Finite Group Databasebeta

← #14424317

Subgroups — #14424317

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92100C211yesmaximal, characteristic
2C2×C3×C5×#137C211yesmaximal, characteristic, complemented
3#11756575C211yesmaximal, characteristic, complemented
4C22×#6717C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11751288C214yesmaximal, complemented
7#11751270C214yesmaximal, complemented
8#11751232C211yesmaximal, characteristic, complemented
9#11751230C211yesmaximal, characteristic, complemented
10#10615713—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic, complemented
13#92100—24no—
14#92100—24no—
15C2×#6821C2212yes—
16C2×D4×D15—22no—
17C2×D4×D15C2212yes—
18C2×#6817—21no—
19C2×#6817C2211yescharacteristic
20C22×D60—21no—
21C22×D60C2211yescharacteristic
22C3×C5×#137—24no—
23C2×C3×C5×Q16C2212yescomplemented
24C2×C3×C5×SD16C2212yescomplemented
25C2×C3×C5×M16C2211yescharacteristic, complemented
26#91836—21no—
27#91835C2214yescomplemented
28#91833—21no—
29C2×#6717—24no—
30C2×#6717C2214yescomplemented
31#91818—22no—
32#91815C2211yescharacteristic, complemented
33C22×C3⋊#168C2211yescharacteristic, complemented
34#91119C2212yescomplemented
35#91116C2212yescomplemented
36#91105C2214yescomplemented
37#91104C2214yescomplemented
38#83982—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×#137D311yescharacteristic, complemented
41#66150—31no—
42C22×#688—31no—
43#65522—31no—
44#65476—34no—
45#65458—34no—
46#65420—31no—
47#65418—31no—
48C23×D15—21no—
49C23×D15—22no—
50C3×C5×#41—22no—
51C3×C5×#41D414yes—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8—22no—
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×Q8C2311yescharacteristic
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4C2311yescharacteristic
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5C2311yescharacteristic
62C2×#1220—22no—
63C2×#1220D412yes—
64#6821—24no—
65#6821—24no—
66D4×D15—216no—
67D4×D15—24no—
68D4×D15—24no—
69#6817—28no—
70#6817—22no—
71#6817—22no—
72C2×D60—28no—
73C2×D60—22no—
74C2×D60—21no—
75C2×D60—22no—
76C2×D60D411yescharacteristic
77C2×D60D414yes—
78C2×D60C2312yes—
79C2×C4×D15—22no—
80C2×C4×D15C2312yes—
81C2×Dic30D411yescharacteristic
82C3×C5×Q16—24no—
83C3×C5×SD16—24no—
84C3×C5×M16—22no—
85C2×C3×C5×C8C2312yes—
86#6719—21no—
87#6717—28no—
88#6714—24no—
89C4⋊Dic15C2312yes—
90#6712—24no—
91C4×Dic15—21no—
92C2×C3⋊#168—22no—
93C2×C3⋊#168C2312yes—
94C2×#1009—51no—
95C2×C3×#137D511yescharacteristic, complemented
96#5816—51no—
97C22×#215—51no—
98#5233—51no—
99#5187—54no—
100#5169—54no—
101#5131—51no—
102#5129—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8C2×D311yescharacteristic, complemented
105#4149—64no—
106#4149—64no—
107C2×#713—32no—
108C2×D4×D5—32no—
109C2×D4×D5—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×D20—31no—
113C22×D20—61no—
114C5×#137—24no—
115C2×C5×Q16C2×D312yescomplemented
116C2×C5×SD16C2×D312yescomplemented
117C2×C5×M16C2×D311yescharacteristic, complemented
118#4093—61no—
119#4092—34no—
120#4090—61no—
121C2×#688—34no—
122C2×#688—64no—
123#4075—62no—
124#4072—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953—34no—
129#3952—34no—
130#3009—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×D15—22no—
134C22×D15—28no—
135C22×D15—24no—
136C22×D15—84no—
137C22×D15—82no—
138C22×D15C2×D412yes—
139C3×C5×Q8—21no—
140C3×C5×Q8—22no—
141C3×C5×Q8—22no—
142C3×C5×Q8—42no—
143C3×C5×Q8C2×D412yes—
144C3×C5×D4—24no—
145C3×C5×D4—21no—
146C3×C5×D4C2×D412yes—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5—24no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—42no—
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5C2×D412yes—
154C2×C3×C4×C5C2411yescharacteristic
155#1220—28no—
156#1220—24no—
157C2×Dic15—22no—
158C2×Dic15C2×D412yes—
159D60—28no—
160D60—22no—
161D60—24no—
162D60—22no—
163D60C2×D414yes—
164C4×D15—28no—
165C4×D15—22no—
166C4×D15—22no—
167Dic30—22no—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C2×C3×#41C2×D511yescharacteristic
171C22×C3×Q8C2×D511yescharacteristic, complemented
172#1009—104no—
173#1009—104no—
174C2×#239—52no—
175C2×D3×D4—52no—
176C2×D3×D4—102no—
177C2×#235—51no—
178C2×#235—101no—
179C22×D12—51no—
180C22×D12—101no—
181C3×#137—24no—
182C2×C3×Q16C2×D512yescomplemented
183C2×C3×SD16C2×D512yescomplemented
184C2×C3×M16C2×D511yescharacteristic, complemented
185#947—101no—
186#946—54no—
187#944—101no—
188C2×#215—54no—
189C2×#215—104no—
190#929—102no—
191#926—51no—
192C22×C3⋊C8—51no—
193#835—52no—
194#832—52no—
195#821—54no—
196(C2×D3)⋊C8—54no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—22no—
200C5×#41#6714yes—
201C2×C5×Q8—21no—
202C2×C5×Q8—22no—
203C2×C5×Q8—22no—
204C2×C5×Q8C22×D312yes—
205C2×C5×Q8C22×D311yescharacteristic
206C2×C5×D4—21no—
207C2×C5×D4C22×D311yescharacteristic
208C22×C4×C5—21no—
209C22×C4×C5—21no—
210C22×C4×C5C22×D311yescharacteristic
211C2×C22⋊D5—32no—
212C2×C22⋊D5—62no—
213#713—64no—
214#713—64no—
215D4×D5—64no—
216D4×D5—616no—
217D4×D5—64no—
218#709—68no—
219#709—62no—
220#709—62no—
221C2×D20—34no—
222C2×D20—32no—
223C2×D20—31no—
224C2×D20—62no—
225C2×D20—68no—
226C2×D20—61no—
227C2×D20—62no—
228C2×C4×D5—32no—
229C2×C4×D5—62no—
230C2×Dic10—31no—
231C5×Q16—24no—
232C5×SD16—24no—
233C5×M16—22no—
234C2×C5×C8C22×D312yes—
235C22⋊Dic5—61no—
236#688—68no—
237#685—64no—
238C4⋊Dic5—32no—
239#683—64no—
240C4×Dic5—61no—
241C2×#168—32no—
242C2×#168—62no—
243C2×H(5,𝔽2)—151no—
244C2×#137D1511yescharacteristic, complemented
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—42no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—22no—
258C2×D15—24no—
259C2×D15—24no—
260C2×D15—44no—
261C2×D15—84no—
262C2×D15—88no—
263C3×C4×C5—22no—
264C3×C4×C5—42no—
265C3×C4×C5#12012yes—
266C3×C4×C5C22×D411yescharacteristic
267C3×C4×C5C22×D411yescharacteristic
268Dic15—22no—
269Dic15—42no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—22no—
273C3×#41C22⋊D514yes—
274C2×C3×Q8—21no—
275C2×C3×Q8—22no—
276C2×C3×Q8—22no—
277C2×C3×Q8C22×D512yes—
278C2×C3×Q8C22×D511yescharacteristic
279C2×C3×D4—21no—
280C2×C3×D4C22×D511yescharacteristic
281C22×C3×C4—21no—
282C22×C3×C4—21no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—52no—
285C2×#67—102no—
286#239—104no—
287#239—104no—
288D3×D4—104no—
289D3×D4—1016no—
290D3×D4—104no—
291#235—108no—
292#235—102no—
293#235—102no—
294C2×D12—54no—
295C2×D12—52no—
296C2×D12—51no—
297C2×D12—102no—
298C2×D12—108no—
299C2×D12—101no—
300C2×D12—102no—
301C2×C4×D3—52no—
302C2×C4×D3—102no—
303C2×Dic6—51no—
304C3×Q16—24no—
305C3×SD16—24no—
306C3×M16—22no—
307C2×C3×C8C22×D512yes—
308#217—101no—
309#215—108no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—52no—
312Dic3⋊C4—104no—
313C4×Dic3—101no—
314C2×C3⋊C8—52no—
315C2×C3⋊C8—102no—
316C23×C5—21no—
317C23×C5D3×D411yescharacteristic
318C22×D5—32no—
319C22×D5—62no—
320C22×D5—68no—
321C22×D5—64no—
322C22×D5—244no—
323C22×D5—242no—
324C5×Q8—21no—
325C5×Q8—22no—
326C5×Q8—22no—
327C5×Q8—42no—
328C5×Q8C2×#6712yes—
329C5×D4—24no—
330C5×D4—21no—
331C5×D4C2×#6712yes—
332C2×C4×C5—21no—
333C2×C4×C5—24no—
334C2×C4×C5—22no—
335C2×C4×C5—42no—
336C2×C4×C5D3×D411yescharacteristic
337C2×C4×C5D3×D412yes—
338C2×C4×C5C2×#6712yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—68no—
341C22⋊D5—64no—
342C2×Dic5—32no—
343C2×Dic5—62no—
344D20—34no—
345D20—68no—
346D20—64no—
347D20—62no—
348D20—62no—
349C4×D5—68no—
350C4×D5—62no—
351C4×D5—62no—
352Dic10—62no—
353C5×C8—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—301no—
360C2×#41C2×D1511yescharacteristic
361C22×Q8C2×D1511yescharacteristic, complemented
362C22×D4—152no—
363C22×D4—151no—
364C22×D4—301no—
365C22×D4—302no—
366#137—24no—
367C2×Q16C2×D1512yescomplemented
368C2×SD16—154no—
369C2×SD16—304no—
370C2×SD16C2×D1512yescomplemented
371C2×M16C2×D1511yescharacteristic, complemented
372C22×C8—151no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—151no—
378#103—152no—
379#102—152no—
380#102—154no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—42no—
387D15—44no—
388C23×C3—21no—
389C23×C3D4×D511yescharacteristic
390C22×D3—52no—
391C22×D3—102no—
392C22×D3—108no—
393C22×D3—104no—
394C22×D3—404no—
395C22×D3—402no—
396C3×Q8—21no—
397C3×Q8—22no—
398C3×Q8—22no—
399C3×Q8—42no—
400C3×Q8C2×C22⋊D512yes—
401C3×D4—24no—
402C3×D4—21no—
403C3×D4C2×C22⋊D512yes—
404C2×C3×C4—21no—
405C2×C3×C4—24no—
406C2×C3×C4—22no—
407C2×C3×C4—42no—
408C2×C3×C4D4×D512yes—
409C2×C3×C4D4×D511yescharacteristic
410C2×C3×C4C2×C22⋊D512yes—
411C2×C3×C4C23×D511yescharacteristic
412#67—108no—
413#67—104no—
414C2×Dic3—52no—
415C2×Dic3—102no—
416D12—54no—
417D12—108no—
418D12—102no—
419D12—104no—
420D12—102no—
421C4×D3—108no—
422C4×D3—102no—
423C4×D3—102no—
424Dic6—102no—
425C3×C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—22no—
429C22×C5—42no—
430C22×C5#93712yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—64no—
433C2×D5—64no—
434C2×D5—62no—
435C2×D5—124no—
436C2×D5—244no—
437C2×D5—248no—
438C4×C5—22no—
439C4×C5—42no—
440C4×C5#93712yes—
441C4×C5C2×D3×D411yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—62no—
444Dic5—122no—
445C24—302no—
446C24—301no—
447#41—22no—
448#41—308no—
449#41—304no—
450#41—304no—
451#41—302no—
452#41—302no—
453#41#122014yes—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—151no—
458C2×Q8C22×D1512yes—
459C2×Q8C22×D1511yescharacteristic
460C2×D4—21no—
461C2×D4—152no—
462C2×D4—154no—
463C2×D4—151no—
464C2×D4—152no—
465C2×D4—302no—
466C2×D4—308no—
467C2×D4—302no—
468C2×D4—304no—
469C2×D4—3016no—
470C2×D4—304no—
471C2×D4—301no—
472C2×D4—302no—
473C2×D4C22×D1511yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C22×D1511yescharacteristic
479Q16—24no—
480SD16—24no—
481SD16—308no—
482M16—22no—
483C2×C8—152no—
484C2×C8—302no—
485C2×C8C22×D1512yes—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—304no—
489C22⋊C4—301no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—42no—
495C22×C3#408312yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—104no—
498C2×D3—104no—
499C2×D3—102no—
500C2×D3—204no—
501C2×D3—404no—
502C2×D3—408no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4#408312yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C22×C22⋊D511yescharacteristic
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5#577812yes—
513C2×C5C2×#93711yescharacteristic
514D5—122no—
515D5—124no—
516C23—21no—
517C23—152no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23D4×D1511yescharacteristic
524Q8—21no—
525Q8—22no—
526Q8—22no—
527Q8—42no—
528Q8—302no—
529Q8C2×#122012yes—
530D4—24no—
531D4—21no—
532D4—154no—
533D4—308no—
534D4—308no—
535D4—304no—
536D4—304no—
537D4—302no—
538D4—302no—
539D4C2×#122012yes—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—152no—
545C2×C4—308no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4D4×D1511yescharacteristic
550C2×C4D4×D1512yes—
551C2×C4C2×#122012yes—
552C2×C4C23×D1511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3#6611212yes—
558C2×C3C2×#408311yescharacteristic
559D3—202no—
560D3—204no—
561C5#8398211yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—42no—
565C22—304no—
566C22—302no—
567C22—304no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9182612yes—
572C22C2×D4×D1511yescharacteristic
573C4—22no—
574C4—42no—
575C4—302no—
576C4—602no—
577C4#9182612yes—
578C4C2×D4×D1511yescharacteristic
579C4C22×#122011yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—602no—
584C2—604no—
585C2#1175653712yes—
586C2C2×#9182611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#137C2
socleC22×C3×C5C22×D4
prime radicalC2×#92100C2
hypercenterC2×#137D15
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#137C2semidirect
3. #11756575C2semidirect
4. C22×#6717C2semidirect
5. #11751334C2semidirect
6. #11751288C2semidirect
7. #11751270C2semidirect
8. #11751232C2semidirect
9. #11751230C2semidirect
12. C22×C3×C5×Q8C22semidirect
23. C2×C3×C5×Q16C22semidirect
24. C2×C3×C5×SD16C22semidirect
25. C2×C3×C5×M16C22semidirect
27. #91835C22semidirect
30. C2×#6717C22semidirect
32. #91815C22semidirect
33. C22×C3⋊#168C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #91105C22semidirect
37. #91104C22semidirect
40. C2×C5×#137D3semidirect
95. C2×C3×#137D5semidirect
104. C22×C5×Q8C2×D3semidirect
115. C2×C5×Q16C2×D3semidirect
116. C2×C5×SD16C2×D3semidirect
117. C2×C5×M16C2×D3semidirect
171. C22×C3×Q8C2×D5semidirect
182. C2×C3×Q16C2×D5semidirect
183. C2×C3×SD16C2×D5semidirect
184. C2×C3×M16C2×D5semidirect
244. C2×#137D15semidirect
361. C22×Q8C2×D15semidirect
367. C2×Q16C2×D15semidirect
370. C2×SD16C2×D15semidirect
371. C2×M16C2×D15semidirect
491. C3×C5#3009semidirect
561. C5#83982semidirect
580. C3#10615713semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C2×C3×C5×#1372
#117565752
C22×#67172
#117513342
#117512882
#117512702
#117512322
#117512302
#106157133
#839825