Finite Group Databasebeta

← #14424373

Subgroups — #14424373

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
2C5×#5825C211yesmaximal, characteristic, complemented
3C2×#91321C211yesmaximal, characteristic, complemented
4C22×#6660C211yesmaximal, characteristic, complemented
5C3×#65523C211yesmaximal, characteristic, complemented
6#11748566C214yesmaximal, complemented
7#11748554C214yesmaximal, complemented
8#11748396C211yesmaximal, characteristic, complemented
9#11748393C211yesmaximal, characteristic, complemented
10#10615135—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12#92100—24no—
13#92100—24no—
14C2×#6821—21no—
15C2×D4×D15—22no—
16C2×D4×D15—21no—
17C2×#6817C2212yes—
18C2×#6817C2211yescharacteristic
19C22×D60C2212yes—
20C22×D60C2211yescharacteristic
21C22×C5×Dic6C2211yescharacteristic, complemented
22C5×#944—21no—
23C5×#936—21no—
24C5×#935—22no—
25C5×#926C2211yescharacteristic, complemented
26C5×#924C2214yescomplemented
27C2×C3×#681C2211yescharacteristic, complemented
28C22×C3×#168C2211yescharacteristic, complemented
29#91321—24no—
30C2×#6663C2212yescomplemented
31C2×#6660—24no—
32C2×#6660C2214yescomplemented
33C2×#6660C2212yescomplemented
34C3×C22⋊#168C2214yescomplemented
35#90975C2212yescomplemented
36#90969C2212yescomplemented
37#90969C2214yescomplemented
38#78671—51nomaximal, Hall
39C2×#4149—31no—
40C5×#536—31no—
41C2×#4088—31no—
42C22×#688—31no—
43#65523D311yescharacteristic, complemented
44#65336—34no—
45#65335—34no—
46#65291—31no—
47#65290—31no—
48C23×D15—22no—
49C23×D15—21no—
50C3×C5×#41—22no—
51C3×C5×#41—22no—
52C2×C3×C5×Q8D411yescharacteristic
53C2×C3×C5×D4D412yes—
54C2×C3×C5×D4D411yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—22no—
58C2×#1220—21no—
59#6821—28no—
60D4×D15—216no—
61D4×D15—28no—
62#6817—22no—
63#6817—24no—
64#6817—24no—
65#6817D414yes—
66C2×D60—22no—
67C2×D60—21no—
68C2×D60—22no—
69C2×D60—24no—
70C2×D60—24no—
71C2×D60D414yes—
72C2×D60C2311yescharacteristic
73C2×D60C2312yes—
74C2×C4×D15—22no—
75C2×C4×D15—21no—
76C2×Dic30C2311yescharacteristic
77C22×C5×Dic3—21no—
78C2×C5×Dic6—22no—
79C2×C5×Dic6—22no—
80C2×C5×Dic6—21no—
81C2×C5×Dic6C2312yes—
82C5×#217—24no—
83C5×#217—21no—
84C5×C4⋊Dic3C2312yes—
85C5×Dic3⋊C4—24no—
86C4×C5×Dic3—21no—
87C3×#681—22no—
88C2×C3×#168—22no—
89C2×C3×#168C2312yes—
90C2×C3×#168C2312yes—
91#6663—24no—
92#6660—28no—
93#6660—24no—
94C2×#1009—51no—
95#5825D511yescharacteristic, complemented
96C2×#909—51no—
97C22×#204—51no—
98C3×#408—51no—
99#5131—51no—
100#5113—51no—
101#4733—54no—
102#4731—54no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8—31no—
105#4149—64no—
106#4149—64no—
107C2×#713—61no—
108C2×D4×D5—62no—
109C2×D4×D5—61no—
110C2×#709—31no—
111C2×#709—32no—
112C22×D20—31no—
113C22×D20—32no—
114C5×C4⋊Q8—61no—
115C5×#124—61no—
116C5×#123—62no—
117C5×C22⋊Q8—34no—
118C5×#117—31no—
119C2×#689—32no—
120#4088—64no—
121C2×#688—32no—
122C2×#688—34no—
123C2×#688—64no—
124C2×#681C2×D311yescharacteristic, complemented
125C22×#168C2×D311yescharacteristic, complemented
126C22⋊#168C2×D314yescomplemented
127#3942—32no—
128#3941—34no—
129#3941—32no—
130#3009—151noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×D15—22no—
134C22×D15—21no—
135C22×D15—28no—
136C22×D15—24no—
137C22×D15—82no—
138C22×D15—84no—
139C3×C5×Q8—22no—
140C3×C5×D4—24no—
141C3×C5×D4—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149#1220—28no—
150#1220—24no—
151C2×Dic15—21no—
152D60—28no—
153D60—24no—
154D60—21no—
155D60—22no—
156D60C2×D412yes—
157D60C2×D414yes—
158C4×D15—28no—
159C4×D15—24no—
160Dic30—21no—
161Dic30C2×D412yes—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—42no—
165C5×Dic6—22no—
166C5×Dic6—22no—
167C5×Dic6—42no—
168C3×#168—22no—
169C3×#168—22no—
170C2×C3×#41C2×D511yescharacteristic
171#1009—104no—
172#1009—104no—
173C2×#239—101no—
174C2×D3×D4—102no—
175C2×D3×D4—101no—
176C2×#235—51no—
177C2×#235—52no—
178C22×D12—51no—
179C22×D12—52no—
180C22×Dic6C2×D511yescharacteristic, complemented
181C2×C3×M16—51no—
182C22×C3×C8—51no—
183#944—21no—
184#936—21no—
185#935—22no—
186#926C2×D511yescharacteristic, complemented
187#924C2×D514yescomplemented
188#909—104no—
189C2×Dic12—52no—
190C2×#204—54no—
191C2×#204—52no—
192C2×#204—104no—
193C3×C22⋊C8—54no—
194#821—54no—
195#821—52no—
196#816—52no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—22no—
200C5×#41—22no—
201C2×C5×Q8—32no—
202C2×C5×Q8—61no—
203C2×C5×Q8—62no—
204C2×C5×Q8—62no—
205C2×C5×Q8D1211yescharacteristic
206C2×C5×D4D1212yes—
207C2×C5×D4D1211yescharacteristic
208C22×C4×C5—61no—
209C22×C4×C5C22×D312yes—
210C22×C4×C5C22×D311yescharacteristic
211C2×C22⋊D5—62no—
212C2×C22⋊D5—61no—
213#713—68no—
214D4×D5—616no—
215D4×D5—68no—
216#709—34no—
217#709—64no—
218#709—64no—
219#709—62no—
220C2×D20—34no—
221C2×D20—32no—
222C2×D20—31no—
223C2×D20—64no—
224C2×D20—64no—
225C2×D20—62no—
226C2×D20—61no—
227C2×D20—62no—
228C2×C4×D5—62no—
229C2×C4×D5—61no—
230C2×Dic10—31no—
231C5×C4⋊C4—32no—
232C5×C4⋊C4—64no—
233C5×C22⋊C4—61no—
234C5×C22⋊C4—64no—
235C42×C5—61no—
236#689—64no—
237#688—68no—
238#688—64no—
239#681—22no—
240C2×#168—22no—
241C2×#168C22×D312yes—
242C2×#168C22×D312yes—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—31no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×D15—24no—
258C2×D15—22no—
259C2×D15—44no—
260C2×D15—42no—
261C2×D15—84no—
262C2×D15—88no—
263C3×C4×C5—22no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268C5×Dic3—42no—
269C5×Dic3—42no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—22no—
273C3×#41—22no—
274C2×C3×Q8C22⋊D511yescharacteristic
275C2×C3×D4C22⋊D512yes—
276C2×C3×D4C22⋊D511yescharacteristic
277C22×C3×C4C22×D512yes—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—102no—
280C2×#67—101no—
281C22×Dic3—21no—
282#239—108no—
283D3×D4—1016no—
284D3×D4—108no—
285#235—54no—
286#235—104no—
287#235—104no—
288#235—102no—
289C2×D12—54no—
290C2×D12—52no—
291C2×D12—51no—
292C2×D12—104no—
293C2×D12—104no—
294C2×D12—102no—
295C2×D12—102no—
296C2×D12—101no—
297C2×C4×D3—102no—
298C2×C4×D3—101no—
299C2×Dic6—22no—
300C2×Dic6—22no—
301C2×Dic6—21no—
302C2×Dic6—51no—
303C2×Dic6C22×D512yes—
304C3×M16—102no—
305C2×C3×C8—52no—
306C2×C3×C8—52no—
307C2×C3×C8—102no—
308#217—24no—
309#217—21no—
310C4⋊Dic3C22×D512yes—
311Dic3⋊C4—24no—
312C4×Dic3—21no—
313Dic12—104no—
314#204—108no—
315#204—104no—
316C23×C5C2×D1212yes—
317C23×C5C2×D1211yescharacteristic
318C22×D5—62no—
319C22×D5—61no—
320C22×D5—68no—
321C22×D5—64no—
322C22×D5—242no—
323C22×D5—244no—
324C5×Q8—22no—
325C5×Q8—62no—
326C5×Q8—62no—
327C5×Q8—122no—
328C5×D4—24no—
329C5×D4—22no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—122no—
335C2×C4×C5C2×D1212yes—
336C2×C4×C5C2×D1211yescharacteristic
337C2×C4×C5D3×D412yes—
338C2×C4×C5D3×D412yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—68no—
341C22⋊D5—64no—
342C2×Dic5—61no—
343D20—34no—
344D20—32no—
345D20—68no—
346D20—64no—
347D20—62no—
348D20—61no—
349C4×D5—68no—
350C4×D5—64no—
351Dic10—32no—
352Dic10—61no—
353#168—22no—
354#168—22no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—301no—
360C2×#41D3×D511yescharacteristic
361C22×Q8—31no—
362C22×D4—151no—
363C22×D4—152no—
364C22×D4—302no—
365C22×D4—301no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—154no—
369C2×SD16—152no—
370C2×SD16—304no—
371C2×M16—51no—
372C22×C8—51no—
373C4⋊Q8—61no—
374#124—61no—
375#123—62no—
376C22⋊Q8—34no—
377#117—31no—
378#103—152no—
379#102—152no—
380#102—154no—
381C22⋊C8—54no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—44no—
387D15—42no—
388C23×C3C2×C22⋊D512yes—
389C23×C3C2×C22⋊D511yescharacteristic
390C22×D3—102no—
391C22×D3—101no—
392C22×D3—108no—
393C22×D3—104no—
394C22×D3—402no—
395C22×D3—404no—
396C3×Q8—22no—
397C3×D4—24no—
398C3×D4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4D4×D512yes—
402C2×C3×C4D4×D512yes—
403C2×C3×C4C2×C22⋊D512yes—
404C2×C3×C4C2×C22⋊D511yescharacteristic
405C2×C3×C4C23×D511yescharacteristic
406#67—108no—
407#67—104no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—42no—
411C2×Dic3—101no—
412D12—54no—
413D12—52no—
414D12—108no—
415D12—102no—
416D12—101no—
417D12—104no—
418C4×D3—108no—
419C4×D3—104no—
420Dic6—22no—
421Dic6—22no—
422Dic6—42no—
423Dic6—52no—
424Dic6—101no—
425C3×C8—102no—
426C3×C8—102no—
427C22×C5—22no—
428C22×C5—22no—
429C22×C5—24no—
430C22×C5#88212yes—
431C22×C5C22×D1211yescharacteristic
432C2×D5—64no—
433C2×D5—62no—
434C2×D5—124no—
435C2×D5—122no—
436C2×D5—244no—
437C2×D5—248no—
438C4×C5—22no—
439C4×C5—122no—
440C4×C5—122no—
441C4×C5#88212yes—
442C4×C5C2×D3×D411yescharacteristic
443C4×C5C2×D3×D411yescharacteristic
444Dic5—62no—
445C24—302no—
446C24—301no—
447#41—22no—
448#41—22no—
449#41—154no—
450#41—304no—
451#41—304no—
452#41—308no—
453#41—302no—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—151no—
459C2×Q8#120311yescharacteristic
460C2×D4—154no—
461C2×D4—152no—
462C2×D4—151no—
463C2×D4—304no—
464C2×D4—304no—
465C2×D4—3016no—
466C2×D4—308no—
467C2×D4—302no—
468C2×D4—301no—
469C2×D4—302no—
470C2×D4—301no—
471C2×D4—302no—
472C2×D4#120312yes—
473C2×D4#120311yescharacteristic
474C22×C4—61no—
475C22×C4—302no—
476C22×C4—301no—
477C22×C4C2×D3×D512yes—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—308no—
481SD16—304no—
482M16—102no—
483C2×C8—52no—
484C2×C8—52no—
485C2×C8—102no—
486C4⋊C4—32no—
487C4⋊C4—64no—
488C22⋊C4—61no—
489C22⋊C4—64no—
490C42—61no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3#408312yes—
496C22×C3C22×C22⋊D511yescharacteristic
497C2×D3—104no—
498C2×D3—102no—
499C2×D3—204no—
500C2×D3—202no—
501C2×D3—404no—
502C2×D3—408no—
503C3×C4—22no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507Dic3—42no—
508Dic3—42no—
509Dic3—102no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#575112yes—
514D5—124no—
515D5—122no—
516C23—308no—
517C23—302no—
518C23—301no—
519C23—304no—
520C23—1202no—
521C23—1204no—
522C23C2×#120312yes—
523C23C2×#120311yescharacteristic
524Q8—22no—
525Q8—62no—
526Q8—62no—
527Q8—122no—
528Q8—152no—
529Q8—301no—
530D4—24no—
531D4—22no—
532D4—154no—
533D4—152no—
534D4—308no—
535D4—308no—
536D4—304no—
537D4—301no—
538D4—302no—
539D4—304no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—122no—
545C2×C4—308no—
546C2×C4—301no—
547C2×C4—304no—
548C2×C4C2×#120312yes—
549C2×C4C2×#120311yescharacteristic
550C2×C4#679012yes—
551C2×C4#679012yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—102no—
554C8—102no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3C2×#408311yescharacteristic
558C2×C3#6617112yes—
559D3—204no—
560D3—202no—
561C5#7867111yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—22no—
565C22—304no—
566C22—302no—
567C22—604no—
568C22—602no—
569C22—1204no—
570C22—1208no—
571C22#9157312yes—
572C22C22×#120311yescharacteristic
573C4—22no—
574C4—122no—
575C4—122no—
576C4—302no—
577C4#9157312yes—
578C4C2×#679011yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061513511yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—604no—
584C2—602no—
585C2#1175530812yes—
586C2C2×#9157311yescharacteristic
subgroupG/H
centerC22C22×#1203
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×#92100C2
hypercenterC2×#41D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C5×#5825C2semidirect
3. C2×#91321C2semidirect
4. C22×#6660C2semidirect
5. C3×#65523C2semidirect
6. #11748566C2semidirect
7. #11748554C2semidirect
8. #11748396C2semidirect
9. #11748393C2semidirect
21. C22×C5×Dic6C22semidirect
25. C5×#926C22semidirect
26. C5×#924C22semidirect
27. C2×C3×#681C22semidirect
28. C22×C3×#168C22semidirect
30. C2×#6663C22semidirect
32. C2×#6660C22semidirect
33. C2×#6660C22semidirect
34. C3×C22⋊#168C22semidirect
35. #90975C22semidirect
36. #90969C22semidirect
37. #90969C22semidirect
43. #65523D3semidirect
95. #5825D5semidirect
124. C2×#681C2×D3semidirect
125. C22×#168C2×D3semidirect
126. C22⋊#168C2×D3semidirect
180. C22×Dic6C2×D5semidirect
186. #926C2×D5semidirect
187. #924C2×D5semidirect
491. C3×C5#3009semidirect
561. C5#78671semidirect
580. C3#10615135semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C5×#58252
C2×#913212
C22×#66602
C3×#655232
#117485662
#117485542
#117483962
#117483932
#106151353
#786715