Finite Group Databasebeta

← #14458199

Subgroups — #14458199

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#11756528C211yesmaximal, characteristic, complemented
2C3×#66103C211yesmaximal, characteristic, complemented
3#11755311C212yesmaximal, complemented
4C5×#5196C211yesmaximal, characteristic, complemented
5#11749445C212yesmaximal, complemented
6#11749421C212yesmaximal, complemented
7#11749418C212yesmaximal, complemented
8#11748088C211yesmaximal, characteristic, complemented
9#11748082C211yesmaximal, characteristic, complemented
10#11748076C211yesmaximal, characteristic, complemented
11#11748005C211yesmaximal, characteristic, complemented
12#10611625—31nomaximal, Hall
13#92100C2212yescomplemented
14C3×#4149C2212yescomplemented
15C3×C5×#135C2211yescharacteristic
16#91838C2212yescomplemented
17#91808—21no—
18#91803—22no—
19D8×D15—21no—
20C2×D120C2211yescharacteristic, complemented
21#91790C2211yescharacteristic, complemented
22C5×#948C2212yescomplemented
23C3×#4095C2212yescomplemented
24C3×#4065—21no—
25C3×#4060—22no—
26C3×D5×D8—21no—
27C2×C3×D40C2211yescharacteristic, complemented
28C3×#4047C2211yescharacteristic, complemented
29#91458C2212yescomplemented
30#91312—22no—
31#91312C2212yescomplemented
32#91311—22no—
33#91311C2211yescharacteristic
34#91301—22no—
35#91301C2212yescomplemented
36C2×#6653—22no—
37C2×#6653C2212yescomplemented
38C5×#837C2212yescomplemented
39C5×#819C2211yescharacteristic, complemented
40C5×C8×Dic3C2211yescharacteristic, complemented
41#90993C2211yescharacteristic, complemented
42#90980C2211yescharacteristic, complemented
43#90977C2211yescharacteristic, complemented
44#90960C2212yescomplemented
45#90958C2212yescomplemented
46#84010—51nomaximal, Hall
47#66103—31no—
48#66103D311yescharacteristic, complemented
49#66086—32no—
50C5×#443—31no—
51#65431—32no—
52#65186—31no—
53#65186—31no—
54#65111—32no—
55#65111—32no—
56#64989—31no—
57#64984—31no—
58C3×C5×#41C2312yes—
59#6821—22no—
60D4×D15—22no—
61D4×D15—42no—
62#6817—22no—
63#6817C2311yescharacteristic
64C2×D60—22no—
65C2×D60C2312yes—
66C3×#713—22no—
67C3×D4×D5—22no—
68C3×D4×D5—42no—
69C3×#709—22no—
70C3×#709C2311yescharacteristic
71C2×C3×D20—22no—
72C2×C3×D20C2312yes—
73C2×#1202—42no—
74C3×C5×Q16D411yescharacteristic, complemented
75C3×C5×SD16D412yescomplemented
76C3×C5×D8D411yescharacteristic, complemented
77C2×C3×C5×C8C2311yescharacteristic
78#6717—22no—
79#6715—22no—
80#6710C2311yescharacteristic
81D120—21no—
82D120—21no—
83#6705—21no—
84C8×D15—21no—
85C4×C5×Dic3C2311yescharacteristic
86C5×#208—22no—
87C5×#208C2312yes—
88C2×C5×C3⋊C8—22no—
89C2×C5×C3⋊C8C2311yescharacteristic
90C3×#688—22no—
91C3×#686—22no—
92C3×#681C2311yescharacteristic
93C3×D40—21no—
94C3×D40—21no—
95C3×#676—21no—
96C3×C8×D5—21no—
97#6662D411yescharacteristic, complemented
98#6658—22no—
99#6658D411yescharacteristic, complemented
100#6657—22no—
101#6657D411yescharacteristic, complemented
102#6653—22no—
103#6653—22no—
104#6653—22no—
105#6653—22no—
106#6653D411yescharacteristic, complemented
107C3×#576—51no—
108#5835—52no—
109#5769—51no—
110#5241—52no—
111#5199—52no—
112#5198—52no—
113#5196D511yescharacteristic, complemented
114#5160—51no—
115#5151—51no—
116#5144—51no—
117#5142—51no—
118#4149—32no—
119#4149C2×D312yescomplemented
120C5×#135C2×D311yescharacteristic
121C5×#131—32no—
122#4095—32no—
123#4095C2×D312yescomplemented
124#4065—21no—
125#4065—61no—
126#4060—22no—
127#4060—62no—
128D5×D8—21no—
129D5×D8—61no—
130#4054—32no—
131#4054—32no—
132#4054—62no—
133#4054—62no—
134#4050—31no—
135#4050—62no—
136C2×D40—32no—
137C2×D40—31no—
138C2×D40—62no—
139C2×D40C2×D311yescharacteristic, complemented
140#4047—31no—
141#4047C2×D311yescharacteristic, complemented
142#4020—32no—
143C5×#108—31no—
144C5×#104—32no—
145C4×C5×C8—31no—
146#3955—32no—
147#3955—32no—
148#3951—31no—
149#3937—31no—
150#3937—31no—
151#3292—151noSylow
152C22×D15—22no—
153C22×D15—42no—
154C22×C3×D5—22no—
155C22×C3×D5—42no—
156C3×C5×Q8C2×D412yes—
157C3×C5×D4—22no—
158C3×C5×D4C2×D412yes—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5C2411yescharacteristic
161#1220—21no—
162#1220—42no—
163D60—22no—
164D60—22no—
165D60—42no—
166D60C2×D411yescharacteristic
167C4×D15—21no—
168C4×D15—42no—
169Dic30C2×D411yescharacteristic
170C2×C5×Dic3—21no—
171C3×C22⋊D5—21no—
172C3×C22⋊D5—42no—
173C3×D20—22no—
174C3×D20—22no—
175C3×D20—42no—
176C3×D20C2×D411yescharacteristic
177C3×C4×D5—21no—
178C3×C4×D5—42no—
179C3×Dic10C2×D411yescharacteristic
180#1202—84no—
181C3×C5×C8C2×D411yescharacteristic
182C3×C5×C8C2×D411yescharacteristic
183C3⋊#168—21no—
184C3×#168—21no—
185C5×C3⋊C8—22no—
186C5×C3⋊C8C2×D411yescharacteristic
187C5×C3⋊C8C2×D411yescharacteristic
188C3×H(5,𝔽2)—52no—
189#1009—52no—
190C3×AGL(1,ℤ/8)—52no—
191C3×AGL(1,ℤ/8)—102no—
192C3×#135—101no—
193C3×#135C2×D511yescharacteristic
194C2×C3×D8—51no—
195C2×C3×D8—101no—
196C3×#131—51no—
197#950—51no—
198#950—102no—
199#949—52no—
200#949—52no—
201#949—102no—
202#948C2×D512yescomplemented
203#940—52no—
204#932—52no—
205#932—102no—
206C2×#213—52no—
207C2×#213—102no—
208#919—101no—
209#914—102no—
210D3×D8—101no—
211C2×D24—51no—
212#901—51no—
213#837—51no—
214#837C2×D512yescomplemented
215#833—52no—
216#825—51no—
217#823—52no—
218#822—51no—
219#819C2×D511yescharacteristic, complemented
220C8×Dic3C2×D511yescharacteristic, complemented
221C5×#41C22×D312yes—
222#713—22no—
223#713—62no—
224D4×D5—22no—
225D4×D5—42no—
226D4×D5—62no—
227D4×D5—122no—
228#709—22no—
229#709—31no—
230#709—62no—
231#709C22×D311yescharacteristic
232C2×D20—22no—
233C2×D20—32no—
234C2×D20—62no—
235C2×D20—122no—
236C2×D20C22×D312yes—
237C5×Q16#6711yescharacteristic, complemented
238C5×SD16#6712yescomplemented
239C5×D8#6711yescharacteristic, complemented
240C5×M16—32no—
241C5×M16—62no—
242C2×C5×C8—31no—
243C2×C5×C8—62no—
244C2×C5×C8C22×D311yescharacteristic
245C42×C5—31no—
246#688—22no—
247#688—62no—
248#686—22no—
249#686—62no—
250#681—31no—
251#681C22×D311yescharacteristic
252Dic20—31no—
253D40—21no—
254D40—21no—
255D40—31no—
256D40—62no—
257D40—62no—
258D40—62no—
259D40—62no—
260D40—61no—
261D40—61no—
262#677—31no—
263#677—31no—
264#677—62no—
265#677—62no—
266#676—21no—
267#676—61no—
268C8×D5—21no—
269C8×D5—61no—
270#576—51no—
271#576—152no—
272#576—151no—
273#493—151no—
274#472—151no—
275#472—151no—
276#472—152no—
277#453—152no—
278#453—152no—
279#443—31no—
280#443—151no—
281C22×C3×C5—22no—
282C22×C3×C5H(5,𝔽2)11yescharacteristic
283C2×D15—21no—
284C2×D15—42no—
285C2×D15—82no—
286C2×D15—162no—
287C2×C3×D5—21no—
288C2×C3×D5—42no—
289C2×C3×D5—82no—
290C2×C3×D5—162no—
291C3×C4×C5—22no—
292C3×C4×C5C22×D411yescharacteristic
293C3×C4×C5C22×D411yescharacteristic
294Dic15—21no—
295C3×Dic5—21no—
296C5×Dic3—22no—
297C3×#41—51no—
298C3×#41—102no—
299C3×#41—102no—
300C3×#41C22×D512yes—
301C2×C3×D4—52no—
302C2×C3×D4—102no—
303C2×C3×D4—102no—
304C2×C3×D4—202no—
305C2×#67—202no—
306#239—102no—
307D3×D4—102no—
308D3×D4—202no—
309#235—51no—
310#235—102no—
311C2×D12—52no—
312C2×D12—102no—
313C3×Q16D2011yescharacteristic, complemented
314C3×SD16—102no—
315C3×SD16D2012yescomplemented
316C3×D8—102no—
317C3×D8—101no—
318C3×D8—101no—
319C3×D8D2011yescharacteristic, complemented
320C3×M16—51no—
321C3×M16—101no—
322C2×C3×C8—101no—
323C2×C3×C8C22×D511yescharacteristic
324#216—51no—
325#215—51no—
326#215—102no—
327#215—102no—
328#214—51no—
329#214—102no—
330#213—51no—
331#213—102no—
332#213—102no—
333#213—102no—
334#213—102no—
335#213—102no—
336C4×Dic3C22×D511yescharacteristic
337#208—22no—
338#208—51no—
339#208C22×D512yes—
340C2×C3⋊C8—22no—
341C2×C3⋊C8C22×D511yescharacteristic
342D24—101no—
343D24—101no—
344ΓL(1,𝔽25)—101no—
345D3×C8—101no—
346C22×D5—22no—
347C22×D5—42no—
348C22×D5—62no—
349C22×D5—122no—
350C5×Q8C2×#6712yes—
351C5×D4—22no—
352C5×D4C2×#6712yes—
353C2×C4×C5—22no—
354C2×C4×C5—61no—
355C2×C4×C5C23×D311yescharacteristic
356C22⋊D5—21no—
357C22⋊D5—42no—
358C22⋊D5—61no—
359C22⋊D5—122no—
360D20—22no—
361D20—22no—
362D20—31no—
363D20—42no—
364D20—62no—
365D20—62no—
366D20—122no—
367D20—244no—
368D20D3×D411yescharacteristic
369C4×D5—21no—
370C4×D5—42no—
371C4×D5—61no—
372C4×D5—122no—
373Dic10—31no—
374Dic10D3×D411yescharacteristic
375C5×C8—31no—
376C5×C8—31no—
377C5×C8—62no—
378C5×C8C2×#6711yescharacteristic
379C5×C8C2×#6711yescharacteristic
380#168—21no—
381#168—61no—
382H(5,𝔽2)—52no—
383H(5,𝔽2)—152no—
384AGL(1,ℤ/8)—52no—
385AGL(1,ℤ/8)—102no—
386AGL(1,ℤ/8)—152no—
387AGL(1,ℤ/8)—152no—
388AGL(1,ℤ/8)—152no—
389AGL(1,ℤ/8)—302no—
390AGL(1,ℤ/8)—302no—
391AGL(1,ℤ/8)—302no—
392#135—101no—
393#135—151no—
394#135—302no—
395#135—301no—
396#135D3×D511yescharacteristic
397C2×D8—51no—
398C2×D8—101no—
399C2×D8—152no—
400C2×D8—151no—
401C2×D8—302no—
402C2×D8—301no—
403#131—32no—
404#131—51no—
405#131—151no—
406#127—152no—
407#108—31no—
408#108—151no—
409#104—32no—
410#104—151no—
411#104—151no—
412#100—152no—
413#100—152no—
414C4×C8—31no—
415C2×C3×C5—21no—
416C2×C3×C5—42no—
417C2×C3×C5D4211yescharacteristic
418D15—41no—
419D15—82no—
420C3×D5—41no—
421C3×D5—82no—
422C23×C3—102no—
423C23×C3—202no—
424C22×D3—102no—
425C22×D3—202no—
426C3×Q8—51no—
427C3×Q8C2×D2012yes—
428C3×D4—22no—
429C3×D4—51no—
430C3×D4—101no—
431C3×D4—102no—
432C3×D4—102no—
433C3×D4—202no—
434C3×D4—202no—
435C3×D4C2×D2012yes—
436C2×C3×C4—22no—
437C2×C3×C4—101no—
438C2×C3×C4—202no—
439C2×C3×C4C23×D511yescharacteristic
440#67—101no—
441#67—202no—
442#67—404no—
443C2×Dic3—21no—
444D12—51no—
445D12—102no—
446D12—102no—
447D12—202no—
448C4×D3—101no—
449C4×D3—202no—
450Dic6—51no—
451C3×C8—101no—
452C3×C8C2×D2011yescharacteristic
453C3×C8C2×D2011yescharacteristic
454C3⋊C8—22no—
455C3⋊C8—101no—
456C3⋊C8D4×D511yescharacteristic
457C3⋊C8D4×D511yescharacteristic
458C22×C5—22no—
459C22×C5#100411yescharacteristic
460C2×D5—21no—
461C2×D5—42no—
462C2×D5—61no—
463C2×D5—82no—
464C2×D5—122no—
465C2×D5—162no—
466C2×D5—242no—
467C2×D5—482no—
468C4×C5—22no—
469C4×C5—62no—
470C4×C5C2×D3×D411yescharacteristic
471C4×C5C22×#6711yescharacteristic
472Dic5—21no—
473Dic5—61no—
474#41—51no—
475#41—102no—
476#41—102no—
477#41—151no—
478#41—302no—
479#41—302no—
480#41C2×D3×D512yes—
481C2×D4—52no—
482C2×D4—102no—
483C2×D4—102no—
484C2×D4—152no—
485C2×D4—202no—
486C2×D4—302no—
487C2×D4—302no—
488C2×D4—602no—
489C2×D4—602no—
490Q16—151no—
491Q16#120211yescharacteristic, complemented
492SD16—102no—
493SD16—151no—
494SD16—151no—
495SD16—302no—
496SD16—302no—
497SD16—302no—
498SD16#120212yescomplemented
499D8—102no—
500D8—101no—
501D8—101no—
502D8—151no—
503D8—302no—
504D8—302no—
505D8—302no—
506D8—302no—
507D8—302no—
508D8—301no—
509D8—301no—
510D8#120211yescharacteristic, complemented
511M16—32no—
512M16—51no—
513M16—62no—
514M16—101no—
515M16—151no—
516M16—301no—
517C2×C8—31no—
518C2×C8—62no—
519C2×C8—101no—
520C2×C8—301no—
521C2×C8C2×D3×D511yescharacteristic
522C42—31no—
523C3×C5#329211yescharacteristic, complemented, Hall
524C22×C3—22no—
525C22×C3—101no—
526C22×C3—202no—
527C22×C3—402no—
528C22×C3—802no—
529C22×C3#414911yescharacteristic
530C2×D3—101no—
531C2×D3—202no—
532C2×D3—402no—
533C2×D3—802no—
534C3×C4—22no—
535C3×C4—101no—
536C3×C4C22×D2011yescharacteristic
537C3×C4C2×D4×D511yescharacteristic
538Dic3—22no—
539Dic3—101no—
540C2×C5—21no—
541C2×C5—42no—
542C2×C5D4×#6711yescharacteristic
543D5—41no—
544D5—82no—
545D5—121no—
546D5—242no—
547C23—102no—
548C23—202no—
549C23—302no—
550C23—602no—
551Q8—51no—
552Q8—151no—
553Q8C2×#120212yes—
554D4—22no—
555D4—51no—
556D4—101no—
557D4—102no—
558D4—102no—
559D4—151no—
560D4—202no—
561D4—202no—
562D4—301no—
563D4—302no—
564D4—302no—
565D4—602no—
566D4—602no—
567D4—1204no—
568D4C2×#120212yes—
569C2×C4—22no—
570C2×C4—61no—
571C2×C4—101no—
572C2×C4—202no—
573C2×C4—301no—
574C2×C4—602no—
575C2×C4C22×D3×D511yescharacteristic
576C8—31no—
577C8—31no—
578C8—62no—
579C8—101no—
580C8—301no—
581C8C2×#120211yescharacteristic
582C8C2×#120211yescharacteristic
583C2×C3—21no—
584C2×C3—42no—
585C2×C3—201no—
586C2×C3—402no—
587C2×C3D4×D2011yescharacteristic
588D3—201no—
589D3—402no—
590C5#8401011yescharacteristic, complemented, Sylow
591C22—22no—
592C22—101no—
593C22—202no—
594C22—301no—
595C22—402no—
596C22—602no—
597C22—802no—
598C22—1202no—
599C22—2402no—
600C22#9202611yescharacteristic
601C4—22no—
602C4—62no—
603C4—101no—
604C4—301no—
605C4D3×D4×D511yescharacteristic
606C4C22×#120211yescharacteristic
607C3#1061162511yescharacteristic, complemented, Sylow
608C2—21no—
609C2—42no—
610C2—201no—
611C2—402no—
612C2—601no—
613C2—1202no—
614C2D4×#120211yescharacteristic
subgroupG/H
centerC2D4×#1202
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#135C22
socleC2×C3×C5D42
hypercenter#135D3×D5
hypocenterC3×C5#3292
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #11756528C2semidirect
2. C3×#66103C2semidirect
3. #11755311C2semidirect
4. C5×#5196C2semidirect
5. #11749445C2semidirect
6. #11749421C2semidirect
7. #11749418C2semidirect
8. #11748088C2semidirect
9. #11748082C2semidirect
10. #11748076C2semidirect
11. #11748005C2semidirect
13. #92100C22semidirect
14. C3×#4149C22semidirect
16. #91838C22semidirect
20. C2×D120C22semidirect
21. #91790C22semidirect
22. C5×#948C22semidirect
23. C3×#4095C22semidirect
27. C2×C3×D40C22semidirect
28. C3×#4047C22semidirect
29. #91458C22semidirect
31. #91312C22semidirect
35. #91301C22semidirect
37. C2×#6653C22semidirect
38. C5×#837C22semidirect
39. C5×#819C22semidirect
40. C5×C8×Dic3C22semidirect
41. #90993C22semidirect
42. #90980C22semidirect
43. #90977C22semidirect
44. #90960C22semidirect
45. #90958C22semidirect
48. #66103D3semidirect
74. C3×C5×Q16D4semidirect
75. C3×C5×SD16D4semidirect
76. C3×C5×D8D4semidirect
97. #6662D4semidirect
99. #6658D4semidirect
101. #6657D4semidirect
106. #6653D4semidirect
113. #5196D5semidirect
119. #4149C2×D3semidirect
123. #4095C2×D3semidirect
139. C2×D40C2×D3semidirect
141. #4047C2×D3semidirect
202. #948C2×D5semidirect
214. #837C2×D5semidirect
219. #819C2×D5semidirect
220. C8×Dic3C2×D5semidirect
237. C5×Q16#67semidirect
238. C5×SD16#67semidirect
239. C5×D8#67semidirect
313. C3×Q16D20semidirect
315. C3×SD16D20semidirect
319. C3×D8D20semidirect
491. Q16#1202semidirect
498. SD16#1202semidirect
510. D8#1202semidirect
523. C3×C5#3292semidirect
590. C5#84010semidirect
607. C3#10611625semidirect

Maximal subgroups

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

subgroupindex
#117565282
C3×#661032
#117553112
C5×#51962
#117494452
#117494212
#117494182
#117480882
#117480822
#117480762
#117480052
#106116253
#840105