Finite Group Databasebeta

← #27314807

Subgroups — #27314807

Order 13568.

1326 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810522C211yesmaximal, characteristic, complemented
2C2×#16810516C211yesmaximal, characteristic
3#20272983C211yesmaximal, characteristic, complemented
4#20272972C211yesmaximal, characteristic, complemented
5#20272968C212yesmaximal, complemented
6#20272968C212yesmaximal, complemented
7#20272964C212yesmaximal
8#20272960C211yesmaximal, characteristic
9#20272960C212yesmaximal
10#20272940C211yesmaximal, characteristic, complemented
11C2×C53×#399C211yesmaximal, characteristic
12C22×C4⋊Dic53C2212yes—
13C22×C4⋊Dic53C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C22⋊Dic53C2211yescharacteristic
16C22×C22⋊Dic53C2211yescharacteristic
17C22×C22⋊Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19#16810522—24no—
20#16810516—24no—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C22×C4⋊C4×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2212yes—
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2212yes—
36C53×#399C418yescomplemented
37C2×#95×C53C2211yescharacteristic
38C23×Dic53C2312yes—
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2311yescharacteristic
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53—24no—
47C2×C4⋊Dic53—42no—
48C2×C4⋊Dic53—42no—
49C2×C4⋊Dic53—42no—
50C2×C4⋊Dic53—42no—
51C2×C22⋊Dic53—22no—
52C2×C22⋊Dic53—22no—
53C2×C22⋊Dic53—42no—
54C2×C22⋊Dic53—42no—
55C2×C22⋊Dic53—42no—
56C2×C22⋊Dic53—42no—
57C2×C22⋊Dic53—42no—
58C2×C4×Dic53—42no—
59C2×C4×Dic53D414yes—
60#13133124—22no—
61#13133124—22no—
62#13133124—22no—
63#13133124—24no—
64#13133124—42no—
65#13133124—42no—
66#13133124—42no—
67#13133124—42no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71C25×C53C2311yescharacteristic
72C23×C4×C53C2311yescharacteristic
73C23×C4×C53C2312yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2311yescharacteristic
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—42no—
79C2×C4⋊C4×C53—42no—
80C2×C4⋊C4×C53C2×C414yes—
81C2×C4⋊C4×C53C2×C418yes—
82C2×C22⋊C4×C53—22no—
83C2×C22⋊C4×C53—22no—
84C2×C22⋊C4×C53—42no—
85C2×C22⋊C4×C53—42no—
86C2×C22⋊C4×C53C2×C414yes—
87C2×C22⋊C4×C53C2×C418yes—
88#95×C53C2×C414yes—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—21no—
95C24×C53—21no—
96C24×C53—21no—
97C24×C53—22no—
98C24×C53—21no—
99C24×C53C22×C412yes—
100C24×C53C2411yescharacteristic
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C414yes—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C414yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4112yes—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—21no—
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—21no—
151C22×Dic53—21no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D412yes—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—48no—
161C22⋊C4×C53—28no—
162C22⋊C4×C53—48no—
163C22⋊Dic53—44no—
164C22⋊Dic53—44no—
165C22⋊Dic53—48no—
166C4⋊Dic53—48no—
167C4⋊Dic53—44no—
168C4⋊Dic53—44no—
169C4×Dic53—44no—
170C23×C53—22no—
171C23×C53—24no—
172C23×C53—22no—
173C23×C53—24no—
174C23×C53—22no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—41no—
184C23×C53—41no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53C23×C411yescharacteristic
194C23×C53C22×D411yescharacteristic
195C23×C53C2×#4111yescharacteristic
196C23×C53H(5,𝔽2)12yes—
197C23×C53H(5,𝔽2)12yes—
198C23×C53H(5,𝔽2)12yes—
199C23×C53H(5,𝔽2)11yescharacteristic
200C23×C532−1+412yes—
201C23×C532−1+411yescharacteristic
202C23×C532−1+411yescharacteristic
203C23×C532−1+411yescharacteristic
204C2×C4×C53—24no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—24no—
213C2×C4×C53—24no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—42no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53C4×D414yes—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—42no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262#14628—531nomaximal, Sylow
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—44no—
267C22×C53—44no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—82no—
272C22×C53—82no—
273C22×C53—81no—
274C22×C53C2×C4×D411yescharacteristic
275C22×C53#51812yes—
276C22×C53#51911yescharacteristic
277C22×C53#52012yes—
278C22×C53#52011yescharacteristic
279C22×C53#53411yescharacteristic
280C22×C53#53511yescharacteristic
281C22×C53#53511yescharacteristic
282C22×C53#53611yescharacteristic
283C22×C53#53611yescharacteristic
284C22×C53#53711yescharacteristic
285C22×C53#53912yes—
286C22×C53#54012yes—
287C22×C53#54111yescharacteristic
288C22×C53#54211yescharacteristic
289C22×C53#56012yes—
290C22×C53#56212yes—
291C22×C53#56212yes—
292C22×C53#56212yes—
293C22×C53#56212yes—
294C22×C53#56312yes—
295C22×C53#56312yes—
296C22×C53#56312yes—
297C4×C53—24no—
298C4×C53—42no—
299C4×C53—44no—
300C4×C53—44no—
301Dic53—42no—
302Dic53—42no—
303Dic53—81no—
304Dic53—81no—
305Dic53—82no—
306Dic53—82no—
307C2×#399D5311yescharacteristic
308C2×#389—531no—
309C2×#388—531no—
310#1445—531no—
311#1442—531no—
312#1441—532no—
313#1441—532no—
314#1440—532no—
315#1439—532no—
316#1439—531no—
317#1434—531no—
318C2×C53—82no—
319C2×C53#232211yescharacteristic
320C2×C53#232311yescharacteristic
321C2×C53#232811yescharacteristic
322C2×C53#237612yes—
323C2×C53#237812yes—
324C2×C53#281311yescharacteristic
325C2×C53#281611yescharacteristic
326C2×C53#282011yescharacteristic
327C2×C53#282111yescharacteristic
328C2×C53#283612yes—
329C2×C53#283712yes—
330C22×C4⋊C4—532no—
331C22×C4⋊C4—531no—
332C22×C4⋊C4—531no—
333C22×C4⋊C4C2×D5311yescharacteristic
334C22×C4⋊C4C2×D5312yes—
335C22×C22⋊C4—531no—
336C22×C22⋊C4—531no—
337C22×C22⋊C4—532no—
338C22×C22⋊C4C2×D5311yescharacteristic
339C22×C22⋊C4C2×D5312yes—
340C22×C42—531no—
341#399Dic5318yescomplemented
342#389—1064no—
343#388—1064no—
344C2×#95—531no—
345C2×#95—532no—
346C2×#95—531no—
347C2×#95—531no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—532no—
351C2×#95—532no—
352C2×#95—532no—
353C2×#95—532no—
354C2×#95—532no—
355C2×#95C2×D5311yescharacteristic
356C53#1462811yescharacteristic, complemented, Sylow
357C25C22×D5311yescharacteristic
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—532no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—532no—
364C23×C4C22×D5311yescharacteristic
365C23×C4C22×D5312yes—
366C23×C4C22×D5312yes—
367C23×C4C22×D5311yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1064no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—2122no—
376C2×C4⋊C4—2122no—
377C2×C4⋊C4—2122no—
378C2×C4⋊C4—2122no—
379C2×C4⋊C4C2×Dic5314yes—
380C2×C4⋊C4C2×Dic5318yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—42no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—1062no—
386C2×C22⋊C4—1062no—
387C2×C22⋊C4—2122no—
388C2×C22⋊C4—2122no—
389C2×C22⋊C4—2122no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C2×C22⋊C4C2×Dic5314yes—
393C2×C22⋊C4C2×Dic5318yes—
394C2×C42—534no—
395C2×C42—2122no—
396#95—1062no—
397#95—1064no—
398#95—1062no—
399#95—1062no—
400#95—2122no—
401#95—2122no—
402#95—2122no—
403#95—2122no—
404#95—2122no—
405#95—2122no—
406#95—2122no—
407#95C2×Dic5314yes—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24—21no—
413C24—22no—
414C24—21no—
415C24—22no—
416C24—21no—
417C24—21no—
418C24C22×Dic5312yes—
419C24C23×D5311yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1061no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1061no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1061no—
472C22×C4D4×D5312yes—
473C22×C4#1171994712yes—
474C22×C4C22×Dic5314yes—
475C22×C4C22×Dic5312yes—
476C22×C4C22×Dic5314yes—
477C22×C4C22×Dic5312yes—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—2124no—
481C4⋊C4—2124no—
482C4⋊C4—2128no—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—2124no—
486C22⋊C4—2124no—
487C22⋊C4—2128no—
488C42—2124no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23C23×Dic5311yescharacteristic
513C23C2×D4×D5311yescharacteristic
514C23C2×#1171994711yescharacteristic
515C23#1313326212yes—
516C23#1313326212yes—
517C23#1313326311yescharacteristic
518C23#1313326312yes—
519C23#1313326411yescharacteristic
520C23#1313326412yes—
521C23#1313326511yescharacteristic
522C23#1313326511yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1062no—
551C2×C4—1062no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2121no—
565C2×C4—2122no—
566C2×C4—2121no—
567C2×C4—2122no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2122no—
572C2×C4—2121no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4D4×Dic5314yes—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22C2×D4×Dic5311yescharacteristic
593C22#1681109812yes—
594C22#1681110111yescharacteristic
595C22#1681110611yescharacteristic
596C22#1681110612yes—
597C22#1681118411yescharacteristic
598C22#1681118611yescharacteristic
599C22#1681118611yescharacteristic
600C22#1681119511yescharacteristic
601C22#1681119611yescharacteristic
602C22#1681120311yescharacteristic
603C22#1681121712yes—
604C22#1681122712yes—
605C22#1681123311yescharacteristic
606C22#1681124611yescharacteristic
607C22#1681136312yes—
608C22#1681136912yes—
609C22#1681136912yes—
610C22#1681137012yes—
611C22#1681137012yes—
612C22#1681137412yes—
613C22#1681137412yes—
614C22#1681137612yes—
615C4—24no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—4242no—
620C4—4242no—
621C4—4241no—
622C4—4241no—
623C4—2122no—
624C4—2122no—
625C2—82no—
626C2#2027738711yescharacteristic
627C2#2027739911yescharacteristic
628C2#2027744111yescharacteristic
629C2#2027786312yes—
630C2#2027788112yes—
631C2#2028271811yescharacteristic
632C2#2028274411yescharacteristic
633C2#2028276911yescharacteristic
634C2#2028277811yescharacteristic
635C2#2028291412yes—
636C2#2028292912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#399C2
socle, cosocleC24×C53C24
prime radicalC25×C53C23
hypercenterC2×#399D53
hypocenterC53#14628
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#16810522C2semidirect
3. #20272983C2semidirect
4. #20272972C2semidirect
5. #20272968C2semidirect
6. #20272968C2semidirect
10. #20272940C2semidirect
36. C53×#399C4semidirect
341. #399Dic53semidirect
356. C53#14628semidirect

Maximal subgroups

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

subgroupindex
C2×#168105222
C2×#168105162
#202729832
#202729722
#202729682
#202729642
#202729602
#202729402
C2×C53×#3992
#1462853