Finite Group Databasebeta

← C2×#20284639 #28251871

Subgroups — C2×#20284639 #28251871

Order 13568.

2650 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811136C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#20284639C2116yesmaximal, direct product factor
4C2×#16810957C212yesmaximal, complemented
5C2×#16810943C212yesmaximal, complemented
6C2×#16810951C211yesmaximal, characteristic, complemented
7C2×#16810937C211yesmaximal, characteristic, complemented
8C2×#16810936C211yesmaximal, characteristic, complemented
9C2×#16810662C212yesmaximal, complemented
10C2×#16810623C212yesmaximal, complemented
11C2×#16810585C211yesmaximal, characteristic, complemented
12C2×C53×#483C211yesmaximal, characteristic, complemented
13C22×#11719949C2211yescharacteristic
14C23×C4×D53—21no—
15C23×C4×D53C2211yescharacteristic
16C22×#11719912C2212yes—
17#16811136—24no—
18#16811136C2218yes—
19C2×#13133202—21no—
20C2×#13133193C2211yescharacteristic
21C2×#13133192C2211yescharacteristic
22C2×#13133191C2211yescharacteristic
23C2×#13133187—21no—
24C2×#13133179—21no—
25C2×#13133177—21no—
26C2×#13133175C2211yescharacteristic
27C22×C4⋊Dic53C2211yescharacteristic
28C22×#11719919—21no—
29C2×C4⋊C4×D53—24no—
30C2×C4⋊C4×D53—21no—
31C2×C4⋊C4×D53—21no—
32C2×C4⋊C4×D53C2212yes—
33C2×C4⋊C4×D53C2212yes—
34C2×C4⋊C4×D53C2218yescomplemented
35#16810957C22116yescomplemented
36#16810951C2218yescomplemented
37#16810943C22116yescomplemented
38#16810937C2218yescomplemented
39#16810936C2218yescomplemented
40#16810662C22116yescomplemented
41C2×#13133151C2212yescomplemented
42C2×#13133150C2212yescomplemented
43C2×#13133148C2212yescomplemented
44#16810623C22116yescomplemented
45C2×#13133141C2212yescomplemented
46C2×#13133139C2212yescomplemented
47C2×#13133137C2212yescomplemented
48#16810585C2218yescomplemented
49C2×C22⋊#88322C2211yescharacteristic
50C2×#13133129C2212yes—
51C2×C22⋊Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#483C2218yescomplemented
54C2×#106×C53C2212yescomplemented
55C2×#103×C53C2212yescomplemented
56C2×C22⋊C8×C53C2211yescharacteristic
57C24×D53D411yescharacteristic
58C2×#11719949—22no—
59C2×#11719949—28no—
60C2×#11719949C2314yes—
61C22×D212—21no—
62C22×D212—21no—
63C22×D212C2311yescharacteristic
64C22×C22⋊D53—21no—
65C23×Dic53—21no—
66C23×Dic53D411yescharacteristic
67C22×C4×D53—28no—
68C22×C4×D53—22no—
69C22×C4×D53—22no—
70C22×C4×D53—28no—
71C22×C4×D53—21no—
72C22×C4×D53—48no—
73C22×C4×D53D411yescharacteristic
74C22×C4×D53D411yescharacteristic
75C22×C4×D53D414yes—
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2314yes—
78C22×C4×D53C2311yescharacteristic
79C2×#11719912—28no—
80C2×#11719912C2318yes—
81C22×#88322C2312yes—
82#13133202—28no—
83#13133193—22no—
84#13133193C2314yes—
85#13133192—22no—
86#13133192C2314yes—
87#13133191—22no—
88#13133191C2314yes—
89#13133187—28no—
90#13133179—28no—
91#13133177—28no—
92#13133175—22no—
93#13133175C2314yes—
94C2×C4⋊Dic53—21no—
95C2×C4⋊Dic53—22no—
96C2×C4⋊Dic53C2312yes—
97C2×C4⋊Dic53C2314yes—
98C2×C4⋊Dic53C2311yescharacteristic
99C2×#11719919—22no—
100C2×#11719919—22no—
101C2×#11719919—28no—
102C2×#11719919—21no—
103C4⋊C4×D53—24no—
104C4⋊C4×D53—24no—
105C4⋊C4×D53—28no—
106C4⋊C4×D53—28no—
107C4⋊C4×D53C2318yes—
108C4⋊C4×D53C2318yes—
109C2×#11719921—21no—
110C2×#11719921—21no—
111C2×#11719921—21no—
112C2×#11719921—21no—
113C2×C4×Dic53—21no—
114#13133151C2318yescomplemented
115#13133150C2318yescomplemented
116#13133148C2318yescomplemented
117#13133141C2318yescomplemented
118#13133139C2318yescomplemented
119#13133137C2318yescomplemented
120C22⋊#88322C2314yes—
121#13133129C2318yes—
122C22×Q8×C53C2311yescharacteristic
123C23×C4×C53—21no—
124C23×C4×C53C2311yescharacteristic
125C22×C8×C53C2312yes—
126C22⋊Q8×C53—22no—
127C22⋊Q8×C53C2314yes—
128C2×C4⋊C4×C53—21no—
129C2×C4⋊C4×C53—22no—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53C2312yes—
132C2×C4⋊C4×C53C2314yes—
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C22⋊C4×C53—21no—
135#106×C53C2318yescomplemented
136#103×C53C2318yescomplemented
137C22⋊C8×C53C2314yes—
138C24×C53C2×D411yescharacteristic
139C23×D53—24no—
140C23×D53—24no—
141C23×D53—24no—
142C23×D53—21no—
143C23×D53C2×D414yes—
144C23×D53C2×D411yescharacteristic
145C23×D53C2×D411yescharacteristic
146C2×Q8×C53—21no—
147C2×Q8×C53—24no—
148C2×Q8×C53C2412yes—
149C22×C4×C53—24no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—24no—
153C22×C4×C53—22no—
154C22×C4×C53—44no—
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2412yes—
158C22×C4×C53C2411yescharacteristic
159C2×C22⋊D53—24no—
160C2×C22⋊D53—44no—
161C22×Dic53—24no—
162C22×Dic53—21no—
163C22×Dic53—24no—
164C22×Dic53—22no—
165C22×Dic53—44no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D414yes—
168C22×Dic53C2×D411yescharacteristic
169#11719949—216no—
170#11719949—48no—
171C2×D212—24no—
172C2×D212—24no—
173C2×D212—24no—
174C2×D212—21no—
175C2×D212—44no—
176C2×D212—44no—
177C2×D212C2412yes—
178C2×C4×D53—24no—
179C2×C4×D53—24no—
180C2×C4×D53—24no—
181C2×C4×D53—216no—
182C2×C4×D53—216no—
183C2×C4×D53—28no—
184C2×C4×D53—28no—
185C2×C4×D53—24no—
186C2×C4×D53—48no—
187C2×C4×D53—48no—
188C2×C4×D53—432no—
189C2×C4×D53—44no—
190C2×C4×D53C2×D414yes—
191C2×C4×D53C2×D412yes—
192C2×C4×D53C2×D414yes—
193C2×C4×D53C2×D412yes—
194C2×C4×D53#4118yes—
195C2×C4×D53C2412yes—
196C2×C4×D53C2412yes—
197C2×C8×C53—24no—
198C2×C8×C53C2414yes—
199C4⋊C4×C53—24no—
200C4⋊C4×C53—22no—
201C4⋊C4×C53—24no—
202C4⋊C4×C53—21no—
203C4⋊C4×C53C2414yes—
204C4⋊C4×C53C2412yes—
205C22⋊C4×C53—24no—
206#11719921—24no—
207#11719921—24no—
208#11719921—24no—
209#11719921—24no—
210C4⋊Dic53—24no—
211C4⋊Dic53—22no—
212C4⋊Dic53—21no—
213C4⋊Dic53C2414yes—
214C4⋊Dic53C2412yes—
215#11719919—28no—
216#11719919—28no—
217#11719919—24no—
218C4×Dic53—24no—
219C2×#88322—24no—
220C2×#88322C2414yes—
221#11719912—216no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53AGL(1,ℤ/8)14yes—
225C23×C53C22×D412yes—
226C23×C53C22×D411yescharacteristic
227C22×D53—232no—
228C22×D53—22no—
229C22×D53—28no—
230C22×D53—22no—
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—28no—
234C22×D53—28no—
235C22×D53—22no—
236C22×D53—42no—
237C22×D53—42no—
238C22×D53—42no—
239C22×D53#12314yes—
240C22×D53#13714yes—
241C22×D53C22×D412yes—
242C22×D53C22×D412yes—
243Q8×C53—24no—
244Q8×C53—42no—
245C2×C4×C53—22no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—42no—
252C2×C4×C53—48no—
253C2×C4×C53—44no—
254C2×C4×C53—42no—
255C2×C4×C53C22×D412yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4114yes—
258C2×C4×C53C2511yescharacteristic
259C22⋊D53—48no—
260C2×Dic53—22no—
261C2×Dic53—28no—
262C2×Dic53—24no—
263C2×Dic53—22no—
264C2×Dic53—48no—
265C2×Dic53—44no—
266C2×Dic53—42no—
267C2×Dic53#12314yes—
268C2×Dic53C22×D412yes—
269C2×Dic53C22×D412yes—
270D212—24no—
271D212—48no—
272D212—48no—
273D212—42no—
274C4×D53—28no—
275C4×D53—24no—
276C4×D53—24no—
277C4×D53—28no—
278C4×D53—416no—
279C4×D53—416no—
280C4×D53—48no—
281C4×D53#12318yes—
282C4×D53C2×#4118yes—
283C8×C53—24no—
284#88322—24no—
285C2×#3092—531nomaximal, Sylow
286C22×C53—28no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53C2×#12312yes—
290C22×C53C2×AGL(1,ℤ/8)14yes—
291C22×C53C2×AGL(1,ℤ/8)12yes—
292C22×C53C2×#13712yes—
293C22×C53C23×D411yescharacteristic
294C2×D53—232no—
295C2×D53—28no—
296C2×D53—28no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—84no—
305C2×D53C2×#12314yes—
306C2×D53C2×#13714yes—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—44no—
310C4×C53—42no—
311C4×C53C2×#12312yes—
312C4×C53C22×#4112yes—
313Dic53—22no—
314Dic53—44no—
315Dic53—42no—
316Dic53C2×#12314yes—
317C2×#525—531no—
318C23×C4⋊C4—531no—
319#3092—5316no—
320C2×#484—532no—
321C2×#481—532no—
322C2×#483—531no—
323C2×#483D5311yescharacteristic, complemented
324C2×#480—531no—
325C2×#480—531no—
326C2×#428—532no—
327C2×#417—532no—
328C2×#407—531no—
329C2×C53—24no—
330C2×C53#309214yescomplemented
331C2×C53C22×#12311yescharacteristic
332C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
333C2×C53C22×#13711yescharacteristic
334D53—24no—
335D53—24no—
336D53—82no—
337C22×#41—531no—
338C24×C4—531no—
339C24×C4—1061no—
340C22×M16—532no—
341#525—538no—
342#525—1064no—
343C2×#123—1061no—
344C2×C22⋊Q8—531no—
345C2×C22⋊Q8C2×D5311yescharacteristic
346C2×#121—531no—
347C2×#121—531no—
348C2×#120—1061no—
349C2×C4×D4—1061no—
350C2×C4×D4—1061no—
351C2×#117—531no—
352C22×C4⋊C4—531no—
353C22×C4⋊C4—532no—
354C22×C4⋊C4—538no—
355C22×C4⋊C4—532no—
356C22×C4⋊C4—1061no—
357C22×C4⋊C4—1061no—
358C22×C4⋊C4—1064no—
359C22×C4⋊C4—1061no—
360C22×C4⋊C4C2×D5311yescharacteristic
361#484—5316no—
362#483—538no—
363#483C2×D5318yescomplemented
364#481—5316no—
365#480—538no—
366#480—538no—
367#428—5316no—
368C2×#107—532no—
369C2×#107—532no—
370C2×#106—532no—
371C2×#106C2×D5312yescomplemented
372#417—5316no—
373C2×#103—532no—
374C2×#103C2×D5312yescomplemented
375C2×#102—532no—
376C2×#102—532no—
377#407—538no—
378C2×C22⋊C8—531no—
379C2×C22⋊C8—532no—
380C2×C22⋊C8C2×D5311yescharacteristic
381C53C2×#309211yescharacteristic, complemented, Sylow
382C25—531no—
383C2×#41—534no—
384C2×#41—1062no—
385C2×#41—1068no—
386C22×Q8C22×D5311yescharacteristic
387C22×D4—531no—
388C22×D4—1061no—
389C22×D4—1061no—
390C22×D4—1061no—
391C23×C4—21no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—534no—
397C23×C4—531no—
398C23×C4—534no—
399C23×C4—1068no—
400C23×C4—1061no—
401C23×C4—1068no—
402C23×C4—1062no—
403C23×C4—1062no—
404C23×C4—1061no—
405C23×C4—2128no—
406C23×C4C22×D5311yescharacteristic
407C2×M16—538no—
408C2×M16—1068no—
409C22×C8—532no—
410C22×C8C22×D5312yes—
411#123—1068no—
412C22⋊Q8—22no—
413C22⋊Q8—534no—
414C22⋊Q8—1062no—
415C22⋊Q8C22×D5314yes—
416#121—534no—
417#121—534no—
418#121—1062no—
419#121—1062no—
420#120—1068no—
421C4×D4—1068no—
422C4×D4—1068no—
423#117—534no—
424#117—1062no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—532no—
429C2×C4⋊C4—534no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—538no—
432C2×C4⋊C4—538no—
433C2×C4⋊C4—1064no—
434C2×C4⋊C4—1068no—
435C2×C4⋊C4—1064no—
436C2×C4⋊C4—1068no—
437C2×C4⋊C4—1068no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1062no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1062no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4C22×D5312yes—
444C2×C4⋊C4C22×D5314yes—
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C42—1061no—
452#107—538no—
453#107—538no—
454#106—538no—
455#106C22×D5318yescomplemented
456#103—538no—
457#103C22×D5318yescomplemented
458#102—538no—
459#102—538no—
460C22⋊C8—534no—
461C22⋊C8—538no—
462C22⋊C8C22×D5314yes—
463C24—531no—
464C24—531no—
465C24—534no—
466C24—1064no—
467C24—1064no—
468C24—1064no—
469C24—1061no—
470C24D4×D5311yescharacteristic
471#41—10616no—
472#41—2128no—
473C2×Q8—21no—
474C2×Q8—24no—
475C2×Q8C23×D5312yes—
476C2×D4—532no—
477C2×D4—1064no—
478C2×D4—1064no—
479C2×D4—1064no—
480C2×D4—1061no—
481C2×D4—1064no—
482C2×D4—2124no—
483C2×D4—2124no—
484C2×D4—2124no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—531no—
492C22×C4—532no—
493C22×C4—538no—
494C22×C4—532no—
495C22×C4—534no—
496C22×C4—531no—
497C22×C4—534no—
498C22×C4—532no—
499C22×C4—534no—
500C22×C4—532no—
501C22×C4—1064no—
502C22×C4—10616no—
503C22×C4—10616no—
504C22×C4—1064no—
505C22×C4—1064no—
506C22×C4—1068no—
507C22×C4—1068no—
508C22×C4—1064no—
509C22×C4—1064no—
510C22×C4—1064no—
511C22×C4—1061no—
512C22×C4—1062no—
513C22×C4—2124no—
514C22×C4—21232no—
515C22×C4—2128no—
516C22×C4—2128no—
517C22×C4—2124no—
518C22×C4D4×D5312yes—
519C22×C4D4×D5311yescharacteristic
520C22×C4C23×D5312yes—
521C22×C4C23×D5311yescharacteristic
522M16—10616no—
523C2×C8—24no—
524C2×C8—534no—
525C2×C8—1064no—
526C2×C8C23×D5314yes—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—534no—
532C4⋊C4—532no—
533C4⋊C4—1068no—
534C4⋊C4—1068no—
535C4⋊C4—1064no—
536C4⋊C4—1064no—
537C4⋊C4—1062no—
538C4⋊C4—1061no—
539C4⋊C4C23×D5314yes—
540C4⋊C4C23×D5312yes—
541C22⋊C4—24no—
542C22⋊C4—1064no—
543C22⋊C4—1064no—
544C22⋊C4—1064no—
545C22⋊C4—1064no—
546C42—1064no—
547C23—22no—
548C23—22no—
549C23—534no—
550C23—534no—
551C23—532no—
552C23—532no—
553C23—1068no—
554C23—1068no—
555C23—10632no—
556C23—1062no—
557C23—1068no—
558C23—1062no—
559C23—1062no—
560C23—1062no—
561C23—1062no—
562C23—2122no—
563C23—2122no—
564C23—2122no—
565C23#1313324414yes—
566C23C2×D4×D5312yes—
567C23C2×D4×D5311yescharacteristic
568Q8—24no—
569Q8—42no—
570D4—1064no—
571D4—2128no—
572D4—2128no—
573D4—2128no—
574D4—2122no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—534no—
586C2×C4—532no—
587C2×C4—538no—
588C2×C4—538no—
589C2×C4—532no—
590C2×C4—1068no—
591C2×C4—1068no—
592C2×C4—1068no—
593C2×C4—1062no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1064no—
598C2×C4—2128no—
599C2×C4—21216no—
600C2×C4—21216no—
601C2×C4—2128no—
602C2×C4—2124no—
603C2×C4—2122no—
604C2×C4C2×D4×D5312yes—
605C2×C4C2×D4×D5311yescharacteristic
606C2×C4#41×D5314yes—
607C2×C4C24×D5311yescharacteristic
608C8—24no—
609C8—1064no—
610C22—22no—
611C22—22no—
612C22—28no—
613C22—534no—
614C22—534no—
615C22—10632no—
616C22—1062no—
617C22—1062no—
618C22—1062no—
619C22—1062no—
620C22—1068no—
621C22—1068no—
622C22—4244no—
623C22—2122no—
624C22—2122no—
625C22—2122no—
626C22#123×D5312yes—
627C22C2×#1313324414yes—
628C22C2×#1313324412yes—
629C22#137×D5312yes—
630C22C22×D4×D5311yescharacteristic
631C4—22no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—534no—
636C4—1062no—
637C4—2124no—
638C4—2122no—
639C4#123×D5312yes—
640C4C2×#41×D5312yes—
641C2—24no—
642C2—1064no—
643C2—1064no—
644C2—4242no—
645C2#2028463914yesdirect product factor
646C2C2×#123×D5311yescharacteristic
647C2C22×#1313324411yescharacteristic
648C2C2×#137×D5311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#483C2
socleC23×C53C22×D4
prime radicalC2×#16811136C2
hypercenterC2×#483D53
hypocenterC53C2×#3092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. #20284639C2direct (N × H)
4. C2×#16810957C2semidirect
5. C2×#16810943C2semidirect
6. C2×#16810951C2semidirect
7. C2×#16810937C2semidirect
8. C2×#16810936C2semidirect
9. C2×#16810662C2semidirect
10. C2×#16810623C2semidirect
11. C2×#16810585C2semidirect
12. C2×C53×#483C2semidirect
34. C2×C4⋊C4×D53C22semidirect
35. #16810957C22semidirect
36. #16810951C22semidirect
37. #16810943C22semidirect
38. #16810937C22semidirect
39. #16810936C22semidirect
40. #16810662C22semidirect
41. C2×#13133151C22semidirect
42. C2×#13133150C22semidirect
43. C2×#13133148C22semidirect
44. #16810623C22semidirect
45. C2×#13133141C22semidirect
46. C2×#13133139C22semidirect
47. C2×#13133137C22semidirect
48. #16810585C22semidirect
53. C53×#483C22semidirect
54. C2×#106×C53C22semidirect
55. C2×#103×C53C22semidirect
114. #13133151C23semidirect
115. #13133150C23semidirect
116. #13133148C23semidirect
117. #13133141C23semidirect
118. #13133139C23semidirect
119. #13133137C23semidirect
135. #106×C53C23semidirect
136. #103×C53C23semidirect
323. C2×#483D53semidirect
330. C2×C53#3092semidirect
363. #483C2×D53semidirect
371. C2×#106C2×D53semidirect
374. C2×#103C2×D53semidirect
381. C53C2×#3092semidirect
455. #106C22×D53semidirect
457. #103C22×D53semidirect
645. C2#20284639direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111362
C22×C4⋊C4×D532
#202846392
C2×#168109572
C2×#168109432
C2×#168109512
C2×#168109372
C2×#168109362
C2×#168106622
C2×#168106232
C2×#168105852
C2×C53×#4832
C2×#309253