Finite Group Databasebeta

← C2×#20288862 #28290993

Subgroups — C2×#20288862 #28290993

Order 13568.

6554 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288862C2132yesmaximal, direct product factor
2C2×C53⋊#564C211yesmaximal, characteristic, complemented
3C2×#16811377C212yesmaximal, complemented
4C2×#16811374C214yesmaximal, complemented
5C2×#16811372C214yesmaximal, complemented
6C2×#16811371C211yesmaximal, characteristic, complemented
7C2×#16811369C214yesmaximal, complemented
8C2×#16811177C214yesmaximal, complemented
9C2×#16811160C214yesmaximal, complemented
10C2×#16811157C212yesmaximal, complemented
11C2×#16811151C214yesmaximal, complemented
12C2×C53×#533C211yesmaximal, characteristic, complemented
13C53⋊#564C22116yescomplemented
14#16811377C22132yescomplemented
15#16811374C22164yescomplemented
16#16811372C22164yescomplemented
17#16811371C22116yescomplemented
18#16811369C22164yescomplemented
19#16811177C22164yescomplemented
20C2×#13133216C2214yes—
21C2×#13133215C2214yes—
22#16811160C22164yescomplemented
23#16811157C22132yescomplemented
24#16811151C22164yescomplemented
25C2×#13133212C2214yes—
26C2×#13133212C2214yes—
27C2×#13133212C2214yes—
28C2×#13133211C2214yes—
29C2×#13133210C2214yes—
30C2×#13133209C2212yes—
31C2×#13133208C2214yes—
32C2×#13133207C2214yes—
33C2×#13133207C2214yes—
34C2×#13133207C2214yes—
35C2×#13133206C2214yes—
36C2×#13133205C2212yes—
37C2×#13133203C2214yes—
38C2×#13133202C2214yes—
39C2×#13133202C2214yes—
40C2×#13133202C2214yes—
41C2×#13133200C2214yes—
42C2×#13133199C2214yes—
43C2×#13133197C2214yes—
44C2×#13133196C2212yes—
45C2×#13133195C2214yes—
46C2×#13133195C2214yes—
47C2×#13133195C2214yes—
48C2×#13133193C2214yes—
49C2×#13133193C2214yes—
50C2×#13133192C2214yes—
51C2×#13133191C2212yes—
52C2×#13133190C2214yes—
53C2×C4×Dic106C2214yes—
54C2×C4×Dic106C2212yes—
55C2×C4×D212C2214yes—
56C2×C4×D212C2212yes—
57C2×C4×C22⋊D53C2214yes—
58C2×#13133174C2214yes—
59C2×#13133174C2214yes—
60C2×C42×D53C2214yes—
61C53×#533C22116yescomplemented
62C2×C4⋊Q8×C53C2212yes—
63C2×#125×C53C2211yescharacteristic
64C2×#125×C53C2211yescharacteristic
65C2×C22⋊Q8×C53C2214yes—
66C2×C4×Q8×C53C2214yes—
67C2×#117×C53C2212yes—
68C22×C42×C53C2211yescharacteristic
69C22×Dic106C2312yes—
70C22×Dic106C2314yes—
71C22×D212C2314yes—
72C22×D212C2312yes—
73C22×C22⋊D53C2314yes—
74C22×C4×D53C2314yes—
75C22×C4×D53C2314yes—
76C22×C4×D53C2314yes—
77#13133216C23132yes—
78#13133215C23132yes—
79#13133212C23132yes—
80#13133212C23132yes—
81#13133212C23132yes—
82#13133211C23132yes—
83#13133210C23132yes—
84#13133209C23116yes—
85#13133208C23132yes—
86#13133207C23132yes—
87#13133207C23132yes—
88#13133207C23132yes—
89#13133206C23132yes—
90#13133205C23116yes—
91#13133203C23132yes—
92#13133202C23132yes—
93#13133202C23132yes—
94#13133202C23132yes—
95#13133200C23132yes—
96#13133199C23132yes—
97#13133197C23132yes—
98#13133196C23116yes—
99#13133195C23132yes—
100#13133195C23132yes—
101#13133195C23132yes—
102#13133193C23132yes—
103#13133193C23132yes—
104#13133192C23132yes—
105#13133191C23116yes—
106#13133190C23132yes—
107C4×Dic106C23132yes—
108C4×Dic106C23116yes—
109C4×D212C23132yes—
110C4×D212C23116yes—
111C4×C22⋊D53C23132yes—
112#13133174C23132yes—
113#13133174C23132yes—
114C2×C4⋊Dic53C2314yes—
115C2×C4⋊Dic53C2312yes—
116C2×C4⋊Dic53C2312yes—
117C2×C4⋊Dic53C2314yes—
118C2×#11719919C2314yes—
119C2×#11719919C2314yes—
120C2×#11719919C2314yes—
121C2×#11719919C2314yes—
122C2×#11719919C2314yes—
123C2×#11719919C2314yes—
124C2×#11719919C2314yes—
125C2×#11719919C2314yes—
126C2×C22⋊Dic53C2314yes—
127C2×#11719921C2314yes—
128C2×#11719921C2314yes—
129C2×#11719921C2314yes—
130C2×#11719921C2314yes—
131C2×#11719921C2314yes—
132C2×#11719921C2314yes—
133C2×#11719921C2314yes—
134C2×#11719921C2314yes—
135C2×C4×Dic53C2314yes—
136C2×C4×Dic53C2314yes—
137C2×C4×Dic53C2314yes—
138C42×D53C23132yes—
139C22×Q8×C53C2314yes—
140C23×C4×C53C2311yescharacteristic
141C23×C4×C53C2312yes—
142C4⋊Q8×C53C23116yes—
143#125×C53C2318yes—
144#125×C53C2318yes—
145C22⋊Q8×C53C23132yes—
146C4×Q8×C53C23132yes—
147#117×C53C23116yes—
148C2×C4⋊C4×C53C2312yes—
149C2×C4⋊C4×C53C2312yes—
150C2×C4⋊C4×C53C2314yes—
151C2×C4⋊C4×C53C2314yes—
152C2×C4⋊C4×C53C2314yes—
153C2×C22⋊C4×C53C2314yes—
154C2×C42×C53C2312yes—
155C2×C42×C53C2311yescharacteristic
156C2×C42×C53C2311yescharacteristic
157C2×C42×C53C2318yes—
158C2×C42×C53C2314yes—
159C24×C53C2411yescharacteristic
160C23×D53C2414yes—
161C2×Q8×C53—48no—
162C2×Q8×C53C24116yes—
163C22×C4×C53—24no—
164C22×C4×C53—44no—
165C22×C4×C53C2411yescharacteristic
166C22×C4×C53C2412yes—
167C22×C4×C53C2412yes—
168C22×C4×C53C2414yes—
169C22×C4×C53C2414yes—
170C22×C4×C53C2414yes—
171C22×C4×C53C2418yes—
172C22×C4×C53C2411yescharacteristic
173C2×C22⋊D53—48no—
174C2×C22⋊D53C24116yes—
175C22×Dic53C2414yes—
176C22×Dic53C2414yes—
177C22×Dic53C2414yes—
178C2×D212—44no—
179C2×D212—48no—
180C2×D212C24116yes—
181C2×D212C2418yes—
182C2×C4×D53—216no—
183C2×C4×D53—48no—
184C2×C4×D53—48no—
185C2×C4×D53C24116yes—
186C2×C4×D53C24116yes—
187C2×C4×D53C24116yes—
188C2×Dic106—44no—
189C2×Dic106—48no—
190C2×Dic106C2418yes—
191C2×Dic106C24116yes—
192C4⋊C4×C53C2418yes—
193C4⋊C4×C53C2418yes—
194C4⋊C4×C53C24116yes—
195C4⋊C4×C53C24116yes—
196C4⋊C4×C53C24116yes—
197C22⋊C4×C53C24116yes—
198C42×C53C2418yes—
199C42×C53C2414yes—
200C42×C53C2414yes—
201C42×C53C24116yes—
202C22⋊Dic53C24116yes—
203#11719921C24116yes—
204#11719921C24116yes—
205#11719921C24116yes—
206#11719921C24116yes—
207#11719921C24116yes—
208#11719921C24116yes—
209#11719921C24116yes—
210#11719921C24116yes—
211C4⋊Dic53C24116yes—
212C4⋊Dic53C2418yes—
213C4⋊Dic53C2418yes—
214C4⋊Dic53C24116yes—
215#11719919C24116yes—
216#11719919C24116yes—
217#11719919C24116yes—
218#11719919C24116yes—
219#11719919C24116yes—
220#11719919C24116yes—
221#11719919C24116yes—
222#11719919C24116yes—
223C4×Dic53C24116yes—
224C4×Dic53C24116yes—
225C4×Dic53C24116yes—
226C23×C53—22no—
227C23×C53—24no—
228C23×C53C2512yes—
229C23×C53C2511yescharacteristic
230C22×D53—28no—
231C22×D53—28no—
232C22×D53—28no—
233C22×D53C2518yes—
234Q8×C53—416no—
235C2×C4×C53—22no—
236C2×C4×C53—28no—
237C2×C4×C53—24no—
238C2×C4×C53—28no—
239C2×C4×C53—24no—
240C2×C4×C53—28no—
241C2×C4×C53—48no—
242C2×C4×C532−1+414yes—
243C2×C4×C53C2518yes—
244C2×C4×C53C2514yes—
245C2×C4×C53C2518yes—
246C2×C4×C53C2514yes—
247C2×C4×C53C2512yes—
248C2×C4×C53C2512yes—
249C22⋊D53—416no—
250C2×Dic53—28no—
251C2×Dic53—28no—
252C2×Dic53—28no—
253C2×Dic53C2518yes—
254C2×Dic53C2518yes—
255C2×Dic53C2518yes—
256D212—48no—
257D212—416no—
258C4×D53—232no—
259C4×D53—416no—
260C4×D53—416no—
261Dic106—48no—
262Dic106—416no—
263C2×#3531—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—24no—
266C22×C53—44no—
267C22×C53C2×2−1+412yes—
268C22×C53#58514yes—
269C22×C53C2611yescharacteristic
270C2×D53—28no—
271C2×D53—28no—
272C2×D53—28no—
273C2×D53—416no—
274C4×C53—22no—
275C4×C53—28no—
276C4×C53—24no—
277C4×C53—28no—
278C4×C53—24no—
279C4×C53C2×2−1+414yes—
280Dic53—28no—
281Dic53—28no—
282Dic53—28no—
283#3531—5332no—
284C2×#564—531no—
285C2×#563—534no—
286C2×#563—532no—
287C2×#563—534no—
288C2×#562—531no—
289C2×#562—534no—
290C2×#533—534no—
291C2×#533D5311yescharacteristic, complemented
292C2×#529—532no—
293C2×#529—534no—
294C2×#529—534no—
295C2×C53—42no—
296C2×C53#353114yescomplemented
297C2×C53C22×2−1+411yescharacteristic
298C2×C53C2×#58512yes—
299D53—48no—
300#564—5316no—
301#563—5364no—
302#563—5332no—
303#563—5364no—
304#562—5316no—
305#562—5364no—
306#533—5364no—
307#533C2×D53116yescomplemented
308C2×C4⋊Q8—534no—
309C2×C4⋊Q8—534no—
310C2×C4⋊Q8C2×D5312yes—
311#529—5364no—
312#529—5364no—
313#529—5332no—
314C2×#126—534no—
315C2×#126—534no—
316C2×#126—534no—
317C2×#126—534no—
318C2×#126—534no—
319C2×#125—534no—
320C2×#125—534no—
321C2×#125—534no—
322C2×#125—534no—
323C2×#125—532no—
324C2×#125C2×D5311yescharacteristic
325C2×#125C2×D5311yescharacteristic
326C2×#124—534no—
327C2×#124—534no—
328C2×#124—532no—
329C2×#123—534no—
330C2×#123—534no—
331C2×#123—534no—
332C2×#123—534no—
333C2×#123—534no—
334C2×C22⋊Q8—534no—
335C2×C22⋊Q8—534no—
336C2×C22⋊Q8—532no—
337C2×C22⋊Q8—534no—
338C2×C22⋊Q8—534no—
339C2×C22⋊Q8—534no—
340C2×C22⋊Q8—534no—
341C2×C22⋊Q8C2×D5314yes—
342C2×#121—532no—
343C2×#121—534no—
344C2×#121—534no—
345C2×C4×Q8—532no—
346C2×C4×Q8—534no—
347C2×C4×Q8C2×D5314yes—
348C2×C4×D4—534no—
349C2×C4×D4—532no—
350C2×C4×D4—534no—
351C2×#117—534no—
352C2×#117—534no—
353C2×#117C2×D5312yes—
354C22×C42—534no—
355C22×C42C2×D5311yescharacteristic
356C53C2×#353111yescharacteristic, complemented, Sylow
357C22×Q8—532no—
358C22×Q8—534no—
359C22×Q8C22×D5314yes—
360C22×D4—534no—
361C22×D4—534no—
362C22×D4—532no—
363C23×C4—534no—
364C23×C4—534no—
365C23×C4—534no—
366C23×C4C22×D5311yescharacteristic
367C23×C4C22×D5312yes—
368C4⋊Q8—5332no—
369C4⋊Q8—5332no—
370C4⋊Q8C22×D53116yes—
371#126—5332no—
372#126—5332no—
373#126—5332no—
374#126—5332no—
375#126—5332no—
376#125—5332no—
377#125—5332no—
378#125—5332no—
379#125—5332no—
380#125—5316no—
381#125C22×D5318yes—
382#125C22×D5318yes—
383#124—5332no—
384#124—5332no—
385#124—5316no—
386#123—5332no—
387#123—5332no—
388#123—5332no—
389#123—5332no—
390#123—5332no—
391C22⋊Q8—5332no—
392C22⋊Q8—5332no—
393C22⋊Q8—5316no—
394C22⋊Q8—5332no—
395C22⋊Q8—5332no—
396C22⋊Q8—5332no—
397C22⋊Q8—5332no—
398C22⋊Q8C22×D53132yes—
399#121—5316no—
400#121—5332no—
401#121—5332no—
402C4×Q8—5332no—
403C4×Q8—5316no—
404C4×Q8C22×D53132yes—
405C4×D4—5332no—
406C4×D4—5332no—
407C4×D4—5316no—
408#117—5332no—
409#117—5332no—
410#117C22×D53116yes—
411C2×C4⋊C4—534no—
412C2×C4⋊C4—534no—
413C2×C4⋊C4—534no—
414C2×C4⋊C4—534no—
415C2×C4⋊C4—534no—
416C2×C4⋊C4—534no—
417C2×C4⋊C4—532no—
418C2×C4⋊C4—534no—
419C2×C4⋊C4—532no—
420C2×C4⋊C4—534no—
421C2×C4⋊C4—534no—
422C2×C4⋊C4—534no—
423C2×C4⋊C4C22×D5312yes—
424C2×C4⋊C4C22×D5312yes—
425C2×C4⋊C4C22×D5314yes—
426C2×C4⋊C4C22×D5314yes—
427C2×C4⋊C4C22×D5314yes—
428C2×C22⋊C4—534no—
429C2×C22⋊C4—534no—
430C2×C22⋊C4—534no—
431C2×C22⋊C4—534no—
432C2×C22⋊C4—534no—
433C2×C22⋊C4—534no—
434C2×C22⋊C4—534no—
435C2×C22⋊C4—534no—
436C2×C22⋊C4—534no—
437C2×C22⋊C4C22×D5314yes—
438C2×C42—534no—
439C2×C42—534no—
440C2×C42—534no—
441C2×C42—5332no—
442C2×C42C22×D5311yescharacteristic
443C2×C42C22×D5311yescharacteristic
444C2×C42C22×D5318yes—
445C2×C42C22×D5312yes—
446C2×C42C22×D5314yes—
447C24—534no—
448C24C23×D5311yescharacteristic
449C2×Q8—48no—
450C2×Q8—538no—
451C2×Q8—5316no—
452C2×Q8—2124no—
453C2×Q8—2128no—
454C2×Q8C23×D53116yes—
455C2×D4—5316no—
456C2×D4—5316no—
457C2×D4—538no—
458C2×D4—2124no—
459C2×D4—2128no—
460C2×D4—2128no—
461C22×C4—24no—
462C22×C4—44no—
463C22×C4—534no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—5316no—
467C22×C4—5316no—
468C22×C4—5316no—
469C22×C4—10616no—
470C22×C4—2128no—
471C22×C4—2128no—
472C22×C4C23×D5314yes—
473C22×C4C23×D5314yes—
474C22×C4C23×D5311yescharacteristic
475C22×C4C23×D5312yes—
476C22×C4C23×D5312yes—
477C22×C4C23×D5314yes—
478C22×C4C23×D5318yes—
479C22×C4C23×D5311yescharacteristic
480C4⋊C4—5316no—
481C4⋊C4—5316no—
482C4⋊C4—5316no—
483C4⋊C4—5316no—
484C4⋊C4—5316no—
485C4⋊C4—5316no—
486C4⋊C4—538no—
487C4⋊C4—5316no—
488C4⋊C4—538no—
489C4⋊C4—5316no—
490C4⋊C4—5316no—
491C4⋊C4—5316no—
492C4⋊C4C23×D5318yes—
493C4⋊C4C23×D5318yes—
494C4⋊C4C23×D53116yes—
495C4⋊C4C23×D53116yes—
496C4⋊C4C23×D53116yes—
497C22⋊C4—5316no—
498C22⋊C4—5316no—
499C22⋊C4—5316no—
500C22⋊C4—5316no—
501C22⋊C4—5316no—
502C22⋊C4—5316no—
503C22⋊C4—5316no—
504C22⋊C4—5316no—
505C22⋊C4—5316no—
506C22⋊C4C23×D53116yes—
507C42—5316no—
508C42—5316no—
509C42—5316no—
510C42C23×D53116yes—
511C42C23×D5318yes—
512C42C23×D5314yes—
513C42C23×D5314yes—
514C23—22no—
515C23—24no—
516C23—538no—
517C23—1068no—
518C23—1068no—
519C23—1068no—
520C23C24×D5312yes—
521C23C24×D5311yescharacteristic
522Q8—416no—
523Q8—2128no—
524Q8—21216no—
525D4—2128no—
526D4—21216no—
527D4—21216no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—48no—
535C2×C4—538no—
536C2×C4—538no—
537C2×C4—538no—
538C2×C4—1068no—
539C2×C4—1068no—
540C2×C4—1068no—
541C2×C4—10632no—
542C2×C4—21216no—
543C2×C4—21216no—
544C2×C4#1313326514yes—
545C2×C4C24×D5318yes—
546C2×C4C24×D5318yes—
547C2×C4C24×D5312yes—
548C2×C4C24×D5314yes—
549C2×C4C24×D5314yes—
550C2×C4C24×D5312yes—
551C22—22no—
552C22—24no—
553C22—44no—
554C22—1068no—
555C22—1068no—
556C22—1068no—
557C22—21216no—
558C22C2×#1313326512yes—
559C22#1681145914yes—
560C22C25×D5311yescharacteristic
561C4—22no—
562C4—24no—
563C4—24no—
564C4—28no—
565C4—28no—
566C4—1068no—
567C4—1068no—
568C4—1068no—
569C4C2×#1313326514yes—
570C2—42no—
571C2—2128no—
572C2#2028886214yesdirect product factor
573C2C22×#1313326511yescharacteristic
574C2C2×#1681145912yes—
subgroupG/H
centerC22×C4C23×D53
commutator, cosocleC22×C53C26
FrattiniC22C25×D53
FittingC2×C53×#533C2
socleC23×C53C25
hypercenterC2×#533D53
hypocenterC53C2×#3531
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. #20288862C2direct (N × H)
2. C2×C53⋊#564C2semidirect
3. C2×#16811377C2semidirect
4. C2×#16811374C2semidirect
5. C2×#16811372C2semidirect
6. C2×#16811371C2semidirect
7. C2×#16811369C2semidirect
8. C2×#16811177C2semidirect
9. C2×#16811160C2semidirect
10. C2×#16811157C2semidirect
11. C2×#16811151C2semidirect
12. C2×C53×#533C2semidirect
13. C53⋊#564C22semidirect
14. #16811377C22semidirect
15. #16811374C22semidirect
16. #16811372C22semidirect
17. #16811371C22semidirect
18. #16811369C22semidirect
19. #16811177C22semidirect
22. #16811160C22semidirect
23. #16811157C22semidirect
24. #16811151C22semidirect
61. C53×#533C22semidirect
291. C2×#533D53semidirect
296. C2×C53#3531semidirect
307. #533C2×D53semidirect
356. C53C2×#3531semidirect
572. C2#20288862direct (N × H)

Maximal subgroups

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

subgroupindex
#202888622
C2×C53⋊#5642
C2×#168113772
C2×#168113742
C2×#168113722
C2×#168113712
C2×#168113692
C2×#168111772
C2×#168111602
C2×#168111572
C2×#168111512
C2×C53×#5332
C2×#353153