Finite Group Databasebeta

← #14424391

Subgroups — #14424391

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×C5×#951C211yesmaximal, characteristic, complemented
3#11754270C211yesmaximal, characteristic, complemented
4C22×#6660C211yesmaximal, characteristic, complemented
5C3×#65522C211yesmaximal, characteristic, complemented
6#11749307C214yesmaximal, complemented
7#11749062C214yesmaximal, complemented
8#11748392C211yesmaximal, characteristic, complemented
9#11748349C211yesmaximal, characteristic, complemented
10#10615713—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12#92100—24no—
13#92100—24no—
14C2×#6821C2212yes—
15C2×D4×D15—22no—
16C2×D4×D15C2212yes—
17C2×#6817—21no—
18C2×#6817C2211yescharacteristic
19C22×D60—21no—
20C22×D60C2211yescharacteristic
21C22×C5×Dic6C2211yescharacteristic, complemented
22C5×#951—24no—
23C2×C5×#216C2212yescomplemented
24C2×C5×#214C2212yescomplemented
25C2×C5×#208C2211yescharacteristic, complemented
26C3×#4072C2211yescharacteristic, complemented
27C22×C3×#168C2211yescharacteristic, complemented
28#91542—22no—
29#91383C2214yescomplemented
30#91376—21no—
31#91375—21no—
32C2×#6660—24no—
33C2×#6660C2214yescomplemented
34C3×#3967C2212yescomplemented
35C3×#3964C2212yescomplemented
36#90968C2214yescomplemented
37#90957C2214yescomplemented
38#78835—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×#137—31no—
41#66150—31no—
42C22×#688—31no—
43#65522D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×#1220—22no—
58C2×#1220D412yes—
59#6821—24no—
60#6821—24no—
61D4×D15—216no—
62D4×D15—24no—
63D4×D15—24no—
64#6817—28no—
65#6817—22no—
66#6817—22no—
67C2×D60—28no—
68C2×D60—22no—
69C2×D60—21no—
70C2×D60—22no—
71C2×D60D411yescharacteristic
72C2×D60D414yes—
73C2×D60C2312yes—
74C2×C4×D15—22no—
75C2×C4×D15C2312yes—
76C2×Dic30D411yescharacteristic
77C22×C5×Dic3—21no—
78C2×C5×Dic6—22no—
79C2×C5×Dic6—22no—
80C2×C5×Dic6—21no—
81C2×C5×Dic6C2312yes—
82C5×#216—24no—
83C5×#214—24no—
84C5×#208—22no—
85C2×C5×C3⋊C8C2312yes—
86C3×C22⋊Dic5—21no—
87C3×C4⋊Dic5C2312yes—
88C3×C4×Dic5—21no—
89C2×C3×#168—22no—
90C2×C3×#168C2312yes—
91#6668—24no—
92#6667—24no—
93#6660—28no—
94C2×#1009—51no—
95C2×#951D511yescharacteristic, complemented
96C22×#204—51no—
97#5526—51no—
98C3×#416—51no—
99#5110—51no—
100#5001—51no—
101#4808—54no—
102#4776—54no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8—31no—
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—64no—
115C2×C5×Q16—32no—
116C2×C5×SD16—32no—
117C2×C5×M16—31no—
118#4093—61no—
119#4092—34no—
120#4090—61no—
121C2×#688—34no—
122C2×#688—64no—
123#4075—62no—
124#4072C2×D311yescharacteristic, complemented
125C22×#168C2×D311yescharacteristic, complemented
126#3967C2×D312yescomplemented
127#3964C2×D312yescomplemented
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×Q8C2×D412yes—
141C3×C5×D4—24no—
142C3×C5×D4—21no—
143C3×C5×D4C2×D412yes—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—24no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2411yescharacteristic
150#1220—28no—
151#1220—24no—
152C2×Dic15C2×D412yes—
153D60—28no—
154D60—22no—
155D60—24no—
156D60—22no—
157D60C2×D414yes—
158C4×D15—28no—
159C4×D15—22no—
160C4×D15—22no—
161Dic30—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—42no—
164C5×Dic6—22no—
165C5×Dic6—22no—
166C5×Dic6—42no—
167C2×C3×Dic5—22no—
168C3×#168—22no—
169C5×C3⋊C8—22no—
170C2×C3×#41C2×D511yescharacteristic
171#1009—104no—
172#1009—104no—
173C2×#239—52no—
174C2×D3×D4—52no—
175C2×D3×D4—102no—
176C2×#235—51no—
177C2×#235—101no—
178C22×D12—51no—
179C22×D12—101no—
180C22×Dic6C2×D511yescharacteristic, complemented
181C22×C3×C8—51no—
182C3×#117—51no—
183#951—24no—
184C2×#216C2×D512yescomplemented
185C2×#214C2×D512yescomplemented
186C2×#208C2×D511yescharacteristic, complemented
187C2×#204—54no—
188C2×#204—104no—
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#103—52no—
194C3×#102—52no—
195(C2×D3)⋊C8—54no—
196#809—54no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—22no—
200C5×#41D1214yes—
201C2×C5×Q8—32no—
202C2×C5×Q8—61no—
203C2×C5×Q8—62no—
204C2×C5×Q8—62no—
205C2×C5×Q8C22×D311yescharacteristic
206C2×C5×D4—21no—
207C2×C5×D4C22×D311yescharacteristic
208C22×C4×C5—21no—
209C22×C4×C5—61no—
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—62no—
227C2×D20—61no—
228C2×C4×D5—32no—
229C2×C4×D5—62no—
230C2×Dic10—31no—
231C5×Q16—64no—
232C5×SD16—64no—
233C5×M16—62no—
234C2×C5×C8—32no—
235C22⋊Dic5—21no—
236#688—68no—
237#685—64no—
238C4⋊Dic5C22×D312yes—
239#683—64no—
240C4×Dic5—21no—
241C2×#168—22no—
242C2×#168C22×D312yes—
243C2×H(5,𝔽2)—151no—
244C2×#137—31no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—51no—
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#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268C3×Dic5—42no—
269C5×Dic3—42no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—22no—
273C3×#41C22⋊D514yes—
274C2×C3×Q8C22×D511yescharacteristic
275C2×C3×D4—21no—
276C2×C3×D4C22×D511yescharacteristic
277C22×C3×C4—21no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—52no—
280C2×#67—102no—
281C22×Dic3—21no—
282#239—104no—
283#239—104no—
284D3×D4—104no—
285D3×D4—1016no—
286D3×D4—104no—
287#235—108no—
288#235—102no—
289#235—102no—
290C2×D12—54no—
291C2×D12—52no—
292C2×D12—51no—
293C2×D12—102no—
294C2×D12—108no—
295C2×D12—101no—
296C2×D12—102no—
297C2×C4×D3—52no—
298C2×C4×D3—102no—
299C2×Dic6—22no—
300C2×Dic6—22no—
301C2×Dic6—21no—
302C2×Dic6—51no—
303C2×Dic6C22×D512yes—
304C2×C3×C8—52no—
305C2×C3×C8—102no—
306C3×C4⋊C4—52no—
307C3×C22⋊C4—101no—
308C3×C42—101no—
309#216—24no—
310#214—24no—
311(C2×D3)⋊C4—104no—
312C4⋊Dic3—104no—
313#208—22no—
314C2×C3⋊C8C22×D512yes—
315#204—108no—
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—62no—
326C5×Q8—62no—
327C5×Q8—122no—
328C5×Q8C2×D1212yes—
329C5×D4—24no—
330C5×D4—21no—
331C5×D4C2×D1212yes—
332C2×C4×C5—21no—
333C2×C4×C5—24no—
334C2×C4×C5—62no—
335C2×C4×C5—122no—
336C2×C4×C5C2×D1212yes—
337C2×C4×C5D3×D411yescharacteristic
338C2×C4×C5D3×D412yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—68no—
341C22⋊D5—64no—
342C2×Dic5—22no—
343C2×Dic5—32no—
344D20—34no—
345D20—68no—
346D20—64no—
347D20—62no—
348D20—62no—
349C4×D5—68no—
350C4×D5—62no—
351C4×D5—62no—
352Dic10—62no—
353C5×C8—62no—
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—152no—
363C22×D4—151no—
364C22×D4—301no—
365C22×D4—302no—
366#137—64no—
367C2×Q16—32no—
368C2×SD16—32no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—31no—
372C22×C8—51no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—51no—
378#103—52no—
379#102—52no—
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×Q8C2×C22⋊D512yes—
398C3×D4—24no—
399C3×D4—21no—
400C3×D4C2×C22⋊D512yes—
401C2×C3×C4—21no—
402C2×C3×C4—24no—
403C2×C3×C4—102no—
404C2×C3×C4D4×D512yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4C2×C22⋊D512yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—108no—
409#67—104no—
410C2×Dic3—22no—
411C2×Dic3—42no—
412C2×Dic3—52no—
413D12—54no—
414D12—108no—
415D12—102no—
416D12—104no—
417D12—102no—
418C4×D3—108no—
419C4×D3—102no—
420C4×D3—102no—
421Dic6—22no—
422Dic6—22no—
423Dic6—42no—
424Dic6—102no—
425C3×C8—102no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—22no—
429C22×C5—42no—
430C22×C5#88212yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—64no—
433C2×D5—62no—
434C2×D5—64no—
435C2×D5—124no—
436C2×D5—244no—
437C2×D5—248no—
438C4×C5—22no—
439C4×C5—122no—
440C4×C5#88212yes—
441C4×C5C22×D1211yescharacteristic
442C4×C5C2×D3×D411yescharacteristic
443Dic5—42no—
444Dic5—62no—
445C24—302no—
446C24—301no—
447#41—22no—
448#41—308no—
449#41—304no—
450#41—304no—
451#41—302no—
452#41—302no—
453#41#120314yes—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—62no—
457C2×Q8—62no—
458C2×Q8—151no—
459C2×Q8C2×D3×D511yescharacteristic
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—302no—
472C2×D4—301no—
473C2×D4C2×D3×D511yescharacteristic
474C22×C4—21no—
475C22×C4—61no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—64no—
480SD16—64no—
481SD16—308no—
482M16—62no—
483C2×C8—32no—
484C2×C8—52no—
485C2×C8—102no—
486C4⋊C4—52no—
487C4⋊C4—304no—
488C22⋊C4—101no—
489C22⋊C4—304no—
490C42—101no—
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—202no—
505C3×C4#408312yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C22×C22⋊D511yescharacteristic
508Dic3—42no—
509Dic3—102no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#577012yes—
514D5—122no—
515D5—124no—
516C23—21no—
517C23—152no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23#679011yescharacteristic
524Q8—21no—
525Q8—62no—
526Q8—62no—
527Q8—122no—
528Q8—302no—
529Q8C2×#120312yes—
530D4—24no—
531D4—21no—
532D4—154no—
533D4—308no—
534D4—308no—
535D4—304no—
536D4—302no—
537D4—304no—
538D4—302no—
539D4C2×#120312yes—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—122no—
545C2×C4—152no—
546C2×C4—308no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4C2×#120312yes—
550C2×C4#679012yes—
551C2×C4#679011yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—102no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3#6611212yes—
558C2×C3C2×#408311yescharacteristic
559D3—202no—
560D3—204no—
561C5#7883511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—42no—
565C22—304no—
566C22—304no—
567C22—302no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9157312yes—
572C22C2×#679011yescharacteristic
573C4—22no—
574C4—122no—
575C4—202no—
576C4—302no—
577C4#9157312yes—
578C4C22×#120311yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—602no—
584C2—604no—
585C2#1175535112yes—
586C2C2×#9157311yescharacteristic
subgroupG/H
centerC22C2×#6790
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 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C5×#951C2semidirect
3. #11754270C2semidirect
4. C22×#6660C2semidirect
5. C3×#65522C2semidirect
6. #11749307C2semidirect
7. #11749062C2semidirect
8. #11748392C2semidirect
9. #11748349C2semidirect
21. C22×C5×Dic6C22semidirect
23. C2×C5×#216C22semidirect
24. C2×C5×#214C22semidirect
25. C2×C5×#208C22semidirect
26. C3×#4072C22semidirect
27. C22×C3×#168C22semidirect
29. #91383C22semidirect
33. C2×#6660C22semidirect
34. C3×#3967C22semidirect
35. C3×#3964C22semidirect
36. #90968C22semidirect
37. #90957C22semidirect
43. #65522D3semidirect
95. C2×#951D5semidirect
124. #4072C2×D3semidirect
125. C22×#168C2×D3semidirect
126. #3967C2×D3semidirect
127. #3964C2×D3semidirect
180. C22×Dic6C2×D5semidirect
184. C2×#216C2×D5semidirect
185. C2×#214C2×D5semidirect
186. C2×#208C2×D5semidirect
491. C3×C5#3009semidirect
561. C5#78835semidirect
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×C5×#9512
#117542702
C22×#66602
C3×#655222
#117493072
#117490622
#117483922
#117483492
#106157133
#788355