Finite Group Databasebeta

← #14424260

Subgroups — #14424260

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic, complemented
2C22×#6715C211yesmaximal, characteristic, complemented
3#11754609C211yesmaximal, characteristic, complemented
4C2×#91301C211yesmaximal, characteristic, complemented
5C5×#5215C211yesmaximal, characteristic, complemented
6#11749026C214yesmaximal, complemented
7#11749022C214yesmaximal, complemented
8#11748364C211yesmaximal, characteristic, complemented
9#11748343C211yesmaximal, characteristic, complemented
10#10610659—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C22×D60C2211yescharacteristic, complemented
13C22×C3×C22⋊D5—22no—
14C3×#4144—24no—
15C3×#4144—24no—
16C2×C3×#711—21no—
17C2×C3×#711C2212yescomplemented
18C2×C3×D4×D5—21no—
19C2×C3×D4×D5C2212yescomplemented
20C2×C3×#709C2211yescharacteristic
21C2×C3×#709C2211yescharacteristic
22C2×#6715—24no—
23C2×#6715C2214yescomplemented
24C22×C3⋊#168C2211yescharacteristic, complemented
25C5×#926C2211yescharacteristic, complemented
26C2×C5×#208C2211yescharacteristic, complemented
27#91572—22no—
28#91458—21no—
29#91450C2214yescomplemented
30#91374—21no—
31C2×#6658C2212yescomplemented
32#91301—24no—
33C2×#6653C2212yescomplemented
34C5×#832C2214yescomplemented
35#90970C2212yescomplemented
36#90964C2212yescomplemented
37#90955C2214yescomplemented
38#84101—51nomaximal, Hall
39C2×#4144D311yescharacteristic, complemented
40C22×#686—31no—
41C2×#4054—31no—
42#65859—31no—
43C5×#418—31no—
44#65418—31no—
45#65291—31no—
46#65070—34no—
47#65062—34no—
48C24×C3×C5—21no—
49C23×D15—21no—
50C23×C3×D5—22no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—22no—
53C2×C3×C5×D4D414yescomplemented
54C2×C3×C5×D4C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C2×D60—22no—
57C2×D60—22no—
58C2×D60—21no—
59C2×D60C2312yes—
60C2×C3×C22⋊D5—216no—
61C2×C3×C22⋊D5—21no—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—22no—
64C2×C3×C22⋊D5D412yes—
65C22×C3×Dic5—22no—
66C3×#711—28no—
67C3×#711—24no—
68C3×#711—24no—
69C3×D4×D5—28no—
70C3×D4×D5—24no—
71C3×D4×D5—24no—
72C3×#709—22no—
73C3×#709—22no—
74C3×#709—22no—
75C3×#709D414yescomplemented
76C2×C3×D20D411yescharacteristic
77C2×C3×D20C2311yescharacteristic
78C2×C3×C4×D5—21no—
79C2×C3×C4×D5C2312yes—
80C2×C3×Dic10D411yescharacteristic
81C2×C3×Dic10C2311yescharacteristic
82C2×#1202—24no—
83#6715—28no—
84C2×C3⋊#168—22no—
85C2×C3⋊#168C2312yes—
86C5×#217—21no—
87C5×C4⋊Dic3C2312yes—
88C4×C5×Dic3—21no—
89C5×#208—22no—
90C2×C5×C3⋊C8C2312yes—
91#6667—24no—
92#6658—24no—
93#6653—24no—
94C2×C3×H(5,𝔽2)—51no—
95#5830—51no—
96C2×#949—51no—
97C22×#213—51no—
98#5238—54no—
99#5234—51no—
100#5233—51no—
101#5217—54no—
102#5215D511yescharacteristic, complemented
103C22×C5×D4C2×D311yescharacteristic
104C22×C22⋊D5—22no—
105#4144—24no—
106#4144—24no—
107C2×#711—21no—
108C2×#711C2×D312yescomplemented
109C2×D4×D5—21no—
110C2×D4×D5C2×D312yescomplemented
111C2×#709C2×D311yescharacteristic
112C2×#709C2×D311yescharacteristic
113C22×D20—31no—
114C2×C5×M16—31no—
115C5×#117—31no—
116C2×#686—34no—
117C2×#686—64no—
118C22×#168—31no—
119#4054—64no—
120C2×D40—32no—
121C2×#677—32no—
122#4041—34no—
123#4028—62no—
124#4021—61no—
125#4020—61no—
126C5×#102—34no—
127#3952—34no—
128#3942—32no—
129#3941—32no—
130#3008—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—24no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D411yescharacteristic
135C22×D15—22no—
136C22×D15—42no—
137C22×D15—82no—
138C22×C3×D5—21no—
139C22×C3×D5—28no—
140C22×C3×D5—84no—
141C22×C3×D5C2×D412yes—
142C3×C5×D4—24no—
143C3×C5×D4—22no—
144C3×C5×D4C2×D414yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149D60—22no—
150D60—22no—
151D60—42no—
152C2×C5×Dic3—22no—
153C3×C22⋊D5—28no—
154C3×C22⋊D5—28no—
155C3×C22⋊D5—24no—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—28no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5C2×D412yes—
160C3×D20—22no—
161C3×D20—21no—
162C3×D20C2×D412yes—
163C3×C4×D5—24no—
164C3×C4×D5—22no—
165C3×C4×D5—22no—
166C3×Dic10—22no—
167C3×Dic10—21no—
168C3×Dic10C2×D412yes—
169#1202—84no—
170C3⋊#168—22no—
171C5×C3⋊C8—22no—
172C3×H(5,𝔽2)—104no—
173C3×H(5,𝔽2)—104no—
174C2×C3×#41—51no—
175C2×C3×#41—52no—
176C2×C3×#41—51no—
177C2×C3×#41—101no—
178C22×C3×D4—52no—
179C22×C3×D4—101no—
180C22×C3×D4—102no—
181C22×C3×D4C2×D511yescharacteristic
182C22×D12—51no—
183#949—104no—
184#947—101no—
185C2×#215—52no—
186#940—101no—
187#937—102no—
188C2×#213—54no—
189C2×#213—52no—
190C2×#213—104no—
191#930—54no—
192#926C2×D511yescharacteristic, complemented
193C2×#208C2×D511yescharacteristic, complemented
194C22×C3⋊C8—51no—
195#835—52no—
196#832—52no—
197#832C2×D514yescomplemented
198#830—54no—
199C24×C5—21no—
200C23×D5—22no—
201C23×D5—61no—
202C2×C5×D4—21no—
203C2×C5×D4—22no—
204C2×C5×D4#6714yescomplemented
205C2×C5×D4C22×D312yes—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—216no—
208C2×C22⋊D5—21no—
209C2×C22⋊D5—22no—
210C2×C22⋊D5—22no—
211C2×C22⋊D5#6712yes—
212C22×Dic5—22no—
213#711—28no—
214#711—24no—
215#711—24no—
216D4×D5—28no—
217D4×D5—24no—
218D4×D5—24no—
219#709—22no—
220#709—22no—
221#709—22no—
222#709#6714yescomplemented
223C2×D20—32no—
224C2×D20—62no—
225C2×D20—62no—
226C2×D20—61no—
227C2×D20—64no—
228C2×D20#6711yescharacteristic
229C2×D20C22×D311yescharacteristic
230C2×C4×D5—21no—
231C2×C4×D5C22×D312yes—
232C2×Dic10#6711yescharacteristic
233C2×Dic10C22×D311yescharacteristic
234C5×M16—62no—
235C2×C5×C8—32no—
236C5×C4⋊C4—32no—
237C5×C22⋊C4—61no—
238C42×C5—61no—
239#686—68no—
240#685—64no—
241C2×#168—32no—
242C2×#168—62no—
243D40—64no—
244#677—64no—
245C2×H(5,𝔽2)—51no—
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—31no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5—24no—
256C22×C3×C5—44no—
257C22×C3×C5—84no—
258C22×C3×C5#12012yes—
259C22×C3×C5C22×D411yescharacteristic
260C2×D15—42no—
261C2×D15—44no—
262C2×D15—84no—
263C2×C3×D5—22no—
264C2×C3×D5—22no—
265C2×C3×D5—24no—
266C2×C3×D5—42no—
267C2×C3×D5—88no—
268C3×C4×C5#12012yes—
269C3×C4×C5C22×D411yescharacteristic
270C3×C4×C5C22×D411yescharacteristic
271C3×Dic5—22no—
272C3×Dic5—22no—
273C5×Dic3—42no—
274C24×C3—21no—
275C24×C3—102no—
276C23×D3—101no—
277C3×#41—54no—
278C3×#41—104no—
279C3×#41—108no—
280C3×#41—104no—
281C3×#41—102no—
282C3×#41—102no—
283C3×#41—102no—
284C2×C3×Q8—51no—
285C2×C3×Q8—51no—
286C2×C3×D4—21no—
287C2×C3×D4—22no—
288C2×C3×D4—52no—
289C2×C3×D4—51no—
290C2×C3×D4—51no—
291C2×C3×D4—102no—
292C2×C3×D4—104no—
293C2×C3×D4—108no—
294C2×C3×D4—102no—
295C2×C3×D4—1016no—
296C2×C3×D4—101no—
297C2×C3×D4—104no—
298C2×C3×D4D2014yescomplemented
299C2×C3×D4C22×D512yes—
300C22×C3×C4—52no—
301C22×C3×C4—102no—
302C22×C3×C4—101no—
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—104no—
305C2×D12—52no—
306C2×D12—102no—
307C2×D12—102no—
308C2×D12—101no—
309#217—21no—
310#215—104no—
311#213—104no—
312#213—108no—
313(C2×D3)⋊C4—104no—
314C4⋊Dic3C22×D512yes—
315C4×Dic3—21no—
316#208—22no—
317C2×C3⋊C8—52no—
318C2×C3⋊C8—102no—
319C2×C3⋊C8C22×D512yes—
320C23×C5—22no—
321C23×C5—24no—
322C23×C5—82no—
323C23×C5C2×#6711yescharacteristic
324C22×D5—21no—
325C22×D5—28no—
326C22×D5—62no—
327C22×D5—84no—
328C22×D5—122no—
329C22×D5—242no—
330C22×D5C2×#6712yes—
331C5×D4—24no—
332C5×D4—22no—
333C5×D4C2×#6714yes—
334C2×C4×C5—62no—
335C2×C4×C5C2×#6712yes—
336C2×C4×C5C2×#6711yescharacteristic
337C2×C4×C5C2×#6712yes—
338C2×C4×C5C23×D311yescharacteristic
339C22⋊D5—28no—
340C22⋊D5—28no—
341C22⋊D5—24no—
342C22⋊D5—24no—
343C2×Dic5—28no—
344C2×Dic5—21no—
345C2×Dic5C2×#6712yes—
346D20—22no—
347D20—21no—
348D20—62no—
349D20—62no—
350D20—122no—
351D20—244no—
352D20C2×#6712yes—
353C4×D5—24no—
354C4×D5—22no—
355C4×D5—22no—
356Dic10—22no—
357Dic10—21no—
358Dic10C2×#6712yes—
359C5×C8—62no—
360#168—62no—
361H(5,𝔽2)—104no—
362H(5,𝔽2)—104no—
363C2×#41—51no—
364C2×#41—52no—
365C2×#41—51no—
366C2×#41—101no—
367C22×D4—52no—
368C22×D4—101no—
369C22×D4—102no—
370C22×D4—151no—
371C22×D4D3×D511yescharacteristic
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—152no—
375C2×D8—154no—
376C2×D8—304no—
377C2×M16—31no—
378C22×C8—151no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—31no—
384#103—152no—
385#102—34no—
386#102—152no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5—44no—
390C2×C3×C5C2×#12011yescharacteristic
391C2×C3×C5#57612yes—
392D15—82no—
393C3×D5—42no—
394C3×D5—42no—
395C23×C3—22no—
396C23×C3—24no—
397C23×C3—52no—
398C23×C3—82no—
399C23×C3—108no—
400C23×C3—101no—
401C23×C3—404no—
402C23×C3D4×D511yescharacteristic
403C22×D3—102no—
404C22×D3—202no—
405C22×D3—402no—
406C3×Q8—52no—
407C3×Q8—101no—
408C3×Q8—102no—
409C3×D4—22no—
410C3×D4—24no—
411C3×D4—52no—
412C3×D4—108no—
413C3×D4—108no—
414C3×D4—104no—
415C3×D4—104no—
416C3×D4—101no—
417C3×D4—102no—
418C3×D4C2×D2014yes—
419C2×C3×C4—52no—
420C2×C3×C4—108no—
421C2×C3×C4—101no—
422C2×C3×C4—102no—
423C2×C3×C4—102no—
424C2×C3×C4—104no—
425C2×C3×C4C2×D2012yes—
426C2×C3×C4D4×D512yes—
427C2×C3×C4D4×D511yescharacteristic
428C2×C3×C4C23×D511yescharacteristic
429#67—404no—
430C2×Dic3—22no—
431D12—102no—
432D12—102no—
433D12—202no—
434C3⋊C8—22no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—44no—
439C22×C5—84no—
440C22×C5#95312yes—
441C22×C5C22×#6711yescharacteristic
442C2×D5—24no—
443C2×D5—22no—
444C2×D5—22no—
445C2×D5—42no—
446C2×D5—88no—
447C2×D5—124no—
448C2×D5—122no—
449C2×D5—244no—
450C4×C5—122no—
451C4×C5#95312yes—
452C4×C5C22×#6711yescharacteristic
453C4×C5C22×#6711yescharacteristic
454Dic5—22no—
455Dic5—22no—
456C24—21no—
457C24—102no—
458C24—301no—
459#41—54no—
460#41—104no—
461#41—108no—
462#41—104no—
463#41—102no—
464#41—102no—
465#41—102no—
466C2×Q8—51no—
467C2×Q8—51no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—52no—
471C2×D4—51no—
472C2×D4—51no—
473C2×D4—102no—
474C2×D4—104no—
475C2×D4—108no—
476C2×D4—102no—
477C2×D4—1016no—
478C2×D4—101no—
479C2×D4—104no—
480C2×D4—152no—
481C2×D4—304no—
482C2×D4—302no—
483C2×D4—302no—
484C2×D4—301no—
485C2×D4#120214yescomplemented
486C2×D4C2×D3×D512yes—
487C22×C4—52no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4C2×D3×D511yescharacteristic
491SD16—304no—
492D8—304no—
493D8—308no—
494M16—62no—
495C2×C8—32no—
496C2×C8—152no—
497C2×C8—302no—
498C4⋊C4—32no—
499C22⋊C4—61no—
500C22⋊C4—304no—
501C42—61no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—44no—
506C22×C3—84no—
507C22×C3—102no—
508C22×C3—102no—
509C22×C3—104no—
510C22×C3—202no—
511C22×C3—408no—
512C22×C3#402812yes—
513C22×C3C2×D4×D511yescharacteristic
514C2×D3—202no—
515C2×D3—204no—
516C2×D3—404no—
517C3×C4—102no—
518C3×C4—102no—
519C3×C4#402812yes—
520C3×C4C22×D2011yescharacteristic
521C3×C4C2×D4×D511yescharacteristic
522Dic3—42no—
523C2×C5—22no—
524C2×C5—44no—
525C2×C5#583512yes—
526C2×C5C2×#95311yescharacteristic
527D5—42no—
528D5—42no—
529D5—242no—
530C23—22no—
531C23—24no—
532C23—52no—
533C23—82no—
534C23—108no—
535C23—101no—
536C23—302no—
537C23—404no—
538C23—602no—
539C23—1202no—
540C23D5×#6711yescharacteristic
541Q8—52no—
542Q8—101no—
543Q8—102no—
544D4—22no—
545D4—24no—
546D4—52no—
547D4—108no—
548D4—108no—
549D4—104no—
550D4—104no—
551D4—101no—
552D4—102no—
553D4—302no—
554D4—302no—
555D4—602no—
556D4—1204no—
557D4C2×#120214yes—
558C2×C4—52no—
559C2×C4—62no—
560C2×C4—108no—
561C2×C4—101no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—104no—
565C2×C4C2×#120212yes—
566C2×C4D5×#6712yes—
567C2×C4D5×#6711yescharacteristic
568C2×C4C22×D3×D511yescharacteristic
569C8—62no—
570C8—302no—
571C2×C3—22no—
572C2×C3—44no—
573C2×C3—202no—
574C2×C3—202no—
575C2×C3C2×#402811yescharacteristic
576C2×C3#6610812yes—
577D3—402no—
578C5#8410111yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—44no—
582C22—84no—
583C22—102no—
584C22—102no—
585C22—104no—
586C22—202no—
587C22—408no—
588C22—602no—
589C22—604no—
590C22—1204no—
591C22#9156412yes—
592C22C2×D5×#6711yescharacteristic
593C4—102no—
594C4—102no—
595C4—122no—
596C4#9156412yes—
597C4C22×#120211yescharacteristic
598C4C2×D5×#6711yescharacteristic
599C3#1061065911yescharacteristic, complemented, Sylow
600C2—22no—
601C2—44no—
602C2—202no—
603C2—202no—
604C2—1202no—
605C2#1175469212yes—
606C2C2×#9156411yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC22×D4D3×D5
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4144C2semidirect
2. C22×#6715C2semidirect
3. #11754609C2semidirect
4. C2×#91301C2semidirect
5. C5×#5215C2semidirect
6. #11749026C2semidirect
7. #11749022C2semidirect
8. #11748364C2semidirect
9. #11748343C2semidirect
12. C22×D60C22semidirect
17. C2×C3×#711C22semidirect
19. C2×C3×D4×D5C22semidirect
23. C2×#6715C22semidirect
24. C22×C3⋊#168C22semidirect
25. C5×#926C22semidirect
26. C2×C5×#208C22semidirect
29. #91450C22semidirect
31. C2×#6658C22semidirect
33. C2×#6653C22semidirect
34. C5×#832C22semidirect
35. #90970C22semidirect
36. #90964C22semidirect
37. #90955C22semidirect
39. C2×#4144D3semidirect
53. C2×C3×C5×D4D4semidirect
75. C3×#709D4semidirect
102. #5215D5semidirect
108. C2×#711C2×D3semidirect
110. C2×D4×D5C2×D3semidirect
192. #926C2×D5semidirect
193. C2×#208C2×D5semidirect
197. #832C2×D5semidirect
204. C2×C5×D4#67semidirect
222. #709#67semidirect
298. C2×C3×D4D20semidirect
485. C2×D4#1202semidirect
502. C3×C5#3008semidirect
578. C5#84101semidirect
599. C3#10610659semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C22×#67152
#117546092
C2×#913012
C5×#52152
#117490262
#117490222
#117483642
#117483432
#106106593
#841015