Finite Group Databasebeta

← #14424400

Subgroups — #14424400

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×#92017C211yesmaximal, characteristic
2C2×C5×#909C211yesmaximal, characteristic, complemented
3#11754270C211yesmaximal, characteristic, complemented
4C22×#6655C211yesmaximal, characteristic, complemented
5#11751230C211yesmaximal, characteristic, complemented
6C3×#65420C211yesmaximal, characteristic, complemented
7#11748382C211yesmaximal, characteristic, complemented
8#11748154C214yesmaximal, complemented
9#11748118C214yesmaximal, complemented
10#10615713—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C22×C5×Dic6C2211yescharacteristic, complemented
14C22×C3×D20C2211yescharacteristic
15C2×D5×#67—22no—
16#92017—24no—
17#92017—24no—
18C2×#6777C2212yes—
19C2×D3×D20—21no—
20C2×#6766C2212yes—
21C2×#6765—21no—
22C2×C3×C5×M16C2211yescharacteristic, complemented
23C22×C3⋊#168C2211yescharacteristic, complemented
24C5×#909—24no—
25C2×C5×Dic12C2212yescomplemented
26C2×C5×#204C2212yescomplemented
27C3×#4072C2211yescharacteristic, complemented
28#91542—22no—
29#91383C2214yescomplemented
30#91376—21no—
31#91375—21no—
32C2×#6655—24no—
33C2×#6655C2214yescomplemented
34#91104C2214yescomplemented
35C3×#3953C2214yescomplemented
36#90972C2212yescomplemented
37#90966C2212yescomplemented
38#78929—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×#137—31no—
41#66150—31no—
42C22×#688—31no—
43#65522—31no—
44#65476—34no—
45#65458—34no—
46#65420D311yescharacteristic, complemented
47#65418—31no—
48C23×C3×D5—21no—
49C22×D3×D5—22no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×#1220D412yes—
52#6817—22no—
53#6817—22no—
54C2×D60D411yescharacteristic
55C2×C4×D15C2312yes—
56C2×Dic30D411yescharacteristic
57C2×C5×#67—21no—
58C22×C5×Dic3—21no—
59C5×#235—22no—
60C5×#235D414yes—
61C2×C5×D12C2311yescharacteristic
62C2×C4×C5×D3—21no—
63C2×C5×Dic6—22no—
64C2×C5×Dic6—22no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6C2312yes—
67C2×C5×Dic6C2311yescharacteristic
68C2×C3×D20—21no—
69C2×C3×D20—22no—
70C2×C3×D20D414yes—
71C2×C3×D20C2312yes—
72D5×#67—216no—
73C2×#1202—22no—
74C2×#1201—22no—
75C2×D5×Dic3—22no—
76#6777—24no—
77#6777—24no—
78D3×D20—28no—
79#6766—24no—
80#6766—24no—
81#6765—28no—
82C3×C5×M16—22no—
83C2×C3×C5×C8C2312yes—
84C2×C3⋊#168—22no—
85C2×C3⋊#168C2312yes—
86C5×Dic12—24no—
87C5×#204—24no—
88C3×C22⋊Dic5—21no—
89C3×C4⋊Dic5C2312yes—
90C3×C4×Dic5—21no—
91#6668—24no—
92#6667—24no—
93#6655—28no—
94C2×#1004—51no—
95C22×#214—51no—
96C2×#909D511yescharacteristic, complemented
97#5526—51no—
98C3×#418—51no—
99#5129—51no—
100#5002—51no—
101#4783—54no—
102#4776—54no—
103C2×C5×#41—31no—
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—61no—
113C22×D20C2×D311yescharacteristic
114C5×#137—64no—
115C2×C5×Q16—32no—
116C2×C5×SD16—32no—
117C2×C5×M16C2×D311yescharacteristic, complemented
118#4093—61no—
119#4092—34no—
120#4090—61no—
121C2×#688—34no—
122C2×#688—64no—
123#4075—62no—
124#4072C2×D311yescharacteristic, complemented
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953C2×D314yescomplemented
129#3952—34no—
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15C2×D412yes—
133C22×C5×D3—21no—
134C22×C3×D5—22no—
135C22×C3×D5—24no—
136C22×C3×D5—82no—
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×Dic15C2×D412yes—
145D60—22no—
146C4×D15—22no—
147C4×D15—22no—
148Dic30—22no—
149C5×#67—24no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—42no—
153C5×D12—21no—
154C5×D12C2×D412yes—
155C4×C5×D3—24no—
156C5×Dic6—21no—
157C5×Dic6—22no—
158C5×Dic6—22no—
159C5×Dic6—42no—
160C5×Dic6C2×D412yes—
161C2×C3×Dic5—22no—
162C3×D20—24no—
163C3×D20—22no—
164C3×D20C2×D414yes—
165#1202—28no—
166#1201—28no—
167D5×Dic3—28no—
168C3×C5×C8—22no—
169C3⋊#168—22no—
170C22×C3×D4—51no—
171C22×#67—102no—
172#1004—104no—
173#1004—104no—
174C2×#237—52no—
175C2×#237—101no—
176C2×D3×D4—52no—
177C2×D3×D4—101no—
178C2×#235—51no—
179C2×#235C2×D511yescharacteristic
180C22×Dic6C2×D511yescharacteristic, complemented
181C2×C3×M16C2×D511yescharacteristic, complemented
182C3×#117—51no—
183C2×#214—54no—
184C2×#214—104no—
185C22×C3⋊C8—51no—
186#909—24no—
187C2×Dic12C2×D512yescomplemented
188C2×#204C2×D512yescomplemented
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#102—54no—
194(C2×D3)⋊C8—54no—
195#810—52no—
196#809—52no—
197C23×D5—21no—
198C23×D5—62no—
199C5×#41—34no—
200C5×#41—62no—
201C2×C5×Q8—31no—
202C2×C5×Q8—32no—
203C2×C5×Q8—61no—
204C2×C5×Q8—62no—
205C2×C5×Q8—62no—
206C2×C5×D4—31no—
207C2×C5×D4—61no—
208C22×C4×C5—61no—
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—21no—
222C2×D20—22no—
223C2×D20—31no—
224C2×D20—68no—
225C2×D20—62no—
226C2×D20D1214yes—
227C2×D20C22×D312yes—
228C2×C4×D5—32no—
229C2×C4×D5—62no—
230C2×Dic10—31no—
231C5×Q16—64no—
232C5×SD16—64no—
233C5×M16—22no—
234C2×C5×C8C22×D312yes—
235C22⋊Dic5—21no—
236#688—68no—
237#685—64no—
238C4⋊Dic5C22×D312yes—
239#683—64no—
240C4×Dic5—21no—
241C2×#168—32no—
242C2×#168—62no—
243C2×H(5,𝔽2)—151no—
244C2×#137—31no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—51no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C2×C5×D3—22no—
258C2×C5×D3—42no—
259C2×C3×D5—24no—
260C2×C3×D5—44no—
261C2×C3×D5—84no—
262D3×D5—88no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—22no—
267C3×Dic5—42no—
268C5×Dic3—22no—
269C5×Dic3—42no—
270C24×C3—101no—
271C23×D3—102no—
272C2×C3×D4—54no—
273C2×C3×D4—52no—
274C2×C3×D4—101no—
275C2×C3×D4—102no—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—21no—
278C2×#67—52no—
279C2×#67—102no—
280C2×#67—102no—
281C2×#67—1016no—
282C22×Dic3—21no—
283C22×Dic3—102no—
284#237—104no—
285#237—108no—
286#237—104no—
287D3×D4—108no—
288D3×D4—104no—
289D3×D4—104no—
290#235—22no—
291#235—102no—
292#235—102no—
293#235C22⋊D514yes—
294C2×D12—51no—
295C2×D12C22×D511yescharacteristic
296C2×C4×D3—21no—
297C2×C4×D3—52no—
298C2×Dic6—22no—
299C2×Dic6—22no—
300C2×Dic6—21no—
301C2×Dic6—51no—
302C2×Dic6C22×D512yes—
303C2×Dic6C22×D511yescharacteristic
304C3×M16—22no—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4—52no—
307C3×C22⋊C4—101no—
308C3×C42—101no—
309#214—108no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—104no—
312C2×C3⋊C8—52no—
313C2×C3⋊C8—102no—
314Dic12—24no—
315#204—24no—
316C23×C5—61no—
317C23×C5D3×D411yescharacteristic
318C22×D5—22no—
319C22×D5—24no—
320C22×D5—32no—
321C22×D5—68no—
322C22×D5—82no—
323C22×D5—244no—
324C5×Q8—32no—
325C5×Q8—61no—
326C5×Q8—62no—
327C5×Q8—62no—
328C5×Q8—122no—
329C5×D4—32no—
330C5×D4—64no—
331C5×D4—61no—
332C2×C4×C5—61no—
333C2×C4×C5—64no—
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—24no—
345D20—22no—
346D20—68no—
347D20—62no—
348D20C2×D1214yes—
349C4×D5—68no—
350C4×D5—62no—
351C4×D5—62no—
352Dic10—62no—
353C5×C8—22no—
354#168—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—31no—
358C2×#41—151no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8—31no—
362C22×D4—51no—
363C22×D4—152no—
364C22×D4—301no—
365C22×D4—302no—
366#137—64no—
367C2×Q16—32no—
368C2×SD16—32no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16D3×D511yescharacteristic, complemented
372C22×C8—151no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—51no—
378#103—152no—
379#102—54no—
380#102—152no—
381C22⋊C8—154no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—42no—
386C3×D5—44no—
387C5×D3—42no—
388C23×C3—102no—
389C23×C3—104no—
390C23×C3—402no—
391C23×C3D4×D511yescharacteristic
392C22×D3—21no—
393C22×D3—52no—
394C22×D3—108no—
395C22×D3—404no—
396C3×D4—54no—
397C3×D4—102no—
398C3×D4—104no—
399C2×C3×C4—102no—
400C2×C3×C4D4×D512yes—
401C2×C3×C4D4×D511yescharacteristic
402C2×C3×C4C2×C22⋊D512yes—
403C2×C3×C4C23×D511yescharacteristic
404#67—24no—
405#67—108no—
406#67—108no—
407#67—104no—
408C2×Dic3—21no—
409C2×Dic3—22no—
410C2×Dic3—42no—
411C2×Dic3—52no—
412C2×Dic3—108no—
413D12—21no—
414D12—102no—
415D12C2×C22⋊D512yes—
416C4×D3—24no—
417C4×D3—102no—
418C4×D3—102no—
419Dic6—21no—
420Dic6—22no—
421Dic6—22no—
422Dic6—42no—
423Dic6—102no—
424Dic6C2×C22⋊D512yes—
425C3×C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—62no—
429C22×C5—122no—
430C22×C5#88212yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—24no—
433C2×D5—44no—
434C2×D5—62no—
435C2×D5—64no—
436C2×D5—84no—
437C2×D5—248no—
438C4×C5—62no—
439C4×C5—122no—
440C4×C5#88212yes—
441C4×C5C22×D1211yescharacteristic
442C4×C5C2×D3×D411yescharacteristic
443Dic5—42no—
444Dic5—62no—
445C24—101no—
446C24—302no—
447#41—34no—
448#41—62no—
449#41—308no—
450#41—304no—
451#41—304no—
452#41—302no—
453#41—302no—
454C2×Q8—31no—
455C2×Q8—32no—
456C2×Q8—61no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8—151no—
460C2×D4—31no—
461C2×D4—54no—
462C2×D4—52no—
463C2×D4—61no—
464C2×D4—101no—
465C2×D4—102no—
466C2×D4—152no—
467C2×D4—151no—
468C2×D4—308no—
469C2×D4—302no—
470C2×D4—302no—
471C2×D4—304no—
472C2×D4—304no—
473C2×D4—3016no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—64no—
480SD16—64no—
481SD16—308no—
482M16—22no—
483C2×C8—152no—
484C2×C8—302no—
485C2×C8C2×D3×D512yes—
486C4⋊C4—52no—
487C4⋊C4—304no—
488C22⋊C4—101no—
489C22⋊C4—304no—
490C42—101no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—104no—
494C22×C3—204no—
495C22×C3—404no—
496C22×C3#408312yes—
497C22×C3C2×D4×D511yescharacteristic
498C2×D3—22no—
499C2×D3—42no—
500C2×D3—104no—
501C2×D3—102no—
502C2×D3—408no—
503C3×C4—202no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C22×C22⋊D511yescharacteristic
507Dic3—22no—
508Dic3—42no—
509Dic3—102no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#577512yes—
514D5—44no—
515D5—122no—
516C23—61no—
517C23—102no—
518C23—104no—
519C23—152no—
520C23—308no—
521C23—402no—
522C23—1204no—
523C23#677711yescharacteristic
524Q8—32no—
525Q8—61no—
526Q8—62no—
527Q8—62no—
528Q8—122no—
529Q8—302no—
530D4—32no—
531D4—54no—
532D4—64no—
533D4—61no—
534D4—104no—
535D4—102no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—61no—
541C2×C4—64no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—122no—
545C2×C4—152no—
546C2×C4—308no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4D5×D1212yes—
550C2×C4#677711yescharacteristic
551C2×C4D3×C22⋊D512yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—22no—
554C8—302no—
555C2×C3—22no—
556C2×C3—204no—
557C2×C3#6611212yes—
558C2×C3C2×#408311yescharacteristic
559D3—42no—
560D3—202no—
561C5#7892911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—62no—
564C22—104no—
565C22—122no—
566C22—204no—
567C22—304no—
568C22—302no—
569C22—404no—
570C22—1208no—
571C22#9147512yes—
572C22C2×#677711yescharacteristic
573C4—62no—
574C4—122no—
575C4—202no—
576C4—302no—
577C4#9147512yes—
578C4C2×D5×D1211yescharacteristic
579C4C2×D3×C22⋊D511yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—204no—
584C2—602no—
585C2#1175403712yes—
586C2C2×#9147511yescharacteristic
subgroupG/H
centerC22C2×#6777
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#92017C2
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×C5×#909C2semidirect
3. #11754270C2semidirect
4. C22×#6655C2semidirect
5. #11751230C2semidirect
6. C3×#65420C2semidirect
7. #11748382C2semidirect
8. #11748154C2semidirect
9. #11748118C2semidirect
13. C22×C5×Dic6C22semidirect
22. C2×C3×C5×M16C22semidirect
23. C22×C3⋊#168C22semidirect
25. C2×C5×Dic12C22semidirect
26. C2×C5×#204C22semidirect
27. C3×#4072C22semidirect
29. #91383C22semidirect
33. C2×#6655C22semidirect
34. #91104C22semidirect
35. C3×#3953C22semidirect
36. #90972C22semidirect
37. #90966C22semidirect
46. #65420D3semidirect
96. C2×#909D5semidirect
117. C2×C5×M16C2×D3semidirect
124. #4072C2×D3semidirect
128. #3953C2×D3semidirect
180. C22×Dic6C2×D5semidirect
181. C2×C3×M16C2×D5semidirect
187. C2×Dic12C2×D5semidirect
188. C2×#204C2×D5semidirect
371. C2×M16D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#78929semidirect
580. C3#10615713semidirect

Maximal subgroups

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

subgroupindex
C2×#920172
C2×C5×#9092
#117542702
C22×#66552
#117512302
C3×#654202
#117483822
#117481542
#117481182
#106157133
#789295