Finite Group Databasebeta

← #14424364

Subgroups — #14424364

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
2#11756575C211yesmaximal, characteristic, complemented
3C2×C5×#942C211yesmaximal, characteristic, complemented
4C22×C3×#688C211yesmaximal, characteristic, complemented
5#11749404C214yesmaximal, complemented
6#11749399C214yesmaximal, complemented
7#11748388C211yesmaximal, characteristic, complemented
8#11748367C211yesmaximal, characteristic, complemented
9#11748349C211yesmaximal, characteristic, complemented
10#10615713—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic, complemented
12C2×#6817C2211yescharacteristic
13C2×C5×#235C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C2×D5×#67—22no—
16#92017—24no—
17#92017—24no—
18C2×#6777C2212yes—
19C2×D3×D20—21no—
20C2×#6766C2212yes—
21C2×#6765—21no—
22#91836—21no—
23#91835C2214yescomplemented
24#91833—21no—
25#91818—22no—
26#91815C2211yescharacteristic, complemented
27C2×C5×#216C2212yescomplemented
28C5×#942—24no—
29C2×C5×#215C2212yescomplemented
30C2×C5×#208C2211yescharacteristic, complemented
31C2×C3×#688—24no—
32C2×C3×#688C2214yescomplemented
33C22×C3×#168C2211yescharacteristic, complemented
34#90973C2212yescomplemented
35#90967C2212yescomplemented
36#90965C2214yescomplemented
37#90957C2214yescomplemented
38#83859—51nomaximal, Hall
39C2×#4149—31no—
40C2×C5×#137—31no—
41#66150—31no—
42C22×#688D311yescharacteristic, complemented
43#65522—31no—
44#65476—34no—
45#65458—34no—
46#65420—31no—
47#65418—31no—
48C23×C3×D5—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×#1220D412yes—
57#6817—22no—
58#6817—22no—
59C2×D60D411yescharacteristic
60C2×C4×D15C2312yes—
61C2×Dic30D411yescharacteristic
62C2×C5×#67—21no—
63C5×#235—22no—
64C5×#235D414yes—
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
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—
82#6719—21no—
83#6714—24no—
84C4⋊Dic15C2312yes—
85#6712—24no—
86C4×Dic15—21no—
87C5×#216—24no—
88C5×#215—24no—
89C5×#208—22no—
90C2×C5×C3⋊C8C2312yes—
91C3×#688—28no—
92C2×C3×#168—22no—
93C2×C3×#168C2312yes—
94C2×#1004—51no—
95C22×C3×SD16—51no—
96#5816—51no—
97C2×#942D511yescharacteristic, complemented
98#5215—51no—
99#5173—54no—
100#5169—54no—
101#5113—51no—
102#5110—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—31no—
111C2×#709—61no—
112C22×D20—61no—
113C22×D20C2×D311yescharacteristic
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—24no—
122C2×#688C2×D314yescomplemented
123#4075—62no—
124#4072—31no—
125C22×#168C2×D311yescharacteristic, complemented
126#3967—32no—
127#3964—32no—
128#3953—34no—
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—
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
148#1220—24no—
149C2×Dic15—22no—
150C2×Dic15C2×D412yes—
151D60—22no—
152C4×D15—22no—
153C4×D15—22no—
154Dic30—22no—
155C5×#67—24no—
156C2×C5×Dic3—21no—
157C5×D12—21no—
158C5×D12C2×D412yes—
159C4×C5×D3—24no—
160C5×Dic6—21no—
161C5×Dic6C2×D412yes—
162C3×D20—24no—
163C3×D20—22no—
164C3×D20C2×D414yes—
165#1202—28no—
166#1201—28no—
167D5×Dic3—28no—
168C3×#168—22no—
169C5×C3⋊C8—22no—
170C22×C3×Q8C2×D511yescharacteristic, complemented
171C22×C3×D4—51no—
172C22×#67—102no—
173#1004—104no—
174#1004—104no—
175C2×#237—52no—
176C2×#237—101no—
177C2×D3×D4—52no—
178C2×D3×D4—101no—
179C2×#235—51no—
180C2×#235C2×D511yescharacteristic
181C2×C3×SD16—54no—
182C2×C3×SD16—104no—
183C22×C3×C8—51no—
184#947—101no—
185#946—54no—
186#944—101no—
187C2×#216C2×D512yescomplemented
188#942—24no—
189C2×#215C2×D512yescomplemented
190#929—102no—
191#926—51no—
192C2×#208C2×D511yescharacteristic, complemented
193#832—54no—
194#821—52no—
195(C2×D3)⋊C8—54no—
196#816—52no—
197C23×D5—21no—
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—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×D20#6714yes—
227C2×D20C22×D312yes—
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—61no—
236#688—28no—
237#685—64no—
238C4⋊Dic5—32no—
239#683—64no—
240C4×Dic5—61no—
241C2×#168—22no—
242C2×#168C22×D312yes—
243C2×H(5,𝔽2)—151no—
244C2×#137—31no—
245C22×SD16—51no—
246#536—151no—
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—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—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—22no—
268Dic15—42no—
269C5×Dic3—22no—
270C24×C3—101no—
271C23×D3—102no—
272C2×C3×Q8—21no—
273C2×C3×Q8—22no—
274C2×C3×Q8—22no—
275C2×C3×Q8C22×D512yes—
276C2×C3×D4—54no—
277C2×C3×D4—52no—
278C2×C3×D4—101no—
279C2×C3×D4—102no—
280C22×C3×C4—21no—
281C22×C3×C4C22×D511yescharacteristic
282C2×#67—21no—
283C2×#67—52no—
284C2×#67—102no—
285C2×#67—102no—
286C2×#67—1016no—
287C22×Dic3—102no—
288#237—104no—
289#237—108no—
290#237—104no—
291D3×D4—108no—
292D3×D4—104no—
293D3×D4—104no—
294#235—22no—
295#235—102no—
296#235—102no—
297#235C22⋊D514yes—
298C2×D12—51no—
299C2×D12C22×D511yescharacteristic
300C2×C4×D3—21no—
301C2×C4×D3—52no—
302C2×Dic6—51no—
303C2×Dic6C22×D511yescharacteristic
304C3×SD16—108no—
305C2×C3×C8—52no—
306C2×C3×C8—102no—
307#217—101no—
308#216—24no—
309#215—24no—
310(C2×D3)⋊C4—104no—
311C4⋊Dic3—52no—
312Dic3⋊C4—104no—
313C4×Dic3—101no—
314#208—22no—
315C2×C3⋊C8C22×D512yes—
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—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×D411yescharacteristic
338C2×C4×C5C2×#6712yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—68no—
341C22⋊D5—64no—
342C2×Dic5—32no—
343C2×Dic5—62no—
344D20—24no—
345D20—22no—
346D20—68no—
347D20—62no—
348D20C2×#6714yes—
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—31no—
358C2×#41—151no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8D3×D511yescharacteristic, complemented
362C22×D4—51no—
363C22×D4—152no—
364C22×D4—301no—
365C22×D4—302no—
366#137—64no—
367C2×Q16—32no—
368C2×SD16—32no—
369C2×SD16—54no—
370C2×SD16—104no—
371C2×M16—31no—
372C22×C8—51no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—151no—
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—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×Q8—22no—
397C3×Q8—22no—
398C3×Q8—42no—
399C3×D4—54no—
400C3×D4—102no—
401C3×D4—104no—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4D4×D512yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4C2×C22⋊D512yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—24no—
409#67—108no—
410#67—108no—
411#67—104no—
412C2×Dic3—21no—
413C2×Dic3—52no—
414C2×Dic3—108no—
415C2×Dic3—102no—
416D12—21no—
417D12—102no—
418D12C2×C22⋊D512yes—
419C4×D3—24no—
420C4×D3—102no—
421C4×D3—102no—
422Dic6—21no—
423Dic6—102no—
424Dic6C2×C22⋊D512yes—
425C3×C8—102no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—62no—
429C22×C5—122no—
430C22×C5#93712yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—24no—
433C2×D5—44no—
434C2×D5—64no—
435C2×D5—62no—
436C2×D5—84no—
437C2×D5—248no—
438C4×C5—42no—
439C4×C5—62no—
440C4×C5#93712yes—
441C4×C5C2×D3×D411yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—62no—
444Dic5—122no—
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—21no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—31no—
458C2×Q8—151no—
459C2×Q8C2×D3×D512yes—
460C2×D4—31no—
461C2×D4—54no—
462C2×D4—52no—
463C2×D4—61no—
464C2×D4—102no—
465C2×D4—101no—
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—21no—
475C22×C4—61no—
476C22×C4—152no—
477C22×C4—302no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—64no—
480SD16—64no—
481SD16—108no—
482M16—62no—
483C2×C8—32no—
484C2×C8—52no—
485C2×C8—102no—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—304no—
489C22⋊C4—301no—
490C42—301no—
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—42no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C22×C22⋊D511yescharacteristic
507Dic3—22no—
508Dic3—102no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5#576212yes—
513C2×C5C2×#93711yescharacteristic
514D5—44no—
515D5—122no—
516C23—61no—
517C23—102no—
518C23—104no—
519C23—152no—
520C23—308no—
521C23—402no—
522C23—1204no—
523C23#679011yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—32no—
527Q8—42no—
528Q8—61no—
529Q8—302no—
530D4—32no—
531D4—54no—
532D4—64no—
533D4—61no—
534D4—102no—
535D4—104no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—61no—
543C2×C4—64no—
544C2×C4—152no—
545C2×C4—308no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4D5×#6712yes—
550C2×C4D3×C22⋊D512yes—
551C2×C4#679011yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—102no—
555C2×C3—22no—
556C2×C3—204no—
557C2×C3#6611212yes—
558C2×C3C2×#408311yescharacteristic
559D3—42no—
560D3—202no—
561C5#8385911yescharacteristic, complemented, Sylow
562C22—22no—
563C22—62no—
564C22—104no—
565C22—122no—
566C22—204no—
567C22—304no—
568C22—302no—
569C22—404no—
570C22—1208no—
571C22#9157712yes—
572C22C2×#679011yescharacteristic
573C4—42no—
574C4—62no—
575C4—302no—
576C4—602no—
577C4#9157712yes—
578C4C2×D5×#6711yescharacteristic
579C4C2×D3×C22⋊D511yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—204no—
584C2—602no—
585C2#1175509712yes—
586C2C2×#9157711yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92017C2
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. #11756575C2semidirect
3. C2×C5×#942C2semidirect
4. C22×C3×#688C2semidirect
5. #11749404C2semidirect
6. #11749399C2semidirect
7. #11748388C2semidirect
8. #11748367C2semidirect
9. #11748349C2semidirect
11. C22×C3×C5×Q8C22semidirect
23. #91835C22semidirect
26. #91815C22semidirect
27. C2×C5×#216C22semidirect
29. C2×C5×#215C22semidirect
30. C2×C5×#208C22semidirect
32. C2×C3×#688C22semidirect
33. C22×C3×#168C22semidirect
34. #90973C22semidirect
35. #90967C22semidirect
36. #90965C22semidirect
37. #90957C22semidirect
42. C22×#688D3semidirect
97. C2×#942D5semidirect
104. C22×C5×Q8C2×D3semidirect
122. C2×#688C2×D3semidirect
125. C22×#168C2×D3semidirect
170. C22×C3×Q8C2×D5semidirect
187. C2×#216C2×D5semidirect
189. C2×#215C2×D5semidirect
192. C2×#208C2×D5semidirect
361. C22×Q8D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#83859semidirect
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
#117565752
C2×C5×#9422
C22×C3×#6882
#117494042
#117493992
#117483882
#117483672
#117483492
#106157133
#838595