Finite Group Databasebeta

← #14424390

Subgroups — #14424390

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×C3×#4149C211yesmaximal, characteristic
2C2×C5×#951C211yesmaximal, characteristic, complemented
3#11754251C211yesmaximal, characteristic, complemented
4C22×#6655C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11749294C214yesmaximal, complemented
7#11749044C214yesmaximal, complemented
8#11748367C211yesmaximal, characteristic, complemented
9#11748343C211yesmaximal, characteristic, complemented
10#10615713—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic, complemented
13C3×#4149—24no—
14C3×#4149—24no—
15C2×C3×#713C2212yes—
16C2×C3×D4×D5—22no—
17C2×C3×D4×D5C2212yes—
18C2×C3×#709—21no—
19C2×C3×#709C2211yescharacteristic
20C22×C3×D20—21no—
21C22×C3×D20C2211yescharacteristic
22#91815C2211yescharacteristic, complemented
23C22×C3⋊#168C2211yescharacteristic, complemented
24C5×#951—24no—
25C2×C5×#216C2212yescomplemented
26C2×C5×#214C2212yescomplemented
27C2×C5×#208C2211yescharacteristic, complemented
28#91536—22no—
29#91385—21no—
30#91368—21no—
31#91359C2214yescomplemented
32C2×#6655—24no—
33C2×#6655C2214yescomplemented
34#91119C2212yescomplemented
35#91116C2212yescomplemented
36#90965C2214yescomplemented
37#90955C2214yescomplemented
38#84105—51nomaximal, Hall
39C2×#4149D311yescharacteristic
40C2×C5×#137—31no—
41#66150—31no—
42C22×#688—31no—
43#65522—31no—
44#65476—34no—
45#65458—34no—
46#65420—31no—
47#65418—31no—
48C23×C3×D5—21no—
49C23×C3×D5—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
57C22×C5×Dic3—21no—
58C2×C5×Dic6—22no—
59C2×C5×Dic6—22no—
60C2×C5×Dic6—21no—
61C2×C5×Dic6C2312yes—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5D412yes—
64C3×#713—24no—
65C3×#713—24no—
66C3×D4×D5—216no—
67C3×D4×D5—24no—
68C3×D4×D5—24no—
69C3×#709—28no—
70C3×#709—22no—
71C3×#709—22no—
72C2×C3×D20—28no—
73C2×C3×D20—22no—
74C2×C3×D20—21no—
75C2×C3×D20—22no—
76C2×C3×D20D411yescharacteristic
77C2×C3×D20D414yes—
78C2×C3×D20C2312yes—
79C2×C3×C4×D5—22no—
80C2×C3×C4×D5C2312yes—
81C2×C3×Dic10D411yescharacteristic
82#6719—21no—
83C4⋊Dic15C2312yes—
84C4×Dic15—21no—
85C2×C3⋊#168—22no—
86C2×C3⋊#168C2312yes—
87C5×#216—24no—
88C5×#214—24no—
89C5×#208—22no—
90C2×C5×C3⋊C8C2312yes—
91#6669—24no—
92#6665—24no—
93#6655—28no—
94C2×C3×H(5,𝔽2)—51no—
95#5825—51no—
96C2×#951D511yescharacteristic, complemented
97C22×#214—51no—
98#5239—54no—
99#5234—51no—
100#5233—51no—
101#5218—54no—
102#5215—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8—31no—
105#4149—24no—
106#4149—24no—
107C2×#713C2×D312yes—
108C2×D4×D5—22no—
109C2×D4×D5C2×D312yes—
110C2×#709—21no—
111C2×#709C2×D311yescharacteristic
112C22×D20—21no—
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—34no—
122C2×#688—64no—
123#4075—62no—
124#4072—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3953—34no—
129#3952—34no—
130#3009—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—22no—
134C22×C3×D5—28no—
135C22×C3×D5—24no—
136C22×C3×D5—84no—
137C22×C3×D5—82no—
138C22×C3×D5C2×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
150C2×Dic15—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—42no—
153C5×Dic6—22no—
154C5×Dic6—22no—
155C5×Dic6—42no—
156C3×C22⋊D5—28no—
157C3×C22⋊D5—24no—
158C2×C3×Dic5C2×D412yes—
159C3×D20—28no—
160C3×D20—22no—
161C3×D20—24no—
162C3×D20—22no—
163C3×D20C2×D414yes—
164C3×C4×D5—28no—
165C3×C4×D5—22no—
166C3×C4×D5—22no—
167C3×Dic10—22no—
168C3⋊#168—22no—
169C5×C3⋊C8—22no—
170C3×H(5,𝔽2)—104no—
171C3×H(5,𝔽2)—104no—
172C2×C3×#41—51no—
173C2×C3×#41—52no—
174C2×C3×#41—101no—
175C2×C3×#41C2×D511yescharacteristic
176C22×C3×D4—52no—
177C22×C3×D4—51no—
178C22×C3×D4—101no—
179C22×C3×D4—102no—
180C22×Dic6C2×D511yescharacteristic, complemented
181#951—24no—
182#944—101no—
183C2×#216C2×D512yescomplemented
184#936—101no—
185#935—102no—
186C2×#214—54no—
187C2×#214—104no—
188C2×#214C2×D512yescomplemented
189#926—51no—
190#924—54no—
191C2×#208C2×D511yescharacteristic, complemented
192C22×C3⋊C8—51no—
193#835—52no—
194#832—52no—
195#832—54no—
196#830—54no—
197C23×D5—21no—
198C23×D5—22no—
199C5×#41—22no—
200C5×#41#6714yes—
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—22no—
212C2×C22⋊D5#6712yes—
213#713—24no—
214#713—24no—
215D4×D5—216no—
216D4×D5—24no—
217D4×D5—24no—
218#709—28no—
219#709—22no—
220#709—22no—
221C2×D20—28no—
222C2×D20—22no—
223C2×D20—21no—
224C2×D20—22no—
225C2×D20#6711yescharacteristic
226C2×D20#6714yes—
227C2×D20C22×D312yes—
228C2×C4×D5—22no—
229C2×C4×D5C22×D312yes—
230C2×Dic10#6711yescharacteristic
231C5×Q16—64no—
232C5×SD16—64no—
233C5×M16—62no—
234C2×C5×C8—32no—
235C22⋊Dic5—61no—
236#688—68no—
237#685—64no—
238C4⋊Dic5—32no—
239#683—64no—
240C4×Dic5—61no—
241C2×#168—32no—
242C2×#168—62no—
243C2×H(5,𝔽2)—51no—
244C2×#137—31no—
245C22×SD16—151no—
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—22no—
254C22×C3×C5—42no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×C3×D5—22no—
258C2×C3×D5—24no—
259C2×C3×D5—24no—
260C2×C3×D5—44no—
261C2×C3×D5—84no—
262C2×C3×D5—88no—
263C3×C4×C5—22no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—42no—
268C3×Dic5—22no—
269C5×Dic3—42no—
270C24×C3—102no—
271C24×C3—101no—
272C3×#41—22no—
273C3×#41—108no—
274C3×#41—104no—
275C3×#41—104no—
276C3×#41—102no—
277C3×#41—102no—
278C3×#41C22⋊D514yes—
279C2×C3×Q8—51no—
280C2×C3×Q8C22×D511yescharacteristic
281C2×C3×D4—21no—
282C2×C3×D4—52no—
283C2×C3×D4—54no—
284C2×C3×D4—51no—
285C2×C3×D4—52no—
286C2×C3×D4—102no—
287C2×C3×D4—108no—
288C2×C3×D4—102no—
289C2×C3×D4—104no—
290C2×C3×D4—1016no—
291C2×C3×D4—104no—
292C2×C3×D4—101no—
293C2×C3×D4—102no—
294C2×C3×D4C22×D511yescharacteristic
295C22×C3×C4—21no—
296C22×C3×C4—52no—
297C22×C3×C4—102no—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—21no—
300C2×Dic6—22no—
301C2×Dic6—22no—
302C2×Dic6—21no—
303C2×Dic6C22×D512yes—
304#217—104no—
305#217—101no—
306#216—24no—
307#214—24no—
308#214—108no—
309C4⋊Dic3—52no—
310Dic3⋊C4—104no—
311C4×Dic3—101no—
312#208—22no—
313C2×C3⋊C8—52no—
314C2×C3⋊C8—102no—
315C2×C3⋊C8C22×D512yes—
316C23×C5—21no—
317C23×C5C2×#6711yescharacteristic
318C22×D5—22no—
319C22×D5—28no—
320C22×D5—24no—
321C22×D5—84no—
322C22×D5—82no—
323C22×D5C2×#6712yes—
324C5×Q8—21no—
325C5×Q8—62no—
326C5×Q8—62no—
327C5×Q8—122no—
328C5×Q8C2×#6712yes—
329C5×D4—24no—
330C5×D4—21no—
331C5×D4C2×#6712yes—
332C2×C4×C5—21no—
333C2×C4×C5—24no—
334C2×C4×C5—62no—
335C2×C4×C5—122no—
336C2×C4×C5C2×#6712yes—
337C2×C4×C5C2×#6711yescharacteristic
338C2×C4×C5C2×#6712yes—
339C2×C4×C5C23×D311yescharacteristic
340C22⋊D5—28no—
341C22⋊D5—24no—
342C2×Dic5—62no—
343C2×Dic5C2×#6712yes—
344D20—28no—
345D20—22no—
346D20—24no—
347D20—22no—
348D20C2×#6714yes—
349C4×D5—28no—
350C4×D5—22no—
351C4×D5—22no—
352Dic10—22no—
353C5×C8—62no—
354#168—62no—
355H(5,𝔽2)—104no—
356H(5,𝔽2)—104no—
357C2×#41—51no—
358C2×#41—52no—
359C2×#41—101no—
360C2×#41D3×D511yescharacteristic
361C22×Q8—31no—
362C22×D4—52no—
363C22×D4—51no—
364C22×D4—101no—
365C22×D4—102no—
366#137—64no—
367C2×Q16—32no—
368C2×SD16—32no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—31no—
372C22×C8—151no—
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×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C3×D5—42no—
387C3×D5—44no—
388C23×C3—21no—
389C23×C3—52no—
390C23×C3—108no—
391C23×C3—102no—
392C23×C3—104no—
393C23×C3—402no—
394C23×C3—404no—
395C23×C3D4×D511yescharacteristic
396C3×Q8—21no—
397C3×Q8—102no—
398C3×Q8C2×C22⋊D512yes—
399C3×D4—24no—
400C3×D4—21no—
401C3×D4—54no—
402C3×D4—108no—
403C3×D4—108no—
404C3×D4—104no—
405C3×D4—102no—
406C3×D4—104no—
407C3×D4—102no—
408C3×D4C2×C22⋊D512yes—
409C2×C3×C4—21no—
410C2×C3×C4—24no—
411C2×C3×C4—52no—
412C2×C3×C4—108no—
413C2×C3×C4—102no—
414C2×C3×C4—102no—
415C2×C3×C4D4×D512yes—
416C2×C3×C4D4×D511yescharacteristic
417C2×C3×C4C2×C22⋊D512yes—
418C2×C3×C4C23×D511yescharacteristic
419C2×Dic3—22no—
420C2×Dic3—42no—
421C2×Dic3—102no—
422Dic6—22no—
423Dic6—22no—
424Dic6—42no—
425C3⋊C8—22no—
426C3⋊C8—102no—
427C22×C5—22no—
428C22×C5—22no—
429C22×C5—42no—
430C22×C5#95312yes—
431C22×C5C22×#6711yescharacteristic
432C2×D5—24no—
433C2×D5—22no—
434C2×D5—24no—
435C2×D5—44no—
436C2×D5—84no—
437C2×D5—88no—
438C4×C5—22no—
439C4×C5—122no—
440C4×C5#95312yes—
441C4×C5C22×#6711yescharacteristic
442C4×C5C22×#6711yescharacteristic
443Dic5—22no—
444Dic5—122no—
445C24—102no—
446C24—101no—
447#41—22no—
448#41—108no—
449#41—104no—
450#41—104no—
451#41—102no—
452#41—102no—
453#41#120114yes—
454C2×Q8—32no—
455C2×Q8—51no—
456C2×Q8—61no—
457C2×Q8—62no—
458C2×Q8—62no—
459C2×Q8C2×D3×D511yescharacteristic
460C2×D4—21no—
461C2×D4—52no—
462C2×D4—54no—
463C2×D4—51no—
464C2×D4—52no—
465C2×D4—102no—
466C2×D4—108no—
467C2×D4—102no—
468C2×D4—104no—
469C2×D4—1016no—
470C2×D4—104no—
471C2×D4—102no—
472C2×D4—101no—
473C2×D4C2×D3×D511yescharacteristic
474C22×C4—21no—
475C22×C4—52no—
476C22×C4—61no—
477C22×C4—102no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—64no—
480SD16—64no—
481SD16—308no—
482M16—62no—
483C2×C8—32no—
484C2×C8—152no—
485C2×C8—302no—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—304no—
489C22⋊C4—301no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—42no—
495C22×C3—104no—
496C22×C3—104no—
497C22×C3—102no—
498C22×C3—204no—
499C22×C3—404no—
500C22×C3—408no—
501C22×C3#408312yes—
502C22×C3C2×D4×D511yescharacteristic
503C3×C4—22no—
504C3×C4—102no—
505C3×C4#408312yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C22×C22⋊D511yescharacteristic
508Dic3—42no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5#583612yes—
513C2×C5C2×#95311yescharacteristic
514D5—42no—
515D5—44no—
516C23—21no—
517C23—52no—
518C23—108no—
519C23—102no—
520C23—104no—
521C23—402no—
522C23—404no—
523C23D5×#6711yescharacteristic
524Q8—21no—
525Q8—62no—
526Q8—62no—
527Q8—102no—
528Q8—122no—
529Q8C2×#120112yes—
530D4—24no—
531D4—21no—
532D4—54no—
533D4—108no—
534D4—108no—
535D4—104no—
536D4—102no—
537D4—104no—
538D4—102no—
539D4C2×#120112yes—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—52no—
543C2×C4—62no—
544C2×C4—108no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—122no—
548C2×C4—302no—
549C2×C4C2×#120112yes—
550C2×C4D5×#6712yes—
551C2×C4D5×#6711yescharacteristic
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—302no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3—202no—
558C2×C3—204no—
559C2×C3#6611212yes—
560C2×C3C2×#408311yescharacteristic
561C5#8410511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—22no—
564C22—42no—
565C22—104no—
566C22—104no—
567C22—102no—
568C22—204no—
569C22—404no—
570C22—408no—
571C22#9156312yes—
572C22C2×D5×#6711yescharacteristic
573C4—22no—
574C4—102no—
575C4—122no—
576C4—602no—
577C4#9156312yes—
578C4C22×#120111yescharacteristic
579C4C2×D5×#6711yescharacteristic
580C3#1061571311yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—202no—
584C2—204no—
585C2#1175533512yes—
586C2C2×#9156311yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4149C2
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. #11754251C2semidirect
4. C22×#6655C2semidirect
5. #11751334C2semidirect
6. #11749294C2semidirect
7. #11749044C2semidirect
8. #11748367C2semidirect
9. #11748343C2semidirect
12. C22×C5×Dic6C22semidirect
22. #91815C22semidirect
23. C22×C3⋊#168C22semidirect
25. C2×C5×#216C22semidirect
26. C2×C5×#214C22semidirect
27. C2×C5×#208C22semidirect
31. #91359C22semidirect
33. C2×#6655C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #90965C22semidirect
37. #90955C22semidirect
96. C2×#951D5semidirect
180. C22×Dic6C2×D5semidirect
183. C2×#216C2×D5semidirect
188. C2×#214C2×D5semidirect
191. C2×#208C2×D5semidirect
491. C3×C5#3009semidirect
561. C5#84105semidirect
580. C3#10615713semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41492
C2×C5×#9512
#117542512
C22×#66552
#117513342
#117492942
#117490442
#117483672
#117483432
#106157133
#841055