Finite Group Databasebeta

← #14424396

Subgroups — #14424396

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×#92018C211yesmaximal, characteristic
2C2×#91798C211yesmaximal, characteristic, complemented
3#11754252C211yesmaximal, characteristic, complemented
4C22×#6659C211yesmaximal, characteristic, complemented
5C5×#5131C211yesmaximal, characteristic, complemented
6C3×#65418C211yesmaximal, characteristic, complemented
7#11748393C211yesmaximal, characteristic, complemented
8#11748164C214yesmaximal, complemented
9#11748115C214yesmaximal, complemented
10#10610660—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C22×Dic30C2211yescharacteristic, complemented
13C22×C5×D12C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×D3×C22⋊D5—22no—
16#92018—24no—
17#92018—24no—
18C2×#6777—21no—
19C2×D5×D12C2212yes—
20C2×#6772C2212yes—
21C2×#6768—21no—
22C2×C3×C5×M16C2211yescharacteristic, complemented
23#91798—24no—
24C2×Dic60C2212yescomplemented
25C2×#6706C2212yescomplemented
26C5×#926C2211yescharacteristic, complemented
27C22×C3×#168C2211yescharacteristic, complemented
28#91537—22no—
29#91389—21no—
30#91364—21no—
31#91360C2214yescomplemented
32C2×#6659—24no—
33C2×#6659C2214yescomplemented
34C5×#821C2214yescomplemented
35C3×#3952C2214yescomplemented
36#90975C2212yescomplemented
37#90969C2212yescomplemented
38#78671—51nomaximal, Hall
39C2×#4144—31no—
40C22×#687—31no—
41C2×#4055—31no—
42#65860—31no—
43C5×#418—31no—
44#65418D311yescharacteristic, complemented
45#65291—31no—
46#65072—34no—
47#65065—34no—
48C23×C5×D3—21no—
49C22×D3×D5—22no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220—21no—
52C22×Dic15—21no—
53#6817—22no—
54#6817D414yes—
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×Dic30—22no—
58C2×Dic30—22no—
59C2×Dic30—21no—
60C2×Dic30C2312yes—
61C2×Dic30C2311yescharacteristic
62C2×C5×D12—21no—
63C2×C5×D12—22no—
64C2×C5×D12D414yes—
65C2×C5×D12C2312yes—
66C2×C3×C22⋊D5D412yes—
67C3×#709—22no—
68C3×#709—22no—
69C2×C3×D20D411yescharacteristic
70C2×C3×C4×D5C2312yes—
71C2×C3×Dic10D411yescharacteristic
72D3×C22⋊D5—216no—
73C2×#1203—22no—
74C2×#1201—22no—
75C2×D3×Dic5—22no—
76#6777—28no—
77D5×D12—24no—
78D5×D12—24no—
79#6772—24no—
80#6772—24no—
81#6768—28no—
82C3×C5×M16—22no—
83C2×C3×C5×C8C2312yes—
84Dic60—24no—
85#6706—24no—
86C5×#217—21no—
87C5×C4⋊Dic3C2312yes—
88C4×C5×Dic3—21no—
89C2×C3×#168—22no—
90C2×C3×#168C2312yes—
91Dic3⋊Dic5—24no—
92#6665—24no—
93#6659—28no—
94C2×#1009—51no—
95#5825—51no—
96C2×#909—51no—
97C22×#204—51no—
98C3×#408—51no—
99#5131D511yescharacteristic, complemented
100#5113—51no—
101#4733—54no—
102#4731—54no—
103C22×C5×D4—31no—
104C22×C22⋊D5—62no—
105#4144—64no—
106#4144—64no—
107C2×#711—32no—
108C2×#711—61no—
109C2×D4×D5—32no—
110C2×D4×D5—61no—
111C2×#709—31no—
112C2×#709C2×D311yescharacteristic
113C22×Dic10—31no—
114C2×C5×M16C2×D311yescharacteristic, complemented
115C5×#117—31no—
116C2×#687—34no—
117C2×#687—64no—
118C22×#168C2×D311yescharacteristic, complemented
119#4055—64no—
120C2×Dic20—32no—
121C2×#677—32no—
122#4043—34no—
123#4032—62no—
124#4021—61no—
125#4015—61no—
126C5×#102—34no—
127#3952C2×D314yescomplemented
128#3942—32no—
129#3941—32no—
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—21no—
133C22×C5×D3—22no—
134C22×C5×D3—24no—
135C22×C5×D3—82no—
136C22×C3×D5C2×D412yes—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2411yescharacteristic
143#1220—24no—
144C2×Dic15—21no—
145C2×Dic15—22no—
146C2×Dic15—42no—
147D60—21no—
148D60C2×D412yes—
149C4×D15—24no—
150Dic30—21no—
151Dic30—22no—
152Dic30—22no—
153Dic30—42no—
154Dic30C2×D412yes—
155C2×C5×Dic3—22no—
156C5×D12—24no—
157C5×D12—22no—
158C5×D12C2×D414yes—
159C3×C22⋊D5—24no—
160C2×C3×Dic5C2×D412yes—
161C3×D20—22no—
162C3×C4×D5—22no—
163C3×C4×D5—22no—
164C3×Dic10—22no—
165#1203—28no—
166#1201—28no—
167D3×Dic5—28no—
168C3×C5×C8—22no—
169C3×#168—22no—
170C2×C3×#41—51no—
171#1009—104no—
172#1009—104no—
173C2×#239—101no—
174C2×D3×D4—101no—
175C2×D3×D4—102no—
176C2×#235—51no—
177C2×#235—52no—
178C22×D12—52no—
179C22×D12C2×D511yescharacteristic
180C22×Dic6—51no—
181C2×C3×M16C2×D511yescharacteristic, complemented
182C22×C3×C8—51no—
183#944—101no—
184#936—101no—
185#935—102no—
186#926C2×D511yescharacteristic, complemented
187#924—54no—
188#909—104no—
189C2×Dic12—52no—
190C2×#204—54no—
191C2×#204—52no—
192C2×#204—104no—
193C3×C22⋊C8—54no—
194#821—52no—
195#821C2×D514yescomplemented
196#816—52no—
197C24×C5—61no—
198C23×D5—62no—
199C2×C5×D4—34no—
200C2×C5×D4—32no—
201C2×C5×D4—61no—
202C2×C5×D4—62no—
203C22×C4×C5C22×D311yescharacteristic
204C2×C22⋊D5—616no—
205C2×C22⋊D5—61no—
206C2×C22⋊D5—62no—
207C2×C22⋊D5—62no—
208C2×C22⋊D5D1212yes—
209C22×Dic5—62no—
210C22×Dic5—61no—
211#711—68no—
212#711—64no—
213#711—64no—
214D4×D5—64no—
215D4×D5—64no—
216D4×D5—68no—
217#709—22no—
218#709—22no—
219#709—34no—
220#709—62no—
221C2×D20—31no—
222C2×D20D1211yescharacteristic
223C2×C4×D5—61no—
224C2×C4×D5C22×D312yes—
225C2×Dic10—31no—
226C2×Dic10—32no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—61no—
230C2×Dic10D1211yescharacteristic
231C5×M16—22no—
232C2×C5×C8C22×D312yes—
233C5×C4⋊C4—32no—
234C5×C22⋊C4—61no—
235C42×C5—61no—
236#687—68no—
237#685—64no—
238C4⋊Dic5—64no—
239C2×#168—22no—
240C2×#168C22×D312yes—
241Dic20—64no—
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—31no—
250#416—151no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—42no—
257C2×C5×D3—24no—
258C2×C5×D3—44no—
259C2×C5×D3—84no—
260C2×C3×D5—22no—
261C2×C3×D5—24no—
262D3×D5—88no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—22no—
267Dic15—42no—
268C3×Dic5—22no—
269C5×Dic3—42no—
270C23×D3—21no—
271C23×D3—102no—
272C3×#41—102no—
273C3×#41—102no—
274C2×C3×Q8—51no—
275C2×C3×D4—52no—
276C2×C3×D4—51no—
277C22×C3×C4—52no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—102no—
280C2×#67—101no—
281C22×Dic3—101no—
282#239—108no—
283D3×D4—108no—
284D3×D4—1016no—
285#235—54no—
286#235—104no—
287#235—102no—
288#235—104no—
289C2×D12—21no—
290C2×D12—22no—
291C2×D12—51no—
292C2×D12—104no—
293C2×D12—102no—
294C2×D12—104no—
295C2×D12D2014yes—
296C2×D12C22×D512yes—
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—52no—
306C2×C3×C8—102no—
307C2×C3×C8C22×D512yes—
308#217—21no—
309#217—104no—
310C4⋊Dic3C22×D512yes—
311Dic3⋊C4—104no—
312C4×Dic3—21no—
313Dic12—104no—
314#204—104no—
315#204—108no—
316C23×C5—62no—
317C23×C5—64no—
318C23×C5—242no—
319C23×C5C2×D1211yescharacteristic
320C22×D5—68no—
321C22×D5—61no—
322C22×D5—244no—
323C22×D5C2×D1212yes—
324C5×D4—34no—
325C5×D4—64no—
326C5×D4—62no—
327C2×C4×C5—62no—
328C2×C4×C5C2×D1211yescharacteristic
329C2×C4×C5D3×D412yes—
330C2×C4×C5D3×D412yes—
331C2×C4×C5C23×D311yescharacteristic
332C22⋊D5—24no—
333C22⋊D5—68no—
334C22⋊D5—68no—
335C22⋊D5—64no—
336C2×Dic5—68no—
337C2×Dic5—61no—
338C2×Dic5—62no—
339C2×Dic5—122no—
340C2×Dic5C2×D1212yes—
341D20—22no—
342D20—32no—
343D20—61no—
344C4×D5—22no—
345C4×D5—22no—
346C4×D5—64no—
347Dic10—22no—
348Dic10—32no—
349Dic10—61no—
350Dic10—62no—
351Dic10—62no—
352Dic10—122no—
353C5×C8—22no—
354#168—22no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—51no—
358C2×#41—152no—
359C2×#41—151no—
360C2×#41—301no—
361C22×Q8—151no—
362C22×D4—31no—
363C22×D4—152no—
364C22×D4—301no—
365C22×D4—302no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—154no—
369C2×SD16—152no—
370C2×SD16—304no—
371C2×M16D3×D511yescharacteristic, complemented
372C22×C8—51no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—31no—
378#103—152no—
379#102—34no—
380#102—152no—
381C22⋊C8—54no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—42no—
386C3×D5—42no—
387C5×D3—44no—
388C23×C3—52no—
389C23×C3D4×D511yescharacteristic
390C22×D3—22no—
391C22×D3—24no—
392C22×D3—82no—
393C22×D3—108no—
394C22×D3—101no—
395C22×D3—404no—
396C3×Q8—102no—
397C3×D4—104no—
398C3×D4—102no—
399C2×C3×C4—52no—
400C2×C3×C4—102no—
401C2×C3×C4—102no—
402C2×C3×C4C2×D2012yes—
403C2×C3×C4D4×D511yescharacteristic
404C2×C3×C4D4×D512yes—
405C2×C3×C4C23×D511yescharacteristic
406#67—108no—
407#67—104no—
408C2×Dic3—22no—
409C2×Dic3—101no—
410C2×Dic3—102no—
411C2×Dic3—202no—
412D12—22no—
413D12—24no—
414D12—52no—
415D12—108no—
416D12—101no—
417D12C2×D2014yes—
418C4×D3—108no—
419C4×D3—104no—
420Dic6—52no—
421Dic6—101no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3×C8—22no—
426C3×C8—102no—
427C22×C5—22no—
428C22×C5—64no—
429C22×C5—124no—
430C22×C5—244no—
431C22×C5#88212yes—
432C22×C5C22×D1211yescharacteristic
433C2×D5—22no—
434C2×D5—24no—
435C2×D5—62no—
436C2×D5—122no—
437C2×D5—248no—
438C4×C5—122no—
439C4×C5#88212yes—
440C4×C5C2×D3×D411yescharacteristic
441C4×C5C2×D3×D411yescharacteristic
442Dic5—22no—
443Dic5—62no—
444Dic5—122no—
445C24—61no—
446C24—302no—
447#41—102no—
448#41—102no—
449#41—154no—
450#41—304no—
451#41—304no—
452#41—308no—
453#41—302no—
454C2×Q8—51no—
455C2×Q8—151no—
456C2×Q8—152no—
457C2×Q8—301no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×D4—34no—
461C2×D4—32no—
462C2×D4—52no—
463C2×D4—51no—
464C2×D4—61no—
465C2×D4—62no—
466C2×D4—151no—
467C2×D4—302no—
468C2×D4—304no—
469C2×D4—304no—
470C2×D4—308no—
471C2×D4—3016no—
472C2×D4—302no—
473C2×D4—301no—
474C22×C4—52no—
475C22×C4—302no—
476C22×C4—301no—
477C22×C4—301no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—304no—
481SD16—308no—
482M16—22no—
483C2×C8—52no—
484C2×C8—102no—
485C2×C8C2×D3×D512yes—
486C4⋊C4—32no—
487C4⋊C4—304no—
488C22⋊C4—61no—
489C22⋊C4—304no—
490C42—61no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—102no—
494C22×C3—104no—
495C22×C3#402812yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—24no—
498C2×D3—44no—
499C2×D3—84no—
500C2×D3—102no—
501C2×D3—202no—
502C2×D3—408no—
503C3×C4—102no—
504C3×C4#402812yes—
505C3×C4C22×D2011yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507Dic3—42no—
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—124no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#575112yes—
514D5—42no—
515D5—122no—
516C23—52no—
517C23—62no—
518C23—64no—
519C23—242no—
520C23—308no—
521C23—301no—
522C23—1204no—
523C23D5×D1211yescharacteristic
524Q8—102no—
525Q8—152no—
526Q8—301no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—34no—
531D4—64no—
532D4—62no—
533D4—104no—
534D4—102no—
535D4—152no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—301no—
540C2×C4—52no—
541C2×C4—62no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—308no—
545C2×C4—301no—
546C2×C4—304no—
547C2×C4—302no—
548C2×C4—602no—
549C2×C4D5×D1211yescharacteristic
550C2×C4D3×D2012yes—
551C2×C4#679012yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—22no—
554C8—102no—
555C2×C3—22no—
556C2×C3—202no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6610912yes—
559D3—44no—
560D3—202no—
561C5#7867111yescharacteristic, complemented, Sylow
562C22—22no—
563C22—64no—
564C22—104no—
565C22—102no—
566C22—124no—
567C22—244no—
568C22—302no—
569C22—602no—
570C22—1208no—
571C22#9147612yes—
572C22C2×D5×D1211yescharacteristic
573C4—102no—
574C4—122no—
575C4—302no—
576C4—602no—
577C4#9147612yes—
578C4C2×D3×D2011yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061066011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—124no—
583C2—202no—
584C2—602no—
585C2#1175402312yes—
586C2C2×#9147611yescharacteristic
subgroupG/H
centerC22C2×D5×D12
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#92018C2
hypercenterC2×M16D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91798C2semidirect
3. #11754252C2semidirect
4. C22×#6659C2semidirect
5. C5×#5131C2semidirect
6. C3×#65418C2semidirect
7. #11748393C2semidirect
8. #11748164C2semidirect
9. #11748115C2semidirect
12. C22×Dic30C22semidirect
22. C2×C3×C5×M16C22semidirect
24. C2×Dic60C22semidirect
25. C2×#6706C22semidirect
26. C5×#926C22semidirect
27. C22×C3×#168C22semidirect
31. #91360C22semidirect
33. C2×#6659C22semidirect
34. C5×#821C22semidirect
35. C3×#3952C22semidirect
36. #90975C22semidirect
37. #90969C22semidirect
44. #65418D3semidirect
99. #5131D5semidirect
114. C2×C5×M16C2×D3semidirect
118. C22×#168C2×D3semidirect
127. #3952C2×D3semidirect
181. C2×C3×M16C2×D5semidirect
186. #926C2×D5semidirect
195. #821C2×D5semidirect
371. C2×M16D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#78671semidirect
580. C3#10610660semidirect

Maximal subgroups

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

subgroupindex
C2×#920182
C2×#917982
#117542522
C22×#66592
C5×#51312
C3×#654182
#117483932
#117481642
#117481152
#106106603
#786715