Finite Group Databasebeta

← #27311105

Subgroups — #27311105

Order 13568.

1590 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810559C211yesmaximal, characteristic
2C2×#16810521C211yesmaximal, characteristic, complemented
3C2×#16810518C212yesmaximal, complemented
4C2×#16810517C212yesmaximal
5#20272984C211yesmaximal, characteristic, complemented
6#20272981C212yesmaximal, complemented
7#20272979C211yesmaximal, characteristic, complemented
8#20272960C211yesmaximal, characteristic
9#20272958C212yesmaximal
10C4×#13133124C211yesmaximal, characteristic, complemented
11C2×C53×#388C211yesmaximal, characteristic
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×C22⋊Dic53C2212yes—
18C22×C22⋊Dic53C2212yes—
19C22×C4×Dic53C2212yes—
20#16810559C418yescomplemented
21#16810521—24no—
22#16810518—28no—
23#16810517—28no—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C22⋊C4×C53C2211yescharacteristic
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C42×C53C2211yescharacteristic
37C53×#388—24no—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2211yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2312yes—
43C23×Dic53C2312yes—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53—42no—
48C2×C4⋊Dic53C2×C414yes—
49C2×#11719919—24no—
50C2×#11719919—24no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919—42no—
56C2×#11719919—42no—
57C2×#11719919C2×C418yes—
58C2×C22⋊Dic53—24no—
59C2×C22⋊Dic53—24no—
60C2×C22⋊Dic53—24no—
61C2×C22⋊Dic53—42no—
62C2×C22⋊Dic53—42no—
63C2×C22⋊Dic53C2×C418yes—
64C2×C4×Dic53—42no—
65C2×C4×Dic53—42no—
66C2×C4×Dic53D418yes—
67#13133124—24no—
68#13133124—22no—
69#13133124—24no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—24no—
74#13133124C2×C414yes—
75C25×C53C2311yescharacteristic
76C23×C4×C53C2311yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C23×C4×C53C2311yescharacteristic
81C23×C4×C53C2311yescharacteristic
82C2×C4⋊C4×C53—24no—
83C2×C4⋊C4×C53—22no—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—24no—
86C2×C22⋊C4×C53—22no—
87C2×C22⋊C4×C53—22no—
88C2×C22⋊C4×C53—22no—
89C2×C22⋊C4×C53—42no—
90C2×C22⋊C4×C53C2×C414yes—
91C2×C42×C53—42no—
92C2×C42×C53D414yes—
93#95×C53—22no—
94#95×C53—22no—
95C24×C53—21no—
96C24×C53—22no—
97C24×C53—21no—
98C24×C53—22no—
99C24×C53—22no—
100C24×C53—21no—
101C24×C53—21no—
102C24×C53—21no—
103C24×C53—22no—
104C24×C53—21no—
105C24×C53C22×C412yes—
106C24×C53C2411yescharacteristic
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—22no—
129C22×C4×C53—21no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D414yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53#4112yes—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53C22×C414yes—
161C22×Dic53C22×C414yes—
162C22×Dic53C2×D414yes—
163C22×Dic53C2×D414yes—
164C22×Dic53#4114yes—
165C4⋊C4×C53—44no—
166C22⋊C4×C53—28no—
167C22⋊C4×C53—44no—
168C22⋊C4×C53—44no—
169C42×C53—44no—
170C22⋊Dic53—216no—
171C22⋊Dic53—48no—
172C4⋊Dic53—28no—
173C4⋊Dic53—44no—
174#11719919—216no—
175#11719919—48no—
176#11719919—48no—
177C4×Dic53—48no—
178C23×C53—24no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—24no—
182C23×C53—22no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—41no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—41no—
199C23×C53—42no—
200C23×C53—41no—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D412yes—
203C23×C53C22×D411yescharacteristic
204C23×C53C2×#4112yes—
205C23×C53C2×#4111yescharacteristic
206C23×C53H(5,𝔽2)12yes—
207C23×C53H(5,𝔽2)11yescharacteristic
208C23×C532−1+411yescharacteristic
209C23×C532−1+411yescharacteristic
210C23×C532−1+412yes—
211C23×C532−1+411yescharacteristic
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—22no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—42no—
239C2×C4×C53—41no—
240C2×C4×C53C4×D414yes—
241C2×C4×C53#12714yes—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—24no—
248C2×Dic53—24no—
249C2×Dic53—24no—
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—24no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—42no—
266C2×Dic53—42no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53C4×D418yes—
270#14322—531nomaximal, Sylow
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—44no—
274C22×C53—44no—
275C22×C53—42no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—82no—
280C22×C53—81no—
281C22×C53—82no—
282C22×C53C2×C4×D412yes—
283C22×C53C2×C4×D411yescharacteristic
284C22×C53#51911yescharacteristic
285C22×C53#52012yes—
286C22×C53#52011yescharacteristic
287C22×C53C2×#12711yescharacteristic
288C22×C53#53211yescharacteristic
289C22×C53#53212yes—
290C22×C53#53412yes—
291C22×C53#53411yescharacteristic
292C22×C53#53612yes—
293C22×C53#53612yes—
294C22×C53#53611yescharacteristic
295C22×C53#53611yescharacteristic
296C22×C53#53712yes—
297C22×C53#53711yescharacteristic
298C22×C53#53912yes—
299C22×C53#53912yes—
300C22×C53#53912yes—
301C22×C53#54112yes—
302C22×C53#54111yescharacteristic
303C22×C53#54212yes—
304C22×C53#54211yescharacteristic
305C4×C53—24no—
306C4×C53—24no—
307C4×C53—42no—
308C4×C53—42no—
309C4×C53—81no—
310C4×C53—81no—
311Dic53—28no—
312Dic53—44no—
313Dic53—44no—
314Dic53—82no—
315C2×#399—531no—
316C2×#389—531no—
317C2×#389—532no—
318C2×#388—532no—
319C2×#388D5311yescharacteristic
320#1445—532no—
321#1445—531no—
322#1444—531no—
323#1439—532no—
324#1439—531no—
325C4×#95—531no—
326C2×C53—82no—
327C2×C53C4×#12711yescharacteristic
328C2×C53#232211yescharacteristic
329C2×C53#232212yes—
330C2×C53#232312yes—
331C2×C53#232311yescharacteristic
332C2×C53#265611yescharacteristic
333C2×C53#265811yescharacteristic
334C2×C53#266311yescharacteristic
335C2×C53#266312yes—
336C2×C53#266411yescharacteristic
337C2×C53#266412yes—
338C22×C4⋊C4—532no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—532no—
341C22×C4⋊C4—532no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4C2×D5311yescharacteristic
344C22×C22⋊C4—532no—
345C22×C22⋊C4—532no—
346C22×C22⋊C4C2×D5311yescharacteristic
347C22×C22⋊C4C2×D5311yescharacteristic
348C22×C22⋊C4C2×D5311yescharacteristic
349C22×C42—532no—
350C22×C42C2×D5311yescharacteristic
351#399—538no—
352#389—1068no—
353#389—1064no—
354#388—24no—
355#388—1068no—
356C2×#95—531no—
357C2×#95—532no—
358C2×#95—531no—
359C2×#95—532no—
360C2×#95—532no—
361C2×#95—532no—
362C2×#95—532no—
363C2×#95—532no—
364C2×#95C2×D5311yescharacteristic
365C2×#95C2×D5311yescharacteristic
366C53#1432211yescharacteristic, complemented, Sylow
367C25C22×D5311yescharacteristic
368C23×C4—532no—
369C23×C4—532no—
370C23×C4—532no—
371C23×C4—532no—
372C23×C4C22×D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C23×C4C22×D5311yescharacteristic
377C23×C4C22×D5311yescharacteristic
378C2×C4⋊C4—24no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—538no—
381C2×C4⋊C4—534no—
382C2×C4⋊C4—1064no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—1062no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1064no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—2122no—
392C2×C4⋊C4—2122no—
393C2×C4⋊C4—2122no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—42no—
400C2×C22⋊C4—538no—
401C2×C22⋊C4—1064no—
402C2×C22⋊C4—1064no—
403C2×C22⋊C4—1064no—
404C2×C22⋊C4—2122no—
405C2×C22⋊C4—2122no—
406C2×C22⋊C4C4×D5314yes—
407C2×C42—42no—
408C2×C42—538no—
409C2×C42—2122no—
410C2×C42—2122no—
411C2×C42C22⋊D5314yes—
412#95—22no—
413#95—22no—
414#95—534no—
415#95—1064no—
416#95—1064no—
417#95—1064no—
418#95—1064no—
419#95—1064no—
420#95—1062no—
421#95—1064no—
422C24—22no—
423C24—21no—
424C24—22no—
425C24—21no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24C2×C4×D5312yes—
433C24C23×D5311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—534no—
458C22×C4—534no—
459C22×C4—534no—
460C22×C4—534no—
461C22×C4—534no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4C2×C4×D5312yes—
486C22×C4C2×C4×D5312yes—
487C22×C4#1171994512yes—
488C22×C4D4×D5314yes—
489C22×C4C2×C22⋊D5312yes—
490C22×C4C2×C22⋊D5312yes—
491C22×C4C2×C22⋊D5312yes—
492C4⋊C4—44no—
493C4⋊C4—10616no—
494C4⋊C4—1068no—
495C4⋊C4—2124no—
496C4⋊C4—2128no—
497C4⋊C4—2128no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—10616no—
502C22⋊C4—2128no—
503C42—44no—
504C42—2128no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23C22×C4×D5311yescharacteristic
529C23C2×D4×D5312yes—
530C23C22×C22⋊D5311yescharacteristic
531C23#41×D5312yes—
532C23C2×#1171994511yescharacteristic
533C23#1313326212yes—
534C23#1313326311yescharacteristic
535C23#1313326411yescharacteristic
536C23#1313326411yescharacteristic
537C23#1313326412yes—
538C23#1313326411yescharacteristic
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—538no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4—2122no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2122no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C2×C4—2122no—
594C2×C4—2122no—
595C2×C4C4×C22⋊D5314yes—
596C2×C4#1313321314yes—
597C22—42no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—81no—
603C22—82no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22C4×D4×D5312yes—
609C22C2×C4×C22⋊D5311yescharacteristic
610C22#1681110011yescharacteristic
611C22#1681110212yes—
612C22#1681110311yescharacteristic
613C22C2×#1313321311yescharacteristic
614C22#1681116911yescharacteristic
615C22#1681117012yes—
616C22#1681118112yes—
617C22#1681118211yescharacteristic
618C22#1681119312yes—
619C22#1681119312yes—
620C22#1681119411yescharacteristic
621C22#1681119411yescharacteristic
622C22#1681120012yes—
623C22#1681120111yescharacteristic
624C22#1681121412yes—
625C22#1681121512yes—
626C22#1681121612yes—
627C22#1681123012yes—
628C22#1681123511yescharacteristic
629C22#1681124112yes—
630C22#1681124411yescharacteristic
631C4—24no—
632C4—24no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—1068no—
638C4—4242no—
639C4—2124no—
640C4—2124no—
641C2—82no—
642C2C4×#1313321311yescharacteristic
643C2#2027738011yescharacteristic
644C2#2027738512yes—
645C2#2027739012yes—
646C2#2027739711yescharacteristic
647C2#2028081911yescharacteristic
648C2#2028082611yescharacteristic
649C2#2028086511yescharacteristic
650C2#2028086712yes—
651C2#2028087711yescharacteristic
652C2#2028088012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#388C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#388D53
hypocenterC53#14322
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#16810521C2semidirect
3. C2×#16810518C2semidirect
5. #20272984C2semidirect
6. #20272981C2semidirect
7. #20272979C2semidirect
10. C4×#13133124C2semidirect
20. #16810559C4semidirect
366. C53#14322semidirect

Maximal subgroups

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

subgroupindex
C2×#168105592
C2×#168105212
C2×#168105182
C2×#168105172
#202729842
#202729812
#202729792
#202729602
#202729582
C4×#131331242
C2×C53×#3882
#1432253