Finite Group Databasebeta

← C2×#20284665 #28251900

Subgroups — C2×#20284665 #28251900

Order 13568.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic
2C2×#117×D53C211yesmaximal, characteristic
3#20284665C2116yesmaximal, direct product factor
4C2×#16810960C212yesmaximal, complemented
5C2×#16810967C212yesmaximal, complemented
6C2×#16810966C212yesmaximal, complemented
7C2×#16810955C212yesmaximal, complemented
8C2×#16810663C211yesmaximal, characteristic, complemented
9C2×#16810633C211yesmaximal, characteristic, complemented
10C2×#16810631C211yesmaximal, characteristic, complemented
11C2×#16810586C211yesmaximal, characteristic
12C2×C53×#428C211yesmaximal, characteristic, complemented
13C23×Dic106C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C22×#11719917C2211yescharacteristic
16C2×#13133197—21no—
17C2×#13133197—24no—
18C2×#13133197—21no—
19C2×#13133197C2212yes—
20C2×#13133197C2212yes—
21C2×#13133197C2218yes—
22C2×#13133176C2211yescharacteristic
23C2×#13133175C2211yescharacteristic
24C2×#13133175C2211yescharacteristic
25C2×#13133174—21no—
26C2×#13133173—21no—
27#117×D53—24no—
28#117×D53C2218yes—
29C22×C4⋊Dic53—21no—
30C2×C4⋊C4×D53C2211yescharacteristic
31C2×C4⋊C4×D53C2211yescharacteristic
32C22×#11719921—21no—
33C2×C22⋊C4×D53—21no—
34C2×C42×D53—21no—
35#16810967C22116yescomplemented
36#16810966C22116yescomplemented
37#16810960C22116yescomplemented
38#16810955C22116yescomplemented
39#16810663C2218yescomplemented
40C2×#13133150C2212yescomplemented
41C2×#13133148C2212yescomplemented
42#16810633C2218yescomplemented
43#16810631C2218yescomplemented
44C2×#13133142C2212yescomplemented
45C2×#13133142C2212yescomplemented
46C2×#13133140C2212yescomplemented
47C2×#13133140C2212yescomplemented
48#16810586C2218yes—
49C2×#13133129C2212yes—
50C2×#13133129C2212yes—
51C22×M16×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C53×#428C2218yescomplemented
55C2×#107×C53C2212yescomplemented
56C2×#106×C53C2212yescomplemented
57C24×D53D411yescharacteristic
58C22×Dic106—24no—
59C22×Dic106—24no—
60C22×Dic106—21no—
61C22×Dic106—22no—
62C22×Dic106C2312yes—
63C22×Dic106C2314yes—
64C23×Dic53—21no—
65C23×Dic53D411yescharacteristic
66C22×C4×D53—24no—
67C22×C4×D53—24no—
68C22×C4×D53—21no—
69C22×C4×D53—21no—
70C22×C4×D53D411yescharacteristic
71C22×C4×D53D414yes—
72C22×C4×D53D411yescharacteristic
73C22×C4×D53C2311yescharacteristic
74C22×C4×D53C2314yes—
75C22×C4×D53C2311yescharacteristic
76C2×#11719917—24no—
77C2×#11719917C2314yes—
78C22×#88322C2312yes—
79#13133197—28no—
80#13133197—24no—
81#13133197—28no—
82#13133197—24no—
83#13133197C2318yes—
84#13133197C2318yes—
85#13133176—22no—
86#13133176C2314yes—
87#13133175—22no—
88#13133175—22no—
89#13133175C2314yes—
90#13133175C2314yes—
91#13133174—28no—
92#13133173—28no—
93C2×C4⋊Dic53—22no—
94C2×C4⋊Dic53—22no—
95C2×C4⋊Dic53—28no—
96C2×C4⋊Dic53C2311yescharacteristic
97C2×C4⋊Dic53C2311yescharacteristic
98C2×#11719919—21no—
99C2×#11719919—21no—
100C4⋊C4×D53—22no—
101C4⋊C4×D53—22no—
102C4⋊C4×D53C2314yes—
103C4⋊C4×D53C2314yes—
104C2×C22⋊Dic53—21no—
105C2×#11719921—21no—
106C2×#11719921—21no—
107C2×#11719921—22no—
108C2×#11719921—28no—
109C2×#11719921—22no—
110C22⋊C4×D53—28no—
111C2×C4×Dic53—21no—
112C2×C4×Dic53—21no—
113C2×C4×Dic53—21no—
114C42×D53—28no—
115#13133150C2318yescomplemented
116#13133148C2318yescomplemented
117#13133142C2318yescomplemented
118#13133142C2318yescomplemented
119#13133140C2318yescomplemented
120#13133140C2318yescomplemented
121#13133129C2318yes—
122#13133129C2318yes—
123C23×C4×C53—21no—
124C23×C4×C53C2311yescharacteristic
125C2×M16×C53—24no—
126C2×M16×C53C2314yes—
127C22×C8×C53C2312yes—
128#117×C53—22no—
129#117×C53C2314yes—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53—22no—
132C2×C4⋊C4×C53C2312yes—
133C2×C4⋊C4×C53C2314yes—
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C4⋊C4×C53C2311yescharacteristic
136C2×C22⋊C4×C53—21no—
137C2×C42×C53—21no—
138#107×C53C2318yescomplemented
139#106×C53C2318yescomplemented
140C24×C53C2×D411yescharacteristic
141C23×D53—24no—
142C23×D53—24no—
143C23×D53—42no—
144C23×D53C2×D411yescharacteristic
145C23×D53C2×D414yes—
146C23×D53C2×D411yescharacteristic
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—24no—
150C22×C4×C53—22no—
151C22×C4×C53—44no—
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D412yes—
154C22×C4×C53#4114yes—
155C22×C4×C53#4114yes—
156C22×C4×C53C2411yescharacteristic
157C22×C4×C53C2412yes—
158C22×Dic53—24no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—24no—
162C22×Dic53—22no—
163C22×Dic53—44no—
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D414yes—
167C2×C4×D53—24no—
168C2×C4×D53—216no—
169C2×C4×D53—24no—
170C2×C4×D53—22no—
171C2×C4×D53—22no—
172C2×C4×D53—28no—
173C2×C4×D53—22no—
174C2×C4×D53—22no—
175C2×C4×D53—28no—
176C2×C4×D53—24no—
177C2×C4×D53—24no—
178C2×C4×D53—44no—
179C2×C4×D53—44no—
180C2×C4×D53C2×D414yes—
181C2×C4×D53C2×D412yes—
182C2×C4×D53C2×D414yes—
183C2×C4×D53C2×D412yes—
184C2×C4×D53C2412yes—
185C2×C4×D53C2412yes—
186C2×Dic106—24no—
187C2×Dic106—28no—
188C2×Dic106—24no—
189C2×Dic106—28no—
190C2×Dic106—24no—
191C2×Dic106—22no—
192C2×Dic106—48no—
193C2×Dic106—44no—
194C2×Dic106C2414yes—
195M16×C53—28no—
196C2×C8×C53—24no—
197C2×C8×C53C2414yes—
198C4⋊C4×C53—24no—
199C4⋊C4×C53—22no—
200C4⋊C4×C53—21no—
201C4⋊C4×C53—21no—
202C4⋊C4×C53C2412yes—
203C4⋊C4×C53C2414yes—
204C4⋊C4×C53C2412yes—
205C22⋊C4×C53—24no—
206C42×C53—24no—
207C22⋊Dic53—24no—
208#11719921—24no—
209#11719921—24no—
210#11719921—28no—
211#11719921—28no—
212C4⋊Dic53—21no—
213C4⋊Dic53—21no—
214C4⋊Dic53—28no—
215C4⋊Dic53—28no—
216C4⋊Dic53C2412yes—
217C4⋊Dic53C2412yes—
218#11719919—24no—
219#11719919—24no—
220C4×Dic53—24no—
221C4×Dic53—24no—
222C4×Dic53—24no—
223#11719917—28no—
224C2×#88322—24no—
225C2×#88322C2414yes—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53#12314yes—
229C23×C53C22×D412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—28no—
234C22×D53—22no—
235C22×D53—22no—
236C22×D53—22no—
237C22×D53—28no—
238C22×D53—44no—
239C22×D53—416no—
240C22×D53#13714yes—
241C22×D53C22×D412yes—
242C22×D53C22×D412yes—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—24no—
246C2×C4×C53—48no—
247C2×C4×C53—42no—
248C2×C4×C53—42no—
249C2×C4×C53—44no—
250C2×C4×C53#12318yes—
251C2×C4×C53#12314yes—
252C2×C4×C53C22×D412yes—
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C2×#4112yes—
255C2×C4×C53C2×#4114yes—
256C2×C4×C53C2×#4112yes—
257C2×C4×C53C2×#4114yes—
258C2×C4×C53C2511yescharacteristic
259C2×Dic53—28no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—24no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—48no—
266C2×Dic53—44no—
267C2×Dic53—42no—
268C2×Dic53—42no—
269C2×Dic53C22×D412yes—
270C2×Dic53C22×D412yes—
271C4×D53—28no—
272C4×D53—24no—
273C4×D53—28no—
274C4×D53—24no—
275C4×D53—22no—
276C4×D53—22no—
277C4×D53—22no—
278C4×D53—22no—
279C4×D53—48no—
280C4×D53—48no—
281Dic106—24no—
282Dic106—24no—
283Dic106—48no—
284Dic106—44no—
285C8×C53—24no—
286#88322—24no—
287C2×#3095—531nomaximal, Sylow
288C22×C53—28no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53C2×#12314yes—
292C22×C53C2×#12312yes—
293C22×C53C2×#13712yes—
294C22×C53#57512yes—
295C22×C53C23×D411yescharacteristic
296C2×D53—28no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—416no—
303C2×D53—44no—
304C2×D53C2×#13714yes—
305C4×C53—42no—
306C4×C53—42no—
307C4×C53—44no—
308C4×C53C2×#12314yes—
309C4×C53C2×#12312yes—
310C4×C53C22×#4111yescharacteristic
311C4×C53C22×#4111yescharacteristic
312Dic53—22no—
313Dic53—22no—
314Dic53—44no—
315Dic53—42no—
316Dic53—42no—
317C22×C22⋊Q8—531no—
318C22×#117—531no—
319#3095—5316no—
320C2×#484—532no—
321C2×#485—532no—
322C2×#485—532no—
323C2×#483—532no—
324C2×#428—531no—
325C2×#428D5311yescharacteristic, complemented
326C2×#419—531no—
327C2×#419—531no—
328C2×#407—531no—
329C2×C53—24no—
330C2×C53#309514yescomplemented
331C2×C53C22×#12311yescharacteristic
332C2×C53C22×#13711yescharacteristic
333C2×C53C2×#57511yescharacteristic
334D53—24no—
335D53—42no—
336C23×Q8—531no—
337C24×C4—531no—
338C22×M16—531no—
339C22×M16C2×D5311yescharacteristic
340C2×C22⋊Q8—532no—
341C2×C22⋊Q8—532no—
342C2×C22⋊Q8—538no—
343C2×C22⋊Q8—1064no—
344C2×C22⋊Q8—1061no—
345C2×C22⋊Q8—1061no—
346C2×#117—531no—
347C2×#117—538no—
348C2×#117—531no—
349C2×#117—531no—
350C2×#117—1061no—
351C2×#117—1061no—
352C2×#117—1064no—
353C2×#117C2×D5311yescharacteristic
354C22×C4⋊C4—531no—
355C22×C4⋊C4—531no—
356C22×C4⋊C4—1061no—
357C22×C4⋊C4C2×D5311yescharacteristic
358C22×C22⋊C4—1061no—
359C22×C22⋊C4—1061no—
360C22×C42—1061no—
361#485—5316no—
362#485—5316no—
363#484—5316no—
364#483—5316no—
365#428—538no—
366#428C2×D5318yescomplemented
367C2×#107—532no—
368C2×#107C2×D5312yescomplemented
369C2×#106—532no—
370C2×#106C2×D5312yescomplemented
371#419—538no—
372#419—538no—
373C2×#103—532no—
374C2×#103—532no—
375C2×#103—532no—
376C2×#103—532no—
377#407—538no—
378C2×C22⋊C8—532no—
379C2×C22⋊C8—532no—
380C53C2×#309511yescharacteristic, complemented, Sylow
381C25—531no—
382C22×Q8—532no—
383C22×Q8—534no—
384C22×Q8—1062no—
385C22×Q8—1061no—
386C22×Q8—1064no—
387C22×Q8—1064no—
388C23×C4—21no—
389C23×C4—531no—
390C23×C4—534no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—531no—
394C23×C4—531no—
395C23×C4—534no—
396C23×C4—1061no—
397C23×C4—1061no—
398C23×C4—1064no—
399C23×C4—1064no—
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C2×M16—24no—
403C2×M16—534no—
404C2×M16—1064no—
405C2×M16C22×D5314yes—
406C22×C8—532no—
407C22×C8C22×D5312yes—
408C22⋊Q8—538no—
409C22⋊Q8—538no—
410C22⋊Q8—1068no—
411C22⋊Q8—1064no—
412C22⋊Q8—1064no—
413C22⋊Q8—1068no—
414#117—22no—
415#117—534no—
416#117—534no—
417#117—534no—
418#117—1068no—
419#117—1068no—
420#117—1062no—
421#117—1062no—
422#117—1062no—
423#117C22×D5314yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—534no—
429C2×C4⋊C4—534no—
430C2×C4⋊C4—1062no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1068no—
435C2×C4⋊C4—1062no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5312yes—
440C2×C4⋊C4C22×D5314yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1068no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1068no—
446C2×C22⋊C4—1062no—
447C2×C22⋊C4—1062no—
448C2×C22⋊C4—1061no—
449C2×C42—21no—
450C2×C42—1068no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42—1061no—
454#107—538no—
455#107C22×D5318yescomplemented
456#106—538no—
457#106C22×D5318yescomplemented
458#103—538no—
459#103—538no—
460#103—538no—
461#103—538no—
462C22⋊C8—538no—
463C22⋊C8—538no—
464C24—531no—
465C24—531no—
466C24—534no—
467C24—1064no—
468C24—1064no—
469C24—2122no—
470C24D4×D5311yescharacteristic
471C2×Q8—534no—
472C2×Q8—1064no—
473C2×Q8—1062no—
474C2×Q8—1064no—
475C2×Q8—1068no—
476C2×Q8—1064no—
477C2×Q8—1068no—
478C2×Q8—2128no—
479C2×Q8—2124no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—534no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—534no—
489C22×C4—532no—
490C22×C4—532no—
491C22×C4—534no—
492C22×C4—532no—
493C22×C4—532no—
494C22×C4—1064no—
495C22×C4—1064no—
496C22×C4—1064no—
497C22×C4—1064no—
498C22×C4—10616no—
499C22×C4—1064no—
500C22×C4—1062no—
501C22×C4—1062no—
502C22×C4—1068no—
503C22×C4—1062no—
504C22×C4—1062no—
505C22×C4—1068no—
506C22×C4—1064no—
507C22×C4—1061no—
508C22×C4—1061no—
509C22×C4—1062no—
510C22×C4—2124no—
511C22×C4—2124no—
512C22×C4—2124no—
513C22×C4#1171994514yes—
514C22×C4D4×D5312yes—
515C22×C4D4×D5311yescharacteristic
516C22×C4#1171994914yes—
517C22×C4C23×D5312yes—
518C22×C4C23×D5311yescharacteristic
519M16—28no—
520M16—1068no—
521C2×C8—24no—
522C2×C8—534no—
523C2×C8—1064no—
524C2×C8C23×D5314yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—532no—
530C4⋊C4—532no—
531C4⋊C4—1064no—
532C4⋊C4—1064no—
533C4⋊C4—1068no—
534C4⋊C4—1068no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4C23×D5312yes—
538C4⋊C4C23×D5312yes—
539C4⋊C4C23×D5314yes—
540C22⋊C4—24no—
541C22⋊C4—1064no—
542C22⋊C4—1064no—
543C22⋊C4—1068no—
544C22⋊C4—1068no—
545C22⋊C4—1064no—
546C42—24no—
547C42—1064no—
548C42—1064no—
549C42—1064no—
550C23—22no—
551C23—22no—
552C23—534no—
553C23—532no—
554C23—532no—
555C23—1068no—
556C23—1062no—
557C23—1062no—
558C23—1068no—
559C23—1062no—
560C23—1062no—
561C23—1062no—
562C23—2124no—
563C23—21216no—
564C23#1313320214yes—
565C23C2×D4×D5312yes—
566C23C2×D4×D5311yescharacteristic
567Q8—1064no—
568Q8—1064no—
569Q8—2128no—
570Q8—2124no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—532no—
579C2×C4—532no—
580C2×C4—1068no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1068no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1068no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1062no—
593C2×C4—1064no—
594C2×C4—2128no—
595C2×C4—2128no—
596C2×C4—2128no—
597C2×C4—2124no—
598C2×C4—2122no—
599C2×C4—2122no—
600C2×C4#1313320218yes—
601C2×C4#1313320214yes—
602C2×C4C2×D4×D5312yes—
603C2×C4C2×D4×D5311yescharacteristic
604C2×C4C2×#1171994512yes—
605C2×C4C2×#1171994514yes—
606C2×C4C2×#1171994912yes—
607C2×C4C2×#1171994914yes—
608C2×C4C24×D5311yescharacteristic
609C8—24no—
610C8—1064no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—534no—
615C22—1062no—
616C22—1062no—
617C22—1068no—
618C22—1062no—
619C22—1062no—
620C22—1062no—
621C22—21216no—
622C22—2124no—
623C22C2×#1313320214yes—
624C22C2×#1313320212yes—
625C22#137×D5312yes—
626C22#1681141912yes—
627C22C22×D4×D5311yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—1062no—
632C4—1062no—
633C4—2122no—
634C4—2122no—
635C4—2124no—
636C4C2×#1313320214yes—
637C4C2×#1313320212yes—
638C4C22×#1171994511yescharacteristic
639C4C22×#1171994911yescharacteristic
640C2—24no—
641C2—1064no—
642C2—2122no—
643C2#2028466514yesdirect product factor
644C2C22×#1313320211yescharacteristic
645C2C2×#137×D5311yescharacteristic
646C2C2×#1681141911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC24×D53D4
hypercenterC2×#428D53
hypocenterC53C2×#3095
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #20284665C2direct (N × H)
4. C2×#16810960C2semidirect
5. C2×#16810967C2semidirect
6. C2×#16810966C2semidirect
7. C2×#16810955C2semidirect
8. C2×#16810663C2semidirect
9. C2×#16810633C2semidirect
10. C2×#16810631C2semidirect
12. C2×C53×#428C2semidirect
35. #16810967C22semidirect
36. #16810966C22semidirect
37. #16810960C22semidirect
38. #16810955C22semidirect
39. #16810663C22semidirect
40. C2×#13133150C22semidirect
41. C2×#13133148C22semidirect
42. #16810633C22semidirect
43. #16810631C22semidirect
44. C2×#13133142C22semidirect
45. C2×#13133142C22semidirect
46. C2×#13133140C22semidirect
47. C2×#13133140C22semidirect
54. C53×#428C22semidirect
55. C2×#107×C53C22semidirect
56. C2×#106×C53C22semidirect
115. #13133150C23semidirect
116. #13133148C23semidirect
117. #13133142C23semidirect
118. #13133142C23semidirect
119. #13133140C23semidirect
120. #13133140C23semidirect
138. #107×C53C23semidirect
139. #106×C53C23semidirect
325. C2×#428D53semidirect
330. C2×C53#3095semidirect
366. #428C2×D53semidirect
368. C2×#107C2×D53semidirect
370. C2×#106C2×D53semidirect
380. C53C2×#3095semidirect
455. #107C22×D53semidirect
457. #106C22×D53semidirect
643. C2#20284665direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331972
C2×#117×D532
#202846652
C2×#168109602
C2×#168109672
C2×#168109662
C2×#168109552
C2×#168106632
C2×#168106332
C2×#168106312
C2×#168105862
C2×C53×#4282
C2×#309553