Finite Group Databasebeta

← C2×#20284649 #28251882

Subgroups — C2×#20284649 #28251882

Order 13568.

2346 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811135C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic, complemented
3#20284649C2116yesmaximal, direct product factor
4C2×#16810967C212yesmaximal, complemented
5C2×#16810966C212yesmaximal, complemented
6C2×#16810950C212yesmaximal, complemented
7C2×#16810949C212yesmaximal, complemented
8C2×#16810666C211yesmaximal, characteristic, complemented
9C2×#16810625C211yesmaximal, characteristic, complemented
10C2×#16810622C211yesmaximal, characteristic, complemented
11C2×#16810586C211yesmaximal, characteristic, complemented
12C2×C53×#428C211yesmaximal, characteristic, complemented
13C22×#11719945C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C22×#11719917C2211yescharacteristic
16#16811135—24no—
17#16811135C2218yes—
18C2×#13133201—21no—
19C2×#13133197C2211yescharacteristic
20C2×#13133197C2211yescharacteristic
21C2×#13133192C2211yescharacteristic
22C2×#13133192C2211yescharacteristic
23C2×#13133187—21no—
24C2×C4×D212—21no—
25C2×C4×D212—21no—
26C2×#13133175—21no—
27C2×#13133175—21no—
28C2×#13133175—24no—
29C2×#13133175C2212yes—
30C2×#13133175C2212yes—
31C2×#13133175C2218yescomplemented
32C22×C4⋊Dic53C2211yescharacteristic
33C22×#11719921—21no—
34C22×C4×Dic53—21no—
35#16810967C22116yescomplemented
36#16810966C22116yescomplemented
37#16810950C22116yescomplemented
38#16810949C22116yescomplemented
39#16810666C2218yescomplemented
40C2×#13133150C2212yescomplemented
41C2×#13133148C2212yescomplemented
42#16810625C2218yescomplemented
43#16810622C2218yescomplemented
44C2×#13133142C2212yescomplemented
45C2×#13133140C2212yescomplemented
46C2×#13133139C2212yescomplemented
47C2×#13133137C2212yescomplemented
48#16810586C2218yescomplemented
49C2×#13133129C2212yes—
50C2×#13133129C2212yes—
51C22×M16×C53C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C53×#428C2218yescomplemented
55C2×#107×C53C2212yescomplemented
56C2×#106×C53C2212yescomplemented
57C24×D53D411yescharacteristic
58C2×#11719945—22no—
59C2×#11719945—24no—
60C2×#11719945—24no—
61C2×#11719945C2314yes—
62C22×Dic106C2311yescharacteristic
63C22×D212—21no—
64C22×D212—21no—
65C22×D212C2311yescharacteristic
66C22×C22⋊D53—21no—
67C23×Dic53—21no—
68C23×Dic53D411yescharacteristic
69C22×C4×D53—24no—
70C22×C4×D53—24no—
71C22×C4×D53—21no—
72C22×C4×D53D411yescharacteristic
73C22×C4×D53D411yescharacteristic
74C22×C4×D53D414yes—
75C22×C4×D53C2311yescharacteristic
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2314yes—
78C2×#11719917—24no—
79C2×#11719917C2314yes—
80C22×#88322C2312yes—
81#13133201—28no—
82#13133197—22no—
83#13133197—22no—
84#13133197C2314yes—
85#13133197C2314yes—
86#13133192—22no—
87#13133192—22no—
88#13133192C2314yes—
89#13133192C2314yes—
90#13133187—28no—
91C4×D212—28no—
92C4×D212—28no—
93#13133175—24no—
94#13133175—24no—
95#13133175—28no—
96#13133175—28no—
97#13133175C2318yes—
98#13133175C2318yes—
99C2×C4⋊Dic53—21no—
100C2×C4⋊Dic53—22no—
101C2×C4⋊Dic53—21no—
102C2×C4⋊Dic53—21no—
103C2×C4⋊Dic53C2312yes—
104C2×C4⋊Dic53C2314yes—
105C2×#11719921—22no—
106C2×#11719921—22no—
107C2×#11719921—28no—
108C2×#11719921—21no—
109C2×#11719921—21no—
110C2×#11719921—21no—
111C2×#11719921—21no—
112C2×C4×Dic53—22no—
113C2×C4×Dic53—22no—
114C2×C4×Dic53—28no—
115#13133150C2318yescomplemented
116#13133148C2318yescomplemented
117#13133142C2318yescomplemented
118#13133140C2318yescomplemented
119#13133139C2318yescomplemented
120#13133137C2318yescomplemented
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—21no—
143C23×D53—42no—
144C23×D53—42no—
145C23×D53C2×D411yescharacteristic
146C23×D53C2×D411yescharacteristic
147C23×D53C2×D414yes—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—24no—
151C22×C4×C53—22no—
152C22×C4×C53—44no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D412yes—
155C22×C4×C53#4114yes—
156C22×C4×C53#4114yes—
157C22×C4×C53C2411yescharacteristic
158C22×C4×C53C2412yes—
159C2×C22⋊D53—24no—
160C2×C22⋊D53—44no—
161C22×Dic53—24no—
162C22×Dic53—24no—
163C22×Dic53—22no—
164C22×Dic53—21no—
165C22×Dic53—44no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D414yes—
169#11719945—28no—
170#11719945—28no—
171#11719945—48no—
172C2×D212—22no—
173C2×D212—22no—
174C2×D212—24no—
175C2×D212—24no—
176C2×D212—21no—
177C2×D212—44no—
178C2×D212—44no—
179C2×D212C2412yes—
180C2×C4×D53—24no—
181C2×C4×D53—24no—
182C2×C4×D53—216no—
183C2×C4×D53—22no—
184C2×C4×D53—22no—
185C2×C4×D53—28no—
186C2×C4×D53—22no—
187C2×C4×D53—22no—
188C2×C4×D53—28no—
189C2×C4×D53—24no—
190C2×C4×D53—44no—
191C2×C4×D53C2×D414yes—
192C2×C4×D53C2×D414yes—
193C2×C4×D53C2×D412yes—
194C2×C4×D53C2×D412yes—
195C2×C4×D53C2412yes—
196C2×C4×D53C2412yes—
197C2×Dic106—22no—
198C2×Dic106—22no—
199C2×Dic106—21no—
200C2×Dic106C2412yes—
201M16×C53—28no—
202C2×C8×C53—24no—
203C2×C8×C53C2414yes—
204C4⋊C4×C53—22no—
205C4⋊C4×C53—24no—
206C4⋊C4×C53—21no—
207C4⋊C4×C53—21no—
208C4⋊C4×C53C2412yes—
209C4⋊C4×C53C2414yes—
210C4⋊C4×C53C2412yes—
211C22⋊C4×C53—24no—
212C42×C53—24no—
213#11719921—28no—
214#11719921—28no—
215#11719921—24no—
216#11719921—24no—
217#11719921—24no—
218#11719921—24no—
219C4⋊Dic53—22no—
220C4⋊Dic53—24no—
221C4⋊Dic53—24no—
222C4⋊Dic53—24no—
223C4⋊Dic53C2414yes—
224C4×Dic53—28no—
225C4×Dic53—28no—
226#11719917—28no—
227C2×#88322—24no—
228C2×#88322C2414yes—
229C23×C53—22no—
230C23×C53—22no—
231C23×C53#12314yes—
232C23×C53C22×D412yes—
233C23×C53C22×D411yescharacteristic
234C22×D53—28no—
235C22×D53—22no—
236C22×D53—22no—
237C22×D53—22no—
238C22×D53—22no—
239C22×D53—22no—
240C22×D53—22no—
241C22×D53—22no—
242C22×D53—44no—
243C22×D53—44no—
244C22×D53—416no—
245C22×D53—42no—
246C22×D53—42no—
247C22×D53—42no—
248C22×D53C22×D412yes—
249C22×D53C22×D412yes—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—24no—
253C2×C4×C53—48no—
254C2×C4×C53—42no—
255C2×C4×C53—42no—
256C2×C4×C53—44no—
257C2×C4×C53#12314yes—
258C2×C4×C53#12318yes—
259C2×C4×C53C22×D411yescharacteristic
260C2×C4×C53C22×D412yes—
261C2×C4×C53C2×#4112yes—
262C2×C4×C53C2×#4114yes—
263C2×C4×C53C2×#4112yes—
264C2×C4×C53C2×#4114yes—
265C2×C4×C53C2511yescharacteristic
266C22⋊D53—48no—
267C2×Dic53—28no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—24no—
272C2×Dic53—48no—
273C2×Dic53—42no—
274C2×Dic53—44no—
275C2×Dic53C22×D412yes—
276C2×Dic53C22×D412yes—
277D212—22no—
278D212—22no—
279D212—48no—
280D212—48no—
281D212—42no—
282C4×D53—24no—
283C4×D53—24no—
284C4×D53—28no—
285C4×D53—28no—
286C4×D53—22no—
287C4×D53—22no—
288C4×D53—22no—
289C4×D53—22no—
290C4×D53—48no—
291Dic106—22no—
292Dic106—22no—
293Dic106—42no—
294C8×C53—24no—
295#88322—24no—
296C2×#3093—531nomaximal, Sylow
297C22×C53—28no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53C2×#12314yes—
301C22×C53C2×#12312yes—
302C22×C53#57512yes—
303C22×C53#57512yes—
304C22×C53C23×D411yescharacteristic
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—22no—
308C2×D53—22no—
309C2×D53—22no—
310C2×D53—22no—
311C2×D53—416no—
312C2×D53—44no—
313C2×D53—44no—
314C2×D53—42no—
315C2×D53—42no—
316C2×D53—42no—
317C2×D53—84no—
318C4×C53—42no—
319C4×C53—42no—
320C4×C53—44no—
321C4×C53C2×#12312yes—
322C4×C53C2×#12314yes—
323C4×C53C22×#4111yescharacteristic
324C4×C53C22×#4111yescharacteristic
325Dic53—22no—
326Dic53—22no—
327Dic53—42no—
328Dic53—44no—
329C2×#525—531no—
330C22×#117—531no—
331#3093—5316no—
332C2×#485—532no—
333C2×#485—532no—
334C2×#482—532no—
335C2×#482—532no—
336C2×#428—531no—
337C2×#428D5311yescharacteristic, complemented
338C2×#417—531no—
339C2×#417—531no—
340C2×#407—531no—
341C2×C53—24no—
342C2×C53#309314yescomplemented
343C2×C53C22×#12311yescharacteristic
344C2×C53C2×#57511yescharacteristic
345C2×C53C2×#57511yescharacteristic
346D53—42no—
347D53—42no—
348D53—82no—
349C22×#41—531no—
350C24×C4—531no—
351C22×M16—531no—
352C22×M16C2×D5311yescharacteristic
353#525—538no—
354#525—1064no—
355C2×#123—1061no—
356C2×C22⋊Q8—531no—
357C2×C22⋊Q8—531no—
358C2×#121—531no—
359C2×#121—531no—
360C2×#120—1061no—
361C2×C4×D4—1061no—
362C2×C4×D4—1061no—
363C2×#117—532no—
364C2×#117—532no—
365C2×#117—538no—
366C2×#117—1061no—
367C2×#117—1061no—
368C2×#117—1064no—
369C2×#117C2×D5311yescharacteristic
370C22×C4⋊C4—531no—
371C22×C4⋊C4C2×D5311yescharacteristic
372C22×C22⋊C4—1061no—
373C22×C42—1061no—
374#485—5316no—
375#485—5316no—
376#482—5316no—
377#482—5316no—
378#428—538no—
379#428C2×D5318yescomplemented
380C2×#107—532no—
381C2×#107C2×D5312yescomplemented
382C2×#106—532no—
383C2×#106C2×D5312yescomplemented
384#417—538no—
385#417—538no—
386C2×#103—532no—
387C2×#103—532no—
388C2×#102—532no—
389C2×#102—532no—
390#407—538no—
391C2×C22⋊C8—532no—
392C2×C22⋊C8—532no—
393C53C2×#309311yescharacteristic, complemented, Sylow
394C25—531no—
395C2×#41—534no—
396C2×#41—1062no—
397C2×#41—1064no—
398C2×#41—1064no—
399C22×Q8—531no—
400C22×D4—531no—
401C22×D4—1061no—
402C22×D4—1061no—
403C22×D4—1061no—
404C23×C4—21no—
405C23×C4—531no—
406C23×C4—531no—
407C23×C4—531no—
408C23×C4—534no—
409C23×C4—531no—
410C23×C4—531no—
411C23×C4—534no—
412C23×C4—1064no—
413C23×C4—1064no—
414C23×C4—1061no—
415C23×C4—1061no—
416C23×C4C22×D5311yescharacteristic
417C2×M16—24no—
418C2×M16—534no—
419C2×M16—1064no—
420C2×M16C22×D5314yes—
421C22×C8—532no—
422C22×C8C22×D5312yes—
423#123—1068no—
424C22⋊Q8—534no—
425C22⋊Q8—534no—
426C22⋊Q8—1062no—
427C22⋊Q8—1062no—
428#121—534no—
429#121—534no—
430#121—1062no—
431#121—1062no—
432#120—1068no—
433C4×D4—1068no—
434C4×D4—1068no—
435#117—22no—
436#117—538no—
437#117—538no—
438#117—1068no—
439#117—1068no—
440#117—1064no—
441#117—1064no—
442#117C22×D5314yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—532no—
446C2×C4⋊C4—534no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4—1062no—
450C2×C4⋊C4—1061no—
451C2×C4⋊C4C22×D5312yes—
452C2×C4⋊C4C22×D5314yes—
453C2×C4⋊C4C22×D5311yescharacteristic
454C2×C4⋊C4C22×D5311yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4—1068no—
459C2×C22⋊C4—1062no—
460C2×C22⋊C4—1062no—
461C2×C22⋊C4—1061no—
462C2×C22⋊C4—1061no—
463C2×C42—21no—
464C2×C42—1068no—
465C2×C42—1062no—
466C2×C42—1062no—
467#107—538no—
468#107C22×D5318yescomplemented
469#106—538no—
470#106C22×D5318yescomplemented
471#103—538no—
472#103—538no—
473#102—538no—
474#102—538no—
475C22⋊C8—538no—
476C22⋊C8—538no—
477C24—531no—
478C24—531no—
479C24—534no—
480C24—1064no—
481C24—1061no—
482C24—2122no—
483C24—2122no—
484C24D4×D5311yescharacteristic
485#41—1068no—
486#41—1068no—
487#41—2128no—
488C2×Q8—532no—
489C2×Q8—1061no—
490C2×Q8—1062no—
491C2×Q8—1062no—
492C2×D4—532no—
493C2×D4—1064no—
494C2×D4—1064no—
495C2×D4—1064no—
496C2×D4—1061no—
497C2×D4—1062no—
498C2×D4—1062no—
499C2×D4—2124no—
500C2×D4—2124no—
501C2×D4—2124no—
502C22×C4—24no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—44no—
507C22×C4—534no—
508C22×C4—531no—
509C22×C4—531no—
510C22×C4—534no—
511C22×C4—534no—
512C22×C4—532no—
513C22×C4—532no—
514C22×C4—532no—
515C22×C4—532no—
516C22×C4—1064no—
517C22×C4—1064no—
518C22×C4—1064no—
519C22×C4—10616no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1068no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1068no—
526C22×C4—1064no—
527C22×C4—1062no—
528C22×C4—1061no—
529C22×C4—1064no—
530C22×C4—2124no—
531C22×C4—2124no—
532C22×C4#1171994514yes—
533C22×C4D4×D5312yes—
534C22×C4D4×D5311yescharacteristic
535C22×C4#1171994914yes—
536C22×C4C23×D5312yes—
537C22×C4C23×D5311yescharacteristic
538M16—28no—
539M16—1068no—
540C2×C8—24no—
541C2×C8—534no—
542C2×C8—1064no—
543C2×C8C23×D5314yes—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—534no—
549C4⋊C4—1064no—
550C4⋊C4—1064no—
551C4⋊C4—1064no—
552C4⋊C4—1062no—
553C4⋊C4C23×D5314yes—
554C4⋊C4C23×D5312yes—
555C4⋊C4C23×D5312yes—
556C22⋊C4—24no—
557C22⋊C4—1064no—
558C22⋊C4—1064no—
559C22⋊C4—1068no—
560C22⋊C4—1068no—
561C22⋊C4—1064no—
562C22⋊C4—1064no—
563C42—24no—
564C42—1068no—
565C42—1068no—
566C23—22no—
567C23—22no—
568C23—532no—
569C23—532no—
570C23—1068no—
571C23—1062no—
572C23—1062no—
573C23—1062no—
574C23—1062no—
575C23—1062no—
576C23—1062no—
577C23—1062no—
578C23—2124no—
579C23—2124no—
580C23—21216no—
581C23—2122no—
582C23—2122no—
583C23—2122no—
584C23#1313320214yes—
585C23C2×D4×D5312yes—
586C23C2×D4×D5311yescharacteristic
587Q8—1062no—
588Q8—1062no—
589Q8—2122no—
590D4—1062no—
591D4—1062no—
592D4—2128no—
593D4—2128no—
594D4—2128no—
595D4—2122no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—532no—
604C2×C4—532no—
605C2×C4—1068no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1068no—
609C2×C4—1068no—
610C2×C4—1064no—
611C2×C4—1064no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1064no—
617C2×C4—1062no—
618C2×C4—2128no—
619C2×C4—2122no—
620C2×C4—2124no—
621C2×C4—2128no—
622C2×C4#1313320214yes—
623C2×C4#1313320218yes—
624C2×C4C2×D4×D5312yes—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4C2×#1171994512yes—
627C2×C4C2×#1171994514yes—
628C2×C4C2×#1171994912yes—
629C2×C4C2×#1171994914yes—
630C2×C4C24×D5311yescharacteristic
631C8—24no—
632C8—1064no—
633C22—22no—
634C22—22no—
635C22—28no—
636C22—1062no—
637C22—1062no—
638C22—1062no—
639C22—1062no—
640C22—1062no—
641C22—1062no—
642C22—4244no—
643C22—21216no—
644C22—2122no—
645C22—2122no—
646C22—2122no—
647C22—2124no—
648C22—2124no—
649C22C2×#1313320214yes—
650C22C2×#1313320212yes—
651C22#1681141912yes—
652C22#1681142212yes—
653C22C22×D4×D5311yescharacteristic
654C4—44no—
655C4—42no—
656C4—42no—
657C4—1062no—
658C4—1062no—
659C4—2124no—
660C4—2122no—
661C4C2×#1313320214yes—
662C4C2×#1313320212yes—
663C4C22×#1171994511yescharacteristic
664C4C22×#1171994911yescharacteristic
665C2—24no—
666C2—4242no—
667C2—2122no—
668C2—2122no—
669C2#2028464914yesdirect product factor
670C2C22×#1313320211yescharacteristic
671C2C2×#1681141911yescharacteristic
672C2C2×#1681142211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC2×#16811135C2
hypercenterC2×#428D53
hypocenterC53C2×#3093
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
2. C22×#13133175C2semidirect
3. #20284649C2direct (N × H)
4. C2×#16810967C2semidirect
5. C2×#16810966C2semidirect
6. C2×#16810950C2semidirect
7. C2×#16810949C2semidirect
8. C2×#16810666C2semidirect
9. C2×#16810625C2semidirect
10. C2×#16810622C2semidirect
11. C2×#16810586C2semidirect
12. C2×C53×#428C2semidirect
31. C2×#13133175C22semidirect
35. #16810967C22semidirect
36. #16810966C22semidirect
37. #16810950C22semidirect
38. #16810949C22semidirect
39. #16810666C22semidirect
40. C2×#13133150C22semidirect
41. C2×#13133148C22semidirect
42. #16810625C22semidirect
43. #16810622C22semidirect
44. C2×#13133142C22semidirect
45. C2×#13133140C22semidirect
46. C2×#13133139C22semidirect
47. C2×#13133137C22semidirect
48. #16810586C22semidirect
54. C53×#428C22semidirect
55. C2×#107×C53C22semidirect
56. C2×#106×C53C22semidirect
115. #13133150C23semidirect
116. #13133148C23semidirect
117. #13133142C23semidirect
118. #13133140C23semidirect
119. #13133139C23semidirect
120. #13133137C23semidirect
138. #107×C53C23semidirect
139. #106×C53C23semidirect
337. C2×#428D53semidirect
342. C2×C53#3093semidirect
379. #428C2×D53semidirect
381. C2×#107C2×D53semidirect
383. C2×#106C2×D53semidirect
393. C53C2×#3093semidirect
468. #107C22×D53semidirect
470. #106C22×D53semidirect
669. C2#20284649direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111352
C22×#131331752
#202846492
C2×#168109672
C2×#168109662
C2×#168109502
C2×#168109492
C2×#168106662
C2×#168106252
C2×#168106222
C2×#168105862
C2×C53×#4282
C2×#309353