Finite Group Databasebeta

← C5×#60305 #11983025

Subgroups — C5×#60305 #11983025

Order 1280.

2004 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3481C211yesmaximal, characteristic
2C5×#2994C214yesmaximal, complemented
3C5×#2990C212yesmaximal, complemented
4C5×#2979C211yesmaximal, characteristic, complemented
5C5×#2977C214yesmaximal, complemented
6C5×#2974C214yesmaximal, complemented
7C5×#2966C214yesmaximal, complemented
8C5×C8×#41C212yesmaximal, complemented
9C5×#2931C211yesmaximal, characteristic, complemented
10C5×#2929C214yesmaximal, complemented
11C2×C5×C8×D4C214yesmaximal, complemented
12C2×C5×#131C2212yescomplemented
13C23×C5×C8C2214yescomplemented
14C5×#559C411yescharacteristic
15C5×#554C412yes—
16C5×#553C414yes—
17C5×#551C414yes—
18C5×#546C414yes—
19C5×D42C411yescharacteristic
20C5×#532C2211yescharacteristic
21C5×#529C2214yes—
22C5×#525C2214yes—
23C4×C5×#41C2212yes—
24C2×C4×C5×D4C2214yes—
25C5×#446C2214yescomplemented
26C5×C8×Q8C2212yescomplemented
27C5×C8×Q8C2212yescomplemented
28C5×#436C2214yescomplemented
29C5×#436C2214yescomplemented
30C5×#435C2218yescomplemented
31C5×#435C2218yescomplemented
32C5×#435C2218yescomplemented
33C5×C8×D4C2214yes—
34C5×C8×D4C2212yes—
35C5×C8×D4C2212yes—
36C5×C8×D4C2218yescomplemented
37C5×C8×D4C2214yescomplemented
38C5×C8×D4C2218yescomplemented
39C5×C8×D4C2214yescomplemented
40C5×C8×D4C2214yescomplemented
41C5×C8×D4C2214yescomplemented
42C5×#433C2218yescomplemented
43C5×#432C2214yescomplemented
44C5×#431C2212yescomplemented
45C5×#431C2214yescomplemented
46C5×#424C2214yescomplemented
47C5×#423C2211yescharacteristic, complemented
48C2×C5×C4⋊C8C2214yescomplemented
49C5×#408C2214yescomplemented
50C5×#408C2212yescomplemented
51C5×#407C2214yescomplemented
52C2×C5×C22⋊C8C2218yescomplemented
53C5×#405C2214yescomplemented
54C2×C4×C5×C8C2214yescomplemented
55C2×C4×C5×C8C2211yescharacteristic, complemented
56#60305C511yesmaximal, characteristic, direct product factor, Sylow
57C2×C5×#41C2×C412yes—
58C2×C5×#41C2312yes—
59C22×C5×D4C2×C414yes—
60C23×C4×C5C2314yes—
61C5×#131—44no—
62C2×C5×M16C2314yes—
63C2×C5×M16C2314yes—
64C2×C5×M16C2312yes—
65C22×C5×C8—48no—
66C22×C5×C8C2314yes—
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2314yes—
69C22×C5×C8C2314yes—
70C22×C5×C8C2311yescharacteristic
71C22×C5×C8C2318yes—
72C22×C5×C8C2314yes—
73C5×C4⋊Q8C2×C411yescharacteristic
74C5×#127C2×C411yescharacteristic
75C5×#126C2×C418yes—
76C5×#125C2×C414yes—
77C5×#124C2×C414yes—
78C5×#124C2×C412yes—
79C5×#123C2×C414yes—
80C5×#123C2×C414yes—
81C5×#123C2×C418yes—
82C5×C22⋊Q8C2×C418yes—
83C5×C22⋊Q8C2×C414yes—
84C5×#121C2×C418yes—
85C5×#121C2×C414yes—
86C5×#121C2×C414yes—
87C5×#120C2×C414yes—
88C5×#120C2×C414yes—
89C4×C5×Q8C2×C412yes—
90C4×C5×Q8C2312yes—
91C4×C5×D4C2×C414yes—
92C4×C5×D4C2×C418yes—
93C4×C5×D4C2×C414yes—
94C4×C5×D4C2×C412yes—
95C4×C5×D4C2×C414yes—
96C4×C5×D4C2312yes—
97C4×C5×D4C2318yes—
98C4×C5×D4C2314yes—
99C4×C5×D4C2314yes—
100C5×#117C2×C414yes—
101C5×#117C2×C412yes—
102C5×#117C2314yes—
103C2×C5×C4⋊C4C2×C414yes—
104C2×C5×C22⋊C4C2×C418yes—
105C2×C42×C5C2314yes—
106C2×C42×C5C2311yescharacteristic
107C5×C4⋊C8C2312yes—
108C5×C4⋊C8C2314yes—
109C5×C4⋊C8C2318yes—
110C5×C4⋊C8C2314yes—
111C5×C4⋊C8C2312yes—
112C5×C4⋊C8C2314yes—
113C5×C22⋊C8C2314yes—
114C5×C22⋊C8C2318yes—
115C5×C22⋊C8C2318yes—
116C5×C22⋊C8C2318yes—
117C5×C22⋊C8C2314yes—
118C5×C22⋊C8C2314yes—
119C5×#97C2318yes—
120C4×C5×C8C2314yes—
121C4×C5×C8C2312yes—
122C4×C5×C8C2314yes—
123C4×C5×C8C2312yes—
124C4×C5×C8C2314yes—
125#3481C2×C511yescharacteristic
126#2994C2×C514yescomplemented
127#2990C2×C512yescomplemented
128#2979C2×C511yescharacteristic, complemented
129#2977C2×C514yescomplemented
130#2974C2×C514yescomplemented
131#2966C2×C514yescomplemented
132C8×#41C2×C512yescomplemented
133#2931C2×C511yescharacteristic, complemented
134#2929C2×C514yescomplemented
135C2×C8×D4C2×C514yescomplemented
136C24×C5C22×C414yes—
137C5×#41—28no—
138C5×#41—42no—
139C5×#41—42no—
140C2×C5×Q8C22×C412yes—
141C2×C5×Q8C22×C412yes—
142C2×C5×D4—216no—
143C2×C5×D4C22×C414yes—
144C2×C5×D4C22×C418yes—
145C2×C5×D4C22×C414yes—
146C2×C5×D4C22×C412yes—
147C2×C5×D4C22×C414yes—
148C2×C5×D4C22×C412yes—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5C22×C414yes—
152C22×C4×C5C22×C414yes—
153C22×C4×C5C22×C412yes—
154C22×C4×C5C22×C418yes—
155C22×C4×C5C22×C414yes—
156C22×C4×C5C2414yes—
157C22×C4×C5C2411yescharacteristic
158C22×C4×C5C2414yes—
159C5×M16—44no—
160C5×M16—42no—
161C5×M16—44no—
162C2×C5×C8—28no—
163C2×C5×C8—28no—
164C2×C5×C8—22no—
165C2×C5×C8—44no—
166C2×C5×C8—48no—
167C2×C5×C8—41no—
168C2×C5×C8—44no—
169C2×C5×C8C2414yes—
170C2×C5×C8C2414yes—
171C2×C5×C8C2412yes—
172C2×C5×C8C2411yescharacteristic
173C2×C5×C8C2414yes—
174C2×C5×C8C2411yescharacteristic
175C5×C4⋊C4C22×C412yes—
176C5×C4⋊C4C22×C414yes—
177C5×C4⋊C4C22×C418yes—
178C5×C4⋊C4C22×C412yes—
179C5×C4⋊C4C22×C414yes—
180C5×C4⋊C4C22×C414yes—
181C5×C4⋊C4C22×C418yes—
182C5×C22⋊C4C22×C414yes—
183C5×C22⋊C4C22×C418yes—
184C5×C22⋊C4C22×C418yes—
185C5×C22⋊C4C22×C418yes—
186C5×C22⋊C4C22×C414yes—
187C5×C22⋊C4C22×C414yes—
188C5×C22⋊C4C22×C414yes—
189C5×C22⋊C4C22×C414yes—
190C42×C5C22×C411yescharacteristic
191C42×C5C22×C414yes—
192C42×C5C22×C414yes—
193C42×C5C22×C411yescharacteristic
194C42×C5C2412yes—
195C42×C5C2414yes—
196C2×#131C22×C512yescomplemented
197C23×C8C22×C514yescomplemented
198#559C4×C511yescharacteristic
199#554C4×C512yes—
200#553C4×C514yes—
201#551C4×C514yes—
202#546C4×C514yes—
203D42C4×C511yescharacteristic
204#532C22×C511yescharacteristic
205#529C22×C514yes—
206#525C22×C514yes—
207C4×#41C22×C512yes—
208C2×C4×D4C22×C514yes—
209#446C22×C514yescomplemented
210C8×Q8C22×C512yescomplemented
211C8×Q8C22×C512yescomplemented
212#436C22×C514yescomplemented
213#436C22×C514yescomplemented
214#435C22×C518yescomplemented
215#435C22×C518yescomplemented
216#435C22×C518yescomplemented
217C8×D4C22×C512yes—
218C8×D4C22×C514yes—
219C8×D4C22×C512yes—
220C8×D4C22×C518yescomplemented
221C8×D4C22×C514yescomplemented
222C8×D4C22×C518yescomplemented
223C8×D4C22×C514yescomplemented
224C8×D4C22×C514yescomplemented
225C8×D4C22×C514yescomplemented
226#433C22×C518yescomplemented
227#432C22×C514yescomplemented
228#431C22×C514yescomplemented
229#431C22×C512yescomplemented
230#424C22×C514yescomplemented
231#423C22×C511yescharacteristic, complemented
232C2×C4⋊C8C22×C514yescomplemented
233#408C22×C514yescomplemented
234#408C22×C512yescomplemented
235#407C22×C514yescomplemented
236C2×C22⋊C8C22×C518yescomplemented
237#405C22×C514yescomplemented
238C2×C4×C8C22×C514yescomplemented
239C2×C4×C8C22×C511yescharacteristic, complemented
240C23×C5—216no—
241C23×C5—44no—
242C23×C5C23×C414yes—
243C23×C5C23×C411yescharacteristic
244C23×C5C23×C414yes—
245C5×Q8—24no—
246C5×Q8—42no—
247C5×D4—216no—
248C5×D4—28no—
249C5×D4—28no—
250C5×D4—44no—
251C5×D4—42no—
252C5×D4#13118yescomplemented
253C2×C4×C5—28no—
254C2×C4×C5—28no—
255C2×C4×C5—24no—
256C2×C4×C5—24no—
257C2×C4×C5—24no—
258C2×C4×C5—21no—
259C2×C4×C5—24no—
260C2×C4×C5—21no—
261C2×C4×C5—24no—
262C2×C4×C5—44no—
263C2×C4×C5—48no—
264C2×C4×C5C23×C412yes—
265C2×C4×C5C23×C414yes—
266C2×C4×C5C23×C411yescharacteristic
267C2×C4×C5C23×C414yes—
268C2×C4×C5C23×C414yes—
269C2×C4×C5C23×C414yes—
270C2×C4×C5C23×C412yes—
271C2×C4×C5C2511yescharacteristic
272C5×C8—24no—
273C5×C8—24no—
274C5×C8—21no—
275C5×C8—22no—
276C5×C8—24no—
277C5×C8H(5,𝔽2)12yes—
278C2×#41C2×C4×C512yes—
279C2×#41C23×C512yes—
280C22×D4C2×C4×C514yes—
281C23×C4C23×C514yes—
282#131—44no—
283C2×M16C23×C514yes—
284C2×M16C23×C512yes—
285C2×M16C23×C514yes—
286C22×C8—48no—
287C22×C8C23×C514yes—
288C22×C8C23×C514yes—
289C22×C8C23×C511yescharacteristic
290C22×C8C23×C518yes—
291C22×C8C23×C514yes—
292C22×C8C23×C514yes—
293C22×C8C23×C511yescharacteristic
294C4⋊Q8C2×C4×C511yescharacteristic
295#127C2×C4×C511yescharacteristic
296#126C2×C4×C518yes—
297#125C2×C4×C514yes—
298#124C2×C4×C514yes—
299#124C2×C4×C512yes—
300#123C2×C4×C514yes—
301#123C2×C4×C514yes—
302#123C2×C4×C518yes—
303C22⋊Q8C2×C4×C518yes—
304C22⋊Q8C2×C4×C514yes—
305#121C2×C4×C518yes—
306#121C2×C4×C514yes—
307#121C2×C4×C514yes—
308#120C2×C4×C514yes—
309#120C2×C4×C514yes—
310C4×Q8C2×C4×C512yes—
311C4×Q8C23×C512yes—
312C4×D4C2×C4×C514yes—
313C4×D4C2×C4×C518yes—
314C4×D4C2×C4×C514yes—
315C4×D4C2×C4×C512yes—
316C4×D4C2×C4×C514yes—
317C4×D4C23×C512yes—
318C4×D4C23×C518yes—
319C4×D4C23×C514yes—
320C4×D4C23×C514yes—
321#117C2×C4×C514yes—
322#117C2×C4×C512yes—
323#117C23×C514yes—
324C2×C4⋊C4C2×C4×C514yes—
325C2×C22⋊C4C2×C4×C518yes—
326C2×C42C23×C514yes—
327C2×C42C23×C511yescharacteristic
328C4⋊C8C23×C512yes—
329C4⋊C8C23×C514yes—
330C4⋊C8C23×C518yes—
331C4⋊C8C23×C514yes—
332C4⋊C8C23×C512yes—
333C4⋊C8C23×C514yes—
334C22⋊C8C23×C514yes—
335C22⋊C8C23×C518yes—
336C22⋊C8C23×C518yes—
337C22⋊C8C23×C518yes—
338C22⋊C8C23×C514yes—
339C22⋊C8C23×C514yes—
340#97C23×C518yes—
341C4×C8C23×C512yes—
342C4×C8C23×C514yes—
343C4×C8C23×C514yes—
344C4×C8C23×C512yes—
345C4×C8C23×C514yes—
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—24no—
349C22×C5—21no—
350C22×C5—22no—
351C22×C5—28no—
352C22×C5—416no—
353C22×C5C2×#13118yes—
354C22×C5C24×C411yescharacteristic
355C4×C5—24no—
356C4×C5—24no—
357C4×C5—21no—
358C4×C5—22no—
359C4×C5—24no—
360C4×C5—24no—
361C4×C5C2×#13114yes—
362C4×C5C2×H(5,𝔽2)11yescharacteristic
363C4×C5#58511yescharacteristic
364C24C22×C4×C514yes—
365#41—28no—
366#41—42no—
367#41—42no—
368C2×Q8C22×C4×C512yes—
369C2×Q8C22×C4×C512yes—
370C2×D4—216no—
371C2×D4C22×C4×C514yes—
372C2×D4C22×C4×C518yes—
373C2×D4C22×C4×C514yes—
374C2×D4C22×C4×C512yes—
375C2×D4C22×C4×C514yes—
376C2×D4C22×C4×C512yes—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4C22×C4×C514yes—
380C22×C4C22×C4×C514yes—
381C22×C4C22×C4×C512yes—
382C22×C4C22×C4×C518yes—
383C22×C4C22×C4×C514yes—
384C22×C4C24×C514yes—
385C22×C4C24×C511yescharacteristic
386C22×C4C24×C514yes—
387M16—42no—
388M16—44no—
389M16—44no—
390C2×C8—28no—
391C2×C8—28no—
392C2×C8—22no—
393C2×C8—44no—
394C2×C8—48no—
395C2×C8—41no—
396C2×C8—44no—
397C2×C8C24×C512yes—
398C2×C8C24×C514yes—
399C2×C8C24×C514yes—
400C2×C8C24×C511yescharacteristic
401C2×C8C24×C514yes—
402C2×C8C24×C511yescharacteristic
403C4⋊C4C22×C4×C512yes—
404C4⋊C4C22×C4×C514yes—
405C4⋊C4C22×C4×C518yes—
406C4⋊C4C22×C4×C512yes—
407C4⋊C4C22×C4×C514yes—
408C4⋊C4C22×C4×C514yes—
409C4⋊C4C22×C4×C518yes—
410C22⋊C4C22×C4×C514yes—
411C22⋊C4C22×C4×C518yes—
412C22⋊C4C22×C4×C518yes—
413C22⋊C4C22×C4×C518yes—
414C22⋊C4C22×C4×C514yes—
415C22⋊C4C22×C4×C514yes—
416C22⋊C4C22×C4×C514yes—
417C22⋊C4C22×C4×C514yes—
418C42C22×C4×C511yescharacteristic
419C42C22×C4×C514yes—
420C42C22×C4×C514yes—
421C42C22×C4×C511yescharacteristic
422C42C24×C512yes—
423C42C24×C514yes—
424C2×C5—28no—
425C2×C5—41no—
426C2×C5—44no—
427C2×C5C4×H(5,𝔽2)11yescharacteristic
428C2×C5C22×#13112yes—
429C23—216no—
430C23—44no—
431C23C23×C4×C514yes—
432C23C23×C4×C511yescharacteristic
433C23C23×C4×C514yes—
434Q8—24no—
435Q8—42no—
436D4—216no—
437D4—28no—
438D4—28no—
439D4—44no—
440D4—42no—
441D4C5×#13118yescomplemented
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4C23×C4×C512yes—
454C2×C4C23×C4×C514yes—
455C2×C4C23×C4×C511yescharacteristic
456C2×C4C23×C4×C514yes—
457C2×C4C23×C4×C514yes—
458C2×C4C23×C4×C514yes—
459C2×C4C23×C4×C512yes—
460C2×C4C25×C511yescharacteristic
461C8—24no—
462C8—24no—
463C8—21no—
464C8—22no—
465C8—24no—
466C8C5×H(5,𝔽2)12yes—
467C5#6030511yescharacteristic, direct product factor, Sylow
468C22—24no—
469C22—24no—
470C22—24no—
471C22—21no—
472C22—22no—
473C22—28no—
474C22—416no—
475C22C2×C5×#13118yes—
476C22C24×C4×C511yescharacteristic
477C4—24no—
478C4—24no—
479C4—21no—
480C4—22no—
481C4—24no—
482C4—24no—
483C4C2×C5×#13114yes—
484C4C2×C5×H(5,𝔽2)11yescharacteristic
485C4C5×#58511yescharacteristic
486C2—28no—
487C2—44no—
488C2—41no—
489C2C4×C5×H(5,𝔽2)11yescharacteristic
490C2C22×C5×#13112yes—
subgroupG/H
centerC2×C5×C8C24
commutatorC22C24×C4×C5
Frattini, cosocleC2×C4C25×C5
socleC22×C5C24×C4
prime radicalC5×#3481C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C5×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2994C2semidirect
3. C5×#2990C2semidirect
4. C5×#2979C2semidirect
5. C5×#2977C2semidirect
6. C5×#2974C2semidirect
7. C5×#2966C2semidirect
8. C5×C8×#41C2semidirect
9. C5×#2931C2semidirect
10. C5×#2929C2semidirect
11. C2×C5×C8×D4C2semidirect
12. C2×C5×#131C22semidirect
13. C23×C5×C8C22semidirect
25. C5×#446C22semidirect
26. C5×C8×Q8C22semidirect
27. C5×C8×Q8C22semidirect
28. C5×#436C22semidirect
29. C5×#436C22semidirect
30. C5×#435C22semidirect
31. C5×#435C22semidirect
32. C5×#435C22semidirect
36. C5×C8×D4C22semidirect
37. C5×C8×D4C22semidirect
38. C5×C8×D4C22semidirect
39. C5×C8×D4C22semidirect
40. C5×C8×D4C22semidirect
41. C5×C8×D4C22semidirect
42. C5×#433C22semidirect
43. C5×#432C22semidirect
44. C5×#431C22semidirect
45. C5×#431C22semidirect
46. C5×#424C22semidirect
47. C5×#423C22semidirect
48. C2×C5×C4⋊C8C22semidirect
49. C5×#408C22semidirect
50. C5×#408C22semidirect
51. C5×#407C22semidirect
52. C2×C5×C22⋊C8C22semidirect
53. C5×#405C22semidirect
54. C2×C4×C5×C8C22semidirect
55. C2×C4×C5×C8C22semidirect
56. #60305C5direct (N × H)
126. #2994C2×C5semidirect
127. #2990C2×C5semidirect
128. #2979C2×C5semidirect
129. #2977C2×C5semidirect
130. #2974C2×C5semidirect
131. #2966C2×C5semidirect
132. C8×#41C2×C5semidirect
133. #2931C2×C5semidirect
134. #2929C2×C5semidirect
135. C2×C8×D4C2×C5semidirect
196. C2×#131C22×C5semidirect
197. C23×C8C22×C5semidirect
209. #446C22×C5semidirect
210. C8×Q8C22×C5semidirect
211. C8×Q8C22×C5semidirect
212. #436C22×C5semidirect
213. #436C22×C5semidirect
214. #435C22×C5semidirect
215. #435C22×C5semidirect
216. #435C22×C5semidirect
220. C8×D4C22×C5semidirect
221. C8×D4C22×C5semidirect
222. C8×D4C22×C5semidirect
223. C8×D4C22×C5semidirect
224. C8×D4C22×C5semidirect
225. C8×D4C22×C5semidirect
226. #433C22×C5semidirect
227. #432C22×C5semidirect
228. #431C22×C5semidirect
229. #431C22×C5semidirect
230. #424C22×C5semidirect
231. #423C22×C5semidirect
232. C2×C4⋊C8C22×C5semidirect
233. #408C22×C5semidirect
234. #408C22×C5semidirect
235. #407C22×C5semidirect
236. C2×C22⋊C8C22×C5semidirect
237. #405C22×C5semidirect
238. C2×C4×C8C22×C5semidirect
239. C2×C4×C8C22×C5semidirect
252. C5×D4#131semidirect
441. D4C5×#131semidirect
467. C5#60305direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34812
C5×#29942
C5×#29902
C5×#29792
C5×#29772
C5×#29742
C5×#29662
C5×C8×#412
C5×#29312
C5×#29292
C2×C5×C8×D42
#603055