Finite Group Databasebeta

← #14458194

Subgroups — #14458194

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#11755330C212yesmaximal, complemented
3#11753971C211yesmaximal, characteristic, complemented
4C5×#5144C211yesmaximal, characteristic, complemented
5C3×#65431C211yesmaximal, characteristic, complemented
6#11749421C212yesmaximal, complemented
7#11748248C212yesmaximal, complemented
8#11748194C212yesmaximal, complemented
9#11748086C211yesmaximal, characteristic, complemented
10#11748068C211yesmaximal, characteristic, complemented
11#11748005C211yesmaximal, characteristic, complemented
12#10611625—31nomaximal, Hall
13#92100C2212yescomplemented
14#92017C2212yescomplemented
15C3×C5×#131C2211yescharacteristic, complemented
16#91797—22no—
17#91797C2212yescomplemented
18#91793—21no—
19#91793C2211yescharacteristic
20C2×D120—21no—
21C2×D120C2211yescharacteristic, complemented
22C5×#950C2211yescharacteristic
23C5×#901C2211yescharacteristic, complemented
24C3×#4095C2212yescomplemented
25C2×C3×D40C2211yescharacteristic, complemented
26#91521—22no—
27#91506—22no—
28#91498—22no—
29#91481—22no—
30#91458C2212yescomplemented
31#91312C2212yescomplemented
32#91305C2212yescomplemented
33C2×#6653C2212yescomplemented
34#91291C2212yescomplemented
35#91273—21no—
36#91254—22no—
37D3×D40—21no—
38C5×#837C2211yescharacteristic, complemented
39C5×#825C2211yescharacteristic, complemented
40C5×#822C2211yescharacteristic, complemented
41C5×C8×Dic3C2211yescharacteristic, complemented
42C3×#3955C2212yescomplemented
43C3×#3951C2211yescharacteristic, complemented
44#90980C2212yescomplemented
45#90960C2212yescomplemented
46#80941—51nomaximal, Hall
47#66103—32no—
48#66086—31no—
49#66086—31no—
50C5×#443—31no—
51#65431—31no—
52#65431D311yescharacteristic, complemented
53#65186—32no—
54#65111—32no—
55#65111—32no—
56#64989—31no—
57#64984—31no—
58C3×C5×#41C2311yescharacteristic
59#6821—22no—
60D4×D15—22no—
61D4×D15—42no—
62#6817—22no—
63#6817C2312yes—
64C2×D60—22no—
65C2×D60C2312yes—
66C5×#235C2311yescharacteristic
67C2×C3×D20C2312yes—
68D5×#67—42no—
69C2×#1202—42no—
70#6777—22no—
71D3×D20—22no—
72#6766—22no—
73#6765—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×#216D411yescharacteristic, complemented
86C5×#215D411yescharacteristic, complemented
87C5×#214D411yescharacteristic, complemented
88C5×#213D411yescharacteristic, complemented
89C4×C5×Dic3C2311yescharacteristic
90C5×#208C2311yescharacteristic
91C2×C5×C3⋊C8C2311yescharacteristic
92C5×ΓL(1,𝔽25)—21no—
93C5×D3×C8—21no—
94C3×#688—22no—
95C3×#686—22no—
96C3×#681C2312yes—
97C3×D40—21no—
98C3×D40—21no—
99#6660—22no—
100#6657—22no—
101#6654—22no—
102#6653—22no—
103#6653—22no—
104#6653—22no—
105#6651—22no—
106#6648—22no—
107#5774—52no—
108#5757—51no—
109#5752—51no—
110C3×#472—51no—
111#5198—52no—
112#5151—51no—
113#5144D511yescharacteristic, complemented
114#4897—51no—
115#4895—51no—
116#4749—52no—
117#4747—52no—
118#4149—32no—
119#4149—32no—
120C5×#135—31no—
121C5×#131—31no—
122C5×#131C2×D311yescharacteristic, complemented
123#4095—32no—
124#4095C2×D312yescomplemented
125#4065—62no—
126#4060—62no—
127#4060—62no—
128D5×D8—62no—
129#4054—32no—
130#4054—32no—
131#4054—62no—
132#4054—62no—
133#4050—31no—
134#4050—61no—
135#4050—61no—
136C2×D40—32no—
137C2×D40—31no—
138C2×D40—61no—
139C2×D40—61no—
140C2×D40C2×D311yescharacteristic, complemented
141#4047—32no—
142#4020—32no—
143C5×#108—31no—
144C5×#104—31no—
145C5×#104—31no—
146C4×C5×C8—31no—
147#3955—32no—
148#3955C2×D312yescomplemented
149#3951C2×D311yescharacteristic, complemented
150#3937—32no—
151#3292—151noSylow
152C22×D15—22no—
153C22×D15—42no—
154C22×C3×D5—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×#67—21no—
171C2×C5×Dic3—21no—
172C5×D12C2×D411yescharacteristic
173C4×C5×D3—21no—
174C5×Dic6C2×D411yescharacteristic
175C3×D20—22no—
176C3×D20—22no—
177#1202—42no—
178#1202—84no—
179#1201—42no—
180D5×Dic3—42no—
181C3×C5×C8—21no—
182C3×C5×C8C2×D411yescharacteristic
183C3×C5×C8C2×D411yescharacteristic
184C3×#168—22no—
185C5×C3⋊C8—21no—
186C5×C3⋊C8C2×D411yescharacteristic
187C5×C3⋊C8C2×D411yescharacteristic
188#1009—52no—
189#1004—52no—
190C3×AGL(1,ℤ/8)—52no—
191C2×C3×D8—51no—
192C3×#131C2×D511yescharacteristic, complemented
193#950C2×D511yescharacteristic
194#940—52no—
195#932—52no—
196C2×#213—52no—
197#916—102no—
198#914—102no—
199#912—101no—
200#911—102no—
201#911—102no—
202D3×D8—101no—
203D3×D8—102no—
204#908—52no—
205#908—52no—
206#908—102no—
207#907—52no—
208#904—51no—
209#904—101no—
210C2×D24—51no—
211C2×D24—101no—
212#901C2×D511yescharacteristic, complemented
213C3×#108—51no—
214C3×#100—52no—
215#837C2×D511yescharacteristic, complemented
216#825—52no—
217#825C2×D511yescharacteristic, complemented
218#823—52no—
219#822C2×D511yescharacteristic, complemented
220C8×Dic3C2×D511yescharacteristic, complemented
221C5×#41—31no—
222C5×#41C22×D311yescharacteristic
223#713—62no—
224#713—62no—
225D4×D5—62no—
226D4×D5—62no—
227D4×D5—122no—
228D4×D5—122no—
229#709—32no—
230#709—62no—
231#709—62no—
232C2×D20—32no—
233C2×D20—62no—
234C2×D20—62no—
235C2×D20—122no—
236C2×D20C22×D312yes—
237C5×Q16—31no—
238C5×SD16—31no—
239C5×SD16—31no—
240C5×D8—31no—
241C5×M16—21no—
242C5×M16—31no—
243C5×M16—61no—
244C5×M16C22×D311yescharacteristic
245C2×C5×C8—21no—
246C2×C5×C8—31no—
247C2×C5×C8—61no—
248C2×C5×C8C22×D311yescharacteristic
249C42×C5—31no—
250#688—22no—
251#688—62no—
252#686—22no—
253#686—62no—
254#681C22×D312yes—
255Dic20—31no—
256D40—21no—
257D40—21no—
258D40—31no—
259D40—62no—
260D40—62no—
261D40—62no—
262D40—62no—
263D40—61no—
264D40—61no—
265#677—32no—
266#677—62no—
267#677—62no—
268#676—62no—
269C8×D5—62no—
270#576—151no—
271#576—152no—
272#576—151no—
273#493—151no—
274#472—51no—
275#472—152no—
276#472—151no—
277#453—152no—
278#453—152no—
279#443—31no—
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—21no—
288C2×C3×D5—42no—
289C2×C3×D5—82no—
290D3×D5—162no—
291C3×C4×C5—21no—
292C3×C4×C5C22×D411yescharacteristic
293C3×C4×C5C22×D411yescharacteristic
294Dic15—22no—
295C5×Dic3—22no—
296C5×Dic3—21no—
297C3×#41C22×D511yescharacteristic
298C2×C3×D4—52no—
299C2×#67—202no—
300C2×#67—202no—
301#239—102no—
302#237—102no—
303#237—102no—
304D3×D4—102no—
305D3×D4—102no—
306D3×D4—102no—
307D3×D4—202no—
308#235—52no—
309#235—102no—
310#235C22×D511yescharacteristic
311C2×D12—52no—
312C2×D12—102no—
313C3×SD16—102no—
314C3×D8—102no—
315C3×D8—101no—
316C3×D8—101no—
317C3×M16—21no—
318C3×M16—52no—
319C3×M16C22×D511yescharacteristic
320C2×C3×C8—21no—
321C2×C3×C8C22×D511yescharacteristic
322#216D2011yescharacteristic, complemented
323#215D2011yescharacteristic, complemented
324#214—102no—
325#214D2011yescharacteristic, complemented
326#213—102no—
327#213—102no—
328#213—102no—
329#213D2011yescharacteristic, complemented
330C4×Dic3C22×D511yescharacteristic
331#208C22×D511yescharacteristic
332C2×C3⋊C8C22×D511yescharacteristic
333Dic12—51no—
334D24—51no—
335D24—102no—
336D24—102no—
337D24—101no—
338D24—101no—
339#204—52no—
340#204—102no—
341#204—102no—
342ΓL(1,𝔽25)—21no—
343ΓL(1,𝔽25)—102no—
344D3×C8—21no—
345D3×C8—102no—
346C22×D5—22no—
347C22×D5—62no—
348C22×D5—122no—
349C22×D5—122no—
350C5×Q8—31no—
351C5×Q8D3×D411yescharacteristic
352C5×D4—21no—
353C5×D4—31no—
354C5×D4—61no—
355C5×D4D3×D411yescharacteristic
356C2×C4×C5—21no—
357C2×C4×C5—61no—
358C2×C4×C5—61no—
359C2×C4×C5C23×D311yescharacteristic
360C22⋊D5—62no—
361C22⋊D5—122no—
362C22⋊D5—122no—
363D20—22no—
364D20—22no—
365D20—32no—
366D20—62no—
367D20—62no—
368D20—122no—
369D20—122no—
370D20—244no—
371C4×D5—62no—
372C4×D5—122no—
373C4×D5—122no—
374Dic10—32no—
375C5×C8—21no—
376C5×C8—31no—
377C5×C8—31no—
378C5×C8—61no—
379C5×C8D3×D411yescharacteristic
380C5×C8D3×D411yescharacteristic
381#168—22no—
382H(5,𝔽2)—152no—
383H(5,𝔽2)—152no—
384AGL(1,ℤ/8)—52no—
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—31no—
393#135—151no—
394#135—301no—
395#135—301no—
396#135—302no—
397C2×D8—51no—
398C2×D8—152no—
399C2×D8—151no—
400C2×D8—301no—
401C2×D8—301no—
402C2×D8—302no—
403#131—31no—
404#131—152no—
405#131D3×D511yescharacteristic, complemented
406#127—152no—
407#108—31no—
408#108—51no—
409#104—31no—
410#104—31no—
411#104—152no—
412#100—52no—
413#100—152no—
414C4×C8—31no—
415C2×C3×C5—21no—
416C2×C3×C5—41no—
417C2×C3×C5D4211yescharacteristic
418D15—42no—
419D15—82no—
420C3×D5—82no—
421C5×D3—41no—
422C23×C3—102no—
423C22×D3—102no—
424C22×D3—202no—
425C22×D3—202no—
426C3×Q8C2×D2011yescharacteristic
427C3×D4—21no—
428C3×D4—102no—
429C3×D4—102no—
430C3×D4C2×D2011yescharacteristic
431C2×C3×C4—21no—
432C2×C3×C4C23×D511yescharacteristic
433#67—21no—
434#67—102no—
435#67—202no—
436#67—202no—
437#67—202no—
438#67—404no—
439C2×Dic3—21no—
440C2×Dic3—202no—
441D12—52no—
442D12—102no—
443D12—102no—
444D12—202no—
445D12C2×D2011yescharacteristic
446C4×D3—21no—
447C4×D3—102no—
448C4×D3—202no—
449Dic6—52no—
450Dic6C2×D2011yescharacteristic
451C3×C8—21no—
452C3×C8—102no—
453C3×C8D4×D511yescharacteristic
454C3×C8D4×D511yescharacteristic
455C3⋊C8—21no—
456C3⋊C8C2×D2011yescharacteristic
457C3⋊C8C2×D2011yescharacteristic
458C22×C5—21no—
459C22×C5—61no—
460C22×C5#100911yescharacteristic
461C2×D5—42no—
462C2×D5—62no—
463C2×D5—82no—
464C2×D5—122no—
465C2×D5—242no—
466C2×D5—482no—
467C2×D5—482no—
468C4×C5—21no—
469C4×C5—62no—
470C4×C5—61no—
471C4×C5C2×D3×D411yescharacteristic
472C4×C5C2×D3×D411yescharacteristic
473Dic5—62no—
474#41—31no—
475#41—152no—
476#41—302no—
477#41—302no—
478#41—302no—
479#41—302no—
480#41C2×D3×D511yescharacteristic
481C2×D4—52no—
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—31no—
491Q16—151no—
492SD16—31no—
493SD16—31no—
494SD16—102no—
495SD16—152no—
496SD16—302no—
497SD16—302no—
498SD16—302no—
499D8—31no—
500D8—102no—
501D8—101no—
502D8—101no—
503D8—151no—
504D8—302no—
505D8—302no—
506D8—302no—
507D8—302no—
508D8—302no—
509D8—301no—
510D8—301no—
511M16—21no—
512M16—31no—
513M16—52no—
514M16—61no—
515M16—302no—
516M16C2×D3×D511yescharacteristic
517C2×C8—21no—
518C2×C8—31no—
519C2×C8—61no—
520C2×C8—302no—
521C2×C8C2×D3×D511yescharacteristic
522C42—31no—
523C3×C5#329211yescharacteristic, complemented, Hall
524C22×C3—21no—
525C22×C3—202no—
526C22×C3—402no—
527C22×C3#414911yescharacteristic
528C2×D3—21no—
529C2×D3—102no—
530C2×D3—202no—
531C2×D3—402no—
532C2×D3—802no—
533C2×D3—802no—
534C3×C4—21no—
535C3×C4C22×D2011yescharacteristic
536C3×C4C2×D4×D511yescharacteristic
537Dic3—22no—
538Dic3—21no—
539Dic3—102no—
540C2×C5—21no—
541C2×C5—41no—
542C2×C5—121no—
543C2×C5#560911yescharacteristic
544D5—82no—
545D5—122no—
546D5—242no—
547C23—102no—
548C23—302no—
549C23—602no—
550C23—602no—
551Q8—31no—
552Q8—152no—
553Q8D3×D2011yescharacteristic
554D4—21no—
555D4—31no—
556D4—61no—
557D4—102no—
558D4—102no—
559D4—152no—
560D4—302no—
561D4—302no—
562D4—302no—
563D4—602no—
564D4—602no—
565D4—602no—
566D4—602no—
567D4—1204no—
568D4D3×D2011yescharacteristic
569C2×C4—21no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—302no—
573C2×C4—602no—
574C2×C4—602no—
575C2×C4C22×D3×D511yescharacteristic
576C8—21no—
577C8—31no—
578C8—31no—
579C8—61no—
580C8—102no—
581C8#679011yescharacteristic
582C8#679011yescharacteristic
583C2×C3—21no—
584C2×C3—41no—
585C2×C3—402no—
586C2×C3D4×D2011yescharacteristic
587D3—41no—
588D3—202no—
589D3—402no—
590C5#8094111yescharacteristic, complemented, Sylow
591C22—21no—
592C22—61no—
593C22—202no—
594C22—302no—
595C22—402no—
596C22—602no—
597C22—1202no—
598C22—2402no—
599C22—2402no—
600C22#9201911yescharacteristic
601C4—21no—
602C4—62no—
603C4—61no—
604C4—302no—
605C4C2×D3×D2011yescharacteristic
606C4C2×#679011yescharacteristic
607C3#1061162511yescharacteristic, complemented, Sylow
608C2—21no—
609C2—41no—
610C2—121no—
611C2—402no—
612C2—602no—
613C2—1202no—
614C2#1175466111yescharacteristic
subgroupG/H
centerC2#11754661
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. #11755330C2semidirect
3. #11753971C2semidirect
4. C5×#5144C2semidirect
5. C3×#65431C2semidirect
6. #11749421C2semidirect
7. #11748248C2semidirect
8. #11748194C2semidirect
9. #11748086C2semidirect
10. #11748068C2semidirect
11. #11748005C2semidirect
13. #92100C22semidirect
14. #92017C22semidirect
15. C3×C5×#131C22semidirect
17. #91797C22semidirect
21. C2×D120C22semidirect
23. C5×#901C22semidirect
24. C3×#4095C22semidirect
25. C2×C3×D40C22semidirect
30. #91458C22semidirect
31. #91312C22semidirect
32. #91305C22semidirect
33. C2×#6653C22semidirect
34. #91291C22semidirect
38. C5×#837C22semidirect
39. C5×#825C22semidirect
40. C5×#822C22semidirect
41. C5×C8×Dic3C22semidirect
42. C3×#3955C22semidirect
43. C3×#3951C22semidirect
44. #90980C22semidirect
45. #90960C22semidirect
52. #65431D3semidirect
78. Dic60D4semidirect
82. D120D4semidirect
84. #6706D4semidirect
85. C5×#216D4semidirect
86. C5×#215D4semidirect
87. C5×#214D4semidirect
88. C5×#213D4semidirect
113. #5144D5semidirect
122. C5×#131C2×D3semidirect
124. #4095C2×D3semidirect
140. C2×D40C2×D3semidirect
148. #3955C2×D3semidirect
149. #3951C2×D3semidirect
192. C3×#131C2×D5semidirect
212. #901C2×D5semidirect
215. #837C2×D5semidirect
217. #825C2×D5semidirect
219. #822C2×D5semidirect
220. C8×Dic3C2×D5semidirect
322. #216D20semidirect
323. #215D20semidirect
325. #214D20semidirect
329. #213D20semidirect
405. #131D3×D5semidirect
523. C3×C5#3292semidirect
590. C5#80941semidirect
607. C3#10611625semidirect

Maximal subgroups

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

subgroupindex
#117565112
#117553302
#117539712
C5×#51442
C3×#654312
#117494212
#117482482
#117481942
#117480862
#117480682
#117480052
#106116253
#809415