Finite Group Databasebeta

← #27312549

Subgroups — #27312549

Order 13568.

1658 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
1C2×#16810516C212yesmaximal, complemented
2#20272964C212yesmaximal, complemented
3#20272960C212yesmaximal, complemented
4#20272956C212yesmaximal
5#20272952C212yesmaximal, complemented
6#20272947C212yesmaximal, complemented
7#20272945C211yesmaximal, characteristic, complemented
8#20272942C211yesmaximal, characteristic, complemented
9C2×C53×#390C211yesmaximal, characteristic
10C22×C4⋊Dic53C2212yes—
11C22×C22⋊Dic53C2212yes—
12C22×C22⋊Dic53C2212yes—
13C22×C22⋊Dic53C2212yes—
14C22×C22⋊Dic53C2212yes—
15C22×C4×Dic53C2212yes—
16#16810516—28no—
17C2×#13133124C2212yes—
18C2×#13133124C2211yescharacteristic
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2211yescharacteristic
27C23×D4×C53C2211yescharacteristic
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C22⋊C4×C53C2212yes—
30C22×C22⋊C4×C53C2212yes—
31C22×C42×C53C2211yescharacteristic
32C53×#390C418yescomplemented
33C23×Dic53C2312yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C2×C4⋊Dic53—24no—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—42no—
40C2×C22⋊Dic53—24no—
41C2×C22⋊Dic53—24no—
42C2×C22⋊Dic53—24no—
43C2×C22⋊Dic53—24no—
44C2×C22⋊Dic53—42no—
45C2×C22⋊Dic53—42no—
46C2×C22⋊Dic53—42no—
47C2×C22⋊Dic53—42no—
48C2×C22⋊Dic53—42no—
49C2×C22⋊Dic53—42no—
50C2×C22⋊Dic53—42no—
51C2×C22⋊Dic53—42no—
52C2×C4×Dic53—24no—
53C2×C4×Dic53—42no—
54C2×C4×Dic53—42no—
55#13133124—24no—
56#13133124—24no—
57#13133124—22no—
58#13133124—24no—
59#13133124—42no—
60#13133124—41no—
61#13133124—42no—
62#13133124—42no—
63#13133124—41no—
64#13133124D414yescomplemented
65C25×C53C2312yes—
66C22×D4×C53—22no—
67C22×D4×C53—22no—
68C22×D4×C53—42no—
69C22×D4×C53—42no—
70C22×D4×C53C2×C414yes—
71C23×C4×C53C2311yescharacteristic
72C23×C4×C53C2312yes—
73C23×C4×C53C2311yescharacteristic
74C23×C4×C53C2311yescharacteristic
75C2×C4⋊C4×C53—42no—
76C2×C4⋊C4×C53C2×C414yes—
77C2×C22⋊C4×C53—24no—
78C2×C22⋊C4×C53—24no—
79C2×C22⋊C4×C53—42no—
80C2×C22⋊C4×C53—42no—
81C2×C22⋊C4×C53C2×C418yes—
82C2×C22⋊C4×C53C2×C418yes—
83C2×C42×C53—42no—
84C2×C42×C53C2×C414yes—
85C24×C53—22no—
86C24×C53—22no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—22no—
90C24×C53—22no—
91C24×C53—22no—
92C24×C53—22no—
93C24×C53—22no—
94C24×C53—22no—
95C24×C53—22no—
96C24×C53—22no—
97C24×C53—22no—
98C24×C53—22no—
99C24×C53C22×C414yes—
100C24×C53C2411yescharacteristic
101C2×D4×C53—28no—
102C2×D4×C53—44no—
103C2×D4×C53—44no—
104C2×D4×C53—48no—
105C2×D4×C53—48no—
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—21no—
119C22×C4×C53—21no—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C414yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×C4×C53#4112yes—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
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×Dic53C2×D414yes—
155C22×Dic53#4114yes—
156C4⋊C4×C53—44no—
157C22⋊C4×C53—216no—
158C22⋊C4×C53—48no—
159C42×C53—44no—
160C22⋊Dic53—48no—
161C22⋊Dic53—48no—
162C22⋊Dic53—48no—
163C22⋊Dic53—48no—
164C4⋊Dic53—48no—
165C4×Dic53—48no—
166C23×C53—24no—
167C23×C53—24no—
168C23×C53—24no—
169C23×C53—24no—
170C23×C53—24no—
171C23×C53—24no—
172C23×C53—24no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
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×C53—42no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53C23×C411yescharacteristic
202C23×C53C22×D411yescharacteristic
203C23×C53C2×#4112yes—
204C23×C53C2×#4111yescharacteristic
205C23×C53C2×#4111yescharacteristic
206C23×C53C2×#4111yescharacteristic
207C23×C53H(5,𝔽2)12yes—
208C23×C53H(5,𝔽2)12yes—
209C23×C53H(5,𝔽2)12yes—
210C23×C53H(5,𝔽2)11yescharacteristic
211C23×C532−1+411yescharacteristic
212D4×C53—416no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—24no—
222C2×C4×C53—22no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—24no—
226C2×C4×C53—22no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53#11714yes—
234C2×C4×C53C4×D414yes—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—24no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
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—
262C2×Dic53—42no—
263#14444—531nomaximal, Sylow
264C22×C53—44no—
265C22×C53—44no—
266C22×C53—44no—
267C22×C53—44no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—82no—
272C22×C53—82no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53—82no—
279C22×C53C2×#11711yescharacteristic
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C4×#4111yescharacteristic
282C22×C53#51811yescharacteristic
283C22×C53#51811yescharacteristic
284C22×C53#51811yescharacteristic
285C22×C53#52011yescharacteristic
286C22×C53#53512yes—
287C22×C53#53812yes—
288C22×C53#53812yes—
289C22×C53#53812yes—
290C22×C53#54012yes—
291C22×C53#54212yes—
292C22×C53#54612yes—
293C22×C53#54711yescharacteristic
294C22×C53#55011yescharacteristic
295C22×C53#55312yes—
296C22×C53#55312yes—
297C22×C53#55511yescharacteristic
298C22×C53#55611yescharacteristic
299C22×C53#55912yes—
300C22×C53#56012yes—
301C22×C53#56212yes—
302C4×C53—24no—
303C4×C53—24no—
304C4×C53—44no—
305C4×C53—42no—
306Dic53—44no—
307Dic53—44no—
308Dic53—82no—
309Dic53—82no—
310C2×#390D5311yescharacteristic
311C2×#388—532no—
312#1440—532no—
313#1439—532no—
314#1438—532no—
315#1437—532no—
316#1436—532no—
317C4⋊#95—531no—
318#1434—531no—
319C2×C53—84no—
320C2×C53#231211yescharacteristic
321C2×C53#232811yescharacteristic
322C2×C53#233711yescharacteristic
323C2×C53#235411yescharacteristic
324C2×C53#235711yescharacteristic
325C2×C53#236411yescharacteristic
326C2×C53#237711yescharacteristic
327C2×C53#271012yes—
328C2×C53#271412yes—
329C2×C53#280012yes—
330C2×C53#280312yes—
331C23×D4C2×D5311yescharacteristic
332C22×C4⋊C4—532no—
333C22×C4⋊C4C2×D5311yescharacteristic
334C22×C22⋊C4—532no—
335C22×C22⋊C4—532no—
336C22×C22⋊C4—532no—
337C22×C22⋊C4—532no—
338C22×C22⋊C4C2×D5312yes—
339C22×C22⋊C4C2×D5312yes—
340C22×C42—532no—
341C22×C42C2×D5311yescharacteristic
342#390Dic5318yescomplemented
343#388—1068no—
344C2×#95—531no—
345C2×#95—532no—
346C2×#95—531no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95—532no—
350C2×#95—531no—
351C2×#95—531no—
352C2×#95—532no—
353C2×#95—532no—
354C53#1444411yescharacteristic, complemented, Sylow
355C25C22×D5312yes—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—42no—
359C22×D4—42no—
360C22×D4C2×Dic5314yes—
361C23×C4—532no—
362C23×C4—532no—
363C23×C4—532no—
364C23×C4—532no—
365C23×C4C22×D5312yes—
366C23×C4C22×D5311yescharacteristic
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C2×C4⋊C4—42no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—2122no—
373C2×C4⋊C4C2×Dic5314yes—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—42no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—1064no—
382C2×C22⋊C4—2122no—
383C2×C22⋊C4—2122no—
384C2×C22⋊C4—2122no—
385C2×C22⋊C4—2122no—
386C2×C22⋊C4—2122no—
387C2×C22⋊C4—2122no—
388C2×C22⋊C4—2122no—
389C2×C22⋊C4—2122no—
390C2×C22⋊C4C2×Dic5318yes—
391C2×C22⋊C4C2×Dic5318yes—
392C2×C42—42no—
393C2×C42—1064no—
394C2×C42—2122no—
395C2×C42—2122no—
396C2×C42C2×Dic5314yes—
397#95—534no—
398#95—1064no—
399#95—1064no—
400#95—1064no—
401#95—1062no—
402#95—2122no—
403#95—2122no—
404#95—2122no—
405#95—2121no—
406#95—2121no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C22×Dic5314yes—
422C24C23×D5311yescharacteristic
423C2×D4—28no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—48no—
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—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—534no—
443C22×C4—534no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
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—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
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#1171994512yes—
471C22×C4#1171994512yes—
472C22×C4D4×D5312yes—
473C22×C4#1171994712yes—
474C22×C4C22×Dic5312yes—
475C22×C4C22×Dic5314yes—
476C22×C4C22×Dic5312yes—
477C22×C4C22×Dic5312yes—
478C4⋊C4—44no—
479C4⋊C4—2128no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—2128no—
483C22⋊C4—2128no—
484C22⋊C4—2128no—
485C22⋊C4—2128no—
486C42—44no—
487C42—2128no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C23×Dic5311yescharacteristic
524C23C2×D4×D5311yescharacteristic
525C23#41×D5312yes—
526C23C2×#1171994711yescharacteristic
527C23C2×#1171994511yescharacteristic
528C23C2×#1171994511yescharacteristic
529C23#1313326212yes—
530C23#1313326212yes—
531C23#1313326212yes—
532C23#1313326311yescharacteristic
533C23#1313326411yescharacteristic
534D4—416no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—1064no—
556C2×C4—1064no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4#1313317614yes—
584C2×C4D4×Dic5314yes—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22C2×#1313317611yescharacteristic
601C22C2×D4×Dic5311yescharacteristic
602C22#41×Dic5311yescharacteristic
603C22#1681109811yescharacteristic
604C22#1681109811yescharacteristic
605C22#1681109811yescharacteristic
606C22#1681110611yescharacteristic
607C22#1681118612yes—
608C22#1681120712yes—
609C22#1681121112yes—
610C22#1681121112yes—
611C22#1681122712yes—
612C22#1681123712yes—
613C22#1681126212yes—
614C22#1681127811yescharacteristic
615C22#1681129411yescharacteristic
616C22#1681131612yes—
617C22#1681132312yes—
618C22#1681133811yescharacteristic
619C22#1681134311yescharacteristic
620C22#1681135612yes—
621C22#1681136112yes—
622C22#1681136812yes—
623C4—24no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—4242no—
628C4—4242no—
629C4—2124no—
630C4—2124no—
631C2—84no—
632C2#2027731311yescharacteristic
633C2#2027744111yescharacteristic
634C2#2027753011yescharacteristic
635C2#2027766411yescharacteristic
636C2#2027768811yescharacteristic
637C2#2027776711yescharacteristic
638C2#2027787211yescharacteristic
639C2#2028145412yes—
640C2#2028150812yes—
641C2#2028261212yes—
642C2#2028264912yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#390C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC2×#390D53
hypocenterC53#14444
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#16810516C2semidirect
2. #20272964C2semidirect
3. #20272960C2semidirect
5. #20272952C2semidirect
6. #20272947C2semidirect
7. #20272945C2semidirect
8. #20272942C2semidirect
32. C53×#390C4semidirect
64. #13133124D4semidirect
342. #390Dic53semidirect
354. C53#14444semidirect

Maximal subgroups

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

subgroupindex
C2×#168105162
#202729642
#202729602
#202729562
#202729522
#202729472
#202729452
#202729422
C2×C53×#3902
#1444453