Finite Group Databasebeta

← D5×#5056 #14437767

Subgroups — D5×#5056 #14437767

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D5×#124C211yesmaximal, characteristic
2D5×#870C211yesmaximal, characteristic, complemented
3#11751157C211yesmaximal, characteristic, complemented
4C5×#5056C211yesmaximal, characteristic, complemented
5#11749331C212yesmaximal, complemented
6D5×#835C212yesmaximal, complemented
7#11749085C212yesmaximal, complemented
8D5×#832C212yesmaximal
9C4×D5×C3⋊C8C211yesmaximal, characteristic, complemented
10#11746896C211yesmaximal, characteristic, complemented
11#11746891C211yesmaximal, characteristic, complemented
12D5×#488—31nomaximal, Hall
13C2×C3×Q8×D5C2211yescharacteristic
14C2×C3×D4×D5C2211yescharacteristic
15C3×C5×#124C2211yescharacteristic
16#91759C2211yescharacteristic, complemented
17C5×#870C2211yescharacteristic, complemented
18C3×#4093C2211yescharacteristic
19C3×#4082C2211yescharacteristic
20C3×#4031—22no—
21C3×D5×C22⋊C4—22no—
22C3×#4021C2211yescharacteristic
23C3×C42×D5C2211yescharacteristic
24D5×C4⋊Dic3C2212yes—
25D5×Dic3⋊C4—22no—
26#91388C2212yescomplemented
27#91378—22no—
28C2×D5×C3⋊C8C2212yes—
29#91119C2212yescomplemented
30#91116C2212yes—
31C4×C3⋊#168C2211yescharacteristic, complemented
32C5×#835C2212yescomplemented
33C5×#832C2212yes—
34C4×C5×C3⋊C8C2211yescharacteristic, complemented
35#90971C2212yescomplemented
36#90970C2212yescomplemented
37#90965C2212yes—
38#90964C2212yes—
39Dic5×C3⋊C8C2212yescomplemented
40C2×#5056—51nomaximal, Hall
41D5×#125—31no—
42D5×#124D311yescharacteristic
43C5×#488—31no—
44#65357—31no—
45#65345—31no—
46D5×#103—32no—
47#65089—32no—
48D5×#102—32no—
49#65057—32no—
50#64986—31no—
51C4×C8×D5—31no—
52C23×C3×D5—21no—
53C2×C3×C5×Q8C2311yescharacteristic
54C2×C3×C5×D4C2311yescharacteristic
55C2×C3×C22⋊D5—21no—
56C3×Q8×D5—24no—
57C3×D4×D5—24no—
58C2×C3×D20C2311yescharacteristic
59C2×C3×C4×D5—21no—
60C2×C3×C4×D5D412yes—
61C2×C3×C4×D5C2311yescharacteristic
62C2×C3×Dic10—21no—
63C2×C3×Dic10C2311yescharacteristic
64C2×D5×Dic3—22no—
65C3×C5×C22⋊C4—22no—
66C3×C42×C5C2311yescharacteristic
67C4⋊Dic15C2312yes—
68#6712—22no—
69C2×C3⋊#168C2312yes—
70C5×C4⋊Dic3C2312yes—
71C5×Dic3⋊C4—22no—
72C2×C5×C3⋊C8C2312yes—
73C3×C22⋊Dic5—22no—
74C3×#685—22no—
75C3×#685—22no—
76C3×C4×Dic5D412yes—
77C3×C4×Dic5C2311yescharacteristic
78Dic3⋊Dic5—22no—
79#6669—22no—
80#6668—22no—
81D5×C3⋊C8—24no—
82C2×C3×#124—51no—
83C2×#870—51no—
84C2×#835—52no—
85C2×#832—52no—
86#5056—51no—
87#5056—51no—
88#5056—52no—
89#5056—52no—
90#5056—51no—
91#5056D511yescharacteristic, direct product factor
92C2×C4×C3⋊C8—51no—
93C2×Q8×D5C2×D311yescharacteristic
94C2×D4×D5C2×D311yescharacteristic
95C5×#125—31no—
96C5×#124C2×D311yescharacteristic
97#4093C2×D311yescharacteristic
98#4082C2×D311yescharacteristic
99C2×C8×D5—32no—
100D5×C4⋊C4—32no—
101D5×C4⋊C4—62no—
102#4036—32no—
103#4035—62no—
104#4031—22no—
105D5×C22⋊C4—22no—
106#4021C2×D311yescharacteristic
107C42×D5C2×D311yescharacteristic
108#4016—31no—
109C5×#103—32no—
110C5×#102—32no—
111C4×C5×C8—31no—
112#3967—32no—
113#3964—32no—
114#3953—32no—
115#3948—32no—
116C8×Dic5—32no—
117#3942—32no—
118#3941—32no—
119C4×#168—31no—
120C2×#488—151noSylow
121C23×C3×C5—21no—
122C22×C3×D5—21no—
123C22×C3×D5—42no—
124C22×C3×D5—44no—
125C22×C3×D5C2×D411yescharacteristic
126C3×C5×Q8—22no—
127C3×C5×D4—22no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5C2×D412yes—
130C2×C3×C4×C5C2411yescharacteristic
131C2×Dic15—22no—
132C2×C5×Dic3—22no—
133C3×C22⋊D5—42no—
134C2×C3×Dic5—21no—
135C2×C3×Dic5C2×D412yes—
136C2×C3×Dic5C2×D411yescharacteristic
137C3×D20—22no—
138C3×C4×D5—24no—
139C3×C4×D5—42no—
140C3×C4×D5#4114yes—
141C3×Dic10—22no—
142C3×Dic10—42no—
143D5×Dic3—44no—
144C3⋊#168—22no—
145C5×C3⋊C8—22no—
146C22×C3×Q8—51no—
147C22×C3×D4—51no—
148C3×#124—51no—
149C3×#124—51no—
150C3×#124—51no—
151C3×#124—102no—
152C3×#124C2×D511yescharacteristic
153C2×C3×C22⋊C4—102no—
154C2×C3×C42—51no—
155C2×C4⋊Dic3—52no—
156C2×Dic3⋊C4—102no—
157C22×C3⋊C8—52no—
158#870—51no—
159#870—52no—
160#870—102no—
161#870C2×D511yescharacteristic, complemented
162#835—52no—
163#835—52no—
164#835—52no—
165#835C2×D512yescomplemented
166#832—52no—
167#832—52no—
168#832—52no—
169#832C2×D512yes—
170C4×C3⋊C8—51no—
171C4×C3⋊C8—52no—
172C4×C3⋊C8C2×D511yescharacteristic, complemented
173C23×D5—21no—
174C2×C5×Q8C22×D311yescharacteristic
175C2×C5×D4C22×D311yescharacteristic
176C2×C22⋊D5—21no—
177Q8×D5—24no—
178D4×D5—24no—
179C2×D20C22×D311yescharacteristic
180C2×C4×D5—21no—
181C2×C4×D5—62no—
182C2×C4×D5#6712yes—
183C2×C4×D5C22×D311yescharacteristic
184C2×Dic10—21no—
185C2×Dic10C22×D311yescharacteristic
186C2×C5×C8—32no—
187C5×C4⋊C4—32no—
188C5×C4⋊C4—62no—
189C5×C22⋊C4—22no—
190C42×C5C22×D311yescharacteristic
191C22⋊Dic5—22no—
192#685—22no—
193#685—22no—
194C4⋊Dic5—32no—
195C4⋊Dic5—62no—
196#683—62no—
197#683—62no—
198#683—62no—
199C4×Dic5#6712yes—
200C4×Dic5C22×D311yescharacteristic
201C2×#168—32no—
202C8×D5—64no—
203C2×#125—151no—
204C2×#124—51no—
205#488—31no—
206#488—151no—
207#488—152no—
208#488—152no—
209#488—151no—
210#488—151no—
211C2×#103—152no—
212C2×#102—152no—
213C2×C4×C8—151no—
214C22×C3×C5—41no—
215C22×C3×C5—42no—
216C22×C3×C5C22×D411yescharacteristic
217C2×C3×D5—41no—
218C2×C3×D5—42no—
219C2×C3×D5—84no—
220C2×C3×D5#12412yes—
221C2×C3×D5#13514yes—
222C3×C4×C5—22no—
223C3×C4×C5—41no—
224C3×C4×C5C2×#4112yes—
225Dic15—42no—
226C3×Dic5—22no—
227C3×Dic5—41no—
228C3×Dic5#12412yes—
229C5×Dic3—42no—
230C24×C3—101no—
231C2×C3×Q8—51no—
232C2×C3×Q8—101no—
233C2×C3×Q8—104no—
234C2×C3×Q8C22×D511yescharacteristic
235C2×C3×D4—51no—
236C2×C3×D4—104no—
237C2×C3×D4—101no—
238C2×C3×D4C22×D511yescharacteristic
239C22×C3×C4—51no—
240C22×C3×C4—52no—
241C22×C3×C4—101no—
242C22×Dic3—102no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—102no—
245C3×C22⋊C4—102no—
246C3×C22⋊C4—102no—
247C3×C42—52no—
248C3×C42—51no—
249C3×C42C22×D511yescharacteristic
250C4⋊Dic3—52no—
251C4⋊Dic3—102no—
252C4⋊Dic3C22×D512yes—
253Dic3⋊C4—22no—
254Dic3⋊C4—102no—
255Dic3⋊C4—102no—
256Dic3⋊C4—102no—
257C2×C3⋊C8—52no—
258C2×C3⋊C8—104no—
259C2×C3⋊C8C22×D512yes—
260C23×C5—21no—
261C22×D5—21no—
262C22×D5—42no—
263C22×D5—44no—
264C22×D5C2×#6711yescharacteristic
265C5×Q8—22no—
266C5×D4—22no—
267C2×C4×C5—21no—
268C2×C4×C5—62no—
269C2×C4×C5C2×#6712yes—
270C2×C4×C5C23×D311yescharacteristic
271C22⋊D5—42no—
272C2×Dic5—21no—
273C2×Dic5—62no—
274C2×Dic5C2×#6712yes—
275C2×Dic5C2×#6711yescharacteristic
276D20—22no—
277C4×D5—24no—
278C4×D5—42no—
279C4×D5—124no—
280C4×D5#23714yes—
281Dic10—22no—
282Dic10—42no—
283C5×C8—62no—
284#168—62no—
285C22×Q8—51no—
286C22×D4—51no—
287C22×C8—152no—
288#125—31no—
289#125—152no—
290#125—151no—
291#125—302no—
292#124—51no—
293#124—51no—
294#124—51no—
295#124—102no—
296#124D3×D511yescharacteristic
297C2×C4⋊C4—152no—
298C2×C4⋊C4—302no—
299C2×C22⋊C4—102no—
300C2×C42—51no—
301#103—32no—
302#103—152no—
303#103—152no—
304#103—152no—
305#102—32no—
306#102—152no—
307#102—152no—
308#102—152no—
309C4×C8—31no—
310C4×C8—152no—
311C4×C8—151no—
312C2×C3×C5—81no—
313C2×C3×C5C2×#12411yescharacteristic
314C2×C3×C5C2×#13512yes—
315C3×D5—81no—
316C3×D5#48814yescomplemented
317C23×C3—21no—
318C23×C3—51no—
319C23×C3—101no—
320C23×C3—202no—
321C23×C3—204no—
322C3×Q8—22no—
323C3×Q8—102no—
324C3×Q8—202no—
325C3×D4—22no—
326C3×D4—102no—
327C3×D4—202no—
328C2×C3×C4—21no—
329C2×C3×C4—51no—
330C2×C3×C4—54no—
331C2×C3×C4—52no—
332C2×C3×C4—104no—
333C2×C3×C4—101no—
334C2×C3×C4—202no—
335C2×C3×C4D4×D512yes—
336C2×C3×C4C23×D511yescharacteristic
337C2×Dic3—22no—
338C2×Dic3—102no—
339C2×Dic3—204no—
340C3⋊C8—22no—
341C3⋊C8—102no—
342C22×C5—41no—
343C22×C5—42no—
344C22×C5C22×#6711yescharacteristic
345C2×D5—41no—
346C2×D5—42no—
347C2×D5—84no—
348C2×D5#93612yes—
349C2×D5#95014yes—
350C4×C5—22no—
351C4×C5—41no—
352C4×C5—122no—
353C4×C5C2×#23712yes—
354Dic5—22no—
355Dic5—41no—
356Dic5—122no—
357Dic5#93612yes—
358C24—101no—
359C2×Q8—51no—
360C2×Q8—101no—
361C2×Q8—104no—
362C2×Q8C2×D3×D511yescharacteristic
363C2×D4—51no—
364C2×D4—104no—
365C2×D4—101no—
366C2×D4C2×D3×D511yescharacteristic
367C22×C4—51no—
368C22×C4—52no—
369C22×C4—101no—
370C22×C4—302no—
371C2×C8—32no—
372C2×C8—152no—
373C2×C8—304no—
374C4⋊C4—32no—
375C4⋊C4—62no—
376C4⋊C4—152no—
377C4⋊C4—302no—
378C4⋊C4—302no—
379C4⋊C4—302no—
380C4⋊C4—302no—
381C22⋊C4—22no—
382C22⋊C4—102no—
383C22⋊C4—102no—
384C22⋊C4—102no—
385C42—52no—
386C42—51no—
387C42C2×D3×D511yescharacteristic
388C3×C5C2×#48811yescharacteristic, complemented, Hall
389C22×C3—42no—
390C22×C3—41no—
391C22×C3—54no—
392C22×C3—52no—
393C22×C3—202no—
394C22×C3—201no—
395C22×C3—404no—
396C22×C3C2×D4×D511yescharacteristic
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—52no—
400C3×C4—102no—
401C3×C4—201no—
402C3×C4D5×#4112yes—
403Dic3—42no—
404Dic3—202no—
405C2×C5—81no—
406C2×C5C2×#93611yescharacteristic
407C2×C5C2×#95012yes—
408D5—81no—
409D5#505614yesdirect product factor
410C23—21no—
411C23—51no—
412C23—101no—
413C23—202no—
414C23—204no—
415Q8—22no—
416Q8—102no—
417Q8—202no—
418D4—22no—
419D4—102no—
420D4—202no—
421C2×C4—21no—
422C2×C4—51no—
423C2×C4—54no—
424C2×C4—52no—
425C2×C4—62no—
426C2×C4—104no—
427C2×C4—101no—
428C2×C4—202no—
429C2×C4—302no—
430C2×C4—604no—
431C2×C4D5×#6712yes—
432C2×C4C22×D3×D511yescharacteristic
433C8—62no—
434C8—302no—
435C2×C3—54no—
436C2×C3—81no—
437C2×C3—401no—
438C2×C3D5×#12411yescharacteristic
439C2×C3D5×#13512yes—
440C5C2×#505611yescharacteristic, complemented, Sylow
441C22—41no—
442C22—42no—
443C22—52no—
444C22—54no—
445C22—201no—
446C22—202no—
447C22—404no—
448C22C2×D5×#6711yescharacteristic
449C4—22no—
450C4—41no—
451C4—52no—
452C4—102no—
453C4—122no—
454C4—201no—
455C4—602no—
456C4D5×#23712yes—
457C3D5×#48811yescharacteristic, complemented, Sylow
458C2—54no—
459C2—81no—
460C2—401no—
461C2D5×#93611yescharacteristic
462C2D5×#95012yes—
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#124C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×D4×D5C22
hypercenter#124D3×D5
hypocenterC3×C5C2×#488
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. D5×#870C2semidirect
3. #11751157C2semidirect
4. C5×#5056C2semidirect
5. #11749331C2semidirect
6. D5×#835C2semidirect
7. #11749085C2semidirect
9. C4×D5×C3⋊C8C2semidirect
10. #11746896C2semidirect
11. #11746891C2semidirect
16. #91759C22semidirect
17. C5×#870C22semidirect
26. #91388C22semidirect
29. #91119C22semidirect
31. C4×C3⋊#168C22semidirect
32. C5×#835C22semidirect
34. C4×C5×C3⋊C8C22semidirect
35. #90971C22semidirect
36. #90970C22semidirect
39. Dic5×C3⋊C8C22semidirect
91. #5056D5direct (N × H)
161. #870C2×D5semidirect
165. #835C2×D5semidirect
172. C4×C3⋊C8C2×D5semidirect
316. C3×D5#488semidirect
388. C3×C5C2×#488semidirect
409. D5#5056direct (N × H)
440. C5C2×#5056semidirect
457. C3D5×#488semidirect

Maximal subgroups

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

subgroupindex
C3×D5×#1242
D5×#8702
#117511572
C5×#50562
#117493312
D5×#8352
#117490852
D5×#8322
C4×D5×C3⋊C82
#117468962
#117468912
D5×#4883
C2×#50565