Finite Group Databasebeta

← C2×#11748160 #14423462

Subgroups — C2×#11748160 #14423462

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×#6773C211yesmaximal, characteristic, complemented
2C22×D120C211yesmaximal, characteristic, complemented
3C2×#91450C211yesmaximal, characteristic, complemented
4C22×#6660C211yesmaximal, characteristic, complemented
5C2×C5×#816C211yesmaximal, characteristic, complemented
6C2×C3×#3952C211yesmaximal, characteristic, complemented
7C2×#90969C211yesmaximal, characteristic, complemented
8#11748160C218yesmaximal, direct product factor
9C2×#65100—31nomaximal, Hall
10C22×D60C2211yescharacteristic
11C22×D60C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C22×C3×C4×D5C2211yescharacteristic
14C22×#1202—21no—
15C22×#1202—21no—
16C22×#1200—21no—
17C2×#6773—24no—
18C2×#6773—24no—
19C2×#6773—22no—
20C2×#6773C2214yescomplemented
21C22×C3×C5×C8C2211yescharacteristic
22C2×D120—24no—
23C2×D120C2214yescomplemented
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C22×C3×#168C2211yescharacteristic
26C2×#6667—21no—
27#91450—22no—
28#91450C2214yescomplemented
29C2×#6660—24no—
30C2×#6660C2214yescomplemented
31C5×#816C2214yescomplemented
32C3×#3952C2214yescomplemented
33#90969C2214yescomplemented
34C2×#4734—51nomaximal, Hall
35C22×#713—31no—
36C22×#688—31no—
37C22×D40—31no—
38C2×#4041—31no—
39C2×C5×#103—31no—
40C2×#3952D311yescharacteristic, complemented
41C2×#3941—31no—
42#65100—38no—
43C23×D15—21no—
44C23×D15—21no—
45C23×C3×D5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C2×D60—21no—
48C2×D60—22no—
49C2×D60—22no—
50C2×D60—22no—
51C2×D60—21no—
52C2×D60D414yescomplemented
53C2×D60C2312yes—
54C2×D60C2312yes—
55C22×C5×Dic3—21no—
56C22×C5×Dic3—21no—
57C2×C5×Dic6—21no—
58C2×C5×Dic6—22no—
59C2×C5×Dic6D414yescomplemented
60C2×C5×Dic6C2312yes—
61C22×C3×Dic5D411yescharacteristic
62C2×C3×C4×D5—22no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5D412yes—
65C2×C3×C4×D5C2312yes—
66C2×#1202—28no—
67C2×#1202—24no—
68C2×#1202—24no—
69C2×#1202—44no—
70C2×#1200—28no—
71C2×#1200—24no—
72#6773—216no—
73#6773—28no—
74#6773—28no—
75C2×C3×C5×C8—22no—
76C2×C3×C5×C8C2312yes—
77D120—28no—
78C5×C4⋊Dic3—21no—
79C5×C4⋊Dic3C2312yes—
80C2×C3×#168—22no—
81C2×C3×#168C2312yes—
82#6667—24no—
83#6660—28no—
84C22×#235—51no—
85C2×#930—51no—
86C22×D24—51no—
87C22×#204—51no—
88C2×C3×C22⋊C8—51no—
89C2×#821—51no—
90C2×#816D511yescharacteristic, complemented
91#4734—58no—
92C22×C5×Q8—31no—
93C2×#713—34no—
94C2×#713—64no—
95C2×#713—64no—
96C2×#713—62no—
97C22×D20—31no—
98C22×D20—31no—
99C22×D20—61no—
100C22×D20—61no—
101C22×C4×D5—61no—
102C22×C4×D5C2×D311yescharacteristic
103C22×C5×C8C2×D311yescharacteristic
104C2×C5×C4⋊C4—31no—
105C2×#688—34no—
106C2×#688—64no—
107C2×#685—61no—
108C22×#168C2×D311yescharacteristic
109C2×D40—34no—
110C2×D40—64no—
111#4041—34no—
112#4041—62no—
113C5×#103—34no—
114#3952C2×D314yescomplemented
115#3941—34no—
116C2×#449—151noSylow
117C23×C3×C5C2×D411yescharacteristic
118C22×D15—24no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—42no—
122C22×D15—42no—
123C22×D15—42no—
124C22×D15—42no—
125C22×D15—42no—
126C22×C3×D5—22no—
127C22×C3×D5—22no—
128C22×C3×D5—22no—
129C22×C3×D5C2×D412yes—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2411yescharacteristic
134D60—24no—
135D60—22no—
136D60—22no—
137D60—22no—
138D60—42no—
139D60C2×D414yescomplemented
140C2×C5×Dic3—24no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—42no—
144C5×Dic6—24no—
145C5×Dic6—22no—
146C5×Dic6C2×D414yescomplemented
147C2×C3×Dic5—22no—
148C2×C3×Dic5C2×D412yes—
149C3×C4×D5—24no—
150C3×C4×D5—22no—
151C3×C4×D5—22no—
152#1202—216no—
153#1202—48no—
154#1200—216no—
155C3×C5×C8—22no—
156C3×#168—22no—
157C23×C3×C4—51no—
158C22×#67—101no—
159C22×#67—101no—
160C2×#235—54no—
161C2×#235—104no—
162C2×#235—102no—
163C2×#235—104no—
164C22×D12—51no—
165C22×D12—51no—
166C22×C4×D3—101no—
167C22×Dic6C2×D511yescharacteristic
168C22×C3×C8—51no—
169C22×C3×C8C2×D511yescharacteristic
170#930—54no—
171#930—102no—
172C2×(C2×D3)⋊C4—101no—
173C2×C4⋊Dic3C2×D511yescharacteristic
174C2×D24—54no—
175C2×D24—104no—
176C2×#204—54no—
177C2×#204—104no—
178C3×C22⋊C8—54no—
179#821—54no—
180#816C2×D514yescomplemented
181C23×D5—61no—
182C23×D5—61no—
183C23×D5D1211yescharacteristic
184C2×C5×Q8—34no—
185C2×C5×Q8—32no—
186C2×C5×Q8—61no—
187C2×C5×Q8—62no—
188C22×C4×C5—61no—
189C22×C4×C5—61no—
190C22×C4×C5C22×D311yescharacteristic
191C22×Dic5D1211yescharacteristic
192#713—68no—
193#713—68no—
194#713—616no—
195C2×D20—34no—
196C2×D20—32no—
197C2×D20—32no—
198C2×D20—68no—
199C2×D20—64no—
200C2×D20—61no—
201C2×D20—62no—
202C2×D20—62no—
203C2×D20—62no—
204C2×D20—64no—
205C2×D20—61no—
206C2×D20—124no—
207C2×C4×D5—22no—
208C2×C4×D5—22no—
209C2×C4×D5—68no—
210C2×C4×D5—64no—
211C2×C4×D5D1212yes—
212C2×C4×D5C22×D312yes—
213C2×C5×C8—22no—
214C2×C5×C8C22×D312yes—
215C5×C4⋊C4—32no—
216C5×C4⋊C4—61no—
217#688—68no—
218#685—64no—
219C2×#168—22no—
220C2×#168C22×D312yes—
221D40—68no—
222C22×#41—151no—
223C22×SD16—151no—
224C22×D8—151no—
225C2×#121—151no—
226#449—158no—
227C2×#103—31no—
228C2×#102—151no—
229C2×C22⋊C8—51no—
230C22×C3×C5#12012yes—
231C22×C3×C5#13512yes—
232C22×C3×C5AGL(1,ℤ/8)12yes—
233C22×C3×C5C22×D411yescharacteristic
234C2×D15—24no—
235C2×D15—42no—
236C2×D15—42no—
237C2×D15—48no—
238C2×D15—42no—
239C2×D15—42no—
240C2×D15—42no—
241C2×D15—84no—
242C2×C3×D5—22no—
243C2×C3×D5—22no—
244C2×C3×D5—22no—
245C2×C3×D5—44no—
246C3×C4×C5#12012yes—
247C3×C4×C5C22×D411yescharacteristic
248C3×C4×C5C22×D411yescharacteristic
249C3×Dic5—22no—
250C5×Dic3—24no—
251C5×Dic3—42no—
252C24×C3—51no—
253C23×D3—101no—
254C23×D3—101no—
255C22×C3×C4—51no—
256C22×C3×C4—52no—
257C22×C3×C4—52no—
258C22×C3×C4—102no—
259C22×C3×C4—102no—
260C22×C3×C4C22×D511yescharacteristic
261C2×#67—104no—
262C2×#67—108no—
263C2×#67—104no—
264C2×#67—204no—
265C22×Dic3—21no—
266C22×Dic3—21no—
267#235—108no—
268#235—1016no—
269#235—108no—
270C2×D12—54no—
271C2×D12—52no—
272C2×D12—52no—
273C2×D12—102no—
274C2×D12—101no—
275C2×D12—102no—
276C2×D12—102no—
277C2×D12—101no—
278C2×C4×D3—104no—
279C2×C4×D3—108no—
280C2×Dic6—21no—
281C2×Dic6—22no—
282C2×Dic6D2014yescomplemented
283C2×Dic6C22×D512yes—
284C2×C3×C8—22no—
285C2×C3×C8—52no—
286C2×C3×C8—102no—
287C2×C3×C8C22×D512yes—
288(C2×D3)⋊C4—104no—
289C4⋊Dic3—21no—
290C4⋊Dic3C22×D512yes—
291D24—108no—
292#204—108no—
293C23×C5C2×D1211yescharacteristic
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—22no—
297C22×D5—64no—
298C22×D5—62no—
299C22×D5—62no—
300C22×D5—122no—
301C22×D5—122no—
302C22×D5—122no—
303C22×D5—122no—
304C22×D5—122no—
305C22×D5C2×D1212yes—
306C5×Q8—34no—
307C5×Q8—64no—
308C5×Q8—62no—
309C2×C4×C5—64no—
310C2×C4×C5—62no—
311C2×C4×C5—62no—
312C2×C4×C5—122no—
313C2×C4×C5C2×D1211yescharacteristic
314C2×C4×C5D3×D412yes—
315C2×C4×C5D3×D412yes—
316C2×C4×C5C23×D311yescharacteristic
317C2×Dic5—22no—
318C2×Dic5C2×D1212yes—
319D20—34no—
320D20—616no—
321D20—64no—
322D20—62no—
323D20—62no—
324D20—62no—
325D20—128no—
326D20—122no—
327C4×D5—24no—
328C4×D5—22no—
329C4×D5—22no—
330C4×D5—616no—
331C5×C8—22no—
332#168—22no—
333C2×#41—154no—
334C2×#41—302no—
335C2×#41—304no—
336C2×#41—304no—
337C22×Q8—31no—
338C22×D4—151no—
339C22×D4—151no—
340C22×D4—301no—
341C22×D4—301no—
342C23×C4—51no—
343C23×C4—301no—
344C2×SD16—154no—
345C2×SD16—304no—
346C2×D8—154no—
347C2×D8—304no—
348C22×C8—51no—
349C22×C8D3×D511yescharacteristic
350#121—154no—
351#121—302no—
352C2×C4⋊C4—31no—
353C2×C22⋊C4—301no—
354#103—34no—
355#102—154no—
356C22⋊C8—54no—
357C2×C3×C5#44914yescomplemented
358C2×C3×C5C2×#12011yescharacteristic
359C2×C3×C5C2×#13511yescharacteristic
360C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
361D15—44no—
362D15—82no—
363C3×D5—42no—
364C23×C3—52no—
365C23×C3—102no—
366C23×C3—102no—
367C23×C3—102no—
368C23×C3D4×D511yescharacteristic
369C22×D3—102no—
370C22×D3—104no—
371C22×D3—102no—
372C22×D3—202no—
373C22×D3—202no—
374C22×D3—202no—
375C22×D3—202no—
376C22×D3—202no—
377C2×C3×C4—52no—
378C2×C3×C4—102no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4—104no—
382C2×C3×C4C2×D2012yes—
383C2×C3×C4D4×D511yescharacteristic
384C2×C3×C4D4×D512yes—
385C2×C3×C4C23×D511yescharacteristic
386#67—1016no—
387#67—208no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—22no—
391C2×Dic3—42no—
392D12—54no—
393D12—102no—
394D12—102no—
395D12—102no—
396D12—104no—
397D12—202no—
398C4×D3—1016no—
399Dic6—22no—
400Dic6—24no—
401Dic6C2×D2014yescomplemented
402C3×C8—22no—
403C3×C8—102no—
404C22×C5#88212yes—
405C22×C5#90412yes—
406C22×C5#90812yes—
407C22×C5C22×D1211yescharacteristic
408C2×D5—22no—
409C2×D5—22no—
410C2×D5—22no—
411C2×D5—44no—
412C2×D5—64no—
413C2×D5—122no—
414C2×D5—122no—
415C2×D5—128no—
416C2×D5—122no—
417C2×D5—122no—
418C2×D5—122no—
419C2×D5—244no—
420C4×C5—64no—
421C4×C5—122no—
422C4×C5#88212yes—
423C4×C5C2×D3×D411yescharacteristic
424C4×C5C2×D3×D411yescharacteristic
425Dic5—22no—
426C24—51no—
427C24—301no—
428C24—301no—
429#41—308no—
430#41—308no—
431#41—3016no—
432C2×Q8—34no—
433C2×Q8—32no—
434C2×Q8—61no—
435C2×Q8—62no—
436C2×D4—154no—
437C2×D4—152no—
438C2×D4—152no—
439C2×D4—304no—
440C2×D4—308no—
441C2×D4—304no—
442C2×D4—301no—
443C2×D4—301no—
444C2×D4—302no—
445C2×D4—302no—
446C2×D4—302no—
447C2×D4—604no—
448C22×C4—51no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—304no—
456C22×C4—308no—
457C22×C4C2×D3×D511yescharacteristic
458SD16—308no—
459D8—308no—
460C2×C8—22no—
461C2×C8—52no—
462C2×C8—102no—
463C2×C8C2×D3×D512yes—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C22⋊C4—304no—
467C3×C5C2×#44911yescharacteristic, complemented, Hall
468C22×C3—102no—
469C22×C3—102no—
470C22×C3—102no—
471C22×C3—204no—
472C22×C3#402812yes—
473C22×C3#406012yes—
474C22×C3#406512yes—
475C22×C3C2×D4×D511yescharacteristic
476C2×D3—104no—
477C2×D3—202no—
478C2×D3—202no—
479C2×D3—208no—
480C2×D3—202no—
481C2×D3—202no—
482C2×D3—202no—
483C2×D3—404no—
484C3×C4—102no—
485C3×C4#402812yes—
486C3×C4C22×D2011yescharacteristic
487C3×C4C2×D4×D511yescharacteristic
488Dic3—24no—
489Dic3—42no—
490C2×C5#473414yescomplemented
491C2×C5C2×#88211yescharacteristic
492C2×C5C2×#90411yescharacteristic
493C2×C5C2×#90811yescharacteristic
494D5—42no—
495D5—124no—
496D5—242no—
497C23—52no—
498C23—102no—
499C23—102no—
500C23—102no—
501C23—304no—
502C23—302no—
503C23—302no—
504C23—602no—
505C23—602no—
506C23—602no—
507C23—602no—
508C23—602no—
509C23D5×D1211yescharacteristic
510Q8—34no—
511Q8—64no—
512Q8—62no—
513D4—154no—
514D4—3016no—
515D4—304no—
516D4—302no—
517D4—302no—
518D4—302no—
519D4—608no—
520D4—602no—
521C2×C4—52no—
522C2×C4—64no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—102no—
526C2×C4—104no—
527C2×C4—102no—
528C2×C4—102no—
529C2×C4—122no—
530C2×C4—3016no—
531C2×C4D5×D1211yescharacteristic
532C2×C4D3×D2012yes—
533C2×C4#679012yes—
534C2×C4C22×D3×D511yescharacteristic
535C8—22no—
536C8—102no—
537C2×C3—202no—
538C2×C3#6510014yescomplemented
539C2×C3C2×#402811yescharacteristic
540C2×C3C2×#406011yescharacteristic
541C2×C3C2×#406511yescharacteristic
542D3—204no—
543D3—402no—
544C5C2×#473411yescharacteristic, complemented, Sylow
545C22—102no—
546C22—102no—
547C22—102no—
548C22—204no—
549C22—304no—
550C22—602no—
551C22—608no—
552C22—602no—
553C22—602no—
554C22—602no—
555C22—602no—
556C22—1204no—
557C22#9124912yes—
558C22#9127112yes—
559C22#9147612yes—
560C22C2×D5×D1211yescharacteristic
561C4—64no—
562C4—102no—
563C4—122no—
564C4#9147612yes—
565C4C2×D3×D2011yescharacteristic
566C4C2×#679011yescharacteristic
567C3C2×#6510011yescharacteristic, complemented, Sylow
568C2—202no—
569C2—604no—
570C2—1202no—
571C2#1174816014yesdirect product factor
572C2C2×#9124911yescharacteristic
573C2C2×#9127111yescharacteristic
574C2C2×#9147611yescharacteristic
subgroupG/H
centerC23D5×D12
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC23×C3×C5C2×D4
hypercenterC22×C8D3×D5
hypocenterC3×C5C2×#449
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×#6773C2semidirect
2. C22×D120C2semidirect
3. C2×#91450C2semidirect
4. C22×#6660C2semidirect
5. C2×C5×#816C2semidirect
6. C2×C3×#3952C2semidirect
7. C2×#90969C2semidirect
8. #11748160C2direct (N × H)
20. C2×#6773C22semidirect
23. C2×D120C22semidirect
28. #91450C22semidirect
30. C2×#6660C22semidirect
31. C5×#816C22semidirect
32. C3×#3952C22semidirect
33. #90969C22semidirect
40. C2×#3952D3semidirect
52. C2×D60D4semidirect
59. C2×C5×Dic6D4semidirect
90. C2×#816D5semidirect
114. #3952C2×D3semidirect
139. D60C2×D4semidirect
146. C5×Dic6C2×D4semidirect
180. #816C2×D5semidirect
282. C2×Dic6D20semidirect
357. C2×C3×C5#449semidirect
401. Dic6C2×D20semidirect
467. C3×C5C2×#449semidirect
490. C2×C5#4734semidirect
538. C2×C3#65100semidirect
544. C5C2×#4734semidirect
567. C3C2×#65100semidirect
571. C2#11748160direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67732
C22×D1202
C2×#914502
C22×#66602
C2×C5×#8162
C2×C3×#39522
C2×#909692
#117481602
C2×#651003
C2×#47345