Finite Group Databasebeta

← D5×#878 #11752826

Subgroups — D5×#878 #11752826

Order 960.

644 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×Dic3C211yesmaximal, characteristic
2C2×D5×Dic6C211yesmaximal, characteristic, complemented
3#91767C211yesmaximal, characteristic, complemented
4C5×#878C211yesmaximal, characteristic, complemented
5C3×D5×C22⋊C4C211yesmaximal, characteristic
6#91576C211yesmaximal, characteristic, complemented
7#91575C211yesmaximal, characteristic, complemented
8D5×#217C211yesmaximal, characteristic
9#91431C211yesmaximal, characteristic, complemented
10#91430C211yesmaximal, characteristic, complemented
11#91422C211yesmaximal, characteristic, complemented
12#91421C211yesmaximal, characteristic, complemented
13D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
14D5×Dic3⋊C4C211yesmaximal, characteristic, complemented
15D5×Dic3⋊C4C211yesmaximal, characteristic, complemented
16D5×C22⋊Q8—31nomaximal, Hall
17C23×C3×D5C2211yescharacteristic
18C22×Dic15C2211yescharacteristic
19C2×Dic30C2211yescharacteristic, complemented
20C22×C5×Dic3C2211yescharacteristic
21C2×C5×Dic6C2211yescharacteristic, complemented
22C2×C3×C4×D5C2211yescharacteristic
23C2×C3×C4×D5C2211yescharacteristic
24C2×#1204C2211yescharacteristic, complemented
25C2×#1204C2211yescharacteristic, complemented
26C2×D5×Dic3—24no—
27C2×D5×Dic3C2211yescharacteristic
28C2×D5×Dic3C2211yescharacteristic
29C2×D5×Dic3C2211yescharacteristic
30C2×D5×Dic3C2211yescharacteristic
31C2×D5×Dic3C2211yescharacteristic
32C2×D5×Dic3C2211yescharacteristic
33D5×Dic6—24no—
34C3×C5×C22⋊C4C2211yescharacteristic
35#6719C2211yescharacteristic
36C4⋊Dic15C2211yescharacteristic, complemented
37#6712C2211yescharacteristic, complemented
38#6712C2211yescharacteristic, complemented
39C5×#217C2211yescharacteristic
40C5×C4⋊Dic3C2211yescharacteristic, complemented
41C5×Dic3⋊C4C2211yescharacteristic, complemented
42C5×Dic3⋊C4C2211yescharacteristic, complemented
43C3×C22⋊Dic5C2211yescharacteristic
44C3×#685C2211yescharacteristic
45C3×#685C2211yescharacteristic
46Dic3⋊Dic5C2211yescharacteristic, complemented
47Dic3⋊Dic5C2211yescharacteristic, complemented
48#6669C2211yescharacteristic, complemented
49#6669C2211yescharacteristic, complemented
50#6668C2211yescharacteristic, complemented
51#6668C2211yescharacteristic, complemented
52#6665C2211yescharacteristic
53#6665C2211yescharacteristic
54C2×#878—51nomaximal, Hall
55C2×Q8×D5—31no—
56C22×C4×D5—31no—
57C5×C22⋊Q8—31no—
58#4092—31no—
59#4070—31no—
60#4043—31no—
61#4042—31no—
62#4042—31no—
63D5×C4⋊C4—31no—
64D5×C4⋊C4—31no—
65D5×C4⋊C4—31no—
66D5×C22⋊C4—31no—
67D5×C22⋊C4D311yescharacteristic
68#4024—31no—
69#4024—31no—
70C23×C3×C5C2311yescharacteristic
71C22×C3×D5—22no—
72C22×C3×D5—22no—
73C22×C3×D5Q814yescomplemented
74C22×C3×D5C2311yescharacteristic
75C22×C3×D5C2311yescharacteristic
76C2×C3×C4×C5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78C2×Dic15—22no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C2×Dic15C2311yescharacteristic
83Dic30—22no—
84C2×C5×Dic3—22no—
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C2×C5×Dic3C2311yescharacteristic
88C2×C5×Dic3C2311yescharacteristic
89C5×Dic6—22no—
90C2×C3×Dic5C2311yescharacteristic
91C2×C3×Dic5C2311yescharacteristic
92C3×C4×D5—22no—
93C3×C4×D5—22no—
94#1204—22no—
95#1204—22no—
96D5×Dic3—22no—
97D5×Dic3—22no—
98D5×Dic3—22no—
99D5×Dic3—22no—
100D5×Dic3—22no—
101D5×Dic3D414yes—
102C23×Dic3—51no—
103C22×Dic6—51no—
104C2×C3×C22⋊C4—51no—
105C2×#217—51no—
106C2×C4⋊Dic3—51no—
107C2×Dic3⋊C4—51no—
108C2×Dic3⋊C4—51no—
109#878—51no—
110#878—51no—
111#878—51no—
112#878—51no—
113#878—51no—
114#878—51no—
115#878—51no—
116#878D511yescharacteristic, direct product factor
117C23×D5C2×D311yescharacteristic
118C2×C5×Q8—31no—
119C22×C4×C5—31no—
120C22×Dic5—31no—
121Q8×D5—64no—
122C2×C4×D5—31no—
123C2×C4×D5—31no—
124C2×C4×D5—31no—
125C2×C4×D5—31no—
126C2×C4×D5—31no—
127C2×C4×D5—31no—
128C2×C4×D5—64no—
129C2×C4×D5C2×D311yescharacteristic
130C2×C4×D5C2×D311yescharacteristic
131C2×Dic10—31no—
132C2×Dic10—31no—
133C2×Dic10—31no—
134C5×C4⋊C4—31no—
135C5×C4⋊C4—31no—
136C5×C4⋊C4—31no—
137C5×C22⋊C4—31no—
138C5×C22⋊C4C2×D311yescharacteristic
139C22⋊Dic5—31no—
140C22⋊Dic5C2×D311yescharacteristic
141#685—31no—
142#685—31no—
143#685C2×D311yescharacteristic
144#685C2×D311yescharacteristic
145C4⋊Dic5—31no—
146C4⋊Dic5—31no—
147C4⋊Dic5—31no—
148#683—31no—
149#683—31no—
150#683—31no—
151#683—31no—
152#683—31no—
153#683—31no—
154C2×C22⋊Q8—151noSylow
155C22×C3×C5—21no—
156C22×C3×C5—21no—
157C22×C3×C5C2×Q812yescomplemented
158C22×C3×C5C2411yescharacteristic
159C2×C3×D5—21no—
160C2×C3×D5—21no—
161C2×C3×D5—28no—
162C2×C3×D5C2×D412yes—
163C2×C3×D5C2×Q812yes—
164C2×C3×D5C2×Q812yescomplemented
165C2×C3×D5#4112yes—
166C3×C4×C5—21no—
167C3×C4×C5—21no—
168Dic15—21no—
169Dic15—21no—
170Dic15—21no—
171Dic15C2×D412yes—
172C3×Dic5—21no—
173C3×Dic5—21no—
174C5×Dic3—21no—
175C5×Dic3—21no—
176C5×Dic3—21no—
177C5×Dic3C2×D412yes—
178C24×C3—51no—
179C22×C3×C4—51no—
180C22×C3×C4—51no—
181C22×Dic3—51no—
182C22×Dic3—51no—
183C22×Dic3—51no—
184C22×Dic3—51no—
185C22×Dic3—51no—
186C22×Dic3—51no—
187C22×Dic3—51no—
188C22×Dic3—104no—
189C22×Dic3C2×D511yescharacteristic
190C2×Dic6—51no—
191C2×Dic6—51no—
192C2×Dic6—51no—
193C2×Dic6—104no—
194C2×Dic6C2×D511yescharacteristic, complemented
195C3×C22⋊C4—51no—
196C3×C22⋊C4—51no—
197C3×C22⋊C4—51no—
198C3×C22⋊C4C2×D511yescharacteristic
199#217—51no—
200#217—51no—
201#217—51no—
202#217C2×D511yescharacteristic
203C4⋊Dic3—51no—
204C4⋊Dic3—51no—
205C4⋊Dic3—51no—
206C4⋊Dic3C2×D511yescharacteristic, complemented
207Dic3⋊C4—51no—
208Dic3⋊C4—51no—
209Dic3⋊C4—51no—
210Dic3⋊C4—51no—
211Dic3⋊C4—51no—
212Dic3⋊C4—51no—
213Dic3⋊C4C2×D511yescharacteristic, complemented
214Dic3⋊C4C2×D511yescharacteristic, complemented
215C23×C5C22×D311yescharacteristic
216C22×D5—22no—
217C22×D5—22no—
218C22×D5Dic614yescomplemented
219C22×D5C22×D311yescharacteristic
220C22×D5C22×D311yescharacteristic
221C5×Q8—62no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—31no—
225C2×C4×C5—31no—
226C2×C4×C5—62no—
227C2×C4×C5C22×D311yescharacteristic
228C2×C4×C5C22×D311yescharacteristic
229C2×Dic5—31no—
230C2×Dic5—31no—
231C2×Dic5—31no—
232C2×Dic5—31no—
233C2×Dic5—62no—
234C2×Dic5C22×D311yescharacteristic
235C2×Dic5C22×D311yescharacteristic
236C4×D5—22no—
237C4×D5—22no—
238C4×D5—34no—
239C4×D5—62no—
240C4×D5—62no—
241C4×D5—62no—
242C4×D5—62no—
243C4×D5—62no—
244Dic10—62no—
245Dic10—62no—
246Dic10—62no—
247C22×Q8—151no—
248C23×C4—151no—
249C22⋊Q8—31no—
250C22⋊Q8—151no—
251C22⋊Q8—151no—
252C22⋊Q8—151no—
253C22⋊Q8—151no—
254C22⋊Q8—151no—
255C22⋊Q8—151no—
256C22⋊Q8—151no—
257C2×C4⋊C4—151no—
258C2×C4⋊C4—151no—
259C2×C4⋊C4—151no—
260C2×C22⋊C4—51no—
261C2×C22⋊C4—151no—
262C2×C3×C5—22no—
263C2×C3×C5C22×D411yescharacteristic
264C2×C3×C5C22×Q811yescharacteristic
265C2×C3×C5C2×#4111yescharacteristic
266C3×D5—22no—
267C3×D5C22⋊Q814yescomplemented
268C23×C3—51no—
269C23×C3—54no—
270C23×C3—51no—
271C23×C3—102no—
272C23×C3—102no—
273C23×C3C22×D511yescharacteristic
274C2×C3×C4—51no—
275C2×C3×C4—51no—
276C2×C3×C4—102no—
277C2×C3×C4—102no—
278C2×C3×C4C22×D511yescharacteristic
279C2×C3×C4C22×D511yescharacteristic
280C2×Dic3—22no—
281C2×Dic3—51no—
282C2×Dic3—54no—
283C2×Dic3—51no—
284C2×Dic3—51no—
285C2×Dic3—51no—
286C2×Dic3—102no—
287C2×Dic3—102no—
288C2×Dic3—102no—
289C2×Dic3—102no—
290C2×Dic3—102no—
291C2×Dic3—102no—
292C2×Dic3C22×D511yescharacteristic
293C2×Dic3C22×D511yescharacteristic
294C2×Dic3C22×D511yescharacteristic
295C2×Dic3C22×D511yescharacteristic
296Dic6—22no—
297Dic6—102no—
298Dic6—102no—
299Dic6—102no—
300C22×C5—21no—
301C22×C5—21no—
302C22×C5C2×Dic612yescomplemented
303C22×C5C23×D311yescharacteristic
304C2×D5—21no—
305C2×D5—21no—
306C2×D5—28no—
307C2×D5C2×Dic612yes—
308C2×D5C2×Dic612yescomplemented
309C2×D5D3×D412yes—
310C2×D5#23712yes—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—32no—
314C4×C5—61no—
315C4×C5—61no—
316C4×C5—61no—
317Dic5—21no—
318Dic5—21no—
319Dic5—32no—
320Dic5—61no—
321Dic5—61no—
322Dic5—61no—
323C24—51no—
324C2×Q8—31no—
325C2×Q8—151no—
326C2×Q8—151no—
327C2×Q8—151no—
328C2×Q8—304no—
329C22×C4—31no—
330C22×C4—51no—
331C22×C4—51no—
332C22×C4—151no—
333C22×C4—151no—
334C22×C4—151no—
335C22×C4—151no—
336C22×C4—151no—
337C22×C4—151no—
338C22×C4—151no—
339C22×C4—304no—
340C4⋊C4—31no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—151no—
344C4⋊C4—151no—
345C4⋊C4—151no—
346C4⋊C4—151no—
347C4⋊C4—151no—
348C4⋊C4—151no—
349C4⋊C4—151no—
350C4⋊C4—151no—
351C4⋊C4—151no—
352C22⋊C4—31no—
353C22⋊C4—51no—
354C22⋊C4—51no—
355C22⋊C4—51no—
356C22⋊C4—151no—
357C22⋊C4—151no—
358C22⋊C4—151no—
359C22⋊C4D3×D511yescharacteristic
360C3×C5C2×C22⋊Q811yescharacteristic, complemented, Hall
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—52no—
364C22×C3—52no—
365C22×C3—52no—
366C22×C3—52no—
367C22×C3—108no—
368C22×C3—101no—
369C22×C3—101no—
370C22×C3Q8×D512yescomplemented
371C22×C3C23×D511yescharacteristic
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—101no—
375C3×C4—101no—
376Dic3—21no—
377Dic3—21no—
378Dic3—21no—
379Dic3—52no—
380Dic3—101no—
381Dic3—101no—
382Dic3—101no—
383Dic3D4×D512yes—
384C2×C5—22no—
385C2×C5C22×Dic611yescharacteristic
386C2×C5C2×D3×D411yescharacteristic
387C2×C5C2×#23711yescharacteristic
388D5—22no—
389D5#87814yesdirect product factor
390C23—51no—
391C23—54no—
392C23—51no—
393C23—102no—
394C23—102no—
395C23C2×D3×D511yescharacteristic
396Q8—62no—
397Q8—302no—
398Q8—302no—
399Q8—302no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—51no—
405C2×C4—51no—
406C2×C4—62no—
407C2×C4—102no—
408C2×C4—102no—
409C2×C4—154no—
410C2×C4—151no—
411C2×C4—151no—
412C2×C4—151no—
413C2×C4—151no—
414C2×C4—302no—
415C2×C4—302no—
416C2×C4—302no—
417C2×C4—302no—
418C2×C4—302no—
419C2×C4—302no—
420C2×C4C2×D3×D511yescharacteristic
421C2×C4C2×D3×D511yescharacteristic
422C2×C3—22no—
423C2×C3—54no—
424C2×C3—102no—
425C2×C3C2×D4×D511yescharacteristic
426C2×C3C2×Q8×D511yescharacteristic
427C2×C3D5×#4111yescharacteristic
428C5C2×#87811yescharacteristic, complemented, Sylow
429C22—21no—
430C22—21no—
431C22—52no—
432C22—52no—
433C22—52no—
434C22—52no—
435C22—108no—
436C22—101no—
437C22—101no—
438C22D5×Dic612yescomplemented
439C22C22×D3×D511yescharacteristic
440C4—21no—
441C4—21no—
442C4—32no—
443C4—61no—
444C4—61no—
445C4—61no—
446C4—101no—
447C4—101no—
448C4—152no—
449C4—301no—
450C4—301no—
451C4—301no—
452C3D5×C22⋊Q811yescharacteristic, complemented, Sylow
453C2—22no—
454C2—54no—
455C2—102no—
456C2C2×D5×Dic611yescharacteristic
457C2D3×D4×D511yescharacteristic
458C2D5×#23711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
prime radicalC23×C3×D5C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5C2×C22⋊Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D5×Dic6C2semidirect
3. #91767C2semidirect
4. C5×#878C2semidirect
6. #91576C2semidirect
7. #91575C2semidirect
9. #91431C2semidirect
10. #91430C2semidirect
11. #91422C2semidirect
12. #91421C2semidirect
13. D5×C4⋊Dic3C2semidirect
14. D5×Dic3⋊C4C2semidirect
15. D5×Dic3⋊C4C2semidirect
19. C2×Dic30C22semidirect
21. C2×C5×Dic6C22semidirect
24. C2×#1204C22semidirect
25. C2×#1204C22semidirect
36. C4⋊Dic15C22semidirect
37. #6712C22semidirect
38. #6712C22semidirect
40. C5×C4⋊Dic3C22semidirect
41. C5×Dic3⋊C4C22semidirect
42. C5×Dic3⋊C4C22semidirect
46. Dic3⋊Dic5C22semidirect
47. Dic3⋊Dic5C22semidirect
48. #6669C22semidirect
49. #6669C22semidirect
50. #6668C22semidirect
51. #6668C22semidirect
73. C22×C3×D5Q8semidirect
116. #878D5direct (N × H)
157. C22×C3×C5C2×Q8semidirect
164. C2×C3×D5C2×Q8semidirect
194. C2×Dic6C2×D5semidirect
206. C4⋊Dic3C2×D5semidirect
213. Dic3⋊C4C2×D5semidirect
214. Dic3⋊C4C2×D5semidirect
218. C22×D5Dic6semidirect
267. C3×D5C22⋊Q8semidirect
302. C22×C5C2×Dic6semidirect
308. C2×D5C2×Dic6semidirect
360. C3×C5C2×C22⋊Q8semidirect
370. C22×C3Q8×D5semidirect
389. D5#878direct (N × H)
428. C5C2×#878semidirect
438. C22D5×Dic6semidirect
452. C3D5×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×D5×Dic32
C2×D5×Dic62
#917672
C5×#8782
C3×D5×C22⋊C42
#915762
#915752
D5×#2172
#914312
#914302
#914222
#914212
D5×C4⋊Dic32
D5×Dic3⋊C42
D5×C22⋊Q83
C2×#8785