Finite Group Databasebeta

← #27400685

Subgroups — #27400685

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811329C211yesmaximal, characteristic
2C2×C8×Dic106C211yesmaximal, characteristic, complemented
3#20277667C211yesmaximal, characteristic, complemented
4#20275678C211yesmaximal, characteristic, complemented
5#20275674C211yesmaximal, characteristic, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275671C211yesmaximal, characteristic, complemented
8#20275661C211yesmaximal, characteristic, complemented
9#20274724C211yesmaximal, characteristic
10#20274722C211yesmaximal, characteristic
11#20274714C211yesmaximal, characteristic
12C4×#13133139C211yesmaximal, characteristic
13#20274529C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1991C211yesmaximal, characteristic, complemented
16#16811329—24no—
17#16811329C418yescomplemented
18C2×#13133216C2211yescharacteristic
19C2×#13133208C2211yescharacteristic
20C2×#13133197—21no—
21C2×#13133195—21no—
22C2×#13133195—21no—
23C2×C4×Dic106C2211yescharacteristic
24C2×C4×D212C2211yescharacteristic
25C2×#13133179C2211yescharacteristic
26C2×#13133179C2211yescharacteristic
27C2×C4⋊C4×D53—21no—
28C8×Dic106—24no—
29C2×C4⋊#88322C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133139C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37C2×C8×Dic53C2211yescharacteristic
38#16810527—21no—
39#16810508—21no—
40#16810505—21no—
41#16810492C2211yescharacteristic
42#16810488C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810474—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#125×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106D411yescharacteristic
64C22×D212C2311yescharacteristic
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53—21no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133216—22no—
71#13133216C2×C414yes—
72#13133208—22no—
73#13133208C2×C414yes—
74#13133197—28no—
75#13133195—28no—
76#13133195—28no—
77C4×Dic106C2×C414yes—
78C4×Dic106D414yes—
79C4×D212—22no—
80C4×D212C2×C414yes—
81#13133179—22no—
82#13133179—22no—
83#13133179C2×C414yes—
84#13133179C2×C414yes—
85C2×C4⋊Dic53—21no—
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919D411yescharacteristic
95C2×#11719919D411yescharacteristic
96C4⋊C4×D53—28no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99C2×#11719921—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53C2311yescharacteristic
104C2×C4×Dic53C2311yescharacteristic
105C4⋊#88322—22no—
106#13133146—22no—
107#13133146—22no—
108#13133139—22no—
109#13133139Q814yescomplemented
110#13133137—22no—
111#13133137—22no—
112C8×Dic53—22no—
113C8×Dic53—22no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121#125×C53—22no—
122#125×C53C2×C414yes—
123C2×C4⋊C4×C53—21no—
124C2×C4⋊C4×C53—21no—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C42×C53C2311yescharacteristic
130C4×C8×C53—22no—
131#95×C53—21no—
132#95×C53—21no—
133C23×D53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212C22×C412yes—
152C2×D212C2×Q812yes—
153C2×D212#4112yes—
154C2×C4×D53—24no—
155C2×C4×D53—24no—
156C2×C4×D53—24no—
157C2×C4×D53—42no—
158C2×C4×D53—42no—
159C2×C4×D53—81no—
160C2×C4×D53—81no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×Dic106—22no—
164C2×Dic106—22no—
165C2×Dic106—24no—
166C2×Dic106—81no—
167C2×Dic106—81no—
168C2×Dic106C22⋊C414yes—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53C2×Q812yes—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53C22×C412yes—
184C4⋊C4×C53C22×C412yes—
185C42×C53C22×C412yes—
186C42×C53C2×D412yes—
187#11719921—24no—
188#11719921—24no—
189#11719921—24no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—24no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C22⋊C414yes—
196C4⋊Dic53C22×C412yes—
197C4⋊Dic53C2×Q812yes—
198#11719919—24no—
199#11719919—24no—
200#11719919—42no—
201#11719919—81no—
202#11719919C22⋊C414yes—
203#11719919C22⋊C414yes—
204C4×Dic53—42no—
205C4×Dic53—81no—
206C4×Dic53—81no—
207C4×Dic53C22×C412yes—
208C4×Dic53C22×C412yes—
209C4×Dic53C2×D412yes—
210C4×Dic53C2×D412yes—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—41no—
224C22×D53—41no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
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—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53C2×C22⋊C412yes—
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C23×C411yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×Q811yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×C4×C532−1+411yescharacteristic
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—22no—
256C2×Dic53—22no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53C2×C22⋊C412yes—
265C2×Dic53C2×C22⋊C412yes—
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53C2×C22⋊C412yes—
268C2×Dic53#13512yes—
269C2×Dic53#13512yes—
270D212—22no—
271D212—22no—
272D212—22no—
273D212C4×Q814yes—
274C4×D53—48no—
275C4×D53—84no—
276C4×D53—84no—
277Dic106—28no—
278Dic106—84no—
279C8×C53—22no—
280C8×C53—41no—
281#88322—41no—
282#88322—41no—
283#22519—531nomaximal, Sylow
284C22×C53C22×C22⋊C411yescharacteristic
285C22×C53#54811yescharacteristic
286C22×C53D4×Q811yescharacteristic
287C22×C53C2×#13511yescharacteristic
288C22×C53C2×#13511yescharacteristic
289C22×C53#57611yescharacteristic
290C22×C53#57711yescharacteristic
291C2×D53—24no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—42no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—42no—
301C4×C53—42no—
302C4×C53—81no—
303C4×C53—81no—
304C4×C53C2×C4×Q811yescharacteristic
305C4×C53#51911yescharacteristic
306C4×C53#54811yescharacteristic
307C4×C53D4×Q811yescharacteristic
308Dic53—22no—
309Dic53—22no—
310Dic53—22no—
311Dic53—42no—
312Dic53—42no—
313Dic53—81no—
314Dic53#41614yes—
315C2×#554—531no—
316C2×C8×Q8—531no—
317#2355—531no—
318#1991—531no—
319#1991D5311yescharacteristic, complemented
320#1990—531no—
321#1989—531no—
322#1989—531no—
323#1987—531no—
324#1804—531no—
325#1804—531no—
326#1802—531no—
327C4×#102—531no—
328C4×#102—531no—
329C4×#102—531no—
330C2×C53Q8×C22⋊C411yescharacteristic
331C2×C53C2×#41611yescharacteristic
332C2×C53#289711yescharacteristic
333C2×C53#336011yescharacteristic
334C2×C53#336111yescharacteristic
335C2×C53#338011yescharacteristic
336C2×C53#338111yescharacteristic
337D53—44no—
338#554—538no—
339#554—1064no—
340C2×C4⋊Q8—531no—
341C2×#125—531no—
342C2×#125C2×D5311yescharacteristic
343C2×C22⋊Q8—1061no—
344C2×C22⋊Q8—1061no—
345C2×C22⋊Q8—1061no—
346C2×C4×Q8—531no—
347C2×C4×D4—531no—
348C2×C4×D4—531no—
349C2×C4×D4—531no—
350C22×C4⋊C4—1061no—
351C8×Q8—1064no—
352C2×C4⋊C8—531no—
353C2×C4⋊C8—531no—
354C2×C4⋊C8—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×#102—531no—
359C2×C4×C8—531no—
360C2×C4×C8—531no—
361C2×C4×C8C2×D5311yescharacteristic
362#391—1061no—
363#387—1061no—
364#387—1061no—
365#384—531no—
366#382—531no—
367#382C2×D5311yescharacteristic
368C4×C4⋊C4—531no—
369C4×C4⋊C4—531no—
370C4×C4⋊C4—531no—
371C4×C4⋊C4—1061no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340C2×D5311yescharacteristic
381#340C2×D5311yescharacteristic
382#340C2×D5311yescharacteristic
383#340C2×D5311yescharacteristic
384C53#2251911yescharacteristic, complemented, Sylow
385C22×Q8—531no—
386C22×Q8—1061no—
387C22×D4—531no—
388C23×C4—1061no—
389C23×C4—1061no—
390C23×C4—1061no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C4⋊Q8—534no—
396C4⋊Q8—1062no—
397#125—22no—
398#125—534no—
399#125—1062no—
400#125C4×D5314yes—
401C22⋊Q8—1068no—
402C22⋊Q8—1068no—
403C22⋊Q8—1068no—
404C4×Q8—534no—
405C4×Q8—534no—
406C4×D4—534no—
407C4×D4—534no—
408C4×D4—534no—
409C4×D4—1062no—
410C4×D4—1062no—
411C4×D4—1062no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—1068no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4C22×D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C22⋊C4—1061no—
431C2×C22⋊C4—1061no—
432C2×C22⋊C4—1061no—
433C2×C42—531no—
434C2×C42—531no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C22×D5311yescharacteristic
439C4⋊C8—1062no—
440C4⋊C8—1062no—
441C4⋊C8—1062no—
442#102—534no—
443#102—1062no—
444#102—1062no—
445#102—1062no—
446C4×C8—22no—
447C4×C8—1062no—
448C4×C8—1062no—
449#95—21no—
450#95—21no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456C24—1061no—
457C2×Q8—534no—
458C2×Q8—1062no—
459C2×Q8—1062no—
460C2×Q8—1064no—
461C2×Q8—4241no—
462C2×Q8—4241no—
463C2×D4—532no—
464C2×D4—532no—
465C2×D4—532no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—1064no—
478C22×C4—1064no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—1064no—
482C22×C4—1061no—
483C22×C4—4241no—
484C22×C4—4241no—
485C22×C4—4241no—
486C22×C4—4241no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4D4×D5311yescharacteristic
490C22×C4D4×D5311yescharacteristic
491C22×C4C23×D5311yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C2×C8—1061no—
502C2×C8—1061no—
503C2×C8Q8×D5312yes—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—534no—
513C4⋊C4—534no—
514C4⋊C4—534no—
515C4⋊C4—532no—
516C4⋊C4—532no—
517C4⋊C4—1064no—
518C4⋊C4—1064no—
519C4⋊C4—1064no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—4241no—
525C4⋊C4—2122no—
526C4⋊C4C2×C4×D5312yes—
527C4⋊C4C2×C4×D5312yes—
528C22⋊C4—1064no—
529C22⋊C4—1064no—
530C22⋊C4—1064no—
531C42—532no—
532C42—532no—
533C42—532no—
534C42—532no—
535C42—4241no—
536C42—4241no—
537C42—2122no—
538C42C2×C4×D5312yes—
539C42D4×D5312yes—
540C23—1062no—
541C23—1062no—
542C23—1062no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23C2×D4×D5311yescharacteristic
548Q8—1068no—
549Q8—4244no—
550D4—534no—
551D4—1062no—
552D4—1062no—
553D4—1062no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—532no—
571C2×C4—532no—
572C2×C4—532no—
573C2×C4—532no—
574C2×C4—532no—
575C2×C4—532no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—4244no—
585C2×C4—4244no—
586C2×C4—2128no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4C22⋊C4×D5312yes—
595C2×C4C22⋊C4×D5312yes—
596C2×C4C22×C4×D5311yescharacteristic
597C2×C4C2×D4×D5311yescharacteristic
598C2×C4C2×Q8×D5311yescharacteristic
599C2×C4#41×D5311yescharacteristic
600C2×C4#1313326411yescharacteristic
601C2×C4#1313326511yescharacteristic
602C8—22no—
603C8—41no—
604C8—2121no—
605C8—2121no—
606C22—1064no—
607C22—2122no—
608C22—2122no—
609C22—2122no—
610C22—2122no—
611C22—2122no—
612C22—2122no—
613C22C2×C22⋊C4×D5311yescharacteristic
614C22#1681128111yescharacteristic
615C22#1681128811yescharacteristic
616C22#135×D5311yescharacteristic
617C22#135×D5311yescharacteristic
618C22#1681142511yescharacteristic
619C22#1681143011yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—534no—
627C4—1062no—
628C4—1062no—
629C4—1062no—
630C4—4241no—
631C4—2122no—
632C4—2122no—
633C4C4×Q8×D5311yescharacteristic
634C4#1681109911yescharacteristic
635C4#1681128311yescharacteristic
636C4#1681128611yescharacteristic
637C2—2124no—
638C2#2027754911yescharacteristic
639C2#416×D5311yescharacteristic
640C2#2028323711yescharacteristic
641C2#2028736111yescharacteristic
642C2#2028737211yescharacteristic
643C2#2028757911yescharacteristic
644C2#2028758711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1991C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1991D53
hypocenterC53#22519
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic106C2semidirect
3. #20277667C2semidirect
4. #20275678C2semidirect
5. #20275674C2semidirect
6. #20275673C2semidirect
7. #20275671C2semidirect
8. #20275661C2semidirect
15. C53×#1991C2semidirect
17. #16811329C4semidirect
109. #13133139Q8semidirect
319. #1991D53semidirect
384. C53#22519semidirect

Maximal subgroups

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

subgroupindex
C2×#168113292
C2×C8×Dic1062
#202776672
#202756782
#202756742
#202756732
#202756712
#202756612
#202747242
#202747222
#202747142
C4×#131331392
#202745292
C53×#19912
#2251953