Finite Group Databasebeta

← #27441278

Subgroups — #27441278

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810916C211yesmaximal, characteristic, complemented
2#20278400C211yesmaximal, characteristic, complemented
3#20277556C211yesmaximal, characteristic
4#20276234C211yesmaximal, characteristic, complemented
5#20276101C211yesmaximal, characteristic, complemented
6#20276099C211yesmaximal, characteristic, complemented
7#20275849C211yesmaximal, characteristic, complemented
8#20274981C211yesmaximal, characteristic, complemented
9#20274971C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic, complemented
11#20274893C211yesmaximal, characteristic, complemented
12#20274744C211yesmaximal, characteristic, complemented
13#20274728C211yesmaximal, characteristic, complemented
14#20274655C211yesmaximal, characteristic
15C2×C53×#416C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19#16810916—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×C22⋊#88322C2211yescharacteristic
31#16810546—21no—
32#16810535—21no—
33#16810529—21no—
34#16810526—21no—
35#16810526—21no—
36#16810526C2211yescharacteristic
37#16810526C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810507—21no—
41#16810488C2211yescharacteristic
42#16810483—21no—
43#16810483C2211yescharacteristic
44#16810483C2211yescharacteristic
45C4×C22⋊Dic53—21no—
46C4×C22⋊Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×#41×C53C2211yescharacteristic
54C23×C8×C53C2211yescharacteristic
55C2×#117×C53C2211yescharacteristic
56C53×#416—24no—
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C22×Dic106—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—41no—
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—41no—
69Q8×Dic53—41no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53D411yescharacteristic
86C2×C22⋊Dic53D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—22no—
92#13133151—41no—
93#13133150—22no—
94#13133148—41no—
95#13133146—22no—
96#13133146—41no—
97#13133141—22no—
98#13133141—22no—
99#13133138—41no—
100#13133138—41no—
101C22⋊#88322—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C2×#41×C53—22no—
111C2×#41×C53—22no—
112C2×#41×C53—22no—
113C2×#41×C53—41no—
114C2×#41×C53—41no—
115C2×#41×C53Q814yescomplemented
116C22×Q8×C53C2311yescharacteristic
117C22×D4×C53—21no—
118C22×D4×C53C2311yescharacteristic
119C23×C4×C53—21no—
120C23×C4×C53C2311yescharacteristic
121C22×C8×C53—22no—
122C22×C8×C53—22no—
123C22×C8×C53—41no—
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126#117×C53—22no—
127#117×C53—41no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C22⋊C4×C53—21no—
131C2×C42×C53—21no—
132#103×C53—22no—
133#103×C53D414yes—
134#102×C53—22no—
135#102×C53—22no—
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×C53C2×D412yes—
146C2×Q8×C53C2×Q812yes—
147C2×Q8×C53#4112yes—
148C2×D4×C53—24no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—21no—
154C2×D4×C53—42no—
155C2×D4×C53—42no—
156C2×D4×C53C2×Q812yes—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—21no—
162C22×C4×C53—21no—
163C22×C4×C53—24no—
164C22×C4×C53—21no—
165C22×C4×C53—42no—
166C22×C4×C53—42no—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×Q812yes—
169C22×C4×C53#4112yes—
170C22×C4×C53C2411yescharacteristic
171C22×Dic53—21no—
172C22×Dic53—21no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C22×Dic53C2×D411yescharacteristic
179C2×Dic106—42no—
180C2×Dic106—42no—
181C2×Dic106—81no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C2×C8×C53—44no—
188C2×C8×C53C2×D412yes—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53C2×D412yes—
193C22⋊C4×C53—42no—
194C42×C53—42no—
195C22⋊Dic53—22no—
196C22⋊Dic53—41no—
197C22⋊Dic53—42no—
198C22⋊Dic53—42no—
199C22⋊Dic53—81no—
200C22⋊Dic53—81no—
201C22⋊Dic53—81no—
202C22⋊Dic53—81no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211C4⋊Dic53—42no—
212C4⋊Dic53—42no—
213C4⋊Dic53—81no—
214#11719919—42no—
215#11719919—42no—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—21no—
220C4×Dic53—42no—
221C4×Dic53—42no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53—22no—
235C23×C53—41no—
236C23×C53—41no—
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—22no—
246Q8×C53C22⋊Q814yes—
247D4×C53—22no—
248D4×C53—22no—
249D4×C53—48no—
250D4×C53—41no—
251D4×C53—41no—
252C2×C4×C53—22no—
253C2×C4×C53—21no—
254C2×C4×C53—22no—
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—22no—
262C2×C4×C53—48no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53C22⋊Q812yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×Q811yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C53H(5,𝔽2)11yescharacteristic
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—21no—
284C2×Dic53—21no—
285C2×Dic53—21no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
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#13512yes—
299Dic106—84no—
300C8×C53—22no—
301C8×C53—41no—
302#88322—41no—
303#88322—41no—
304#25959—531nomaximal, Sylow
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—42no—
313C22×C53—42no—
314C22×C53—42no—
315C22×C53—81no—
316C22×C53—81no—
317C22×C53—81no—
318C22×C53—81no—
319C22×C53C2×C22⋊Q811yescharacteristic
320C22×C53D4211yescharacteristic
321C22×C53#54611yescharacteristic
322C22×C53C2×#13511yescharacteristic
323C22×C53#57511yescharacteristic
324C22×C53#57611yescharacteristic
325C22×C53#57811yescharacteristic
326C4×C53—22no—
327C4×C53—24no—
328C4×C53—42no—
329C4×C53—81no—
330C4×C53C2×C22⋊Q811yescharacteristic
331C4×C53#54311yescharacteristic
332C4×C53#54611yescharacteristic
333C4×C53#55111yescharacteristic
334Dic53—22no—
335Dic53—41no—
336Dic53—41no—
337Dic53—41no—
338Dic53—81no—
339Dic53—81no—
340Dic53—81no—
341Dic53—81no—
342C2×#477—531no—
343C2×#416D5311yescharacteristic
344#2439—531no—
345#2341—531no—
346#2092—531no—
347#2067—531no—
348#2067—531no—
349#2023—531no—
350#1865—531no—
351#1863—531no—
352#1854—531no—
353#1850—531no—
354#1808—531no—
355#1805—531no—
356#1788—531no—
357C2×C53—42no—
358C2×C53—82no—
359C2×C53#243711yescharacteristic
360C2×C53#307611yescharacteristic
361C2×C53#308311yescharacteristic
362C2×C53#326811yescharacteristic
363C2×C53#328511yescharacteristic
364C2×C53#330511yescharacteristic
365C2×C53#331311yescharacteristic
366C22×#41C2×D5311yescharacteristic
367C23×C8C2×D5311yescharacteristic
368C2×C4⋊Q8—531no—
369C2×C4×Q8—531no—
370C2×C4×Q8—531no—
371C2×#117C2×D5311yescharacteristic
372#477—1064no—
373C2×#107—531no—
374C2×#107—531no—
375C2×#107—531no—
376C2×#106—531no—
377C2×C4⋊C8—531no—
378C2×C4⋊C8—531no—
379#416—24no—
380C2×#103—531no—
381C2×#103—531no—
382C2×#103C2×D5311yescharacteristic
383C2×#103C2×D5311yescharacteristic
384C2×#102—531no—
385C2×#102—531no—
386C2×#102C2×D5311yescharacteristic
387C2×#102C2×D5311yescharacteristic
388C2×C22⋊C8—531no—
389#396—1061no—
390#393—1061no—
391#391—1061no—
392#390—531no—
393#390—531no—
394#390—1061no—
395#390—1061no—
396#389—531no—
397#389—531no—
398#387—1061no—
399#382—531no—
400#380—531no—
401#380—531no—
402#380—1061no—
403C4×C22⋊C4—531no—
404C4×C22⋊C4—1061no—
405#340—531no—
406#340—531no—
407#340—531no—
408#340—531no—
409#336—531no—
410#336—531no—
411C53#2595911yescharacteristic, complemented, Sylow
412C2×#41—22no—
413C2×#41—22no—
414C2×#41—22no—
415C2×#41—41no—
416C2×#41—41no—
417C2×#41Dic10614yescomplemented
418C22×Q8—1061no—
419C22×Q8C22×D5311yescharacteristic
420C22×D4—21no—
421C22×D4C22×D5311yescharacteristic
422C23×C4—21no—
423C23×C4C22×D5311yescharacteristic
424C22×C8—22no—
425C22×C8—22no—
426C22×C8—41no—
427C22×C8—531no—
428C22×C8—531no—
429C22×C8C22×D5311yescharacteristic
430C22×C8C22×D5311yescharacteristic
431C4⋊Q8—1062no—
432C4⋊Q8—2121no—
433C4×Q8—1062no—
434C4×Q8—1062no—
435C4×Q8—2121no—
436C4×Q8—2121no—
437#117—22no—
438#117—41no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—531no—
442C2×C4⋊C4—531no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—531no—
452C2×C22⋊C4—531no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4—1061no—
459C2×C42—21no—
460C2×C42—531no—
461C2×C42—531no—
462C2×C42—1061no—
463C2×C42—1061no—
464#107—1062no—
465#107—1062no—
466#107—2121no—
467#106—2121no—
468C4⋊C8—1062no—
469C4⋊C8—2121no—
470#103—22no—
471#103—1062no—
472#103—1062no—
473#103C22⋊D5314yes—
474#102—22no—
475#102—22no—
476#102—2121no—
477#102—2121no—
478C22⋊C8—1062no—
479#95—531no—
480#95—531no—
481#95—1061no—
482#95—1061no—
483#95—1061no—
484#95—1061no—
485#95—1061no—
486#95—1061no—
487C24—21no—
488C24D4×D5311yescharacteristic
489#41—28no—
490#41—44no—
491#41—44no—
492#41—44no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—4241no—
497C2×Q8—2122no—
498C2×Q8—2122no—
499C2×Q8C2×Dic10612yes—
500C2×Q8#1171994512yes—
501C2×Q8C2×C22⋊D5312yes—
502C2×D4—24no—
503C2×D4—21no—
504C2×D4—21no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×Dic10612yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—24no—
518C22×C4—21no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—531no—
522C22×C4—531no—
523C22×C4—531no—
524C22×C4—531no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4C2×Dic10612yes—
530C22×C4D4×D5311yescharacteristic
531C22×C4#1171994712yes—
532C22×C4C23×D5311yescharacteristic
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—44no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8C2×C22⋊D5312yes—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—4241no—
558C4⋊C4—2122no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C4⋊C4—2122no—
562C4⋊C4C2×C22⋊D5312yes—
563C22⋊C4—42no—
564C22⋊C4—1062no—
565C22⋊C4—4241no—
566C22⋊C4—4241no—
567C22⋊C4—4241no—
568C22⋊C4—4241no—
569C22⋊C4—2122no—
570C22⋊C4—2121no—
571C22⋊C4—2122no—
572C42—42no—
573C42—1061no—
574C42—1061no—
575C42—1061no—
576C42—1061no—
577C42—2122no—
578C42—2122no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23#1313319412yes—
593C23C2×D4×D5311yescharacteristic
594Q8—22no—
595Q8—22no—
596Q8—22no—
597Q8—4244no—
598Q8#1313319614yes—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—532no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1061no—
626C2×C4—1061no—
627C2×C4—1061no—
628C2×C4—1061no—
629C2×C4—1061no—
630C2×C4—1061no—
631C2×C4—1061no—
632C2×C4—2121no—
633C2×C4—2121no—
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#1313319412yes—
645C2×C4C22×C22⋊D5311yescharacteristic
646C2×C4C22×Dic10611yescharacteristic
647C2×C4C2×#1171994711yescharacteristic
648C2×C4C2×#1171994511yescharacteristic
649C2×C4#1313326211yescharacteristic
650C2×C4#1313326211yescharacteristic
651C8—22no—
652C8—41no—
653C8—2121no—
654C8—2121no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C2×#1313319411yescharacteristic
670C22D4×C22⋊D5311yescharacteristic
671C22#1681126311yescharacteristic
672C22#135×D5311yescharacteristic
673C22#1681142011yescharacteristic
674C22#1681142511yescharacteristic
675C22#1681143711yescharacteristic
676C4—22no—
677C4—24no—
678C4—42no—
679C4—81no—
680C4—1062no—
681C4—4241no—
682C4—4241no—
683C4—4241no—
684C4—4241no—
685C4—2121no—
686C4—2121no—
687C4—2121no—
688C4C2×#1313319611yescharacteristic
689C4#1681124911yescharacteristic
690C4#1681126711yescharacteristic
691C4#1681129911yescharacteristic
692C2—42no—
693C2—82no—
694C2#2027837611yescharacteristic
695C2#2028449911yescharacteristic
696C2#2028456411yescharacteristic
697C2#2028625511yescharacteristic
698C2#2028645011yescharacteristic
699C2#2028666111yescharacteristic
700C2#2028677711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#416C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#416D53
hypocenterC53#25959
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
1. C2×#16810916C2semidirect
2. #20278400C2semidirect
4. #20276234C2semidirect
5. #20276101C2semidirect
6. #20276099C2semidirect
7. #20275849C2semidirect
8. #20274981C2semidirect
9. #20274971C2semidirect
10. #20274914C2semidirect
11. #20274893C2semidirect
12. #20274744C2semidirect
13. #20274728C2semidirect
115. C2×#41×C53Q8semidirect
411. C53#25959semidirect
417. C2×#41Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168109162
#202784002
#202775562
#202762342
#202761012
#202760992
#202758492
#202749812
#202749712
#202749142
#202748932
#202747442
#202747282
#202746552
C2×C53×#4162
#2595953