Finite Group Databasebeta

← #27557145

Subgroups — #27557145

Order 13568.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283025C212yesmaximal, complemented
2#20283002C212yesmaximal
3#20280829C211yesmaximal, characteristic, complemented
4#20280829C211yesmaximal, characteristic, complemented
5#20280520C212yesmaximal
6#20279989C214yesmaximal, complemented
7#20279799C214yesmaximal, complemented
8#20279433C214yesmaximal, complemented
9#20278736C214yesmaximal, complemented
10#20277448C212yesmaximal, complemented
11#20277313C212yesmaximal
12C4×#13133196C212yesmaximal, complemented
13C2×C4×C22⋊C4×C53C211yesmaximal, characteristic
14C2×#13133209C2212yescomplemented
15C2×#13133196C2212yes—
16C2×#13133196C2212yes—
17C2×#13133196C2212yes—
18C2×#13133196C2212yes—
19C2×C4×Dic106C2212yescomplemented
20#16810559C2214yes—
21#16810555C2214yescomplemented
22#16810555C2214yescomplemented
23#16810546C2214yes—
24#16810546C2214yes—
25#16810544C2214yescomplemented
26#16810544C2214yescomplemented
27#16810535C2214yes—
28#16810529C2212yescomplemented
29#16810527C2214yescomplemented
30#16810521C2212yescomplemented
31#16810521C2212yescomplemented
32#16810521C2212yescomplemented
33#16810520C2214yescomplemented
34#16810516C2214yes—
35#16810516C2214yes—
36#16810516C2214yes—
37#16810516C2214yes—
38#16810510C2214yescomplemented
39#16810507C2214yescomplemented
40#16810507C2214yescomplemented
41#16810507C2214yescomplemented
42#16810506C2214yescomplemented
43#16810496C2214yes—
44#16810496C2214yes—
45#16810496C2214yes—
46#16810496C2214yes—
47#16810489C2212yescomplemented
48#16810483C2212yes—
49#16810483C2212yes—
50#16810483C2212yes—
51#16810483C2212yes—
52#16810483C2212yes—
53#16810483C2212yes—
54#16810479C2214yes—
55C4×#11719919C2214yescomplemented
56C4×C4⋊Dic53C2212yescomplemented
57C4×C22⋊Dic53C2214yes—
58C24×C4×C53C2211yescharacteristic
59C22×C22⋊C4×C53C2211yescharacteristic
60C22×C22⋊C4×C53C2211yescharacteristic
61C22×C42×C53C2211yescharacteristic
62C22×C42×C53C2211yescharacteristic
63C4×C22⋊C4×C53C2214yes—
64C4×C22⋊C4×C53C2214yes—
65C2×#95×C53C2211yescharacteristic
66C2×#95×C53C2211yescharacteristic
67C22×Dic106C2312yes—
68C22×Dic106C2312yes—
69#13133209—44no—
70#13133196—28no—
71#13133196—28no—
72#13133196—44no—
73#13133196—44no—
74C4×Dic106—44no—
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2312yes—
77C2×C4⋊Dic53C2312yes—
78C2×C4⋊Dic53C2312yes—
79C2×C4⋊Dic53C2312yes—
80C2×C4⋊Dic53C2312yes—
81C2×C4⋊Dic53C2312yes—
82C2×C4⋊Dic53C2312yes—
83C2×C4⋊Dic53C2312yes—
84C2×C4⋊Dic53C2312yes—
85C2×#11719919C2314yes—
86C2×#11719919C2314yes—
87C2×#11719919C2314yes—
88C2×#11719919C2314yes—
89C2×#11719919C2314yes—
90C2×#11719919C2314yes—
91C2×C22⋊Dic53C2314yes—
92C2×C22⋊Dic53C2314yes—
93C2×C22⋊Dic53C2314yes—
94C2×C22⋊Dic53C2314yes—
95C2×C22⋊Dic53C2314yes—
96C2×C22⋊Dic53C2314yes—
97C2×C4×Dic53C2314yes—
98C2×C4×Dic53C2314yes—
99#13133124C2314yes—
100#13133124C2314yes—
101#13133124C2314yes—
102#13133124C2314yes—
103#13133124C2314yes—
104#13133124C2314yes—
105#13133124C2314yes—
106#13133124C2314yes—
107#13133124C2314yes—
108#13133124C2314yes—
109C25×C53C2311yescharacteristic
110C23×C4×C53—24no—
111C23×C4×C53—42no—
112C23×C4×C53—42no—
113C23×C4×C53C2311yescharacteristic
114C23×C4×C53C2312yes—
115C23×C4×C53C2312yes—
116C23×C4×C53C2311yescharacteristic
117C23×C4×C53C2311yescharacteristic
118C23×C4×C53C2311yescharacteristic
119C23×C4×C53C2311yescharacteristic
120C23×C4×C53C2311yescharacteristic
121C2×C22⋊C4×C53—24no—
122C2×C22⋊C4×C53Q818yescomplemented
123C2×C22⋊C4×C53C2314yes—
124C2×C22⋊C4×C53C2314yes—
125C2×C42×C53—42no—
126C2×C42×C53—42no—
127C2×C42×C53C2312yes—
128C2×C42×C53C2312yes—
129C2×C42×C53C2312yes—
130C2×C42×C53C2312yes—
131#95×C53C2314yes—
132#95×C53C2314yes—
133C24×C53—22no—
134C24×C53—22no—
135C24×C53—41no—
136C24×C53—41no—
137C24×C53—41no—
138C24×C53—41no—
139C24×C53C2×Q814yes—
140C24×C53C2412yes—
141C24×C53C2411yescharacteristic
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—24no—
147C22×C4×C53—24no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—41no—
151C22×C4×C53—41no—
152C22×C4×C53—41no—
153C22×C4×C53—41no—
154C22×C4×C53—41no—
155C22×C4×C53—41no—
156C22×C4×C53—42no—
157C22×C4×C53—42no—
158C22×C4×C53—41no—
159C22×C4×C53—41no—
160C22×C4×C53—41no—
161C22×C4×C53—41no—
162C22×C4×C53—416no—
163C22×C4×C53C2×Q814yes—
164C22×C4×C53C2×Q814yes—
165C22×C4×C53#4114yes—
166C22×C4×C53#4114yes—
167C22×C4×C53C2412yes—
168C22×C4×C53C2412yes—
169C22×C4×C53C2412yes—
170C22×C4×C53C2412yes—
171C22×C4×C53C2412yes—
172C22×C4×C53C2411yescharacteristic
173C22×C4×C53C2411yescharacteristic
174C22×Dic53C2414yes—
175C22×Dic53C2414yes—
176C22×Dic53C2414yes—
177C22×Dic53C2414yes—
178C2×Dic106—24no—
179C2×Dic106—24no—
180C2×Dic106—42no—
181C2×Dic106—42no—
182C2×Dic106—42no—
183C2×Dic106—42no—
184C22⋊C4×C53—28no—
185C22⋊C4×C53C2×Q8116yes—
186C42×C53—24no—
187C42×C53—24no—
188C42×C53—42no—
189C42×C53—42no—
190C22⋊Dic53—28no—
191C22⋊Dic53—28no—
192C22⋊Dic53—44no—
193C22⋊Dic53—44no—
194C22⋊Dic53—44no—
195C22⋊Dic53—44no—
196C4⋊Dic53—24no—
197C4⋊Dic53—24no—
198C4⋊Dic53—42no—
199C4⋊Dic53—42no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53—42no—
205C4⋊Dic53—42no—
206#11719919—28no—
207#11719919—28no—
208#11719919—28no—
209#11719919—28no—
210#11719919—44no—
211#11719919—44no—
212C4×Dic53—44no—
213C4×Dic53—44no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—28no—
219C23×C53—22no—
220C23×C53—22no—
221C23×C53—22no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—44no—
226C23×C53—44no—
227C23×C53—44no—
228C23×C53—44no—
229C23×C53—44no—
230C23×C53—44no—
231C23×C53C22×Q814yes—
232C23×C53C22×Q812yes—
233C23×C53C2×#4112yes—
234C23×C53C2×#4112yes—
235C23×C53C2511yescharacteristic
236C2×C4×C53—28no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—44no—
253C2×C4×C53—44no—
254C2×C4×C53—44no—
255C2×C4×C53—44no—
256C2×C4×C53—48no—
257C2×C4×C53—48no—
258C2×C4×C53—44no—
259C2×C4×C53C22×Q814yes—
260C2×C4×C53C22×Q814yes—
261C2×C4×C53C2×#4114yes—
262C2×C4×C53C2×#4114yes—
263C2×C4×C53H(5,𝔽2)12yes—
264C2×C4×C53H(5,𝔽2)12yes—
265C2×Dic53—24no—
266C2×Dic53—24no—
267C2×Dic53—24no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—24no—
274C2×Dic53—24no—
275C2×Dic53—24no—
276C2×Dic53—24no—
277Dic106—48no—
278Dic106—84no—
279#35012—531nomaximal, Sylow
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—24no—
287C22×C53—24no—
288C22×C53—24no—
289C22×C53—416no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53#53314yes—
295C22×C53C23×Q811yescharacteristic
296C22×C53C22×#4111yescharacteristic
297C22×C53C22×#4111yescharacteristic
298C22×C53C2×H(5,𝔽2)11yescharacteristic
299C22×C53C2×H(5,𝔽2)11yescharacteristic
300C22×C53#58511yescharacteristic
301C22×C53#58511yescharacteristic
302C4×C53—24no—
303C4×C53—24no—
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—24no—
307C4×C53—24no—
308C4×C53—42no—
309Dic53—44no—
310Dic53—44no—
311Dic53—44no—
312Dic53—44no—
313#2851—532no—
314#2848—532no—
315#2659—531no—
316#2659—531no—
317#2626—532no—
318#2574—534no—
319#2556—534no—
320#2522—534no—
321#2465—534no—
322#2329—532no—
323#2312—532no—
324C4×C22⋊Q8—532no—
325C2×C4×C22⋊C4D5311yescharacteristic
326C2×C53—24no—
327C2×C53—44no—
328C2×C53C2×#53311yescharacteristic
329C2×C53#345911yescharacteristic
330C2×C53#346011yescharacteristic
331C2×C53#349011yescharacteristic
332C2×C53#349311yescharacteristic
333C2×C53#350111yescharacteristic
334C2×C53#350111yescharacteristic
335C24×C4C2×D5311yescharacteristic
336C2×#125—532no—
337C2×C22⋊Q8—532no—
338C2×C22⋊Q8—532no—
339C2×C22⋊Q8—532no—
340C2×C22⋊Q8—532no—
341C2×C4×Q8—532no—
342C22×C22⋊C4C2×D5311yescharacteristic
343C22×C22⋊C4C2×D5311yescharacteristic
344C22×C42C2×D5311yescharacteristic
345C22×C42C2×D5311yescharacteristic
346#399—534no—
347#398—534no—
348#398—534no—
349#396—534no—
350#396—534no—
351#395—534no—
352#395—534no—
353#393—534no—
354#391—532no—
355#391—534no—
356#389—532no—
357#389—532no—
358#389—532no—
359#389—534no—
360#388—534no—
361#388—534no—
362#388—534no—
363#388—534no—
364#387—534no—
365#387—534no—
366#387—534no—
367#387—534no—
368#387—534no—
369#385—534no—
370#385—534no—
371#385—534no—
372#385—534no—
373#382—532no—
374#380—532no—
375#380—532no—
376#380—532no—
377#380—532no—
378#380—532no—
379#380—532no—
380#379—534no—
381C4×C4⋊C4—534no—
382C4×C4⋊C4—532no—
383C4×C22⋊C4—534no—
384C4×C22⋊C4C2×D5314yes—
385C4×C22⋊C4C2×D5314yes—
386C2×#95C2×D5311yescharacteristic
387C2×#95C2×D5311yescharacteristic
388C53#3501211yescharacteristic, complemented, Sylow
389C25C22×D5311yescharacteristic
390C22×Q8—532no—
391C22×Q8—532no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4C22×D5311yescharacteristic
396C23×C4C22×D5312yes—
397C23×C4C22×D5312yes—
398C23×C4C22×D5311yescharacteristic
399C23×C4C22×D5311yescharacteristic
400C23×C4C22×D5311yescharacteristic
401C23×C4C22×D5311yescharacteristic
402C23×C4C22×D5311yescharacteristic
403#125—2124no—
404C22⋊Q8—1068no—
405C22⋊Q8—1068no—
406C22⋊Q8—2124no—
407C22⋊Q8—2124no—
408C4×Q8—2124no—
409C2×C4⋊C4—534no—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—534no—
412C2×C4⋊C4—534no—
413C2×C4⋊C4—532no—
414C2×C4⋊C4—532no—
415C2×C4⋊C4—532no—
416C2×C4⋊C4—532no—
417C2×C4⋊C4—532no—
418C2×C4⋊C4—534no—
419C2×C4⋊C4—532no—
420C2×C4⋊C4—532no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—532no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—534no—
427C2×C22⋊C4—534no—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—534no—
430C2×C22⋊C4—534no—
431C2×C22⋊C4—534no—
432C2×C22⋊C4Dic10618yescomplemented
433C2×C22⋊C4C22×D5314yes—
434C2×C22⋊C4C22×D5314yes—
435C2×C42—42no—
436C2×C42—42no—
437C2×C42—534no—
438C2×C42—534no—
439C2×C42C22×D5312yes—
440C2×C42C22×D5312yes—
441C2×C42C22×D5312yes—
442C2×C42C22×D5312yes—
443#95—534no—
444#95—534no—
445#95—534no—
446#95—534no—
447#95—534no—
448#95—534no—
449#95—534no—
450#95—534no—
451#95—534no—
452#95—534no—
453#95C22×D5314yes—
454#95C22×D5314yes—
455C24—22no—
456C24—22no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24C2×Dic10614yes—
462C24C23×D5312yes—
463C24C23×D5311yescharacteristic
464C2×Q8—1064no—
465C2×Q8—1064no—
466C2×Q8—2122no—
467C2×Q8—2122no—
468C2×Q8—2122no—
469C2×Q8—2122no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—416no—
491C22×C4—534no—
492C22×C4—534no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4C2×Dic10614yes—
496C22×C4C2×Dic10614yes—
497C22×C4#1171994514yes—
498C22×C4#1171994514yes—
499C22×C4C23×D5312yes—
500C22×C4C23×D5312yes—
501C22×C4C23×D5312yes—
502C22×C4C23×D5312yes—
503C22×C4C23×D5312yes—
504C22×C4C23×D5311yescharacteristic
505C22×C4C23×D5311yescharacteristic
506C4⋊C4—1068no—
507C4⋊C4—1068no—
508C4⋊C4—1064no—
509C4⋊C4—1064no—
510C4⋊C4—1068no—
511C4⋊C4—1068no—
512C4⋊C4—2122no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2124no—
516C4⋊C4—2122no—
517C4⋊C4—2122no—
518C4⋊C4—2124no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C22⋊C4—28no—
523C22⋊C4—1068no—
524C22⋊C4—1068no—
525C22⋊C4—2124no—
526C22⋊C4—2124no—
527C22⋊C4—2124no—
528C22⋊C4—2124no—
529C22⋊C4C2×Dic106116yes—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—2124no—
535C42—2124no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C22×Dic10614yes—
554C23C22×Dic10612yes—
555C23C2×#1171994512yes—
556C23C2×#1171994512yes—
557C23C24×D5311yescharacteristic
558Q8—4244no—
559Q8—2128no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4C22×Dic10614yes—
596C2×C4C22×Dic10614yes—
597C2×C4C2×#1171994514yes—
598C2×C4C2×#1171994514yes—
599C2×C4#1313326212yes—
600C2×C4#1313326212yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1681118014yes—
616C22C23×Dic10611yescharacteristic
617C22C22×#1171994511yescharacteristic
618C22C22×#1171994511yescharacteristic
619C22C2×#1313326211yescharacteristic
620C22C2×#1313326211yescharacteristic
621C22#1681145911yescharacteristic
622C22#1681145911yescharacteristic
623C4—24no—
624C4—24no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—24no—
629C4—42no—
630C4—2124no—
631C4—2124no—
632C4—2124no—
633C4—2124no—
634C2—24no—
635C2—44no—
636C2C2×#1681118011yescharacteristic
637C2#2028812411yescharacteristic
638C2#2028813711yescharacteristic
639C2#2028840411yescharacteristic
640C2#2028843511yescharacteristic
641C2#2028851811yescharacteristic
642C2#2028851811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D53
commutator, socle, cosocleC23×C53C25
FittingC2×C4×C22⋊C4×C53C2
prime radicalC25×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#35012
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20283025C2semidirect
3. #20280829C2semidirect
4. #20280829C2semidirect
6. #20279989C2semidirect
7. #20279799C2semidirect
8. #20279433C2semidirect
9. #20278736C2semidirect
10. #20277448C2semidirect
12. C4×#13133196C2semidirect
14. C2×#13133209C22semidirect
19. C2×C4×Dic106C22semidirect
21. #16810555C22semidirect
22. #16810555C22semidirect
25. #16810544C22semidirect
26. #16810544C22semidirect
28. #16810529C22semidirect
29. #16810527C22semidirect
30. #16810521C22semidirect
31. #16810521C22semidirect
32. #16810521C22semidirect
33. #16810520C22semidirect
38. #16810510C22semidirect
39. #16810507C22semidirect
40. #16810507C22semidirect
41. #16810507C22semidirect
42. #16810506C22semidirect
47. #16810489C22semidirect
55. C4×#11719919C22semidirect
56. C4×C4⋊Dic53C22semidirect
122. C2×C22⋊C4×C53Q8semidirect
388. C53#35012semidirect
432. C2×C22⋊C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202830252
#202830022
#202808292
#202805202
#202799892
#202797992
#202794332
#202787362
#202774482
#202773132
C4×#131331962
C2×C4×C22⋊C4×C532
#3501253