Finite Group Databasebeta

← #14458460

Subgroups — #14458460

Order 1920.

924 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756338C211yesmaximal, characteristic, complemented
2D4×D60C211yesmaximal, characteristic
3C3×C4×C5×SD16C211yesmaximal, characteristic, complemented
4#11751270C212yesmaximal, complemented
5#11751128C211yesmaximal, characteristic, complemented
6#11751109C211yesmaximal, characteristic, complemented
7#11750964C212yesmaximal, complemented
8#11750908C212yesmaximal, complemented
9D15×#102C212yesmaximal, complemented
10#11750795C211yesmaximal, characteristic, complemented
11C8×D60C211yesmaximal, characteristic, complemented
12#10611407—31nomaximal, Hall
13C2×D4×D15C2212yes—
14C22×D60—21no—
15C2×C3×C5×SD16C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic, complemented
17C3×C4×C5×D4C2211yescharacteristic
18#91835C2212yescomplemented
19C2×#6717C2212yescomplemented
20#91819—21no—
21C2×C8×D15C2212yescomplemented
22#91785—22no—
23#91784C2212yes—
24C4⋊C4×D15C2212yescomplemented
25#91771—22no—
26#91763C2211yescharacteristic
27C4×D60—21no—
28C4×D60C2211yescharacteristic
29#91759—21no—
30#91758C2211yescharacteristic, complemented
31C3×C5×#106C2211yescharacteristic, complemented
32C3×C5×#103C2211yescharacteristic, complemented
33C3×C5×#102C2211yescharacteristic
34C3×C4×C5×C8C2211yescharacteristic, complemented
35#91116C2212yes—
36#91105C2212yescomplemented
37#91105C2212yescomplemented
38#91104C2212yescomplemented
39#91093C2212yescomplemented
40#91092C2212yescomplemented
41C4⋊(C3⋊#168)C2211yescharacteristic, complemented
42#79676—51nomaximal, Hall
43#65913—31no—
44D4×D20—31no—
45C4×C5×SD16D311yescharacteristic, complemented
46#65458—32no—
47#65316—31no—
48#65297—31no—
49#65152—32no—
50#65096—32no—
51D5×#102—32no—
52#64983—31no—
53C8×D20—31no—
54C23×D15—22no—
55C2×C3×C5×Q8C2311yescharacteristic
56C2×C3×C5×D4C2311yescharacteristic
57C22×C3×C4×C5—21no—
58C2×#1220—22no—
59D4×D15—24no—
60D4×D15—42no—
61C2×D60—22no—
62C2×D60—21no—
63C2×D60—21no—
64C2×D60—44no—
65C2×D60D411yescharacteristic
66C2×D60C2312yes—
67C2×C4×D15—22no—
68C2×C4×D15C2312yes—
69C2×Dic30—21no—
70C3×C5×SD16—22no—
71C2×C3×C5×C8C2311yescharacteristic
72C2×C3×C5×C8C2311yescharacteristic
73C3×C5×C4⋊C4—21no—
74C3×C5×C4⋊C4C2311yescharacteristic
75C3×C5×C4⋊C4C2311yescharacteristic
76C3×C5×C22⋊C4—21no—
77C3×C42×C5—21no—
78C3×C42×C5C2311yescharacteristic
79#6717—42no—
80#6714—22no—
81#6714—22no—
82#6714D412yes—
83C4⋊Dic15—21no—
84C4⋊Dic15D411yescharacteristic
85C4⋊Dic15C2312yes—
86#6712—22no—
87#6712—22no—
88C2×C3⋊#168C2312yes—
89C8×D15—42no—
90#5579—51no—
91D4×D12—51no—
92C3×C4×SD16D511yescharacteristic, complemented
93#5169—52no—
94#5027—51no—
95#5008—51no—
96#4863—52no—
97#4807—52no—
98D3×#102—52no—
99#4694—51no—
100C8×D12—51no—
101C2×D4×D5—32no—
102C22×D20—61no—
103C2×C5×SD16C2×D311yescharacteristic
104C4×C5×Q8C2×D311yescharacteristic, complemented
105C4×C5×D4C2×D311yescharacteristic
106#4092—32no—
107C2×#688—32no—
108#4076—61no—
109C2×C8×D5—32no—
110#4042—62no—
111#4041—32no—
112D5×C4⋊C4—32no—
113#4028—62no—
114#4020—31no—
115C4×D20—31no—
116C4×D20—61no—
117#4016—61no—
118#4015—31no—
119C5×#106C2×D311yescharacteristic, complemented
120C5×#103C2×D311yescharacteristic, complemented
121C5×#102C2×D311yescharacteristic
122C4×C5×C8C2×D311yescharacteristic, complemented
123#3964—32no—
124#3953—32no—
125#3953—32no—
126#3952—32no—
127#3941—32no—
128#3940—32no—
129C4⋊#168—31no—
130#3295—151noSylow
131C23×C3×C5—21no—
132C22×D15—22no—
133C22×D15—44no—
134C22×D15—44no—
135C22×D15—82no—
136C22×D15C2×D412yes—
137C3×C5×Q8—21no—
138C3×C5×Q8—21no—
139C3×C5×D4—21no—
140C3×C5×D4#4112yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—42no—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2411yescharacteristic
148#1220—44no—
149C2×Dic15—22no—
150C2×Dic15C2×D412yes—
151D60—22no—
152D60—22no—
153D60—44no—
154D60—42no—
155D60—81no—
156D60SD1614yescomplemented
157C4×D15—22no—
158C4×D15—22no—
159C4×D15—82no—
160Dic30—81no—
161C3×C5×C8—21no—
162C3×C5×C8—21no—
163C3⋊#168—22no—
164C2×D3×D4—52no—
165C22×D12—101no—
166C2×C3×SD16C2×D511yescharacteristic
167C3×C4×Q8C2×D511yescharacteristic, complemented
168C3×C4×D4C2×D511yescharacteristic
169#946—52no—
170C2×#215—52no—
171#930—101no—
172C2×D3×C8—52no—
173#896—102no—
174#895—52no—
175D3×C4⋊C4—52no—
176#882—102no—
177#874—51no—
178C4×D12—51no—
179C4×D12—101no—
180#870—101no—
181#869—51no—
182C3×#106C2×D511yescharacteristic, complemented
183C3×#103C2×D511yescharacteristic, complemented
184C3×#102C2×D511yescharacteristic
185C3×C4×C8C2×D511yescharacteristic, complemented
186#832—52no—
187#821—52no—
188#821—52no—
189(C2×D3)⋊C8—52no—
190#809—52no—
191#808—52no—
192C4⋊(C3⋊C8)—51no—
193C23×D5—62no—
194C2×C5×Q8C22×D311yescharacteristic
195C2×C5×D4C22×D311yescharacteristic
196C22×C4×C5—21no—
197C2×C22⋊D5—62no—
198D4×D5—64no—
199D4×D5—122no—
200C2×D20—31no—
201C2×D20—32no—
202C2×D20—62no—
203C2×D20—61no—
204C2×D20—61no—
205C2×D20—124no—
206C2×C4×D5—32no—
207C2×C4×D5—62no—
208C2×Dic10—61no—
209C5×SD16—22no—
210C2×C5×C8C22×D311yescharacteristic
211C2×C5×C8C22×D311yescharacteristic
212C5×C4⋊C4—21no—
213C5×C4⋊C4C22×D311yescharacteristic
214C5×C4⋊C4C22×D311yescharacteristic
215C5×C22⋊C4—21no—
216C42×C5—21no—
217C42×C5C22×D311yescharacteristic
218#688—122no—
219#685—32no—
220#685—62no—
221#685—62no—
222C4⋊Dic5—31no—
223C4⋊Dic5—32no—
224C4⋊Dic5—61no—
225#683—62no—
226#683—62no—
227C2×#168—32no—
228C8×D5—122no—
229#554—151no—
230D42—151no—
231#486—151no—
232#481—152no—
233#476—151no—
234#465—152no—
235#460—151no—
236#450—152no—
237C4×SD16D1511yescharacteristic, complemented
238C8×D4—151no—
239C2×#102—152no—
240C22×C3×C5—22no—
241C22×C3×C5—41no—
242C22×C3×C5—41no—
243C22×C3×C5C22×D411yescharacteristic
244C2×D15—22no—
245C2×D15—22no—
246C2×D15—42no—
247C2×D15—42no—
248C2×D15—42no—
249C2×D15—88no—
250C2×D15C2×SD1614yes—
251C3×C4×C5—21no—
252C3×C4×C5—41no—
253C3×C4×C5—41no—
254C3×C4×C5—41no—
255C3×C4×C5C2×SD1612yes—
256C3×C4×C5C2×#4111yescharacteristic
257C3×C4×C5H(5,𝔽2)11yescharacteristic
258Dic15—22no—
259Dic15—42no—
260C23×D3—102no—
261C2×C3×Q8C22×D511yescharacteristic
262C2×C3×D4C22×D511yescharacteristic
263C22×C3×C4—21no—
264C2×#67—102no—
265D3×D4—104no—
266D3×D4—202no—
267C2×D12—51no—
268C2×D12—52no—
269C2×D12—102no—
270C2×D12—101no—
271C2×D12—101no—
272C2×D12—204no—
273C2×C4×D3—52no—
274C2×C4×D3—102no—
275C2×Dic6—101no—
276C3×SD16—22no—
277C2×C3×C8C22×D511yescharacteristic
278C2×C3×C8C22×D511yescharacteristic
279C3×C4⋊C4—21no—
280C3×C4⋊C4C22×D511yescharacteristic
281C3×C4⋊C4C22×D511yescharacteristic
282C3×C22⋊C4—21no—
283C3×C42—21no—
284C3×C42C22×D511yescharacteristic
285#215—202no—
286(C2×D3)⋊C4—52no—
287(C2×D3)⋊C4—102no—
288(C2×D3)⋊C4—102no—
289C4⋊Dic3—51no—
290C4⋊Dic3—52no—
291C4⋊Dic3—101no—
292Dic3⋊C4—102no—
293Dic3⋊C4—102no—
294C2×C3⋊C8—52no—
295D3×C8—202no—
296C23×C5—21no—
297C22×D5—32no—
298C22×D5—62no—
299C22×D5—124no—
300C22×D5—124no—
301C22×D5—242no—
302C5×Q8—21no—
303C5×Q8—21no—
304C5×D4—21no—
305C5×D4#23512yes—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—42no—
310C2×C4×C5D3×D411yescharacteristic
311C2×C4×C5D3×D411yescharacteristic
312C2×C4×C5C23×D311yescharacteristic
313C22⋊D5—124no—
314C2×Dic5—32no—
315C2×Dic5—62no—
316D20—34no—
317D20—62no—
318D20—62no—
319D20—124no—
320D20—122no—
321D20—241no—
322C4×D5—62no—
323C4×D5—62no—
324C4×D5—242no—
325Dic10—241no—
326C5×C8—21no—
327C5×C8—21no—
328#168—62no—
329C22×D4—152no—
330C22×D4—301no—
331C2×SD16—152no—
332C2×SD16C2×D1511yescharacteristic
333C22×C8—152no—
334C4⋊Q8—151no—
335#127—151no—
336#125—301no—
337C22⋊Q8—152no—
338C22⋊Q8—302no—
339#121—152no—
340#121—301no—
341#120—302no—
342C4×Q8C2×D1511yescharacteristic, complemented
343C4×D4—151no—
344C4×D4—301no—
345C4×D4C2×D1511yescharacteristic
346C2×C4⋊C4—152no—
347#106—152no—
348#106C2×D1511yescharacteristic, complemented
349C4⋊C8—151no—
350#103C2×D1511yescharacteristic, complemented
351#102—152no—
352#102—152no—
353#102—152no—
354#102—152no—
355#102C2×D1511yescharacteristic
356C22⋊C8—152no—
357C4×C8C2×D1511yescharacteristic, complemented
358C2×C3×C5—42no—
359C2×C3×C5#54611yescharacteristic
360C2×C3×C5C22×SD1611yescharacteristic
361C2×C3×C5#57611yescharacteristic
362D15—24no—
363D15—82no—
364C23×C3—21no—
365C22×D3—52no—
366C22×D3—102no—
367C22×D3—204no—
368C22×D3—204no—
369C22×D3—402no—
370C3×Q8—21no—
371C3×Q8—21no—
372C3×D4—21no—
373C3×D4#70912yes—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—42no—
378C2×C3×C4D4×D511yescharacteristic
379C2×C3×C4D4×D511yescharacteristic
380C2×C3×C4C23×D511yescharacteristic
381#67—204no—
382C2×Dic3—52no—
383C2×Dic3—102no—
384D12—54no—
385D12—102no—
386D12—102no—
387D12—204no—
388D12—202no—
389D12—401no—
390C4×D3—102no—
391C4×D3—102no—
392C4×D3—402no—
393Dic6—401no—
394C3×C8—21no—
395C3×C8—21no—
396C3⋊C8—102no—
397C22×C5—22no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5C2×D3×D411yescharacteristic
401C2×D5—34no—
402C2×D5—62no—
403C2×D5—62no—
404C2×D5—122no—
405C2×D5—122no—
406C2×D5—122no—
407C2×D5—248no—
408C4×C5—21no—
409C4×C5—41no—
410C4×C5—41no—
411C4×C5—41no—
412C4×C5D3×SD1612yes—
413C4×C5C2×#23511yescharacteristic
414C4×C5#100911yescharacteristic
415Dic5—62no—
416Dic5—122no—
417C24—302no—
418C2×Q8—301no—
419C2×Q8C22×D1511yescharacteristic
420C2×D4—151no—
421C2×D4—152no—
422C2×D4—302no—
423C2×D4—302no—
424C2×D4—304no—
425C2×D4—301no—
426C2×D4—301no—
427C2×D4—604no—
428C2×D4—602no—
429C2×D4C22×D1511yescharacteristic
430C22×C4—21no—
431C22×C4—152no—
432C22×C4—302no—
433SD16—22no—
434SD16—602no—
435C2×C8—152no—
436C2×C8—602no—
437C2×C8C22×D1511yescharacteristic
438C2×C8C22×D1511yescharacteristic
439C4⋊C4—21no—
440C4⋊C4—151no—
441C4⋊C4—152no—
442C4⋊C4—302no—
443C4⋊C4—302no—
444C4⋊C4—301no—
445C4⋊C4C22×D1511yescharacteristic
446C4⋊C4C22×D1511yescharacteristic
447C22⋊C4—21no—
448C22⋊C4—152no—
449C22⋊C4—302no—
450C22⋊C4—302no—
451C42—21no—
452C42C22×D1511yescharacteristic
453C3×C5#329511yescharacteristic, complemented, Hall
454C22×C3—22no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3C2×D4×D511yescharacteristic
458C2×D3—54no—
459C2×D3—102no—
460C2×D3—102no—
461C2×D3—202no—
462C2×D3—202no—
463C2×D3—202no—
464C2×D3—408no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4D5×SD1612yes—
470C3×C4C2×#70911yescharacteristic
471C3×C4#414911yescharacteristic
472Dic3—102no—
473Dic3—202no—
474C2×C5—42no—
475C2×C5#555011yescharacteristic
476C2×C5C2×D3×SD1611yescharacteristic
477C2×C5#576911yescharacteristic
478D5—64no—
479D5—242no—
480C23—21no—
481C23—152no—
482C23—302no—
483C23—604no—
484C23—604no—
485C23—1202no—
486Q8—21no—
487Q8—21no—
488Q8—1201no—
489D4—21no—
490D4—154no—
491D4—302no—
492D4—302no—
493D4—604no—
494D4—604no—
495D4—602no—
496D4—1201no—
497D4#681712yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—152no—
503C2×C4—302no—
504C2×C4—302no—
505C2×C4—302no—
506C2×C4—1202no—
507C2×C4D4×D1511yescharacteristic
508C2×C4D4×D1511yescharacteristic
509C2×C4C23×D1511yescharacteristic
510C8—21no—
511C8—21no—
512C8—302no—
513C2×C3—42no—
514C2×C3#6588411yescharacteristic
515C2×C3C2×D5×SD1611yescharacteristic
516C2×C3#6610311yescharacteristic
517D3—104no—
518D3—402no—
519C5#7967611yescharacteristic, complemented, Sylow
520C22—22no—
521C22—41no—
522C22—41no—
523C22—154no—
524C22—302no—
525C22—302no—
526C22—602no—
527C22—602no—
528C22—602no—
529C22—1208no—
530C22C2×D4×D1511yescharacteristic
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—302no—
536C4—602no—
537C4SD16×D1512yes—
538C4C2×#681711yescharacteristic
539C4#9210011yescharacteristic
540C3#1061140711yescharacteristic, complemented, Sylow
541C2—42no—
542C2—304no—
543C2—1202no—
544C2#1175630911yescharacteristic
545C2C2×SD16×D1511yescharacteristic
546C2#1175652811yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C4×C5×SD16C2
socleC22×C3×C5C22×D4
prime radicalD4×D60C2
hypercenterC4×SD16D15
hypocenterC3×C5#3295
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #11756338C2semidirect
3. C3×C4×C5×SD16C2semidirect
4. #11751270C2semidirect
5. #11751128C2semidirect
6. #11751109C2semidirect
7. #11750964C2semidirect
8. #11750908C2semidirect
9. D15×#102C2semidirect
10. #11750795C2semidirect
11. C8×D60C2semidirect
16. C3×C4×C5×Q8C22semidirect
18. #91835C22semidirect
19. C2×#6717C22semidirect
21. C2×C8×D15C22semidirect
24. C4⋊C4×D15C22semidirect
30. #91758C22semidirect
31. C3×C5×#106C22semidirect
32. C3×C5×#103C22semidirect
34. C3×C4×C5×C8C22semidirect
36. #91105C22semidirect
37. #91105C22semidirect
38. #91104C22semidirect
39. #91093C22semidirect
40. #91092C22semidirect
41. C4⋊(C3⋊#168)C22semidirect
45. C4×C5×SD16D3semidirect
92. C3×C4×SD16D5semidirect
104. C4×C5×Q8C2×D3semidirect
119. C5×#106C2×D3semidirect
120. C5×#103C2×D3semidirect
122. C4×C5×C8C2×D3semidirect
156. D60SD16semidirect
167. C3×C4×Q8C2×D5semidirect
182. C3×#106C2×D5semidirect
183. C3×#103C2×D5semidirect
185. C3×C4×C8C2×D5semidirect
237. C4×SD16D15semidirect
342. C4×Q8C2×D15semidirect
348. #106C2×D15semidirect
350. #103C2×D15semidirect
357. C4×C8C2×D15semidirect
453. C3×C5#3295semidirect
519. C5#79676semidirect
540. C3#10611407semidirect

Maximal subgroups

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

subgroupindex
#117563382
D4×D602
C3×C4×C5×SD162
#117512702
#117511282
#117511092
#117509642
#117509082
D15×#1022
#117507952
C8×D602
#106114073
#796765