Finite Group Databasebeta

← #27542691

Subgroups — #27542691

Order 13568.

2390 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D53C211yesmaximal, characteristic
2#20277325C211yesmaximal, characteristic, complemented
3#20277324C211yesmaximal, characteristic, complemented
4#20277323C211yesmaximal, characteristic, complemented
5#20277322C212yesmaximal, complemented
6#20277320C212yesmaximal, complemented
7#20277314C214yesmaximal, complemented
8#20277309C214yesmaximal
9#20277209C214yesmaximal, complemented
10#20277207C214yesmaximal, complemented
11#20277121C211yesmaximal, characteristic, complemented
12#20277120C211yesmaximal, characteristic, complemented
13#20277097C212yesmaximal
14C4×#13133175C212yesmaximal, complemented
15C53×#2314C211yesmaximal, characteristic, complemented
16C23×C4×D53C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C2×#13133175C2212yes—
19C2×#13133175C2212yes—
20C2×#13133175C2212yes—
21C2×#13133174C2211yescharacteristic
22C2×#13133174C2211yescharacteristic
23C2×#13133173C2212yes—
24#117×D53C4116yescomplemented
25C2×C4⋊C4×D53C2212yes—
26C2×C22⋊C4×D53C2212yes—
27C2×C42×D53C2211yescharacteristic
28C2×C42×D53C2211yescharacteristic
29#16810517C2214yes—
30#16810516C2214yes—
31#16810515C2214yes—
32#16810514C2214yes—
33#16810513C2214yes—
34#16810512C2214yes—
35#16810511C2214yes—
36#16810510C2214yescomplemented
37#16810509C2214yescomplemented
38#16810508C2214yescomplemented
39#16810507C2214yescomplemented
40#16810506C2214yescomplemented
41#16810505C2214yescomplemented
42#16810504C2214yescomplemented
43#16810493C2211yescharacteristic, complemented
44#16810492C2212yescomplemented
45#16810489C2211yescharacteristic, complemented
46#16810488C2212yescomplemented
47#16810485C2214yes—
48#16810483C2212yes—
49#16810482C2214yes—
50#16810480C2214yes—
51#16810480C2212yes—
52#16810480C2212yes—
53#16810478C2214yes—
54C4⋊C4×Dic53C2212yescomplemented
55C4⋊C4×Dic53C2212yescomplemented
56C4⋊C4×Dic53C2214yescomplemented
57C4×C4⋊Dic53C2212yescomplemented
58#16810474C2214yescomplemented
59C22⋊C4×Dic53C2214yes—
60C4×#11719921C2214yes—
61#16810470C2214yes—
62#16810470C2214yes—
63#16810468C2214yescomplemented
64C42×Dic53C2214yescomplemented
65C2×#117×C53C2211yescharacteristic
66C53×#388C2214yes—
67C53×#387C2214yescomplemented
68C53×#384C2211yescharacteristic, complemented
69C53×#382C2211yescharacteristic, complemented
70C53×#380C2212yes—
71C4×C4⋊C4×C53C2212yescomplemented
72C24×D53C2311yescharacteristic
73C23×Dic53C2311yescharacteristic
74C22×C4×D53—24no—
75C22×C4×D53—24no—
76C22×C4×D53C2×C418yes—
77C22×C4×D53C2311yescharacteristic
78C22×C4×D53C2311yescharacteristic
79C22×C4×D53C2311yescharacteristic
80C22×C4×D53C2312yes—
81C22×C4×D53C2312yes—
82C22×C4×D53C2311yescharacteristic
83#13133176C2×C418yes—
84#13133175—28no—
85#13133175—28no—
86#13133175C2×C4116yes—
87#13133174C2×C418yes—
88#13133174C2×C418yes—
89#13133173C2×C4116yes—
90C2×C4⋊Dic53C2312yes—
91C2×C4⋊Dic53C2312yes—
92C2×C4⋊Dic53C2314yes—
93C2×C4⋊Dic53C2312yes—
94C2×#11719919C2314yes—
95C2×#11719919C2314yes—
96C2×#11719919C2312yes—
97C2×#11719919C2312yes—
98C4⋊C4×D53C2×C4116yes—
99C2×C22⋊Dic53C2314yes—
100C2×C22⋊Dic53C2312yes—
101C2×#11719921C2314yes—
102C2×#11719921C2314yes—
103C2×#11719921C2314yes—
104C2×#11719921C2314yes—
105C2×#11719921C2312yes—
106C2×#11719921C2312yes—
107C22⋊C4×D53C2×C4116yes—
108C2×C4×Dic53C2314yes—
109C2×C4×Dic53C2314yes—
110C2×C4×Dic53C2314yes—
111C2×C4×Dic53C2314yes—
112C2×C4×Dic53C2314yes—
113C2×C4×Dic53C2312yes—
114C2×C4×Dic53C2312yes—
115C2×C4×Dic53C2311yescharacteristic
116C2×C4×Dic53C2311yescharacteristic
117C42×D53C2×C418yes—
118C42×D53C2×C418yes—
119#13133124C2314yes—
120#13133124C2314yes—
121#13133124C2314yes—
122#13133124C2314yes—
123#13133124C2314yes—
124#13133124C2314yes—
125#13133124C2314yes—
126#13133124C2314yes—
127#13133124C2314yes—
128C23×C4×C53C2311yescharacteristic
129#117×C53C2×C418yes—
130C2×C4⋊C4×C53C2312yes—
131C2×C4⋊C4×C53C2312yes—
132C2×C4⋊C4×C53C2314yes—
133C2×C4⋊C4×C53C2312yes—
134C2×C22⋊C4×C53C2314yes—
135C2×C22⋊C4×C53C2312yes—
136C2×C42×C53C2314yes—
137C2×C42×C53C2311yescharacteristic
138C2×C42×C53C2311yescharacteristic
139#95×C53C2314yes—
140#95×C53C2314yes—
141#95×C53C2314yes—
142C24×C53C2411yescharacteristic
143C23×D53—22no—
144C23×D53—22no—
145C23×D53—42no—
146C23×D53—42no—
147C23×D53C22×C414yes—
148C23×D53C2411yescharacteristic
149C23×D53C2411yescharacteristic
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53C22×C414yes—
153C22×C4×C53C2414yes—
154C22×C4×C53C2414yes—
155C22×C4×C53C2411yescharacteristic
156C22×C4×C53C2412yes—
157C22×C4×C53C2412yes—
158C22×C4×C53C2411yescharacteristic
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53C22×C414yes—
162C22×Dic53C2414yes—
163C22×Dic53C2414yes—
164C22×Dic53C2411yescharacteristic
165C22×Dic53C2412yes—
166C22×Dic53C2412yes—
167C22×Dic53C2411yescharacteristic
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—22no—
171C2×C4×D53—22no—
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—232no—
175C2×C4×D53—42no—
176C2×C4×D53—42no—
177C2×C4×D53—42no—
178C2×C4×D53—42no—
179C2×C4×D53C22×C414yes—
180C2×C4×D53C22×C414yes—
181C2×C4×D53C22×C414yes—
182C2×C4×D53C22×C418yes—
183C2×C4×D53C22×C418yes—
184C2×C4×D53C22×C414yes—
185C2×C4×D53#4114yes—
186C2×C4×D53#4114yes—
187C4⋊C4×C53—24no—
188C4⋊C4×C53—24no—
189C4⋊C4×C53—44no—
190C4⋊C4×C53C22×C418yes—
191C22⋊C4×C53—44no—
192C22⋊C4×C53C22×C418yes—
193C42×C53—44no—
194C42×C53C22×C414yes—
195C42×C53C22×C414yes—
196C22⋊Dic53—44no—
197C22⋊Dic53C22×C418yes—
198#11719921—28no—
199#11719921—28no—
200#11719921—44no—
201#11719921—44no—
202#11719921C22×C418yes—
203#11719921C22×C418yes—
204C4⋊Dic53—24no—
205C4⋊Dic53—24no—
206C4⋊Dic53—44no—
207C4⋊Dic53C22×C418yes—
208#11719919—44no—
209#11719919—44no—
210#11719919C22×C418yes—
211#11719919C22×C418yes—
212C4×Dic53—28no—
213C4×Dic53—28no—
214C4×Dic53—44no—
215C4×Dic53—44no—
216C4×Dic53—44no—
217C4×Dic53C22×C418yes—
218C4×Dic53C22×C418yes—
219C4×Dic53C22×C414yes—
220C4×Dic53C22×C414yes—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53C23×C412yes—
226C23×C53C2511yescharacteristic
227C22×D53—22no—
228C22×D53—22no—
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—28no—
234C22×D53—48no—
235C22×D53—44no—
236C22×D53—48no—
237C22×D53—44no—
238C22×D53C23×C412yes—
239C22×D53C23×C412yes—
240C22×D53C2×#4112yes—
241C22×D53C2×#4112yes—
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—28no—
249C2×C4×C53—24no—
250C2×C4×C53—24no—
251C2×C4×C53—24no—
252C2×C4×C53—24no—
253C2×C4×C53—24no—
254C2×C4×C53—24no—
255C2×C4×C53C23×C412yes—
256C2×C4×C53C23×C414yes—
257C2×C4×C53C23×C414yes—
258C2×C4×C53C23×C412yes—
259C2×C4×C53C2×#4112yes—
260C2×C4×C53C2×#4112yes—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—28no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—24no—
271C2×Dic53—24no—
272C2×Dic53—24no—
273C2×Dic53—24no—
274C2×Dic53C23×C412yes—
275C2×Dic53C23×C414yes—
276C2×Dic53C23×C414yes—
277C2×Dic53C23×C412yes—
278C2×Dic53C2×#4112yes—
279C2×Dic53C2×#4112yes—
280C4×D53—28no—
281C4×D53—28no—
282C4×D53—28no—
283C4×D53—48no—
284C4×D53—48no—
285C4×D53#117116yes—
286#34048—531nomaximal, Sylow
287C22×C53—22no—
288C22×C53—24no—
289C22×C53—41no—
290C22×C53—42no—
291C22×C53—41no—
292C22×C53C24×C411yescharacteristic
293C22×C53C22×#4111yescharacteristic
294C22×C53C22×#4111yescharacteristic
295C22×C53#58512yes—
296C22×C53#58512yes—
297C2×D53—24no—
298C2×D53—22no—
299C2×D53—24no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—24no—
303C2×D53—416no—
304C2×D53—41no—
305C2×D53—42no—
306C2×D53—41no—
307C2×D53C2×#11714yes—
308C4×C53—22no—
309C4×C53—24no—
310C4×C53—24no—
311C4×C53—44no—
312C4×C53—44no—
313C4×C53C2×#11714yes—
314Dic53—22no—
315Dic53—24no—
316Dic53—24no—
317Dic53—44no—
318Dic53—44no—
319Dic53C2×#11714yes—
320C22×#117—531no—
321#2314—531no—
322#2314—531no—
323#2314—532no—
324#2314—532no—
325#2314—531no—
326#2314D5311yescharacteristic, complemented
327#2313—534no—
328#2312—534no—
329#2298—534no—
330#2298—534no—
331#2285—531no—
332#2285—531no—
333#2280—532no—
334C4×#117—532no—
335C2×C53—42no—
336C2×C53C22×#11711yescharacteristic
337C2×C53#342812yes—
338C2×C53#345511yescharacteristic
339C2×C53#345512yes—
340C2×C53#345511yescharacteristic
341D53—24no—
342D53—42no—
343C24×C4—531no—
344C2×#117—531no—
345C2×#117—5316no—
346C2×#117—532no—
347C2×#117—532no—
348C2×#117—532no—
349C2×#117—532no—
350C2×#117—531no—
351C2×#117—531no—
352C2×#117C2×D5311yescharacteristic
353C22×C4⋊C4—532no—
354C22×C22⋊C4—532no—
355C22×C42—531no—
356C22×C42—531no—
357#388—534no—
358#388—534no—
359#388—534no—
360#388—534no—
361#388—534no—
362#388—534no—
363#388—534no—
364#388C2×D5314yes—
365#387—534no—
366#387—534no—
367#387—534no—
368#387—534no—
369#387—534no—
370#387—534no—
371#387—534no—
372#387C2×D5314yescomplemented
373#384—532no—
374#384—531no—
375#384C2×D5311yescharacteristic, complemented
376#382—532no—
377#382—531no—
378#382C2×D5311yescharacteristic, complemented
379#381—534no—
380#380—532no—
381#380—534no—
382#380—532no—
383#380—532no—
384#380—534no—
385#380C2×D5312yes—
386#379—534no—
387C4×C4⋊C4—532no—
388C4×C4⋊C4—532no—
389C4×C4⋊C4—534no—
390C4×C4⋊C4—532no—
391C4×C4⋊C4—534no—
392C4×C4⋊C4C2×D5312yescomplemented
393C4×C22⋊C4—534no—
394C4×C22⋊C4—534no—
395C4×C22⋊C4—534no—
396C4×C22⋊C4—534no—
397#376—534no—
398C43—534no—
399C53#3404811yescharacteristic, complemented, Sylow
400C25—531no—
401C23×C4—531no—
402C23×C4—531no—
403C23×C4—538no—
404C23×C4—531no—
405C23×C4—531no—
406C23×C4—531no—
407C23×C4—532no—
408C23×C4—532no—
409C23×C4—1064no—
410C23×C4—1064no—
411C23×C4C22×D5311yescharacteristic
412#117—538no—
413#117—5316no—
414#117—5316no—
415#117—538no—
416#117—538no—
417#117—1068no—
418#117—1068no—
419#117C4×D5318yes—
420C2×C4⋊C4—532no—
421C2×C4⋊C4—532no—
422C2×C4⋊C4—532no—
423C2×C4⋊C4—5316no—
424C2×C4⋊C4—532no—
425C2×C4⋊C4—532no—
426C2×C4⋊C4—534no—
427C2×C4⋊C4—534no—
428C2×C4⋊C4—534no—
429C2×C4⋊C4C22×D5312yes—
430C2×C4⋊C4C22×D5312yes—
431C2×C4⋊C4C22×D5314yes—
432C2×C4⋊C4C22×D5312yes—
433C2×C22⋊C4—532no—
434C2×C22⋊C4—532no—
435C2×C22⋊C4—532no—
436C2×C22⋊C4—5316no—
437C2×C22⋊C4—534no—
438C2×C22⋊C4—534no—
439C2×C22⋊C4—534no—
440C2×C22⋊C4—534no—
441C2×C22⋊C4—534no—
442C2×C22⋊C4C22×D5314yes—
443C2×C22⋊C4C22×D5312yes—
444C2×C42—532no—
445C2×C42—532no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—538no—
449C2×C42—538no—
450C2×C42—534no—
451C2×C42—534no—
452C2×C42—534no—
453C2×C42—534no—
454C2×C42—534no—
455C2×C42C22×D5314yes—
456C2×C42C22×D5311yescharacteristic
457C2×C42C22×D5311yescharacteristic
458#95—534no—
459#95—534no—
460#95—534no—
461#95—534no—
462#95—534no—
463#95—534no—
464#95—534no—
465#95—534no—
466#95—534no—
467#95C22×D5314yes—
468#95C22×D5314yes—
469#95C22×D5314yes—
470C24—531no—
471C24—531no—
472C24—534no—
473C24—1062no—
474C24—1062no—
475C24—2122no—
476C24—2122no—
477C24C23×D5311yescharacteristic
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—531no—
481C22×C4—534no—
482C22×C4—534no—
483C22×C4—534no—
484C22×C4—534no—
485C22×C4—531no—
486C22×C4—532no—
487C22×C4—532no—
488C22×C4—534no—
489C22×C4—534no—
490C22×C4—534no—
491C22×C4—538no—
492C22×C4—538no—
493C22×C4—534no—
494C22×C4—534no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—10632no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1062no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4—2122no—
508C22×C4C2×C4×D5314yes—
509C22×C4C23×D5314yes—
510C22×C4C23×D5314yes—
511C22×C4C23×D5311yescharacteristic
512C22×C4C23×D5312yes—
513C22×C4C23×D5312yes—
514C22×C4C23×D5311yescharacteristic
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—538no—
519C4⋊C4—538no—
520C4⋊C4—538no—
521C4⋊C4—1064no—
522C4⋊C4—1064no—
523C4⋊C4—2124no—
524C4⋊C4—2124no—
525C4⋊C4—2124no—
526C4⋊C4C2×C4×D5318yes—
527C22⋊C4—44no—
528C22⋊C4—538no—
529C22⋊C4—538no—
530C22⋊C4—538no—
531C22⋊C4—1068no—
532C22⋊C4—1068no—
533C22⋊C4—2124no—
534C22⋊C4—2124no—
535C22⋊C4—2124no—
536C22⋊C4C2×C4×D5318yes—
537C42—44no—
538C42—538no—
539C42—538no—
540C42—534no—
541C42—534no—
542C42—1068no—
543C42—1068no—
544C42—2124no—
545C42—2124no—
546C42—2124no—
547C42C2×C4×D5314yes—
548C42C2×C4×D5314yes—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—532no—
554C23—532no—
555C23—532no—
556C23—532no—
557C23—1068no—
558C23—1062no—
559C23—1062no—
560C23—1061no—
561C23—1061no—
562C23—1062no—
563C23—1062no—
564C23—2128no—
565C23—2124no—
566C23—2128no—
567C23—2124no—
568C23C22×C4×D5312yes—
569C23C24×D5311yescharacteristic
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—532no—
584C2×C4—532no—
585C2×C4—532no—
586C2×C4—5316no—
587C2×C4—534no—
588C2×C4—534no—
589C2×C4—532no—
590C2×C4—1068no—
591C2×C4—1068no—
592C2×C4—1068no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1068no—
600C2×C4—1064no—
601C2×C4—1064no—
602C2×C4—1064no—
603C2×C4—1064no—
604C2×C4—1064no—
605C2×C4—1064no—
606C2×C4—2128no—
607C2×C4—2128no—
608C2×C4C22×C4×D5314yes—
609C2×C4C22×C4×D5314yes—
610C2×C4C22×C4×D5312yes—
611C2×C4C22×C4×D5312yes—
612C2×C4#41×D5312yes—
613C2×C4#41×D5312yes—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—534no—
620C22—1062no—
621C22—1064no—
622C22—1062no—
623C22—1064no—
624C22—1062no—
625C22—1064no—
626C22—21216no—
627C22—2121no—
628C22—2122no—
629C22—2121no—
630C22C23×C4×D5311yescharacteristic
631C22C2×#41×D5311yescharacteristic
632C22C2×#41×D5311yescharacteristic
633C22#1681145812yes—
634C22#1681146012yes—
635C4—22no—
636C4—24no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—534no—
641C4—1062no—
642C4—1064no—
643C4—1064no—
644C4—2124no—
645C4—2124no—
646C4#117×D5314yes—
647C2—42no—
648C2—1064no—
649C2—2122no—
650C2C2×#117×D5311yescharacteristic
651C2#2028791512yes—
652C2#2028809911yescharacteristic
653C2#2028810212yes—
654C2#2028810611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingC53×#2314C2
socle, cosocleC23×C53C25
prime radicalC24×D53C23
hypercenter#2314D53
hypocenterC53#34048
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #20277325C2semidirect
3. #20277324C2semidirect
4. #20277323C2semidirect
5. #20277322C2semidirect
6. #20277320C2semidirect
7. #20277314C2semidirect
9. #20277209C2semidirect
10. #20277207C2semidirect
11. #20277121C2semidirect
12. #20277120C2semidirect
14. C4×#13133175C2semidirect
15. C53×#2314C2semidirect
24. #117×D53C4semidirect
36. #16810510C22semidirect
37. #16810509C22semidirect
38. #16810508C22semidirect
39. #16810507C22semidirect
40. #16810506C22semidirect
41. #16810505C22semidirect
42. #16810504C22semidirect
43. #16810493C22semidirect
44. #16810492C22semidirect
45. #16810489C22semidirect
46. #16810488C22semidirect
54. C4⋊C4×Dic53C22semidirect
55. C4⋊C4×Dic53C22semidirect
56. C4⋊C4×Dic53C22semidirect
57. C4×C4⋊Dic53C22semidirect
58. #16810474C22semidirect
63. #16810468C22semidirect
64. C42×Dic53C22semidirect
67. C53×#387C22semidirect
68. C53×#384C22semidirect
69. C53×#382C22semidirect
71. C4×C4⋊C4×C53C22semidirect
326. #2314D53semidirect
372. #387C2×D53semidirect
375. #384C2×D53semidirect
378. #382C2×D53semidirect
392. C4×C4⋊C4C2×D53semidirect
399. C53#34048semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D532
#202773252
#202773242
#202773232
#202773222
#202773202
#202773142
#202773092
#202772092
#202772072
#202771212
#202771202
#202770972
C4×#131331752
C53×#23142
#3404853