Finite Group Databasebeta

← #27441239

Subgroups — #27441239

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic, complemented
2C2×#16810619C211yesmaximal, characteristic
3#20278428C211yesmaximal, characteristic, complemented
4#20276253C211yesmaximal, characteristic, complemented
5#20276110C211yesmaximal, characteristic, complemented
6#20276096C211yesmaximal, characteristic, complemented
7#20275866C211yesmaximal, characteristic, complemented
8#20274962C211yesmaximal, characteristic, complemented
9#20274962C211yesmaximal, characteristic, complemented
10#20274912C211yesmaximal, characteristic, complemented
11#20274895C211yesmaximal, characteristic, complemented
12#20274742C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic, complemented
14#20274655C211yesmaximal, characteristic
15C53×#2339C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133209C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19#16810923—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26#16810619—24no—
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×#13133138C2211yescharacteristic
35C2×C22⋊#88322C2211yescharacteristic
36#16810559—21no—
37#16810546—21no—
38#16810521C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810520—21no—
41#16810506—21no—
42#16810493C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C22×#41×C53C2211yescharacteristic
51C2×C4×D4×C53C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#391C2211yescharacteristic
55C53×#386—21no—
56C53×#386—21no—
57C53×#381—21no—
58C53×#380C2211yescharacteristic
59C4×C22⋊C4×C53C2211yescharacteristic
60C2×#95×C53—21no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133209—22no—
69#13133209—41no—
70#13133176—22no—
71#13133176—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C4×Dic53—21no—
85#13133150—22no—
86#13133150—41no—
87#13133150—41no—
88#13133148—22no—
89C4⋊#88322—22no—
90C4⋊#88322—41no—
91#13133141—22no—
92#13133141—22no—
93#13133141—41no—
94#13133141—41no—
95#13133138—22no—
96#13133138—22no—
97#13133138—22no—
98#13133138D414yes—
99C22⋊#88322—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C2×#41×C53—22no—
104C2×#41×C53—22no—
105C2×#41×C53—22no—
106C2×#41×C53—41no—
107C2×#41×C53—41no—
108C2×#41×C53Q814yes—
109C22×Q8×C53C2311yescharacteristic
110C22×D4×C53—21no—
111C22×D4×C53C2311yescharacteristic
112C23×C4×C53—21no—
113C23×C4×C53—21no—
114C23×C4×C53—21no—
115C23×C4×C53C2311yescharacteristic
116C4×D4×C53—22no—
117C4×D4×C53—22no—
118C4×D4×C53—41no—
119C4×D4×C53—41no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C22⋊C4×C53—21no—
125C2×C22⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C42×C53C2311yescharacteristic
128C2×C42×C53C2311yescharacteristic
129#95×C53—21no—
130#95×C53—21no—
131#95×C53—21no—
132#95×C53—21no—
133#95×C53—21no—
134#95×C53D411yescharacteristic
135#95×C53D411yescharacteristic
136C24×C53—21no—
137C24×C53C2×D411yescharacteristic
138#41×C53—28no—
139#41×C53—44no—
140#41×C53—44no—
141#41×C53—44no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53C2×Q812yes—
148C2×D4×C53—24no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—42no—
153C2×D4×C53—42no—
154C2×D4×C53C2×D412yes—
155C2×D4×C53C2×Q812yes—
156C2×D4×C53#4112yes—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—21no—
162C22×C4×C53—24no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—42no—
166C22×C4×C53—42no—
167C22×C4×C53—42no—
168C22×C4×C53—42no—
169C22×C4×C53—42no—
170C22×C4×C53—42no—
171C22×C4×C53—81no—
172C22×C4×C53—81no—
173C22×C4×C53C2×D411yescharacteristic
174C22×C4×C53C2×D411yescharacteristic
175C22×C4×C53C2×D411yescharacteristic
176C22×C4×C53C2×D411yescharacteristic
177C22×C4×C53C2×D411yescharacteristic
178C22×C4×C53C2×Q812yes—
179C22×C4×C53#4112yes—
180C22×C4×C53C2411yescharacteristic
181C22×Dic53—21no—
182C22×Dic53—21no—
183C22×Dic53—21no—
184C22×Dic53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C22⋊C4×C53—22no—
190C22⋊C4×C53—41no—
191C22⋊C4×C53—42no—
192C22⋊C4×C53—42no—
193C22⋊C4×C53—81no—
194C42×C53—21no—
195C42×C53—21no—
196C42×C53—21no—
197C42×C53—21no—
198C22⋊Dic53—42no—
199C22⋊Dic53—81no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—42no—
208C4⋊Dic53—81no—
209C4⋊Dic53C2×D412yes—
210#11719919—42no—
211#11719919—42no—
212#11719919—42no—
213#11719919—81no—
214C4×Dic53—42no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—44no—
227C2×#88322C2×D412yes—
228C23×C53—22no—
229C23×C53—21no—
230C23×C53—22no—
231C23×C53—21no—
232C23×C53—22no—
233C23×C53—21no—
234C23×C53—21no—
235C23×C53—21no—
236C23×C53—21no—
237C23×C53—41no—
238C23×C53—41no—
239C23×C53—41no—
240C23×C53—41no—
241C23×C53C22⋊Q812yes—
242C23×C53C22×D411yescharacteristic
243Q8×C53—22no—
244Q8×C53—22no—
245Q8×C53—41no—
246Q8×C53—41no—
247D4×C53—22no—
248D4×C53—22no—
249D4×C53—22no—
250D4×C53—48no—
251D4×C53C22⋊Q814yes—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—22no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—21no—
267C2×C4×C53—21no—
268C2×C4×C53—21no—
269C2×C4×C53—21no—
270C2×C4×C53—48no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53—41no—
275C2×C4×C53—41no—
276C2×C4×C53—41no—
277C2×C4×C53—41no—
278C2×C4×C53—41no—
279C2×C4×C53—84no—
280C2×C4×C53—84no—
281C2×C4×C53C22⋊Q812yes—
282C2×C4×C53#13512yes—
283C2×C4×C53C22×D411yescharacteristic
284C2×C4×C53C22×Q811yescharacteristic
285C2×C4×C53C2×#4111yescharacteristic
286C2×C4×C53C2×#4111yescharacteristic
287C2×C4×C53H(5,𝔽2)11yescharacteristic
288C2×C4×C53H(5,𝔽2)11yescharacteristic
289C2×Dic53—22no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—41no—
295C2×Dic53—41no—
296C2×Dic53—41no—
297C2×Dic53—41no—
298C2×Dic53—41no—
299C2×Dic53—41no—
300C2×Dic53—41no—
301#88322—22no—
302#88322—41no—
303#88322—41no—
304#88322—41no—
305#25956—531nomaximal, Sylow
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—22no—
309C22×C53—24no—
310C22×C53—42no—
311C22×C53—42no—
312C22×C53—42no—
313C22×C53—41no—
314C22×C53—41no—
315C22×C53—42no—
316C22×C53—42no—
317C22×C53—41no—
318C22×C53—41no—
319C22×C53—42no—
320C22×C53C2×C22⋊Q811yescharacteristic
321C22×C53D4211yescharacteristic
322C22×C53#54611yescharacteristic
323C22×C53C2×#13511yescharacteristic
324C22×C53#57511yescharacteristic
325C22×C53#57611yescharacteristic
326C22×C53#57811yescharacteristic
327C4×C53—22no—
328C4×C53—22no—
329C4×C53—42no—
330C4×C53—41no—
331C4×C53—41no—
332C4×C53—41no—
333C4×C53—81no—
334C4×C53—81no—
335C4×C53C2×C22⋊Q811yescharacteristic
336C4×C53#54311yescharacteristic
337C4×C53#54611yescharacteristic
338C4×C53#55111yescharacteristic
339Dic53—42no—
340Dic53—81no—
341Dic53—81no—
342Dic53—81no—
343C2×#478—531no—
344C2×#416—531no—
345#2441—531no—
346#2339D5311yescharacteristic
347#2095—531no—
348#2068—531no—
349#2066—531no—
350#2026—531no—
351#1862—531no—
352#1862—531no—
353#1854—531no—
354#1850—531no—
355#1808—531no—
356#1802—531no—
357#1788—531no—
358C2×C53—42no—
359C2×C53—44no—
360C2×C53#243711yescharacteristic
361C2×C53#307611yescharacteristic
362C2×C53#308311yescharacteristic
363C2×C53#326811yescharacteristic
364C2×C53#328311yescharacteristic
365C2×C53#329711yescharacteristic
366C2×C53#331511yescharacteristic
367C22×#41C2×D5311yescharacteristic
368C23×C8—531no—
369C2×#125—531no—
370C2×C4×D4C2×D5311yescharacteristic
371C2×C4×D4C2×D5311yescharacteristic
372C2×#117—531no—
373#478—1064no—
374C2×#107—531no—
375C2×#107—531no—
376C2×#107—531no—
377C2×#106—531no—
378C2×C4⋊C8—531no—
379C2×C4⋊C8—531no—
380#416—1064no—
381C2×#103—531no—
382C2×#103—531no—
383C2×#103—531no—
384C2×#103—531no—
385C2×#102—531no—
386C2×#102—531no—
387C2×#102—531no—
388C2×#102—531no—
389C2×C22⋊C8—531no—
390#399—1061no—
391#396—1061no—
392#391C2×D5311yescharacteristic
393#391C2×D5311yescharacteristic
394#389—531no—
395#389—531no—
396#389—1061no—
397#387—1061no—
398#386—21no—
399#386—21no—
400#384—531no—
401#381—21no—
402#380—531no—
403#380C2×D5311yescharacteristic
404C4×C22⋊C4C2×D5311yescharacteristic
405C2×#95—21no—
406#340—531no—
407#340—531no—
408#340—531no—
409#340—531no—
410#336—531no—
411#336—531no—
412C53#2595611yescharacteristic, complemented, Sylow
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—22no—
416C2×#41—41no—
417C2×#41—41no—
418C2×#41Dic10614yes—
419C22×Q8C22×D5311yescharacteristic
420C22×D4—21no—
421C22×D4C22×D5311yescharacteristic
422C23×C4—21no—
423C23×C4—21no—
424C23×C4—21no—
425C23×C4C22×D5311yescharacteristic
426C22×C8—531no—
427C22×C8—531no—
428C22×C8—531no—
429C22×C8—531no—
430C22×C8—1062no—
431C22×C8—1062no—
432C22×C8—2121no—
433#125—1062no—
434#125—2121no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—41no—
438C4×D4—41no—
439#117—1062no—
440#117—2121no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—531no—
444C2×C4⋊C4—531no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1061no—
450C2×C4⋊C4—1061no—
451C2×C4⋊C4C22×D5311yescharacteristic
452C2×C4⋊C4C22×D5311yescharacteristic
453C2×C22⋊C4—21no—
454C2×C22⋊C4—21no—
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4C22⋊D5311yescharacteristic
459C2×C22⋊C4C22⋊D5311yescharacteristic
460C2×C42—1061no—
461C2×C42C22×D5311yescharacteristic
462C2×C42C22×D5311yescharacteristic
463#107—1062no—
464#107—2121no—
465#107—2121no—
466#106—1062no—
467C4⋊C8—1062no—
468C4⋊C8—2121no—
469#103—1062no—
470#103—1062no—
471#103—2121no—
472#103—2121no—
473#102—534no—
474#102—1062no—
475#102—1062no—
476#102—1062no—
477C22⋊C8—1062no—
478#95—21no—
479#95—21no—
480#95—21no—
481#95—21no—
482#95—21no—
483#95—1061no—
484#95—1061no—
485#95—1061no—
486#95C22⋊D5311yescharacteristic
487#95C22⋊D5311yescharacteristic
488C24—21no—
489C24C2×C22⋊D5311yescharacteristic
490#41—28no—
491#41—44no—
492#41—44no—
493#41—44no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8C2×Dic10612yes—
500C2×D4—24no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4C2×Dic10612yes—
507C2×D4D4×D5312yes—
508C2×D4#1171994712yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—24no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—81no—
524C22×C4—81no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4C2×Dic10612yes—
530C22×C4#1171994512yes—
531C22×C4C2×C22⋊D5311yescharacteristic
532C22×C4C2×C22⋊D5311yescharacteristic
533C22×C4C2×C22⋊D5311yescharacteristic
534C22×C4C2×C22⋊D5311yescharacteristic
535C22×C4C2×C22⋊D5311yescharacteristic
536C22×C4C23×D5311yescharacteristic
537C2×C8—532no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8—1061no—
546C2×C8—1061no—
547C2×C8—1061no—
548C2×C8—1061no—
549C2×C8—2124no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—532no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—1061no—
558C4⋊C4—1061no—
559C4⋊C4—1061no—
560C4⋊C4—1061no—
561C4⋊C4—1061no—
562C4⋊C4—4241no—
563C4⋊C4—4241no—
564C4⋊C4—2122no—
565C4⋊C4—2122no—
566C4⋊C4—2122no—
567C4⋊C4—2122no—
568C22⋊C4—22no—
569C22⋊C4—41no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—81no—
573C22⋊C4—4241no—
574C22⋊C4—2122no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—2122no—
580C23—22no—
581C23—21no—
582C23—22no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—21no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23#1313319612yes—
594C23C22×C22⋊D5311yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—22no—
602D4—48no—
603D4#1313319414yes—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—1062no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4—2121no—
643C2×C4—2121no—
644C2×C4—2121no—
645C2×C4#1313319612yes—
646C2×C4#1313323712yes—
647C2×C4C2×D4×D5311yescharacteristic
648C2×C4C22×Dic10611yescharacteristic
649C2×C4C2×#1171994711yescharacteristic
650C2×C4C2×#1171994511yescharacteristic
651C2×C4#1313326211yescharacteristic
652C2×C4#1313326211yescharacteristic
653C8—1062no—
654C8—2121no—
655C8—2121no—
656C8—2121no—
657C22—22no—
658C22—22no—
659C22—24no—
660C22—22no—
661C22—42no—
662C22—41no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—41no—
668C22—42no—
669C22—41no—
670C22—42no—
671C22C2×#1313319611yescharacteristic
672C22D4×C22⋊D5311yescharacteristic
673C22#1681126711yescharacteristic
674C22C2×#1313323711yescharacteristic
675C22#1681142111yescharacteristic
676C22#1681142611yescharacteristic
677C22#1681143811yescharacteristic
678C4—22no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—42no—
683C4—41no—
684C4—81no—
685C4—81no—
686C4—4241no—
687C4—4241no—
688C4—4241no—
689C4—2122no—
690C4C2×#1313319411yescharacteristic
691C4#1681124911yescharacteristic
692C4#1681126311yescharacteristic
693C4#1681129911yescharacteristic
694C2—42no—
695C2—44no—
696C2#2027837611yescharacteristic
697C2#2028450311yescharacteristic
698C2#2028456611yescharacteristic
699C2#2028625111yescharacteristic
700C2#2028643311yescharacteristic
701C2#2028656511yescharacteristic
702C2#2028681111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2339C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#2339D53
hypocenterC53#25956
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2339

Complements

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

NH ≅ G/Ntype
1. C2×#16810923C2semidirect
3. #20278428C2semidirect
4. #20276253C2semidirect
5. #20276110C2semidirect
6. #20276096C2semidirect
7. #20275866C2semidirect
8. #20274962C2semidirect
9. #20274962C2semidirect
10. #20274912C2semidirect
11. #20274895C2semidirect
12. #20274742C2semidirect
13. #20274715C2semidirect
412. C53#25956semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
C2×#168106192
#202784282
#202762532
#202761102
#202760962
#202758662
#202749622
#202749122
#202748952
#202747422
#202747152
#202746552
C53×#23392
#2595653