Finite Group Databasebeta

← C2×#11748511 #14423451

Subgroups — C2×#11748511 #14423451

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×#709C211yesmaximal, characteristic, complemented
2C2×C5×#930C211yesmaximal, characteristic, complemented
3C22×#6656C211yesmaximal, characteristic, complemented
4C22×#6652C211yesmaximal, characteristic, complemented
5C2×#91114C211yesmaximal, characteristic, complemented
6#11748511C218yesmaximal, direct product factor
7C2×#90970C211yesmaximal, characteristic, complemented
8C2×#90964C211yesmaximal, characteristic, complemented
9C2×#65326—31nomaximal, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C22×C5×#67—21no—
12C22×C5×D12C2211yescharacteristic
13C22×C3×C22⋊D5—21no—
14C2×C3×#709—22no—
15C2×C3×#709—24no—
16C2×C3×#709—24no—
17C2×C3×#709C2214yescomplemented
18C22×C3×D20C2211yescharacteristic
19C22×C3×C4×D5—21no—
20C22×C3×Dic10C2211yescharacteristic
21C22×C3⋊#168C2211yescharacteristic
22C22×C3⋊#168C2211yescharacteristic
23C5×#930—22no—
24C5×#930C2214yescomplemented
25C2×C5×(C2×D3)⋊C4—21no—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×#6656—24no—
28C2×#6656C2214yescomplemented
29C2×#6652—24no—
30C2×#6652C2214yescomplemented
31#91114C2214yescomplemented
32#90970C2214yescomplemented
33#90964C2214yescomplemented
34C2×#5238—51nomaximal, Hall
35C22×#709D311yescharacteristic, complemented
36C2×C5×#121—31no—
37C22×#687—31no—
38C22×#686—31no—
39C2×C22⋊#168—31no—
40#65326—38no—
41C2×#3942—31no—
42C2×#3941—31no—
43C24×C3×C5D411yescharacteristic
44C23×C5×D3—21no—
45C23×C3×D5—21no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5D411yescharacteristic
49C22×C3×C4×C5D412yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C2×C5×#67—24no—
53C2×C5×#67—44no—
54C22×C5×Dic3—21no—
55C2×C5×D12—22no—
56C2×C5×D12—22no—
57C2×C5×D12—21no—
58C2×C5×D12C2312yes—
59C2×C3×C22⋊D5—28no—
60C2×C3×C22⋊D5—24no—
61C22×C3×Dic5—21no—
62C3×#709—216no—
63C3×#709—28no—
64C3×#709—28no—
65C2×C3×D20—21no—
66C2×C3×D20—22no—
67C2×C3×D20D414yescomplemented
68C2×C3×D20C2312yes—
69C2×C3×C4×D5—28no—
70C2×C3×C4×D5—24no—
71C2×C3×Dic10—21no—
72C2×C3×Dic10—22no—
73C2×C3×Dic10D414yescomplemented
74C2×C3×Dic10C2312yes—
75C2×C3⋊#168—22no—
76C2×C3⋊#168—22no—
77C2×C3⋊#168C2312yes—
78C2×C3⋊#168C2312yes—
79C5×(C2×D3)⋊C4—24no—
80C5×C4⋊Dic3—21no—
81C5×C4⋊Dic3C2312yes—
82#6656—28no—
83#6652—28no—
84C22×C3×#41—51no—
85C22×#215—51no—
86C22×#213—51no—
87C2×#930D511yescharacteristic, complemented
88#5238—58no—
89C2×#835—51no—
90C2×#832—51no—
91C2×#830—51no—
92C22×C5×D4—31no—
93C22×C5×D4—61no—
94C23×C4×C5C2×D311yescharacteristic
95C22×C22⋊D5—21no—
96C2×#709—22no—
97C2×#709—24no—
98C2×#709—24no—
99C2×#709C2×D314yescomplemented
100C22×D20C2×D311yescharacteristic
101C22×C4×D5—21no—
102C22×Dic10C2×D311yescharacteristic
103C5×#121—34no—
104C5×#121—62no—
105C2×C5×C4⋊C4—31no—
106C2×C5×C22⋊C4—61no—
107C2×#687—34no—
108C2×#687—64no—
109C2×#686—34no—
110C2×#686—64no—
111C22×#168—31no—
112C22×#168—31no—
113C22⋊#168—34no—
114#3942—34no—
115#3941—34no—
116C2×#449—151noSylow
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5C2×D412yes—
121C23×C3×C5C2×D411yescharacteristic
122C22×C5×D3—22no—
123C22×C5×D3—42no—
124C22×C5×D3—42no—
125C22×C5×D3—42no—
126C22×C3×D5—24no—
127C22×C3×D5—22no—
128C22×C3×D5—42no—
129C22×C3×D5—42no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—24no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C2×D412yes—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D412yes—
137C2×C3×C4×C5C2×D412yes—
138C2×C3×C4×C5C2411yescharacteristic
139C5×#67—48no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—42no—
142C5×D12—22no—
143C5×D12—22no—
144C5×D12—42no—
145C3×C22⋊D5—216no—
146C2×C3×Dic5—24no—
147C2×C3×Dic5—22no—
148C3×D20—24no—
149C3×D20—22no—
150C3×D20C2×D414yescomplemented
151C3×C4×D5—216no—
152C3×Dic10—24no—
153C3×Dic10—22no—
154C3×Dic10C2×D414yescomplemented
155C3⋊#168—22no—
156C3⋊#168—22no—
157C2×C3×#41—54no—
158C2×C3×#41—104no—
159C2×C3×#41—104no—
160C2×C3×#41—102no—
161C22×C3×Q8—51no—
162C22×C3×D4—51no—
163C22×C3×D4—101no—
164C23×C3×C4—101no—
165C23×C3×C4C2×D511yescharacteristic
166C22×#67—21no—
167C22×D12C2×D511yescharacteristic
168C2×#215—54no—
169C2×#215—104no—
170C2×#213—54no—
171C2×#213—104no—
172#930—22no—
173#930C2×D514yescomplemented
174C2×(C2×D3)⋊C4—21no—
175C2×C4⋊Dic3C2×D511yescharacteristic
176C22×C3⋊C8—51no—
177C22×C3⋊C8—51no—
178#835—54no—
179#832—54no—
180#830—54no—
181C24×C5—61no—
182C24×C5#6711yescharacteristic
183C23×D5—21no—
184C2×C5×D4—32no—
185C2×C5×D4—64no—
186C2×C5×D4—62no—
187C2×C5×D4—62no—
188C2×C5×D4—61no—
189C2×C5×D4—124no—
190C22×C4×C5—22no—
191C22×C4×C5—22no—
192C22×C4×C5—61no—
193C22×C4×C5#6711yescharacteristic
194C22×C4×C5#6712yes—
195C22×C4×C5C22×D312yes—
196C22×C4×C5C22×D311yescharacteristic
197C2×C22⋊D5—28no—
198C2×C22⋊D5—24no—
199C22×Dic5—21no—
200#709—216no—
201#709—28no—
202#709—28no—
203C2×D20—21no—
204C2×D20—22no—
205C2×D20#6714yescomplemented
206C2×D20C22×D312yes—
207C2×C4×D5—28no—
208C2×C4×D5—24no—
209C2×Dic10—21no—
210C2×Dic10—22no—
211C2×Dic10#6714yescomplemented
212C2×Dic10C22×D312yes—
213C5×C4⋊C4—32no—
214C5×C4⋊C4—61no—
215C5×C22⋊C4—64no—
216#687—68no—
217#686—68no—
218C2×#168—32no—
219C2×#168—32no—
220C2×#168—62no—
221C2×#168—62no—
222C22×#41—51no—
223C22×SD16—151no—
224C22×D8—151no—
225C2×#121—31no—
226#449—158no—
227C2×#103—151no—
228C2×#102—151no—
229C2×C22⋊C8—151no—
230C22×C3×C5—22no—
231C22×C3×C5—22no—
232C22×C3×C5—22no—
233C22×C3×C5—44no—
234C22×C3×C5#12012yes—
235C22×C3×C5#13512yes—
236C22×C3×C5AGL(1,ℤ/8)12yes—
237C22×C3×C5C22×D411yescharacteristic
238C2×C5×D3—42no—
239C2×C5×D3—42no—
240C2×C5×D3—42no—
241C2×C5×D3—84no—
242C2×C3×D5—24no—
243C2×C3×D5—42no—
244C2×C3×D5—42no—
245C2×C3×D5—48no—
246C3×C4×C5—22no—
247C3×C4×C5#12012yes—
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C3×Dic5—24no—
251C5×Dic3—42no—
252C24×C3—101no—
253C24×C3C22⋊D511yescharacteristic
254C23×D3—21no—
255C3×#41—108no—
256C3×#41—108no—
257C3×#41—1016no—
258C2×C3×Q8—54no—
259C2×C3×Q8—52no—
260C2×C3×Q8—102no—
261C2×C3×Q8—101no—
262C2×C3×D4—54no—
263C2×C3×D4—52no—
264C2×C3×D4—108no—
265C2×C3×D4—104no—
266C2×C3×D4—102no—
267C2×C3×D4—101no—
268C22×C3×C4—22no—
269C22×C3×C4—22no—
270C22×C3×C4—108no—
271C22×C3×C4—101no—
272C22×C3×C4—104no—
273C22×C3×C4C22⋊D511yescharacteristic
274C22×C3×C4C22⋊D512yes—
275C22×C3×C4C22×D512yes—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—24no—
278C2×#67—44no—
279C22×Dic3—21no—
280C2×D12—22no—
281C2×D12—22no—
282C2×D12—21no—
283C2×D12C22×D512yes—
284#215—108no—
285#213—108no—
286(C2×D3)⋊C4—24no—
287C4⋊Dic3—21no—
288C4⋊Dic3C22×D512yes—
289C2×C3⋊C8—52no—
290C2×C3⋊C8—52no—
291C2×C3⋊C8—102no—
292C2×C3⋊C8—102no—
293C23×C5—22no—
294C23×C5—22no—
295C23×C5—22no—
296C23×C5—62no—
297C23×C5—122no—
298C23×C5—122no—
299C23×C5—122no—
300C23×C5C2×#6712yes—
301C23×C5C2×#6711yescharacteristic
302C22×D5—24no—
303C22×D5—22no—
304C22×D5—42no—
305C22×D5—42no—
306C5×D4—62no—
307C5×D4—62no—
308C5×D4—128no—
309C5×D4—122no—
310C2×C4×C5—22no—
311C2×C4×C5—24no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—62no—
315C2×C4×C5—122no—
316C2×C4×C5C2×#6712yes—
317C2×C4×C5C2×#6711yescharacteristic
318C2×C4×C5C2×#6712yes—
319C2×C4×C5C2×#6712yes—
320C2×C4×C5C23×D311yescharacteristic
321C22⋊D5—216no—
322C2×Dic5—24no—
323C2×Dic5—22no—
324D20—24no—
325D20—22no—
326D20C2×#6714yescomplemented
327C4×D5—216no—
328Dic10—24no—
329Dic10—22no—
330Dic10C2×#6714yescomplemented
331#168—62no—
332#168—62no—
333C2×#41—54no—
334C2×#41—104no—
335C2×#41—104no—
336C2×#41—102no—
337C22×Q8—51no—
338C22×D4—31no—
339C22×D4—51no—
340C22×D4—61no—
341C22×D4—101no—
342C23×C4—101no—
343C23×C4D3×D511yescharacteristic
344C2×SD16—154no—
345C2×SD16—304no—
346C2×D8—154no—
347C2×D8—304no—
348C22×C8—151no—
349C22×C8—151no—
350#121—34no—
351#121—62no—
352C2×C4⋊C4—31no—
353C2×C22⋊C4—61no—
354#103—154no—
355#102—154no—
356C22⋊C8—154no—
357C2×C3×C5—42no—
358C2×C3×C5#44914yescomplemented
359C2×C3×C5C2×#12011yescharacteristic
360C2×C3×C5C2×#13511yescharacteristic
361C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
362C3×D5—44no—
363C5×D3—82no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—102no—
368C23×C3—104no—
369C23×C3—202no—
370C23×C3—202no—
371C23×C3C2×C22⋊D512yes—
372C23×C3C2×C22⋊D511yescharacteristic
373C22×D3—22no—
374C22×D3—42no—
375C22×D3—42no—
376C22×D3—42no—
377C3×Q8—54no—
378C3×Q8—102no—
379C3×Q8—104no—
380C3×D4—54no—
381C3×D4—1016no—
382C3×D4—102no—
383C3×D4—104no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—24no—
388C2×C3×C4—1016no—
389C2×C3×C4—102no—
390C2×C3×C4—104no—
391C2×C3×C4D4×D512yes—
392C2×C3×C4D4×D512yes—
393C2×C3×C4C2×C22⋊D512yes—
394C2×C3×C4C2×C22⋊D511yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—48no—
397C2×Dic3—22no—
398C2×Dic3—42no—
399D12—22no—
400D12—22no—
401D12—42no—
402C3⋊C8—102no—
403C3⋊C8—102no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—44no—
408C22×C5—122no—
409C22×C5—122no—
410C22×C5—122no—
411C22×C5—244no—
412C22×C5#94912yes—
413C22×C5#95012yes—
414C22×C5#95312yes—
415C22×C5C22×#6711yescharacteristic
416C2×D5—24no—
417C2×D5—48no—
418C2×D5—42no—
419C2×D5—42no—
420C4×C5—22no—
421C4×C5—122no—
422C4×C5#95312yes—
423C4×C5C22×#6711yescharacteristic
424C4×C5C22×#6711yescharacteristic
425Dic5—24no—
426C24—61no—
427C24—101no—
428C24#120111yescharacteristic
429#41—108no—
430#41—108no—
431#41—1016no—
432C2×Q8—54no—
433C2×Q8—52no—
434C2×Q8—101no—
435C2×Q8—102no—
436C2×D4—32no—
437C2×D4—54no—
438C2×D4—52no—
439C2×D4—64no—
440C2×D4—61no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4—108no—
444C2×D4—104no—
445C2×D4—101no—
446C2×D4—102no—
447C2×D4—124no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—61no—
451C22×C4—108no—
452C22×C4—101no—
453C22×C4—104no—
454C22×C4#120112yes—
455C22×C4#120111yescharacteristic
456C22×C4C2×D3×D512yes—
457C22×C4C2×D3×D511yescharacteristic
458SD16—308no—
459D8—308no—
460C2×C8—152no—
461C2×C8—152no—
462C2×C8—302no—
463C2×C8—302no—
464C4⋊C4—32no—
465C4⋊C4—61no—
466C22⋊C4—64no—
467C3×C5C2×#44911yescharacteristic, complemented, Hall
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—44no—
472C22×C3—104no—
473C22×C3—208no—
474C22×C3—202no—
475C22×C3—202no—
476C22×C3#407812yes—
477C22×C3#408312yes—
478C22×C3#409612yes—
479C22×C3C22×C22⋊D511yescharacteristic
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—84no—
484C3×C4—22no—
485C3×C4—104no—
486C3×C4#408312yes—
487C3×C4C2×D4×D511yescharacteristic
488C3×C4C2×D4×D511yescharacteristic
489Dic3—42no—
490C2×C5—42no—
491C2×C5—242no—
492C2×C5#523814yescomplemented
493C2×C5C2×#94911yescharacteristic
494C2×C5C2×#95011yescharacteristic
495C2×C5C2×#95311yescharacteristic
496D5—44no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—62no—
501C23—102no—
502C23—104no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—202no—
507C23—202no—
508C23C2×#120112yes—
509C23C2×#120111yescharacteristic
510Q8—54no—
511Q8—102no—
512Q8—104no—
513D4—54no—
514D4—62no—
515D4—62no—
516D4—1016no—
517D4—102no—
518D4—104no—
519D4—128no—
520D4—122no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—62no—
526C2×C4—1016no—
527C2×C4—102no—
528C2×C4—104no—
529C2×C4—122no—
530C2×C4C2×#120112yes—
531C2×C4C2×#120111yescharacteristic
532C2×C4D5×#6712yes—
533C2×C4D5×#6712yes—
534C2×C4C22×D3×D511yescharacteristic
535C8—302no—
536C8—302no—
537C2×C3—42no—
538C2×C3—204no—
539C2×C3#6532614yescomplemented
540C2×C3C2×#407811yescharacteristic
541C2×C3C2×#408311yescharacteristic
542C2×C3C2×#409611yescharacteristic
543D3—82no—
544C5C2×#523811yescharacteristic, complemented, Sylow
545C22—22no—
546C22—22no—
547C22—22no—
548C22—44no—
549C22—104no—
550C22—122no—
551C22—122no—
552C22—122no—
553C22—202no—
554C22—202no—
555C22—208no—
556C22—244no—
557C22#9129712yes—
558C22#9129912yes—
559C22#9156312yes—
560C22C22×#120111yescharacteristic
561C4—22no—
562C4—104no—
563C4—122no—
564C4#9156312yes—
565C4C2×D5×#6711yescharacteristic
566C4C2×D5×#6711yescharacteristic
567C3C2×#6532611yescharacteristic, complemented, Sylow
568C2—42no—
569C2—204no—
570C2—242no—
571C2#1174851114yesdirect product factor
572C2C2×#9129711yescharacteristic
573C2C2×#9129911yescharacteristic
574C2C2×#9156311yescharacteristic
subgroupG/H
centerC23C2×#1201
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C2×D4
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#449
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#709C2semidirect
2. C2×C5×#930C2semidirect
3. C22×#6656C2semidirect
4. C22×#6652C2semidirect
5. C2×#91114C2semidirect
6. #11748511C2direct (N × H)
7. C2×#90970C2semidirect
8. C2×#90964C2semidirect
17. C2×C3×#709C22semidirect
24. C5×#930C22semidirect
28. C2×#6656C22semidirect
30. C2×#6652C22semidirect
31. #91114C22semidirect
32. #90970C22semidirect
33. #90964C22semidirect
35. C22×#709D3semidirect
67. C2×C3×D20D4semidirect
73. C2×C3×Dic10D4semidirect
87. C2×#930D5semidirect
99. C2×#709C2×D3semidirect
150. C3×D20C2×D4semidirect
154. C3×Dic10C2×D4semidirect
173. #930C2×D5semidirect
205. C2×D20#67semidirect
211. C2×Dic10#67semidirect
326. D20C2×#67semidirect
330. Dic10C2×#67semidirect
358. C2×C3×C5#449semidirect
467. C3×C5C2×#449semidirect
492. C2×C5#5238semidirect
539. C2×C3#65326semidirect
544. C5C2×#5238semidirect
567. C3C2×#65326semidirect
571. C2#11748511direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7092
C2×C5×#9302
C22×#66562
C22×#66522
C2×#911142
#117485112
C2×#909702
C2×#909642
C2×#653263
C2×#52385