Finite Group Databasebeta

← #14424365

Subgroups — #14424365

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×#92019C211yesmaximal, characteristic
2C22×#6717C211yesmaximal, characteristic, complemented
3C2×C5×#942C211yesmaximal, characteristic, complemented
4C3×#66150C211yesmaximal, characteristic, complemented
5#11749406C214yesmaximal, complemented
6#11749397C214yesmaximal, complemented
7#11748392C211yesmaximal, characteristic, complemented
8#11748382C211yesmaximal, characteristic, complemented
9#11748343C211yesmaximal, characteristic, complemented
10#10615713—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic, complemented
12C22×D60C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×#6790—22no—
16#92019—24no—
17#92019—24no—
18C2×D3×D20—21no—
19C2×D5×D12C2212yes—
20C2×#6773C2212yes—
21C2×#6769—21no—
22C2×#6717—24no—
23C2×#6717C2214yescomplemented
24C22×C3⋊#168C2211yescharacteristic, complemented
25C2×C5×#216C2212yescomplemented
26C5×#942—24no—
27C2×C5×#215C2212yescomplemented
28C2×C5×#208C2211yescharacteristic, complemented
29C3×#4093—21no—
30C3×#4092C2214yescomplemented
31C3×#4090—21no—
32C3×#4075—22no—
33C3×#4072C2211yescharacteristic, complemented
34#90972C2212yescomplemented
35#90968C2214yescomplemented
36#90966C2212yescomplemented
37#90955C2214yescomplemented
38#83404—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×#137—31no—
41#66150D311yescharacteristic, complemented
42C22×#688—31no—
43#65522—31no—
44#65476—34no—
45#65458—34no—
46#65420—31no—
47#65418—31no—
48C23×D15—21no—
49C22×D3×D5—22no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8C2312yes—
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×D60—21no—
57C2×D60—22no—
58C2×D60D414yes—
59C2×D60C2312yes—
60C2×C5×#67—21no—
61C5×#235—22no—
62C5×#235D414yes—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3—21no—
65C2×C5×Dic6C2311yescharacteristic
66C2×C3×C22⋊D5D412yes—
67C3×#709—22no—
68C3×#709—22no—
69C2×C3×D20D411yescharacteristic
70C2×C3×C4×D5C2312yes—
71C2×C3×Dic10D411yescharacteristic
72#6790—216no—
73C2×#1203—22no—
74C2×#1202—22no—
75C2×#1200—22no—
76D3×D20—28no—
77D5×D12—24no—
78D5×D12—24no—
79#6773—24no—
80#6773—24no—
81#6769—28no—
82#6717—28no—
83C2×C3⋊#168—22no—
84C2×C3⋊#168C2312yes—
85C5×#216—24no—
86C5×#215—24no—
87C5×#208—22no—
88C2×C5×C3⋊C8C2312yes—
89C3×C22⋊Dic5—21no—
90C3×#685—24no—
91C3×C4⋊Dic5C2312yes—
92C3×#683—24no—
93C3×C4×Dic5—21no—
94C2×#1009—51no—
95C3×#536—51no—
96C2×#942D511yescharacteristic, complemented
97C22×#215—51no—
98#5234—51no—
99#5047—54no—
100#5046—54no—
101#5002—51no—
102#5001—51no—
103C2×C5×#41—31no—
104C22×C5×Q8C2×D311yescharacteristic, complemented
105#4149—64no—
106#4149—64no—
107C2×#713—32no—
108C2×D4×D5—32no—
109C2×D4×D5—62no—
110C2×#709—61no—
111C2×#709C2×D311yescharacteristic
112C22×D20—31no—
113C22×D20—61no—
114C5×#137—64no—
115C2×C5×Q16—32no—
116C2×C5×SD16—32no—
117C2×C5×M16—31no—
118#4093—21no—
119#4092C2×D314yescomplemented
120#4090—21no—
121C2×#688—34no—
122C2×#688—64no—
123#4075—22no—
124#4072C2×D311yescharacteristic, complemented
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953—34no—
129#3952—34no—
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—22no—
133C22×D15—24no—
134C22×D15—82no—
135C22×C5×D3—21no—
136C22×C3×D5C2×D412yes—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C3×C5×Q8—22no—
140C3×C5×Q8—22no—
141C3×C5×Q8—42no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—42no—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2411yescharacteristic
148D60—24no—
149D60—22no—
150D60C2×D414yes—
151C5×#67—24no—
152C2×C5×Dic3—21no—
153C5×D12—21no—
154C5×D12C2×D412yes—
155C4×C5×D3—24no—
156C5×Dic6—21no—
157C5×Dic6C2×D412yes—
158C3×C22⋊D5—24no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5C2×D412yes—
161C3×D20—22no—
162C3×C4×D5—22no—
163C3×C4×D5—22no—
164C3×Dic10—22no—
165#1203—28no—
166#1202—28no—
167#1200—28no—
168C3⋊#168—22no—
169C5×C3⋊C8—22no—
170C2×C3×#41—51no—
171C22×C3×Q8C2×D511yescharacteristic, complemented
172#1009—104no—
173#1009—104no—
174C2×#239—101no—
175C2×D3×D4—101no—
176C2×D3×D4—102no—
177C2×#235—52no—
178C2×#235C2×D511yescharacteristic
179C22×D12—51no—
180C22×D12—52no—
181C3×C4⋊Q8—101no—
182C3×#124—101no—
183C3×#123—102no—
184C3×C22⋊Q8—54no—
185C3×#117—51no—
186C2×#216C2×D512yescomplemented
187#942—24no—
188C2×#215—54no—
189C2×#215—104no—
190C2×#215C2×D512yescomplemented
191C2×#208C2×D511yescharacteristic, complemented
192C22×C3⋊C8—51no—
193#830—54no—
194#810—52no—
195#809—52no—
196#809—54no—
197C23×D5—61no—
198C23×D5—62no—
199C5×#41—34no—
200C5×#41—62no—
201C2×C5×Q8—21no—
202C2×C5×Q8—22no—
203C2×C5×Q8—22no—
204C2×C5×Q8—31no—
205C2×C5×Q8C22×D312yes—
206C2×C5×D4—31no—
207C2×C5×D4—61no—
208C22×C4×C5—21no—
209C22×C4×C5—61no—
210C22×C4×C5C22×D311yescharacteristic
211C2×C22⋊D5—62no—
212C2×C22⋊D5#6712yes—
213#713—64no—
214#713—64no—
215D4×D5—64no—
216D4×D5—64no—
217D4×D5—616no—
218#709—22no—
219#709—22no—
220#709—68no—
221C2×D20—34no—
222C2×D20—32no—
223C2×D20—68no—
224C2×D20—62no—
225C2×D20—62no—
226C2×D20—61no—
227C2×D20#6711yescharacteristic
228C2×C4×D5—62no—
229C2×C4×D5C22×D312yes—
230C2×Dic10#6711yescharacteristic
231C5×Q16—64no—
232C5×SD16—64no—
233C5×M16—62no—
234C2×C5×C8—32no—
235C22⋊Dic5—21no—
236#688—68no—
237#685—24no—
238C4⋊Dic5C22×D312yes—
239#683—24no—
240C4×Dic5—21no—
241C2×#168—32no—
242C2×#168—62no—
243C2×H(5,𝔽2)—151no—
244C2×#137—31no—
245C22×SD16—151no—
246#536—51no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—24no—
256C2×D15—44no—
257C2×D15—84no—
258C2×C5×D3—22no—
259C2×C5×D3—42no—
260C2×C3×D5—22no—
261C2×C3×D5—24no—
262D3×D5—88no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267C3×Dic5—22no—
268C3×Dic5—42no—
269C5×Dic3—22no—
270C23×D3—102no—
271C23×D3—101no—
272C3×#41—102no—
273C3×#41—102no—
274C2×C3×Q8—21no—
275C2×C3×Q8—22no—
276C2×C3×Q8—22no—
277C2×C3×Q8—51no—
278C2×C3×Q8C22×D512yes—
279C2×C3×D4—52no—
280C2×C3×D4—51no—
281C22×C3×C4—21no—
282C22×C3×C4—52no—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—21no—
285C2×#67—102no—
286#239—108no—
287D3×D4—108no—
288D3×D4—1016no—
289#235—22no—
290#235—104no—
291#235—104no—
292#235C22⋊D514yes—
293C2×D12—54no—
294C2×D12—52no—
295C2×D12—104no—
296C2×D12—102no—
297C2×D12—101no—
298C2×D12—102no—
299C2×D12—104no—
300C2×D12C22×D511yescharacteristic
301C2×C4×D3—21no—
302C2×C4×D3—102no—
303C2×Dic6C22×D511yescharacteristic
304C3×C4⋊C4—52no—
305C3×C4⋊C4—104no—
306C3×C22⋊C4—104no—
307C3×C22⋊C4—101no—
308C3×C42—101no—
309#216—24no—
310#215—24no—
311#215—108no—
312#208—22no—
313C2×C3⋊C8—52no—
314C2×C3⋊C8—102no—
315C2×C3⋊C8C22×D512yes—
316C23×C5—61no—
317C23×C5C2×#6711yescharacteristic
318C22×D5—68no—
319C22×D5—62no—
320C22×D5—64no—
321C22×D5—244no—
322C22×D5—242no—
323C22×D5C2×#6712yes—
324C5×Q8—22no—
325C5×Q8—22no—
326C5×Q8—32no—
327C5×Q8—42no—
328C5×Q8—61no—
329C5×D4—32no—
330C5×D4—64no—
331C5×D4—61no—
332C2×C4×C5—22no—
333C2×C4×C5—42no—
334C2×C4×C5—61no—
335C2×C4×C5—64no—
336C2×C4×C5D3×D412yes—
337C2×C4×C5D3×D412yes—
338C2×C4×C5C2×#6711yescharacteristic
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—24no—
341C22⋊D5—68no—
342C2×Dic5—22no—
343C2×Dic5C2×#6712yes—
344D20—22no—
345D20—34no—
346D20—68no—
347D20—64no—
348D20—62no—
349C4×D5—22no—
350C4×D5—22no—
351C4×D5—68no—
352Dic10—22no—
353C5×C8—62no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—31no—
358C2×#41—51no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8D3×D511yescharacteristic, complemented
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—151no—
373C4⋊Q8—101no—
374#124—101no—
375#123—102no—
376C22⋊Q8—54no—
377#117—51no—
378#103—152no—
379#102—154no—
380#102—152no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—44no—
386C3×D5—42no—
387C5×D3—42no—
388C23×C3—52no—
389C23×C3D4×D511yescharacteristic
390C22×D3—21no—
391C22×D3—108no—
392C22×D3—102no—
393C22×D3—104no—
394C22×D3—404no—
395C22×D3—402no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—42no—
399C3×Q8—102no—
400C3×D4—104no—
401C3×D4—102no—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4—52no—
405C2×C3×C4—102no—
406C2×C3×C4—102no—
407C2×C3×C4—102no—
408C2×C3×C4D4×D511yescharacteristic
409C2×C3×C4D4×D512yes—
410C2×C3×C4C2×C22⋊D512yes—
411C2×C3×C4C23×D511yescharacteristic
412#67—24no—
413#67—108no—
414C2×Dic3—21no—
415D12—21no—
416D12—54no—
417D12—108no—
418D12—102no—
419D12—104no—
420D12C2×C22⋊D512yes—
421C4×D3—24no—
422C4×D3—108no—
423Dic6—21no—
424Dic6C2×C22⋊D512yes—
425C3⋊C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—62no—
429C22×C5—122no—
430C22×C5#93712yes—
431C22×C5C22×#6711yescharacteristic
432C2×D5—22no—
433C2×D5—24no—
434C2×D5—64no—
435C2×D5—124no—
436C2×D5—248no—
437C2×D5—244no—
438C4×C5—42no—
439C4×C5—62no—
440C4×C5#93712yes—
441C4×C5C2×D3×D411yescharacteristic
442C4×C5C2×D3×D411yescharacteristic
443Dic5—22no—
444Dic5—42no—
445C24—302no—
446C24—301no—
447#41—34no—
448#41—62no—
449#41—102no—
450#41—102no—
451#41—308no—
452#41—304no—
453#41—304no—
454C2×Q8—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—31no—
458C2×Q8—51no—
459C2×Q8C2×D3×D512yes—
460C2×D4—31no—
461C2×D4—52no—
462C2×D4—51no—
463C2×D4—61no—
464C2×D4—154no—
465C2×D4—152no—
466C2×D4—302no—
467C2×D4—308no—
468C2×D4—302no—
469C2×D4—304no—
470C2×D4—304no—
471C2×D4—3016no—
472C2×D4—301no—
473C2×D4—302no—
474C22×C4—21no—
475C22×C4—52no—
476C22×C4—61no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—64no—
480SD16—64no—
481SD16—308no—
482M16—62no—
483C2×C8—32no—
484C2×C8—152no—
485C2×C8—302no—
486C4⋊C4—52no—
487C4⋊C4—104no—
488C22⋊C4—104no—
489C22⋊C4—101no—
490C42—101no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—104no—
494C22×C3—102no—
495C22×C3#408312yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—22no—
498C2×D3—42no—
499C2×D3—104no—
500C2×D3—204no—
501C2×D3—408no—
502C2×D3—404no—
503C3×C4—42no—
504C3×C4—102no—
505C3×C4—202no—
506C3×C4#408312yes—
507C3×C4C2×D4×D511yescharacteristic
508C3×C4C22×C22⋊D511yescharacteristic
509Dic3—22no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5C2×#93711yescharacteristic
513C2×C5#583712yes—
514D5—42no—
515D5—124no—
516C23—52no—
517C23—61no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23D5×#6711yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—32no—
527Q8—42no—
528Q8—61no—
529Q8—102no—
530D4—32no—
531D4—64no—
532D4—61no—
533D4—104no—
534D4—102no—
535D4—154no—
536D4—308no—
537D4—308no—
538D4—302no—
539D4—304no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—52no—
543C2×C4—61no—
544C2×C4—64no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—102no—
548C2×C4—308no—
549C2×C4D5×#6711yescharacteristic
550C2×C4D3×C22⋊D512yes—
551C2×C4#679012yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—302no—
555C2×C3—22no—
556C2×C3—202no—
557C2×C3#6611212yes—
558C2×C3C2×#408311yescharacteristic
559D3—42no—
560D3—204no—
561C5#8340411yescharacteristic, complemented, Sylow
562C22—22no—
563C22—62no—
564C22—104no—
565C22—102no—
566C22—122no—
567C22—304no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9157712yes—
572C22C2×D5×#6711yescharacteristic
573C4—42no—
574C4—62no—
575C4—102no—
576C4—202no—
577C4#9157712yes—
578C4C2×D3×C22⋊D511yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—202no—
584C2—604no—
585C2#1175508612yes—
586C2C2×#9157711yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92019C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C22×#6717C2semidirect
3. C2×C5×#942C2semidirect
4. C3×#66150C2semidirect
5. #11749406C2semidirect
6. #11749397C2semidirect
7. #11748392C2semidirect
8. #11748382C2semidirect
9. #11748343C2semidirect
11. C22×C3×C5×Q8C22semidirect
23. C2×#6717C22semidirect
24. C22×C3⋊#168C22semidirect
25. C2×C5×#216C22semidirect
27. C2×C5×#215C22semidirect
28. C2×C5×#208C22semidirect
30. C3×#4092C22semidirect
33. C3×#4072C22semidirect
34. #90972C22semidirect
35. #90968C22semidirect
36. #90966C22semidirect
37. #90955C22semidirect
41. #66150D3semidirect
96. C2×#942D5semidirect
104. C22×C5×Q8C2×D3semidirect
119. #4092C2×D3semidirect
124. #4072C2×D3semidirect
171. C22×C3×Q8C2×D5semidirect
186. C2×#216C2×D5semidirect
190. C2×#215C2×D5semidirect
191. C2×#208C2×D5semidirect
361. C22×Q8D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#83404semidirect
580. C3#10615713semidirect

Maximal subgroups

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

subgroupindex
C2×#920192
C22×#67172
C2×C5×#9422
C3×#661502
#117494062
#117493972
#117483922
#117483822
#117483432
#106157133
#834045