Finite Group Databasebeta

← C2×#20284634 #28251866

Subgroups — C2×#20284634 #28251866

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×#16811133C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#20284634C2116yesmaximal, direct product factor
4C2×#16810960C212yesmaximal, complemented
5C2×#16810940C212yesmaximal, complemented
6C2×#16810954C211yesmaximal, characteristic, complemented
7C2×#16810953C211yesmaximal, characteristic, complemented
8C2×#16810934C211yesmaximal, characteristic, complemented
9C2×#16810664C212yesmaximal, complemented
10C2×#16810621C212yesmaximal, complemented
11C2×#16810585C211yesmaximal, characteristic, complemented
12C2×C53×#480C211yesmaximal, characteristic, complemented
13C22×#11719947C2211yescharacteristic
14C23×C4×D53—21no—
15C23×C4×D53C2211yescharacteristic
16C22×#11719912C2212yes—
17#16811133—24no—
18#16811133C2218yes—
19C2×#13133199—21no—
20C2×#13133197C2211yescharacteristic
21C2×#13133196C2211yescharacteristic
22C2×#13133188C2211yescharacteristic
23C2×#13133185—21no—
24C2×D4×Dic53—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#16810960C22116yescomplemented
36#16810954C2218yescomplemented
37#16810953C2218yescomplemented
38#16810940C22116yescomplemented
39#16810934C2218yescomplemented
40#16810664C22116yescomplemented
41C2×#13133150C2212yescomplemented
42C2×#13133149C2212yescomplemented
43C2×#13133148C2212yescomplemented
44#16810621C22116yescomplemented
45C2×#13133142C2212yescomplemented
46C2×#13133140C2212yescomplemented
47C2×#13133138C2212yescomplemented
48#16810585C2218yescomplemented
49C2×C22⋊#88322C2211yescharacteristic
50C2×#13133129C2212yes—
51C2×#121×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#480C2218yescomplemented
54C2×#107×C53C2212yescomplemented
55C2×#102×C53C2212yescomplemented
56C2×C22⋊C8×C53C2211yescharacteristic
57C24×D53D411yescharacteristic
58C2×#11719947—22no—
59C2×#11719947—28no—
60C2×#11719947C2314yes—
61C22×Dic106C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C22⋊D53—21no—
64C23×Dic53—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—48no—
72C22×C4×D53D411yescharacteristic
73C22×C4×D53D411yescharacteristic
74C22×C4×D53D414yes—
75C22×C4×D53C2311yescharacteristic
76C22×C4×D53C2314yes—
77C22×C4×D53C2311yescharacteristic
78C2×#11719912—28no—
79C2×#11719912C2318yes—
80C22×#88322C2312yes—
81#13133199—28no—
82#13133197—22no—
83#13133197C2314yes—
84#13133196—22no—
85#13133196C2314yes—
86#13133188—22no—
87#13133188C2314yes—
88#13133185—28no—
89D4×Dic53—28no—
90#13133177—28no—
91#13133175—22no—
92#13133175C2314yes—
93C2×C4⋊Dic53—21no—
94C2×C4⋊Dic53—22no—
95C2×C4⋊Dic53—21no—
96C2×C4⋊Dic53C2312yes—
97C2×C4⋊Dic53C2314yes—
98C2×C4⋊Dic53C2311yescharacteristic
99C2×#11719919—21no—
100C2×#11719919—22no—
101C2×#11719919—22no—
102C2×#11719919—28no—
103C4⋊C4×D53—24no—
104C4⋊C4×D53—24no—
105C4⋊C4×D53—28no—
106C4⋊C4×D53—28no—
107C4⋊C4×D53C2318yes—
108C4⋊C4×D53C2318yes—
109C2×C22⋊Dic53—21no—
110C2×C22⋊Dic53—21no—
111C2×#11719921—21no—
112C2×#11719921—21no—
113C2×C4×Dic53—21no—
114#13133150C2318yescomplemented
115#13133149C2318yescomplemented
116#13133148C2318yescomplemented
117#13133142C2318yescomplemented
118#13133140C2318yescomplemented
119#13133138C2318yescomplemented
120C22⋊#88322C2314yes—
121#13133129C2318yes—
122C22×D4×C53—21no—
123C22×D4×C53C2311yescharacteristic
124C23×C4×C53—21no—
125C23×C4×C53C2311yescharacteristic
126C22×C8×C53C2312yes—
127#121×C53—22no—
128#121×C53C2314yes—
129C2×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53—22no—
131C2×C4⋊C4×C53C2312yes—
132C2×C4⋊C4×C53C2314yes—
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C22⋊C4×C53—21no—
135#107×C53C2318yescomplemented
136#102×C53C2318yescomplemented
137C22⋊C8×C53C2314yes—
138C24×C53—21no—
139C24×C53C2×D411yescharacteristic
140C23×D53—24no—
141C23×D53—24no—
142C23×D53—24no—
143C23×D53C2×D414yes—
144C23×D53C2×D411yescharacteristic
145C23×D53C2×D411yescharacteristic
146C2×D4×C53—24no—
147C2×D4×C53—21no—
148C2×D4×C53—24no—
149C2×D4×C53—44no—
150C2×D4×C53C2412yes—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—24no—
154C22×C4×C53—22no—
155C22×C4×C53—44no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2412yes—
159C22×C4×C53C2411yescharacteristic
160C2×C22⋊D53—24no—
161C2×C22⋊D53—24no—
162C2×C22⋊D53—44no—
163C2×C22⋊D53—44no—
164C22×Dic53—24no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—24no—
168C22×Dic53—22no—
169C22×Dic53—24no—
170C22×Dic53—44no—
171C22×Dic53—44no—
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D414yes—
174C22×Dic53C2×D411yescharacteristic
175#11719947—216no—
176#11719947—48no—
177C2×C4×D53—24no—
178C2×C4×D53—24no—
179C2×C4×D53—24no—
180C2×C4×D53—216no—
181C2×C4×D53—216no—
182C2×C4×D53—28no—
183C2×C4×D53—28no—
184C2×C4×D53—48no—
185C2×C4×D53—48no—
186C2×C4×D53—432no—
187C2×C4×D53C2×D414yes—
188C2×C4×D53C2×D412yes—
189C2×C4×D53C2×D414yes—
190C2×C4×D53C2×D412yes—
191C2×C4×D53#4118yes—
192C2×C4×D53C2412yes—
193C2×C4×D53C2412yes—
194C2×Dic106—24no—
195C2×Dic106—21no—
196C2×Dic106C2412yes—
197C2×C8×C53—24no—
198C2×C8×C53C2414yes—
199C4⋊C4×C53—24no—
200C4⋊C4×C53—22no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53C2414yes—
203C4⋊C4×C53C2412yes—
204C22⋊C4×C53—24no—
205C22⋊Dic53—24no—
206C22⋊Dic53—24no—
207#11719921—24no—
208#11719921—24no—
209C4⋊Dic53—24no—
210C4⋊Dic53—22no—
211C4⋊Dic53—21no—
212C4⋊Dic53—24no—
213C4⋊Dic53C2414yes—
214C4⋊Dic53C2412yes—
215#11719919—24no—
216#11719919—28no—
217#11719919—28no—
218C4×Dic53—24no—
219C2×#88322—24no—
220C2×#88322C2414yes—
221#11719912—216no—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—42no—
226C23×C53—42no—
227C23×C53—42no—
228C23×C53AGL(1,ℤ/8)14yes—
229C23×C53C22×D412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—232no—
232C22×D53—22no—
233C22×D53—28no—
234C22×D53—22no—
235C22×D53—22no—
236C22×D53—22no—
237C22×D53—28no—
238C22×D53—28no—
239C22×D53#12314yes—
240C22×D53#13714yes—
241C22×D53C22×D412yes—
242C22×D53C22×D412yes—
243D4×C53—24no—
244D4×C53—48no—
245D4×C53—42no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—24no—
250C2×C4×C53—22no—
251C2×C4×C53—48no—
252C2×C4×C53—44no—
253C2×C4×C53—42no—
254C2×C4×C53C22×D412yes—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4114yes—
257C2×C4×C53C2511yescharacteristic
258C22⋊D53—48no—
259C22⋊D53—48no—
260C2×Dic53—22no—
261C2×Dic53—28no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—42no—
266C2×Dic53—48no—
267C2×Dic53—44no—
268C2×Dic53—42no—
269C2×Dic53—48no—
270C2×Dic53#12314yes—
271C2×Dic53C22×D412yes—
272C2×Dic53C22×D412yes—
273C4×D53—28no—
274C4×D53—24no—
275C4×D53—24no—
276C4×D53—28no—
277C4×D53—416no—
278C4×D53—416no—
279C4×D53#12318yes—
280C4×D53C2×#4118yes—
281Dic106—24no—
282Dic106—42no—
283C8×C53—24no—
284#88322—24no—
285C2×#3092—531nomaximal, Sylow
286C22×C53—28no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—84no—
293C22×C53C2×#12312yes—
294C22×C53C2×AGL(1,ℤ/8)14yes—
295C22×C53C2×AGL(1,ℤ/8)12yes—
296C22×C53C2×#13712yes—
297C22×C53C23×D411yescharacteristic
298C2×D53—232no—
299C2×D53—28no—
300C2×D53—28no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53C2×#12314yes—
306C2×D53C2×#13714yes—
307C4×C53—22no—
308C4×C53—44no—
309C4×C53—42no—
310C4×C53C2×#12312yes—
311C4×C53C22×#4112yes—
312Dic53—22no—
313Dic53—42no—
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×#483—531no—
324C2×#480—531no—
325C2×#480D5311yescharacteristic, complemented
326C2×#428—532no—
327C2×#417—532no—
328C2×#407—531no—
329C2×C53—24no—
330C2×C53—82no—
331C2×C53#309214yescomplemented
332C2×C53C22×#12311yescharacteristic
333C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
334C2×C53C22×#13711yescharacteristic
335D53—24no—
336D53—24no—
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⋊Q8—531no—
346C2×#121—531no—
347C2×#121C2×D5311yescharacteristic
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#483—538no—
364#481—5316no—
365#480—538no—
366#480C2×D5318yescomplemented
367#428—5316no—
368C2×#107—532no—
369C2×#107C2×D5312yescomplemented
370C2×#106—532no—
371C2×#106—532no—
372#417—5316no—
373C2×#103—532no—
374C2×#103—532no—
375C2×#102—532no—
376C2×#102C2×D5312yescomplemented
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×Q8—531no—
387C22×D4—21no—
388C22×D4—1061no—
389C22×D4—1061no—
390C22×D4C22×D5311yescharacteristic
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—1068no—
401C23×C4—1062no—
402C23×C4—1062no—
403C23×C4—1061no—
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—534no—
413C22⋊Q8—534no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416#121—22no—
417#121—534no—
418#121—1062no—
419#121C22×D5314yes—
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—532no—
428C2×C4⋊C4—534no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—538no—
431C2×C4⋊C4—538no—
432C2×C4⋊C4—1064no—
433C2×C4⋊C4—1068no—
434C2×C4⋊C4—1064no—
435C2×C4⋊C4—1068no—
436C2×C4⋊C4—1068no—
437C2×C4⋊C4—1062no—
438C2×C4⋊C4—1062no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1062no—
441C2×C4⋊C4—1061no—
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#107C22×D5318yescomplemented
454#106—538no—
455#106—538no—
456#103—538no—
457#103—538no—
458#102—538no—
459#102C22×D5318yescomplemented
460C22⋊C8—534no—
461C22⋊C8—538no—
462C22⋊C8C22×D5314yes—
463C24—21no—
464C24—531no—
465C24—531no—
466C24—534no—
467C24—1064no—
468C24—1064no—
469C24—1064no—
470C24D4×D5311yescharacteristic
471#41—10616no—
472#41—2128no—
473C2×Q8—532no—
474C2×Q8—1061no—
475C2×Q8—1064no—
476C2×D4—24no—
477C2×D4—21no—
478C2×D4—24no—
479C2×D4—44no—
480C2×D4—1064no—
481C2×D4—1064no—
482C2×D4—2124no—
483C2×D4—2124no—
484C2×D4C23×D5312yes—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—531no—
491C22×C4—532no—
492C22×C4—538no—
493C22×C4—532no—
494C22×C4—534no—
495C22×C4—531no—
496C22×C4—534no—
497C22×C4—532no—
498C22×C4—534no—
499C22×C4—532no—
500C22×C4—1064no—
501C22×C4—10616no—
502C22×C4—10616no—
503C22×C4—1064no—
504C22×C4—1068no—
505C22×C4—1068no—
506C22×C4—1064no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1064no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1062no—
513C22×C4—21232no—
514C22×C4—2128no—
515C22×C4—2128no—
516C22×C4—2124no—
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—21no—
530C4⋊C4—534no—
531C4⋊C4—532no—
532C4⋊C4—1068no—
533C4⋊C4—1068no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1062no—
537C4⋊C4—1061no—
538C4⋊C4—1064no—
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—22no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—534no—
554C23—534no—
555C23—532no—
556C23—532no—
557C23—1068no—
558C23—1068no—
559C23—10632no—
560C23—1062no—
561C23—1068no—
562C23—1062no—
563C23—1062no—
564C23—1062no—
565C23#1313324114yes—
566C23C2×D4×D5312yes—
567C23C2×D4×D5311yescharacteristic
568Q8—1064no—
569Q8—2122no—
570D4—24no—
571D4—48no—
572D4—42no—
573D4—2128no—
574D4—2128no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—534no—
584C2×C4—532no—
585C2×C4—538no—
586C2×C4—538no—
587C2×C4—532no—
588C2×C4—1068no—
589C2×C4—1068no—
590C2×C4—1068no—
591C2×C4—1062no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1064no—
597C2×C4—21216no—
598C2×C4—21216no—
599C2×C4—2128no—
600C2×C4—2122no—
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—42no—
614C22—42no—
615C22—42no—
616C22—84no—
617C22—534no—
618C22—534no—
619C22—10632no—
620C22—1062no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—1068no—
625C22—1068no—
626C22#123×D5312yes—
627C22C2×#1313324114yes—
628C22C2×#1313324112yes—
629C22#137×D5312yes—
630C22C22×D4×D5311yescharacteristic
631C4—22no—
632C4—44no—
633C4—42no—
634C4—534no—
635C4—1062no—
636C4—2124no—
637C4—2122no—
638C4—2122no—
639C4#123×D5312yes—
640C4C2×#41×D5312yes—
641C2—24no—
642C2—82no—
643C2—1064no—
644C2—1064no—
645C2#2028463414yesdirect product factor
646C2C2×#123×D5311yescharacteristic
647C2C22×#1313324111yescharacteristic
648C2C2×#137×D5311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#480C2
socleC23×C53C22×D4
prime radicalC2×#16811133C2
hypercenterC2×#480D53
hypocenterC53C2×#3092
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. #20284634C2direct (N × H)
4. C2×#16810960C2semidirect
5. C2×#16810940C2semidirect
6. C2×#16810954C2semidirect
7. C2×#16810953C2semidirect
8. C2×#16810934C2semidirect
9. C2×#16810664C2semidirect
10. C2×#16810621C2semidirect
11. C2×#16810585C2semidirect
12. C2×C53×#480C2semidirect
34. C2×C4⋊C4×D53C22semidirect
35. #16810960C22semidirect
36. #16810954C22semidirect
37. #16810953C22semidirect
38. #16810940C22semidirect
39. #16810934C22semidirect
40. #16810664C22semidirect
41. C2×#13133150C22semidirect
42. C2×#13133149C22semidirect
43. C2×#13133148C22semidirect
44. #16810621C22semidirect
45. C2×#13133142C22semidirect
46. C2×#13133140C22semidirect
47. C2×#13133138C22semidirect
48. #16810585C22semidirect
53. C53×#480C22semidirect
54. C2×#107×C53C22semidirect
55. C2×#102×C53C22semidirect
114. #13133150C23semidirect
115. #13133149C23semidirect
116. #13133148C23semidirect
117. #13133142C23semidirect
118. #13133140C23semidirect
119. #13133138C23semidirect
135. #107×C53C23semidirect
136. #102×C53C23semidirect
325. C2×#480D53semidirect
331. C2×C53#3092semidirect
366. #480C2×D53semidirect
369. C2×#107C2×D53semidirect
376. C2×#102C2×D53semidirect
381. C53C2×#3092semidirect
453. #107C22×D53semidirect
459. #102C22×D53semidirect
645. C2#20284634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111332
C22×C4⋊C4×D532
#202846342
C2×#168109602
C2×#168109402
C2×#168109542
C2×#168109532
C2×#168109342
C2×#168106642
C2×#168106212
C2×#168105852
C2×C53×#4802
C2×#309253