2240611yescharacteristic, complemented, Sylow
343C2×#41—24no—
344C2×#41—24no—
345C2×#41—24no—
346C2×#41—24no—
347C2×#41—44no—
348C22×Q8C2×Dic5312yes—
349C22×D4C2×Dic5312yes—
350C22×D4C2×Dic5311yescharacteristic
351C22×D4C22×D5312yes—
352C22×D4C22×D5311yescharacteristic
353C23×C4C2×Dic5312yes—
354C23×C4C2×Dic5312yes—
355C23×C4C2×Dic5311yescharacteristic
356C23×C4C22×D5312yes—
357C23×C4C22×D5311yescharacteristic
358C4⋊Q8—22no—
359C4⋊Q8—22no—
360#123—24no—
361#123—24no—
362#123—24no—
363#123—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4—21no—
375#117—1068no—
376#117—2124no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4C2×Dic5311yescharacteristic
383C2×C4⋊C4C2×Dic5312yes—
384C2×C4⋊C4C2×Dic5311yescharacteristic
385C2×C4⋊C4C2×Dic5312yes—
386C2×C4⋊C4C2×Dic5311yescharacteristic
387C2×C4⋊C4C2×Dic5311yescharacteristic
388C2×C4⋊C4C2×Dic5312yes—
389C2×C4⋊C4C22⋊D5312yes—
390C2×C4⋊C4C22⋊D5314yes—
391C2×C4⋊C4C22⋊D5312yes—
392C2×C22⋊C4—532no—
393C2×C22⋊C4—534no—
394C2×C22⋊C4—532no—
395C2×C22⋊C4C2×Dic5312yes—
396C2×C22⋊C4C2×Dic5311yescharacteristic
397C2×C22⋊C4C2×Dic5312yes—
398C2×C22⋊C4C2×Dic5312yes—
399C2×C22⋊C4C22×D5311yescharacteristic
400C2×C42—1064no—
401C2×C42—1062no—
402C2×C42C2×Dic5311yescharacteristic
403#99—534no—
404#99—1068no—
405#99—1064no—
406#99—1062no—
407#99—2124no—
408#95—1062no—
409#95—1064no—
410#95—1062no—
411#95—1064no—
412C24C22×Dic5312yes—
413C24C22×Dic5311yescharacteristic
414C24C23×D5311yescharacteristic
415#41—48no—
416#41—48no—
417#41—48no—
418#41—48no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4D4×D5312yes—
435C2×D4#1171994712yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1064no—
454C22×C4C22×Dic5312yes—
455C22×C4C22×Dic5311yescharacteristic
456C22×C4C22×Dic5312yes—
457C22×C4C22×Dic5312yes—
458C22×C4C22×Dic5311yescharacteristic
459C22×C4C22×Dic5311yescharacteristic
460C22×C4C22×Dic5311yescharacteristic
461C22×C4C22×Dic5311yescharacteristic
462C22×C4C2×C22⋊D5314yes—
463C22×C4C2×C22⋊D5314yes—
464C22×C4C2×C22⋊D5312yes—
465C22×C4C2×C22⋊D5312yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—2128no—
473C4⋊C4—2124no—
474C4⋊C4C22⋊Dic5312yes—
475C4⋊C4C22⋊Dic5311yescharacteristic
476C4⋊C4C22⋊Dic5311yescharacteristic
477C4⋊C4C22⋊Dic5312yes—
478C4⋊C4C22⋊Dic5312yes—
479C4⋊C4C22⋊Dic5314yes—
480C4⋊C4C22⋊Dic5314yes—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—534no—
488C22⋊C4—1062no—
489C22⋊C4—1064no—
490C22⋊C4—1062no—
491C22⋊C4—1062no—
492C22⋊C4—1064no—
493C42—21no—
494C42—21no—
495C42—2128no—
496C42—2124no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23D4×Dic5312yes—
509C23C23×Dic5311yescharacteristic
510C23C2×D4×D5311yescharacteristic
511C23C22×C22⋊D5312yes—
512C23C2×#1171994711yescharacteristic
513C23#1313326212yes—
514Q8—48no—
515D4—24no—
516D4—24no—
517D4—48no—
518D4—42no—
519D4—44no—
520D4—44no—
521D4—42no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—1064no—
550C2×C4—2122no—
551C2×C4—2124no—
552C2×C4—2122no—
553C2×C4—2124no—
554C2×C4—2122no—
555C2×C4—2124no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4C2×C22⋊Dic5311yescharacteristic
559C2×C4C2×C22⋊Dic5312yes—
560C2×C4C2×C22⋊Dic5312yes—
561C2×C4C2×C22⋊Dic5312yes—
562C2×C4C2×C22⋊Dic5314yes—
563C2×C4C2×C22⋊Dic5311yescharacteristic
564C2×C4D4×Dic5312yes—
565C22—21no—
566C22—22no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22C22×C22⋊Dic5311yescharacteristic
583C22C2×D4×Dic5311yescharacteristic
584C22#1681109811yescharacteristic
585C22D4×C22⋊D5312yes—
586C22#1681126712yes—
587C4—22no—
588C4—22no—
589C4—24no—
590C4—22no—
591C4—21no—
592C4—22no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—4244no—
597C4—4242no—
598C4—2124no—
599C4#1681059712yescomplemented
600C2—22no—
601C2—22no—
602C2—21no—
603C2—21no—
604C2—42no—
605C2—82no—
606C2D4×C22⋊Dic5311yescharacteristic
607C2C2×#1681059711yescharacteristic
608C2#2028317911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC22×C53C22×C22⋊C4
cosocleC24×C53C24
hypercenterC2×#547D53
hypocenterC53#22406
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810597C2semidirect
2. #20277584C2semidirect
3. #20275552C2semidirect
4. #20275174C2semidirect
5. #20274703C2semidirect
6. #20274633C2semidirect
7. C4×#13133132C2semidirect
28. C2×C4⋊Q8×C53C4semidirect
29. C2×#123×C53C4semidirect
31. C2×C22⋊Q8×C53C4semidirect
32. C2×#121×C53C4semidirect
34. C2×C4×D4×C53C4semidirect
36. C22×C4⋊C4×C53C4semidirect
51. #13133132D4semidirect
292. C4×C53#410semidirect
317. C2×C4⋊Q8Dic53semidirect
318. C2×#123Dic53semidirect
320. C2×C22⋊Q8Dic53semidirect
321. C2×#121Dic53semidirect
323. C2×C4×D4Dic53semidirect
326. C22×C4⋊C4Dic53semidirect
342. C53#22406semidirect
599. C4#16810597semidirect

Maximal subgroups

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

subgroupindex
C2×#168105972
#202775842
#202755522
#202751742
#202747032
#202746332
C4×#131331322
C2×C53×#5472
#2240653