Finite Group Databasebeta

← #14424326

Subgroups — #14424326

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
2#11756584C211yesmaximal, characteristic, complemented
3C2×#91798C211yesmaximal, characteristic, complemented
4C22×#6706C211yesmaximal, characteristic, complemented
5C3×C5×#408C211yesmaximal, characteristic, complemented
6#11751232C211yesmaximal, characteristic, complemented
7#11751214C211yesmaximal, characteristic, complemented
8#11750834C214yesmaximal, complemented
9#11750832C214yesmaximal, complemented
10#10610402—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×Dic30C2211yescharacteristic, complemented
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C22×C3×C5×C8C2211yescharacteristic, complemented
24#91833—21no—
25#91825—21no—
26#91824—22no—
27#91815C2211yescharacteristic, complemented
28#91813C2214yescomplemented
29#91798—24no—
30C2×Dic60C2212yescomplemented
31C2×#6706—24no—
32C2×#6706C2212yescomplemented
33C2×#6706C2214yescomplemented
34C3×C5×C22⋊C8C2214yescomplemented
35#91105C2212yescomplemented
36#91105C2214yescomplemented
37#91100C2212yescomplemented
38#78671—51nomaximal, Hall
39C2×#4149—31no—
40#66159—31no—
41C2×#4055—31no—
42C22×#677—31no—
43C5×#408D311yescharacteristic, complemented
44#65420—31no—
45#65402—31no—
46#65022—34no—
47#65020—34no—
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—
59C22×Dic15—21no—
60#6821—28no—
61D4×D15—216no—
62D4×D15—28no—
63#6817—22no—
64#6817—24no—
65#6817—24no—
66#6817D414yes—
67C2×D60—22no—
68C2×D60—21no—
69C2×D60—22no—
70C2×D60—24no—
71C2×D60—24no—
72C2×D60D414yes—
73C2×D60C2311yescharacteristic
74C2×D60C2312yes—
75C2×C4×D15—22no—
76C2×C4×D15—21no—
77C2×Dic30—22no—
78C2×Dic30—22no—
79C2×Dic30—21no—
80C2×Dic30C2312yes—
81C2×Dic30C2311yescharacteristic
82C3×C5×M16—22no—
83C2×C3×C5×C8—22no—
84C2×C3×C5×C8C2312yes—
85C2×C3×C5×C8C2312yes—
86#6719—24no—
87#6719—21no—
88C4⋊Dic15C2312yes—
89#6712—24no—
90C4×Dic15—21no—
91Dic60—24no—
92#6706—28no—
93#6706—24no—
94C2×#1009—51no—
95#5825—51no—
96C2×#909—51no—
97C22×#204—51no—
98C3×#408D511yescharacteristic, complemented
99#5131—51no—
100#5113—51no—
101#4733—54no—
102#4731—54no—
103C2×C5×#41C2×D311yescharacteristic
104#4149—64no—
105#4149—64no—
106C2×#713—61no—
107C2×D4×D5—62no—
108C2×D4×D5—61no—
109C2×#709—31no—
110C2×#709—32no—
111C22×D20—31no—
112C22×D20—32no—
113C22×Dic10—31no—
114C2×C5×M16C2×D311yescharacteristic, complemented
115C22×C5×C8C2×D311yescharacteristic, complemented
116#4090—61no—
117#4082—61no—
118#4081—62no—
119#4072—31no—
120#4070—34no—
121#4055—64no—
122C2×Dic20—32no—
123C2×#677—34no—
124C2×#677—32no—
125C2×#677—64no—
126C5×C22⋊C8C2×D314yescomplemented
127#3953—32no—
128#3953—34no—
129#3948—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—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—42no—
155D60—28no—
156D60—24no—
157D60—21no—
158D60—22no—
159D60C2×D412yes—
160D60C2×D414yes—
161C4×D15—28no—
162C4×D15—24no—
163Dic30—21no—
164Dic30—22no—
165Dic30—22no—
166Dic30—42no—
167Dic30C2×D412yes—
168C3×C5×C8—22no—
169C3×C5×C8—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×Dic6—51no—
181C2×C3×M16C2×D511yescharacteristic, complemented
182C22×C3×C8C2×D511yescharacteristic, complemented
183#944—101no—
184#936—101no—
185#935—102no—
186#926—51no—
187#924—54no—
188#909—104no—
189C2×Dic12—52no—
190C2×#204—54no—
191C2×#204—52no—
192C2×#204—104no—
193C3×C22⋊C8C2×D514yescomplemented
194#821—54no—
195#821—52no—
196#816—52no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—22no—
200C5×#41—22no—
201C2×C5×Q8D1211yescharacteristic
202C2×C5×D4D1212yes—
203C2×C5×D4D1211yescharacteristic
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—62no—
207C2×C22⋊D5—61no—
208C22×Dic5—61no—
209#713—68no—
210D4×D5—616no—
211D4×D5—68no—
212#709—34no—
213#709—64no—
214#709—64no—
215#709—62no—
216C2×D20—34no—
217C2×D20—32no—
218C2×D20—31no—
219C2×D20—64no—
220C2×D20—64no—
221C2×D20—62no—
222C2×D20—61no—
223C2×D20—62no—
224C2×C4×D5—62no—
225C2×C4×D5—61no—
226C2×Dic10—31no—
227C2×Dic10—32no—
228C2×Dic10—62no—
229C2×Dic10—62no—
230C2×Dic10—61no—
231C5×M16—22no—
232C2×C5×C8—22no—
233C2×C5×C8C22×D312yes—
234C2×C5×C8C22×D312yes—
235C22⋊Dic5—64no—
236C22⋊Dic5—61no—
237C4⋊Dic5—32no—
238#683—64no—
239C4×Dic5—61no—
240Dic20—64no—
241#677—68no—
242#677—64no—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408D1511yescharacteristic, complemented
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—
268Dic15—42no—
269Dic15—42no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—22no—
273C3×#41—22no—
274C2×C3×Q8D2011yescharacteristic
275C2×C3×D4D2012yes—
276C2×C3×D4D2011yescharacteristic
277C22×C3×C4C22×D512yes—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—102no—
280C2×#67—101no—
281C22×Dic3—101no—
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—101no—
296C2×D12—102no—
297C2×C4×D3—102no—
298C2×C4×D3—101no—
299C2×Dic6—51no—
300C2×Dic6—52no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304C3×M16—22no—
305C2×C3×C8—22no—
306C2×C3×C8C22×D512yes—
307C2×C3×C8C22×D512yes—
308#217—104no—
309#217—101no—
310C4⋊Dic3—52no—
311Dic3⋊C4—104no—
312C4×Dic3—101no—
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×D4—24no—
326C5×D4—22no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5C2×D1212yes—
330C2×C4×C5C2×D1211yescharacteristic
331C2×C4×C5D3×D412yes—
332C2×C4×C5D3×D412yes—
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—68no—
335C22⋊D5—64no—
336C2×Dic5—61no—
337C2×Dic5—62no—
338C2×Dic5—62no—
339C2×Dic5—122no—
340D20—34no—
341D20—32no—
342D20—68no—
343D20—64no—
344D20—62no—
345D20—61no—
346C4×D5—68no—
347C4×D5—64no—
348Dic10—32no—
349Dic10—61no—
350Dic10—62no—
351Dic10—62no—
352Dic10—122no—
353C5×C8—22no—
354C5×C8—22no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—151no—
358C2×#41—152no—
359C2×#41—301no—
360C2×#41C2×D1511yescharacteristic
361C22×Q8—151no—
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×M16C2×D1511yescharacteristic, complemented
372C22×C8C2×D1511yescharacteristic, complemented
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⋊C8C2×D1514yescomplemented
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—44no—
387D15—42no—
388C23×C3C2×D2012yes—
389C23×C3C2×D2011yescharacteristic
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×C4C2×D2012yes—
402C2×C3×C4C2×D2011yescharacteristic
403C2×C3×C4D4×D512yes—
404C2×C3×C4D4×D512yes—
405C2×C3×C4C23×D511yescharacteristic
406#67—108no—
407#67—104no—
408C2×Dic3—101no—
409C2×Dic3—102no—
410C2×Dic3—102no—
411C2×Dic3—202no—
412D12—54no—
413D12—52no—
414D12—108no—
415D12—102no—
416D12—101no—
417D12—104no—
418C4×D3—108no—
419C4×D3—104no—
420Dic6—52no—
421Dic6—101no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3×C8—22no—
426C3×C8—22no—
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#88212yes—
440C4×C5C2×D3×D411yescharacteristic
441C4×C5C2×D3×D411yescharacteristic
442Dic5—62no—
443Dic5—122no—
444Dic5—122no—
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—151no—
455C2×Q8—152no—
456C2×Q8—301no—
457C2×Q8—302no—
458C2×Q8—302no—
459C2×Q8D6011yescharacteristic
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×D4D6012yes—
473C2×D4D6011yescharacteristic
474C22×C4—302no—
475C22×C4—301no—
476C22×C4—301no—
477C22×C4C22×D1512yes—
478C22×C4C22×D1511yescharacteristic
479Q16—304no—
480SD16—308no—
481SD16—304no—
482M16—22no—
483C2×C8—22no—
484C2×C8C22×D1512yes—
485C2×C8C22×D1512yes—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—301no—
489C22⋊C4—304no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3#402812yes—
496C22×C3C22×D2011yescharacteristic
497C2×D3—104no—
498C2×D3—102no—
499C2×D3—204no—
500C2×D3—202no—
501C2×D3—404no—
502C2×D3—408no—
503C3×C4—22no—
504C3×C4#402812yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507Dic3—102no—
508Dic3—202no—
509Dic3—202no—
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×D6012yes—
523C23C2×D6011yescharacteristic
524Q8—22no—
525Q8—152no—
526Q8—301no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—24no—
531D4—22no—
532D4—154no—
533D4—152no—
534D4—308no—
535D4—308no—
536D4—304no—
537D4—304no—
538D4—301no—
539D4—302no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—308no—
543C2×C4—301no—
544C2×C4—304no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—602no—
548C2×C4C2×D6012yes—
549C2×C4C2×D6011yescharacteristic
550C2×C4D4×D1512yes—
551C2×C4D4×D1512yes—
552C2×C4C23×D1511yescharacteristic
553C8—22no—
554C8—22no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6608512yes—
559D3—204no—
560D3—202no—
561C5#7867111yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—24no—
565C22—304no—
566C22—302no—
567C22—604no—
568C22—602no—
569C22—1204no—
570C22—1208no—
571C22#9177112yes—
572C22C22×D6011yescharacteristic
573C4—22no—
574C4—302no—
575C4—602no—
576C4—602no—
577C4#9177112yes—
578C4C2×D4×D1511yescharacteristic
579C4C2×D4×D1511yescharacteristic
580C3#1061040211yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—604no—
584C2—602no—
585C2C2×#9177111yescharacteristic
586C2#1175651012yes—
subgroupG/H
centerC22C22×D60
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#408C2
socleC22×C3×C5C22×D4
prime radicalC2×#92100C2
hypercenter#408D15
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #11756584C2semidirect
3. C2×#91798C2semidirect
4. C22×#6706C2semidirect
5. C3×C5×#408C2semidirect
6. #11751232C2semidirect
7. #11751214C2semidirect
8. #11750834C2semidirect
9. #11750832C2semidirect
21. C22×Dic30C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C22×C3×C5×C8C22semidirect
27. #91815C22semidirect
28. #91813C22semidirect
30. C2×Dic60C22semidirect
32. C2×#6706C22semidirect
33. C2×#6706C22semidirect
34. C3×C5×C22⋊C8C22semidirect
35. #91105C22semidirect
36. #91105C22semidirect
37. #91100C22semidirect
43. C5×#408D3semidirect
98. C3×#408D5semidirect
114. C2×C5×M16C2×D3semidirect
115. C22×C5×C8C2×D3semidirect
126. C5×C22⋊C8C2×D3semidirect
181. C2×C3×M16C2×D5semidirect
182. C22×C3×C8C2×D5semidirect
193. C3×C22⋊C8C2×D5semidirect
251. #408D15semidirect
371. C2×M16C2×D15semidirect
372. C22×C8C2×D15semidirect
381. C22⋊C8C2×D15semidirect
491. C3×C5#3009semidirect
561. C5#78671semidirect
580. C3#10610402semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
#117565842
C2×#917982
C22×#67062
C3×C5×#4082
#117512322
#117512142
#117508342
#117508322
#106104023
#786715