Finite Group Databasebeta

← #14458196

Subgroups — #14458196

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756511C211yesmaximal, characteristic, complemented
2#11755332C212yesmaximal, complemented
3#11753961C211yesmaximal, characteristic, complemented
4C5×#5142C211yesmaximal, characteristic, complemented
5C3×#65433C211yesmaximal, characteristic, complemented
6#11749422C212yesmaximal, complemented
7#11748249C212yesmaximal, complemented
8#11748190C212yesmaximal, complemented
9#11748082C211yesmaximal, characteristic, complemented
10#11748064C211yesmaximal, characteristic, complemented
11#11747996C211yesmaximal, characteristic, complemented
12#10612672—31nomaximal, Hall
13#92100C2212yescomplemented
14#92018C2212yescomplemented
15C3×C5×#131C2211yescharacteristic, complemented
16#91797—22no—
17#91797C2212yescomplemented
18#91793—21no—
19#91793C2211yescharacteristic
20C2×D120—21no—
21C2×D120C2211yescharacteristic, complemented
22C5×#949C2212yescomplemented
23C2×C5×D24C2211yescharacteristic, complemented
24C3×#4096C2211yescharacteristic
25C3×#4047C2211yescharacteristic, complemented
26#91523—22no—
27#91506—22no—
28#91500—22no—
29#91481—22no—
30#91451C2212yescomplemented
31#91304C2212yescomplemented
32C2×#6654C2212yescomplemented
33#91301C2212yescomplemented
34#91290C2212yescomplemented
35#91270—21no—
36#91250—22no—
37D5×D24—21no—
38C5×#823C2212yescomplemented
39C5×#819C2211yescharacteristic, complemented
40C3×#3969C2211yescharacteristic, complemented
41C3×#3957C2211yescharacteristic, complemented
42C3×#3954C2211yescharacteristic, complemented
43C3×C8×Dic5C2211yescharacteristic, complemented
44#90981C2212yescomplemented
45#90960C2212yescomplemented
46#79894—51nomaximal, Hall
47#66108—32no—
48#66091—31no—
49#66086—31no—
50C5×#472—31no—
51#65487—32no—
52#65440—31no—
53#65433D311yescharacteristic, complemented
54#65186—31no—
55#65184—31no—
56#65038—32no—
57#65036—32no—
58C3×C5×#41C2311yescharacteristic
59#6821—22no—
60D4×D15—22no—
61D4×D15—42no—
62#6817—22no—
63#6817C2312yes—
64C2×D60—22no—
65C2×D60C2312yes—
66C2×C5×D12C2312yes—
67C3×#709C2311yescharacteristic
68D3×C22⋊D5—42no—
69C2×#1203—42no—
70#6777—22no—
71D5×D12—22no—
72#6772—22no—
73#6768—22no—
74C3×C5×M16—21no—
75C3×C5×M16C2311yescharacteristic
76C2×C3×C5×C8—21no—
77C2×C3×C5×C8C2311yescharacteristic
78Dic60D411yescharacteristic, complemented
79D120—22no—
80D120—21no—
81D120—21no—
82D120D411yescharacteristic, complemented
83#6706—22no—
84#6706D412yescomplemented
85C5×#215—22no—
86C5×#213—22no—
87C5×#208C2312yes—
88C5×D24—21no—
89C5×D24—21no—
90C3×#689D411yescharacteristic, complemented
91C3×#688D411yescharacteristic, complemented
92C3×#687D411yescharacteristic, complemented
93C3×#686D411yescharacteristic, complemented
94C3×C4×Dic5C2311yescharacteristic
95C3×#681C2311yescharacteristic
96C2×C3×#168C2311yescharacteristic
97C3×#676—21no—
98C3×C8×D5—21no—
99#6659—22no—
100#6658—22no—
101#6654—22no—
102#6654—22no—
103#6654—22no—
104#6653—22no—
105#6651—22no—
106#6648—22no—
107#5769—52no—
108#5752—51no—
109#5752—51no—
110C3×#443—51no—
111#5142—51no—
112#5142D511yescharacteristic, complemented
113#4897—52no—
114#4822—52no—
115#4822—52no—
116#4700—51no—
117#4695—51no—
118#4149—32no—
119#4144—32no—
120C5×AGL(1,ℤ/8)—32no—
121C2×C5×D8—31no—
122C5×#131C2×D311yescharacteristic, complemented
123#4096C2×D311yescharacteristic
124#4086—32no—
125#4078—32no—
126C2×#686—32no—
127#4062—62no—
128#4060—62no—
129#4058—61no—
130#4057—62no—
131#4057—62no—
132D5×D8—61no—
133D5×D8—62no—
134#4054—32no—
135#4054—32no—
136#4054—62no—
137#4053—32no—
138#4050—31no—
139#4050—61no—
140C2×D40—31no—
141C2×D40—61no—
142#4047C2×D311yescharacteristic, complemented
143C5×#108—31no—
144C5×#100—32no—
145#3969C2×D311yescharacteristic, complemented
146#3957—32no—
147#3957C2×D311yescharacteristic, complemented
148#3955—32no—
149#3954C2×D311yescharacteristic, complemented
150C8×Dic5C2×D311yescharacteristic, complemented
151#3292—151noSylow
152C22×D15—22no—
153C22×D15—42no—
154C22×C5×D3—22no—
155C2×D3×D5—42no—
156C3×C5×Q8C2×D411yescharacteristic
157C3×C5×D4—21no—
158C3×C5×D4C2×D411yescharacteristic
159C2×C3×C4×C5—21no—
160C2×C3×C4×C5C2411yescharacteristic
161#1220—22no—
162#1220—42no—
163D60—22no—
164D60—22no—
165D60—42no—
166D60C2×D412yes—
167C4×D15—22no—
168C4×D15—42no—
169Dic30C2×D412yes—
170C5×D12—22no—
171C5×D12—22no—
172C3×C22⋊D5—21no—
173C2×C3×Dic5—21no—
174C3×D20C2×D411yescharacteristic
175C3×C4×D5—21no—
176C3×Dic10C2×D411yescharacteristic
177#1203—42no—
178#1203—84no—
179#1201—42no—
180D3×Dic5—42no—
181C3×C5×C8—21no—
182C3×C5×C8C2×D411yescharacteristic
183C3×C5×C8C2×D411yescharacteristic
184C3×#168—21no—
185C3×#168C2×D411yescharacteristic
186C3×#168C2×D411yescharacteristic
187C5×C3⋊C8—22no—
188#1009—52no—
189#1009—52no—
190C3×#135—51no—
191C3×#131—51no—
192C3×#131C2×D511yescharacteristic, complemented
193#949—52no—
194#949C2×D512yescomplemented
195#919—102no—
196#914—102no—
197#914—102no—
198D3×D8—102no—
199#908—52no—
200#908—52no—
201#908—102no—
202#908—102no—
203#904—51no—
204#904—101no—
205#904—101no—
206C2×D24—52no—
207C2×D24—51no—
208C2×D24—101no—
209C2×D24—101no—
210C2×D24C2×D511yescharacteristic, complemented
211#901—52no—
212#874—52no—
213C3×#108—51no—
214C3×#104—51no—
215C3×#104—51no—
216C3×C4×C8—51no—
217#823—52no—
218#823C2×D512yescomplemented
219#819C2×D511yescharacteristic, complemented
220#805—52no—
221C5×#41C22×D311yescharacteristic
222C2×C5×D4—32no—
223C2×C22⋊D5—122no—
224C2×C22⋊D5—122no—
225#713—62no—
226#711—62no—
227#711—62no—
228D4×D5—62no—
229D4×D5—62no—
230D4×D5—62no—
231D4×D5—122no—
232#709—32no—
233#709—62no—
234#709C22×D311yescharacteristic
235C2×D20—32no—
236C2×D20—62no—
237C5×SD16—62no—
238C5×D8—62no—
239C5×D8—61no—
240C5×D8—61no—
241C5×M16—21no—
242C5×M16—32no—
243C5×M16C22×D311yescharacteristic
244C2×C5×C8—21no—
245C2×C5×C8C22×D311yescharacteristic
246#689D1211yescharacteristic, complemented
247#688D1211yescharacteristic, complemented
248#687—62no—
249#687D1211yescharacteristic, complemented
250#686—62no—
251#686—62no—
252#686—62no—
253#686D1211yescharacteristic, complemented
254C4×Dic5C22×D311yescharacteristic
255#681C22×D311yescharacteristic
256C2×#168C22×D311yescharacteristic
257Dic20—31no—
258D40—31no—
259D40—62no—
260D40—62no—
261D40—61no—
262D40—61no—
263#677—32no—
264#677—62no—
265#677—62no—
266#676—21no—
267#676—62no—
268C8×D5—21no—
269C8×D5—62no—
270#576—151no—
271#576—152no—
272#576—151no—
273#493—151no—
274#472—31no—
275#472—152no—
276#472—151no—
277#453—152no—
278#453—152no—
279#443—51no—
280#443—151no—
281C22×C3×C5—21no—
282C22×C3×C5H(5,𝔽2)11yescharacteristic
283C2×D15—22no—
284C2×D15—42no—
285C2×D15—82no—
286C2×D15—162no—
287C2×C5×D3—42no—
288C2×C5×D3—82no—
289C2×C3×D5—21no—
290D3×D5—162no—
291C3×C4×C5—21no—
292C3×C4×C5C22×D411yescharacteristic
293C3×C4×C5C22×D411yescharacteristic
294Dic15—22no—
295C3×Dic5—22no—
296C3×Dic5—21no—
297C3×#41—51no—
298C3×#41C22×D511yescharacteristic
299#239—102no—
300#239—102no—
301D3×D4—102no—
302D3×D4—102no—
303D3×D4—202no—
304D3×D4—202no—
305#235—52no—
306#235—102no—
307#235—102no—
308C2×D12—52no—
309C2×D12—102no—
310C2×D12—102no—
311C2×D12—202no—
312C2×D12C22×D512yes—
313C3×Q16—51no—
314C3×SD16—51no—
315C3×SD16—51no—
316C3×D8—51no—
317C3×M16—21no—
318C3×M16—51no—
319C3×M16—101no—
320C3×M16C22×D511yescharacteristic
321C2×C3×C8—21no—
322C2×C3×C8—51no—
323C2×C3×C8—101no—
324C2×C3×C8C22×D511yescharacteristic
325C3×C42—51no—
326#215—22no—
327#215—102no—
328#213—22no—
329#213—102no—
330#208C22×D512yes—
331Dic12—51no—
332D24—21no—
333D24—21no—
334D24—51no—
335D24—102no—
336D24—102no—
337D24—102no—
338D24—102no—
339D24—101no—
340D24—101no—
341#204—52no—
342#204—102no—
343#204—102no—
344ΓL(1,𝔽25)—102no—
345D3×C8—102no—
346C23×C5—62no—
347C22×D5—62no—
348C22×D5—122no—
349C22×D5—122no—
350C5×Q8C2×D1211yescharacteristic
351C5×D4—21no—
352C5×D4—62no—
353C5×D4—62no—
354C5×D4C2×D1211yescharacteristic
355C2×C4×C5—21no—
356C2×C4×C5C23×D311yescharacteristic
357C22⋊D5—21no—
358C22⋊D5—62no—
359C22⋊D5—122no—
360C22⋊D5—122no—
361C22⋊D5—122no—
362C22⋊D5—244no—
363C2×Dic5—21no—
364C2×Dic5—122no—
365D20—32no—
366D20—62no—
367D20—62no—
368D20—122no—
369D20C2×D1211yescharacteristic
370C4×D5—21no—
371C4×D5—62no—
372C4×D5—122no—
373Dic10—32no—
374Dic10C2×D1211yescharacteristic
375C5×C8—21no—
376C5×C8—62no—
377C5×C8D3×D411yescharacteristic
378C5×C8D3×D411yescharacteristic
379#168—21no—
380#168C2×D1211yescharacteristic
381#168C2×D1211yescharacteristic
382H(5,𝔽2)—152no—
383H(5,𝔽2)—152no—
384AGL(1,ℤ/8)—32no—
385AGL(1,ℤ/8)—152no—
386AGL(1,ℤ/8)—152no—
387AGL(1,ℤ/8)—152no—
388AGL(1,ℤ/8)—302no—
389AGL(1,ℤ/8)—302no—
390AGL(1,ℤ/8)—302no—
391AGL(1,ℤ/8)—302no—
392#135—51no—
393#135—151no—
394#135—301no—
395#135—301no—
396#135—302no—
397C2×D8—31no—
398C2×D8—152no—
399C2×D8—151no—
400C2×D8—301no—
401C2×D8—301no—
402C2×D8—302no—
403#131—51no—
404#131—152no—
405#131D3×D511yescharacteristic, complemented
406#127—152no—
407#108—31no—
408#108—51no—
409#104—51no—
410#104—51no—
411#104—152no—
412#100—32no—
413#100—152no—
414C4×C8—51no—
415C2×C3×C5—21no—
416C2×C3×C5—41no—
417C2×C3×C5D4211yescharacteristic
418D15—42no—
419D15—82no—
420C3×D5—41no—
421C5×D3—82no—
422C22×D3—22no—
423C22×D3—102no—
424C22×D3—202no—
425C22×D3—202no—
426C3×Q8—51no—
427C3×Q8D4×D511yescharacteristic
428C3×D4—21no—
429C3×D4—51no—
430C3×D4—101no—
431C3×D4D4×D511yescharacteristic
432C2×C3×C4—21no—
433C2×C3×C4—101no—
434C2×C3×C4—101no—
435C2×C3×C4C23×D511yescharacteristic
436#67—102no—
437#67—202no—
438#67—202no—
439D12—22no—
440D12—22no—
441D12—52no—
442D12—102no—
443D12—102no—
444D12—202no—
445D12—202no—
446D12—404no—
447C4×D3—102no—
448C4×D3—202no—
449C4×D3—202no—
450Dic6—52no—
451C3×C8—21no—
452C3×C8—51no—
453C3×C8—51no—
454C3×C8—101no—
455C3×C8D4×D511yescharacteristic
456C3×C8D4×D511yescharacteristic
457C3⋊C8—22no—
458C22×C5—21no—
459C22×C5—122no—
460C22×C5—242no—
461C22×C5#100911yescharacteristic
462C2×D5—21no—
463C2×D5—62no—
464C2×D5—122no—
465C2×D5—242no—
466C2×D5—482no—
467C2×D5—482no—
468C4×C5—21no—
469C4×C5C22×D1211yescharacteristic
470C4×C5C2×D3×D411yescharacteristic
471Dic5—22no—
472Dic5—21no—
473Dic5—62no—
474#41—51no—
475#41—152no—
476#41—302no—
477#41—302no—
478#41—302no—
479#41—302no—
480#41C2×D3×D511yescharacteristic
481C2×D4—32no—
482C2×D4—152no—
483C2×D4—302no—
484C2×D4—302no—
485C2×D4—302no—
486C2×D4—302no—
487C2×D4—602no—
488C2×D4—602no—
489C2×D4—602no—
490Q16—51no—
491Q16—151no—
492SD16—51no—
493SD16—51no—
494SD16—62no—
495SD16—152no—
496SD16—302no—
497SD16—302no—
498SD16—302no—
499D8—51no—
500D8—62no—
501D8—61no—
502D8—61no—
503D8—151no—
504D8—302no—
505D8—302no—
506D8—302no—
507D8—302no—
508D8—302no—
509D8—301no—
510D8—301no—
511M16—21no—
512M16—32no—
513M16—51no—
514M16—101no—
515M16—302no—
516M16C2×D3×D511yescharacteristic
517C2×C8—21no—
518C2×C8—51no—
519C2×C8—101no—
520C2×C8—302no—
521C2×C8C2×D3×D511yescharacteristic
522C42—51no—
523C3×C5#329211yescharacteristic, complemented, Hall
524C22×C3—21no—
525C22×C3—101no—
526C22×C3#414911yescharacteristic
527C2×D3—42no—
528C2×D3—82no—
529C2×D3—102no—
530C2×D3—202no—
531C2×D3—402no—
532C2×D3—802no—
533C2×D3—802no—
534C3×C4—21no—
535C3×C4—102no—
536C3×C4—101no—
537C3×C4C2×D4×D511yescharacteristic
538C3×C4C2×D4×D511yescharacteristic
539Dic3—102no—
540C2×C5—21no—
541C2×C5—41no—
542C2×C5—242no—
543C2×C5D4×D1211yescharacteristic
544D5—41no—
545D5—122no—
546D5—242no—
547C23—62no—
548C23—302no—
549C23—602no—
550C23—602no—
551Q8—51no—
552Q8—152no—
553Q8D5×D1211yescharacteristic
554D4—21no—
555D4—51no—
556D4—62no—
557D4—62no—
558D4—101no—
559D4—152no—
560D4—302no—
561D4—302no—
562D4—302no—
563D4—602no—
564D4—602no—
565D4—602no—
566D4—602no—
567D4—1204no—
568D4D5×D1211yescharacteristic
569C2×C4—21no—
570C2×C4—101no—
571C2×C4—101no—
572C2×C4—302no—
573C2×C4—602no—
574C2×C4—602no—
575C2×C4C22×D3×D511yescharacteristic
576C8—21no—
577C8—51no—
578C8—51no—
579C8—62no—
580C8—101no—
581C8#679011yescharacteristic
582C8#679011yescharacteristic
583C2×C3—21no—
584C2×C3—41no—
585C2×C3—201no—
586C2×C3#6594311yescharacteristic
587D3—82no—
588D3—202no—
589D3—402no—
590C5#7989411yescharacteristic, complemented, Sylow
591C22—21no—
592C22—101no—
593C22—122no—
594C22—242no—
595C22—302no—
596C22—602no—
597C22—1202no—
598C22—2402no—
599C22—2402no—
600C22#9201911yescharacteristic
601C4—21no—
602C4—102no—
603C4—101no—
604C4—302no—
605C4C2×D5×D1211yescharacteristic
606C4C2×#679011yescharacteristic
607C3#1061267211yescharacteristic, complemented, Sylow
608C2—21no—
609C2—41no—
610C2—201no—
611C2—242no—
612C2—602no—
613C2—1202no—
614C2#1175464711yescharacteristic
subgroupG/H
centerC2#11754647
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#131C22
socleC2×C3×C5D42
hypercenter#131D3×D5
hypocenterC3×C5#3292
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #11756511C2semidirect
2. #11755332C2semidirect
3. #11753961C2semidirect
4. C5×#5142C2semidirect
5. C3×#65433C2semidirect
6. #11749422C2semidirect
7. #11748249C2semidirect
8. #11748190C2semidirect
9. #11748082C2semidirect
10. #11748064C2semidirect
11. #11747996C2semidirect
13. #92100C22semidirect
14. #92018C22semidirect
15. C3×C5×#131C22semidirect
17. #91797C22semidirect
21. C2×D120C22semidirect
22. C5×#949C22semidirect
23. C2×C5×D24C22semidirect
25. C3×#4047C22semidirect
30. #91451C22semidirect
31. #91304C22semidirect
32. C2×#6654C22semidirect
33. #91301C22semidirect
34. #91290C22semidirect
38. C5×#823C22semidirect
39. C5×#819C22semidirect
40. C3×#3969C22semidirect
41. C3×#3957C22semidirect
42. C3×#3954C22semidirect
43. C3×C8×Dic5C22semidirect
44. #90981C22semidirect
45. #90960C22semidirect
53. #65433D3semidirect
78. Dic60D4semidirect
82. D120D4semidirect
84. #6706D4semidirect
90. C3×#689D4semidirect
91. C3×#688D4semidirect
92. C3×#687D4semidirect
93. C3×#686D4semidirect
112. #5142D5semidirect
122. C5×#131C2×D3semidirect
142. #4047C2×D3semidirect
145. #3969C2×D3semidirect
147. #3957C2×D3semidirect
149. #3954C2×D3semidirect
150. C8×Dic5C2×D3semidirect
192. C3×#131C2×D5semidirect
194. #949C2×D5semidirect
210. C2×D24C2×D5semidirect
218. #823C2×D5semidirect
219. #819C2×D5semidirect
246. #689D12semidirect
247. #688D12semidirect
249. #687D12semidirect
253. #686D12semidirect
405. #131D3×D5semidirect
523. C3×C5#3292semidirect
590. C5#79894semidirect
607. C3#10612672semidirect

Maximal subgroups

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

subgroupindex
#117565112
#117553322
#117539612
C5×#51422
C3×#654332
#117494222
#117482492
#117481902
#117480822
#117480642
#117479962
#106126723
#798945