Finite Group Databasebeta

← C2×#11749272 #14423474

Subgroups — C2×#11749272 #14423474

Order 1920.

1624 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
1C22×C3×#713C211yesmaximal, characteristic, complemented
2C22×#6717C211yesmaximal, characteristic, complemented
3C2×#91450C211yesmaximal, characteristic, complemented
4C22×#6653C211yesmaximal, characteristic, complemented
5C2×C5×#835C211yesmaximal, characteristic, complemented
6#11749272C218yesmaximal, direct product factor
7C2×#90964C211yesmaximal, characteristic, complemented
8C2×#90955C211yesmaximal, characteristic, complemented
9C2×#65100—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×D60C2211yescharacteristic
12C2×C3×#713—24no—
13C2×C3×#713—24no—
14C2×C3×#713—22no—
15C2×C3×#713C2214yescomplemented
16C22×C3×D20—21no—
17C22×C3×D20C2211yescharacteristic
18C22×C3×C4×D5—21no—
19C22×C3×C4×D5C2211yescharacteristic
20C22×#1202—21no—
21C2×#6717—24no—
22C2×#6717C2214yescomplemented
23C22×C3⋊#168C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C22×C5×C3⋊C8C2211yescharacteristic
26C2×#6667—21no—
27#91450—22no—
28#91450C2214yescomplemented
29C2×#6653—24no—
30C2×#6653C2214yescomplemented
31C5×#835C2214yescomplemented
32#90964C2214yescomplemented
33#90955C2214yescomplemented
34C2×#5238—51nomaximal, Hall
35C22×#713D311yescharacteristic, complemented
36C22×#688—31no—
37C22×D40—31no—
38C2×#4041—31no—
39C2×C5×#103—31no—
40C2×#3952—31no—
41C2×#3941—31no—
42#65100—38no—
43C23×D15—21no—
44C23×C3×D5—21no—
45C23×C3×D5D411yescharacteristic
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—21no—
48C2×C3×C5×Q8D414yescomplemented
49C2×C3×C5×Q8C2312yes—
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5C2311yescharacteristic
52C2×D60—22no—
53C2×D60—22no—
54C2×D60—21no—
55C2×D60C2312yes—
56C22×C5×Dic3—21no—
57C22×C3×Dic5D411yescharacteristic
58C3×#713—216no—
59C3×#713—28no—
60C3×#713—28no—
61C2×C3×D20—28no—
62C2×C3×D20—24no—
63C2×C3×D20—21no—
64C2×C3×D20—22no—
65C2×C3×D20D414yescomplemented
66C2×C3×D20C2312yes—
67C2×C3×C4×D5—28no—
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5—24no—
71C2×C3×C4×D5D412yes—
72C2×C3×C4×D5C2312yes—
73C2×#1202—24no—
74C2×#1202—44no—
75#6717—28no—
76C2×C3⋊#168—22no—
77C2×C3⋊#168C2312yes—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3C2312yes—
80C2×C5×C3⋊C8—22no—
81C2×C5×C3⋊C8C2312yes—
82#6667—24no—
83#6653—28no—
84C22×C3×#41—51no—
85C22×#215—51no—
86C22×#213—51no—
87C2×#930—51no—
88#5238—58no—
89C2×#835D511yescharacteristic, complemented
90C2×#832—51no—
91C2×#830—51no—
92C22×C5×Q8C2×D311yescharacteristic
93C2×#713—24no—
94C2×#713—24no—
95C2×#713—22no—
96C2×#713C2×D314yescomplemented
97C22×D20—21no—
98C22×D20—31no—
99C22×D20—61no—
100C22×D20C2×D311yescharacteristic
101C22×C4×D5—21no—
102C22×C4×D5C2×D311yescharacteristic
103C22×C5×C8—31no—
104C2×C5×C4⋊C4—31no—
105C2×#688—34no—
106C2×#688—64no—
107C2×#685—61no—
108C22×#168—31no—
109C2×D40—34no—
110C2×D40—64no—
111#4041—34no—
112#4041—62no—
113C5×#103—34no—
114#3952—34no—
115#3941—34no—
116C2×#449—151noSylow
117C23×C3×C5C2×D411yescharacteristic
118C22×D15—22no—
119C22×D15—42no—
120C22×D15—42no—
121C22×D15—42no—
122C22×C3×D5—22no—
123C22×C3×D5—24no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—42no—
128C22×C3×D5—42no—
129C22×C3×D5C2×D412yes—
130C3×C5×Q8—24no—
131C3×C5×Q8—22no—
132C3×C5×Q8C2×D414yescomplemented
133C2×C3×C4×C5—24no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2411yescharacteristic
139D60—22no—
140D60—22no—
141D60—42no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—42no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5C2×D412yes—
146C3×D20—216no—
147C3×D20—24no—
148C3×D20—22no—
149C3×D20C2×D414yescomplemented
150C3×C4×D5—216no—
151C3×C4×D5—24no—
152C3×C4×D5—22no—
153C3×C4×D5—22no—
154#1202—48no—
155C3⋊#168—22no—
156C5×C3⋊C8—22no—
157C2×C3×#41—54no—
158C2×C3×#41—102no—
159C2×C3×#41—104no—
160C2×C3×#41—104no—
161C22×C3×Q8C2×D511yescharacteristic
162C22×C3×D4—51no—
163C22×C3×D4—101no—
164C23×C3×C4—51no—
165C23×C3×C4—101no—
166C22×#67—101no—
167C22×D12—51no—
168C2×#215—54no—
169C2×#215—104no—
170C2×#213—54no—
171C2×#213—104no—
172#930—54no—
173#930—102no—
174C2×(C2×D3)⋊C4—101no—
175C2×C4⋊Dic3C2×D511yescharacteristic
176C22×C3⋊C8—51no—
177C22×C3⋊C8C2×D511yescharacteristic
178#835C2×D514yescomplemented
179#832—54no—
180#830—54no—
181C23×D5—21no—
182C23×D5—61no—
183C23×D5#6711yescharacteristic
184C2×C5×Q8—22no—
185C2×C5×Q8—21no—
186C2×C5×Q8#6714yescomplemented
187C2×C5×Q8C22×D312yes—
188C22×C4×C5—21no—
189C22×C4×C5—61no—
190C22×C4×C5C22×D311yescharacteristic
191C22×Dic5#6711yescharacteristic
192#713—216no—
193#713—28no—
194#713—28no—
195C2×D20—28no—
196C2×D20—24no—
197C2×D20—22no—
198C2×D20—21no—
199C2×D20—32no—
200C2×D20—62no—
201C2×D20—62no—
202C2×D20—64no—
203C2×D20—61no—
204C2×D20—124no—
205C2×D20#6714yescomplemented
206C2×D20C22×D312yes—
207C2×C4×D5—28no—
208C2×C4×D5—22no—
209C2×C4×D5—22no—
210C2×C4×D5—24no—
211C2×C4×D5#6712yes—
212C2×C4×D5C22×D312yes—
213C2×C5×C8—32no—
214C2×C5×C8—62no—
215C5×C4⋊C4—32no—
216C5×C4⋊C4—61no—
217#688—68no—
218#685—64no—
219C2×#168—32no—
220C2×#168—62no—
221D40—68no—
222C22×#41—51no—
223C22×SD16—151no—
224C22×D8—151no—
225C2×#121—151no—
226#449—158no—
227C2×#103—31no—
228C2×#102—151no—
229C2×C22⋊C8—151no—
230C22×C3×C5#12012yes—
231C22×C3×C5#13512yes—
232C22×C3×C5AGL(1,ℤ/8)12yes—
233C22×C3×C5C22×D411yescharacteristic
234C2×D15—42no—
235C2×D15—42no—
236C2×D15—42no—
237C2×D15—84no—
238C2×C3×D5—22no—
239C2×C3×D5—24no—
240C2×C3×D5—22no—
241C2×C3×D5—22no—
242C2×C3×D5—42no—
243C2×C3×D5—42no—
244C2×C3×D5—44no—
245C2×C3×D5—48no—
246C3×C4×C5—24no—
247C3×C4×C5#12012yes—
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C3×Dic5—22no—
251C5×Dic3—42no—
252C24×C3—51no—
253C24×C3—101no—
254C23×D3—101no—
255C3×#41—108no—
256C3×#41—1016no—
257C3×#41—108no—
258C2×C3×Q8—21no—
259C2×C3×Q8—22no—
260C2×C3×Q8D2014yescomplemented
261C2×C3×Q8C22×D512yes—
262C2×C3×D4—54no—
263C2×C3×D4—52no—
264C2×C3×D4—104no—
265C2×C3×D4—108no—
266C2×C3×D4—102no—
267C2×C3×D4—101no—
268C22×C3×C4—21no—
269C22×C3×C4—51no—
270C22×C3×C4—52no—
271C22×C3×C4—52no—
272C22×C3×C4—104no—
273C22×C3×C4—108no—
274C22×C3×C4—102no—
275C22×C3×C4—102no—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—104no—
278C2×#67—204no—
279C22×Dic3—21no—
280C2×D12—52no—
281C2×D12—102no—
282C2×D12—102no—
283C2×D12—101no—
284#215—108no—
285#213—108no—
286(C2×D3)⋊C4—104no—
287C4⋊Dic3—21no—
288C4⋊Dic3C22×D512yes—
289C2×C3⋊C8—22no—
290C2×C3⋊C8—52no—
291C2×C3⋊C8—102no—
292C2×C3⋊C8C22×D512yes—
293C23×C5C2×#6711yescharacteristic
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—22no—
297C22×D5—24no—
298C22×D5—22no—
299C22×D5—42no—
300C22×D5—42no—
301C22×D5—62no—
302C22×D5—122no—
303C22×D5—122no—
304C22×D5—122no—
305C22×D5C2×#6712yes—
306C5×Q8—24no—
307C5×Q8—22no—
308C5×Q8C2×#6714yescomplemented
309C2×C4×C5—24no—
310C2×C4×C5—22no—
311C2×C4×C5—62no—
312C2×C4×C5—122no—
313C2×C4×C5C2×#6712yes—
314C2×C4×C5C2×#6711yescharacteristic
315C2×C4×C5C2×#6712yes—
316C2×C4×C5C23×D311yescharacteristic
317C2×Dic5—22no—
318C2×Dic5C2×#6712yes—
319D20—216no—
320D20—24no—
321D20—22no—
322D20—62no—
323D20—62no—
324D20—128no—
325D20—122no—
326D20C2×#6714yescomplemented
327C4×D5—216no—
328C4×D5—24no—
329C4×D5—22no—
330C4×D5—22no—
331C5×C8—62no—
332#168—62no—
333C2×#41—54no—
334C2×#41—102no—
335C2×#41—104no—
336C2×#41—104no—
337C22×Q8D3×D511yescharacteristic
338C22×D4—51no—
339C22×D4—101no—
340C22×D4—151no—
341C22×D4—301no—
342C23×C4—51no—
343C23×C4—101no—
344C2×SD16—154no—
345C2×SD16—304no—
346C2×D8—154no—
347C2×D8—304no—
348C22×C8—31no—
349C22×C8—151no—
350#121—154no—
351#121—302no—
352C2×C4⋊C4—31no—
353C2×C22⋊C4—301no—
354#103—34no—
355#102—154no—
356C22⋊C8—154no—
357C2×C3×C5#44914yescomplemented
358C2×C3×C5C2×#12011yescharacteristic
359C2×C3×C5C2×#13511yescharacteristic
360C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
361D15—82no—
362C3×D5—42no—
363C3×D5—44no—
364C23×C3—52no—
365C23×C3—102no—
366C23×C3—104no—
367C23×C3—102no—
368C23×C3—102no—
369C23×C3—102no—
370C23×C3—202no—
371C23×C3—202no—
372C23×C3D4×D511yescharacteristic
373C22×D3—102no—
374C22×D3—202no—
375C22×D3—202no—
376C22×D3—202no—
377C3×Q8—22no—
378C3×Q8—24no—
379C3×Q8C2×D2014yescomplemented
380C3×D4—54no—
381C3×D4—1016no—
382C3×D4—102no—
383C3×D4—104no—
384C2×C3×C4—22no—
385C2×C3×C4—24no—
386C2×C3×C4—52no—
387C2×C3×C4—1016no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—104no—
392C2×C3×C4C2×D2012yes—
393C2×C3×C4D4×D512yes—
394C2×C3×C4D4×D511yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—208no—
397C2×Dic3—22no—
398C2×Dic3—42no—
399D12—102no—
400D12—102no—
401D12—202no—
402C3⋊C8—22no—
403C3⋊C8—102no—
404C22×C5#94912yes—
405C22×C5#95012yes—
406C22×C5#95312yes—
407C22×C5C22×#6711yescharacteristic
408C2×D5—22no—
409C2×D5—22no—
410C2×D5—22no—
411C2×D5—24no—
412C2×D5—44no—
413C2×D5—48no—
414C2×D5—42no—
415C2×D5—42no—
416C2×D5—122no—
417C2×D5—122no—
418C2×D5—122no—
419C2×D5—244no—
420C4×C5—24no—
421C4×C5—122no—
422C4×C5#95312yes—
423C4×C5C22×#6711yescharacteristic
424C4×C5C22×#6711yescharacteristic
425Dic5—22no—
426C24—51no—
427C24—101no—
428C24—301no—
429#41—108no—
430#41—1016no—
431#41—108no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8#120214yescomplemented
435C2×Q8C2×D3×D512yes—
436C2×D4—54no—
437C2×D4—52no—
438C2×D4—104no—
439C2×D4—108no—
440C2×D4—102no—
441C2×D4—101no—
442C2×D4—152no—
443C2×D4—304no—
444C2×D4—302no—
445C2×D4—302no—
446C2×D4—301no—
447C2×D4—604no—
448C22×C4—21no—
449C22×C4—51no—
450C22×C4—52no—
451C22×C4—52no—
452C22×C4—61no—
453C22×C4—104no—
454C22×C4—108no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4C2×D3×D511yescharacteristic
458SD16—308no—
459D8—308no—
460C2×C8—32no—
461C2×C8—62no—
462C2×C8—152no—
463C2×C8—302no—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C22⋊C4—304no—
467C3×C5C2×#44911yescharacteristic, complemented, Hall
468C22×C3—104no—
469C22×C3—102no—
470C22×C3—102no—
471C22×C3—102no—
472C22×C3—202no—
473C22×C3—202no—
474C22×C3—204no—
475C22×C3—208no—
476C22×C3#402812yes—
477C22×C3#406012yes—
478C22×C3#406512yes—
479C22×C3C2×D4×D511yescharacteristic
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—202no—
483C2×D3—404no—
484C3×C4—24no—
485C3×C4—102no—
486C3×C4#402812yes—
487C3×C4C22×D2011yescharacteristic
488C3×C4C2×D4×D511yescharacteristic
489Dic3—42no—
490C2×C5#523814yescomplemented
491C2×C5C2×#94911yescharacteristic
492C2×C5C2×#95011yescharacteristic
493C2×C5C2×#95311yescharacteristic
494D5—42no—
495D5—44no—
496D5—242no—
497C23—52no—
498C23—102no—
499C23—102no—
500C23—102no—
501C23—104no—
502C23—102no—
503C23—202no—
504C23—202no—
505C23—302no—
506C23—602no—
507C23—602no—
508C23—602no—
509C23D5×#6711yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8C2×#120214yescomplemented
513D4—54no—
514D4—1016no—
515D4—102no—
516D4—104no—
517D4—302no—
518D4—302no—
519D4—608no—
520D4—602no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—52no—
524C2×C4—62no—
525C2×C4—1016no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—104no—
530C2×C4—122no—
531C2×C4C2×#120212yes—
532C2×C4D5×#6712yes—
533C2×C4D5×#6711yescharacteristic
534C2×C4C22×D3×D511yescharacteristic
535C8—62no—
536C8—302no—
537C2×C3—202no—
538C2×C3—204no—
539C2×C3#6510014yescomplemented
540C2×C3C2×#402811yescharacteristic
541C2×C3C2×#406011yescharacteristic
542C2×C3C2×#406511yescharacteristic
543D3—402no—
544C5C2×#523811yescharacteristic, complemented, Sylow
545C22—102no—
546C22—102no—
547C22—104no—
548C22—102no—
549C22—202no—
550C22—202no—
551C22—204no—
552C22—208no—
553C22—602no—
554C22—602no—
555C22—602no—
556C22—1204no—
557C22#9150212yes—
558C22#9151612yes—
559C22#9156412yes—
560C22C2×D5×#6711yescharacteristic
561C4—24no—
562C4—102no—
563C4—122no—
564C4#9156412yes—
565C4C22×#120211yescharacteristic
566C4C2×D5×#6711yescharacteristic
567C3C2×#6510011yescharacteristic, complemented, Sylow
568C2—202no—
569C2—204no—
570C2—1202no—
571C2#1174927214yesdirect product factor
572C2C2×#9150211yescharacteristic
573C2C2×#9151611yescharacteristic
574C2C2×#9156411yescharacteristic
subgroupG/H
centerC23D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#449
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#713C2semidirect
2. C22×#6717C2semidirect
3. C2×#91450C2semidirect
4. C22×#6653C2semidirect
5. C2×C5×#835C2semidirect
6. #11749272C2direct (N × H)
7. C2×#90964C2semidirect
8. C2×#90955C2semidirect
15. C2×C3×#713C22semidirect
22. C2×#6717C22semidirect
28. #91450C22semidirect
30. C2×#6653C22semidirect
31. C5×#835C22semidirect
32. #90964C22semidirect
33. #90955C22semidirect
35. C22×#713D3semidirect
48. C2×C3×C5×Q8D4semidirect
65. C2×C3×D20D4semidirect
89. C2×#835D5semidirect
96. C2×#713C2×D3semidirect
132. C3×C5×Q8C2×D4semidirect
149. C3×D20C2×D4semidirect
178. #835C2×D5semidirect
186. C2×C5×Q8#67semidirect
205. C2×D20#67semidirect
260. C2×C3×Q8D20semidirect
308. C5×Q8C2×#67semidirect
326. D20C2×#67semidirect
357. C2×C3×C5#449semidirect
379. C3×Q8C2×D20semidirect
434. C2×Q8#1202semidirect
467. C3×C5C2×#449semidirect
490. C2×C5#5238semidirect
512. Q8C2×#1202semidirect
539. C2×C3#65100semidirect
544. C5C2×#5238semidirect
567. C3C2×#65100semidirect
571. C2#11749272direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C22×#67172
C2×#914502
C22×#66532
C2×C5×#8352
#117492722
C2×#909642
C2×#909552
C2×#651003
C2×#52385