Finite Group Databasebeta

← C5×#60317 #11983037

Subgroups — C5×#60317 #11983037

Order 1280.

1892 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×#2993C214yesmaximal, complemented
3C5×#2991C212yesmaximal, complemented
4C5×#2979C211yesmaximal, characteristic, complemented
5C5×#2978C214yesmaximal, complemented
6C5×#2975C214yesmaximal, complemented
7C5×#2969C214yesmaximal, complemented
8C5×#2965C212yesmaximal, complemented
9C5×D4×M16C214yesmaximal, complemented
10C5×#2931C211yesmaximal, characteristic, complemented
11C5×#2930C214yesmaximal, complemented
12C2×C5×#131C2212yescomplemented
13C22×C5×M16C2214yescomplemented
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×#446C2212yescomplemented
26C5×#446C2212yescomplemented
27C5×C8×Q8C2214yescomplemented
28C5×#436C2214yes—
29C5×#436C2212yes—
30C5×#436C2212yes—
31C5×#436C2218yescomplemented
32C5×#436C2218yescomplemented
33C5×#436C2214yescomplemented
34C5×#436C2218yescomplemented
35C5×#436C2214yescomplemented
36C5×#435C2218yescomplemented
37C5×#435C2214yescomplemented
38C5×#435C2218yescomplemented
39C5×#435C2214yescomplemented
40C5×C8×D4C2214yescomplemented
41C5×C8×D4C2214yescomplemented
42C5×#433C2218yescomplemented
43C5×#431C2212yescomplemented
44C5×#431C2214yescomplemented
45C5×#431C2214yescomplemented
46C5×#424C2214yescomplemented
47C5×#423C2211yescharacteristic, complemented
48C5×#423C2214yescomplemented
49C5×#408C2214yescomplemented
50C5×#408C2212yescomplemented
51C5×#407C2214yescomplemented
52C5×#407C2218yescomplemented
53C5×#405C2214yescomplemented
54C4×C5×M16C2214yescomplemented
55C2×C4×C5×C8C2211yescharacteristic, complemented
56#60317C511yesmaximal, characteristic, direct product factor, Sylow
57C2×C5×#41C2×C412yes—
58C2×C5×#41C2312yes—
59C22×C5×D4C2×C414yes—
60C23×C4×C5C2314yes—
61C5×#131—44no—
62C2×C5×M16—48no—
63C2×C5×M16C2314yes—
64C2×C5×M16C2314yes—
65C2×C5×M16C2314yes—
66C2×C5×M16C2318yes—
67C2×C5×M16C2314yes—
68C2×C5×M16C2312yes—
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2314yes—
71C22×C5×C8C2311yescharacteristic
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×D4C2314yes—
97C4×C5×D4C2318yes—
98C4×C5×D4C2314yes—
99C4×C5×D4C2312yes—
100C5×#117C2×C414yes—
101C5×#117C2×C412yes—
102C5×#117C2314yes—
103C2×C5×C4⋊C4C2×C414yes—
104C2×C5×C22⋊C4C2×C418yes—
105C2×C42×C5C2311yescharacteristic
106C2×C42×C5C2314yes—
107C5×C4⋊C8C2312yes—
108C5×C4⋊C8C2314yes—
109C5×C4⋊C8C2318yes—
110C5×C4⋊C8C2314yes—
111C5×C4⋊C8C2312yes—
112C5×C4⋊C8C2314yes—
113C5×C22⋊C8C2314yes—
114C5×C22⋊C8C2314yes—
115C5×C22⋊C8C2318yes—
116C5×C22⋊C8C2318yes—
117C5×C22⋊C8C2318yes—
118C5×C22⋊C8C2314yes—
119C5×#97C2314yes—
120C5×#97C2314yes—
121C4×C5×C8C2312yes—
122C4×C5×C8C2312yes—
123C4×C5×C8C2314yes—
124C4×C5×C8C2318yes—
125#3481C2×C511yescharacteristic
126#2993C2×C514yescomplemented
127#2991C2×C512yescomplemented
128#2979C2×C511yescharacteristic, complemented
129#2978C2×C514yescomplemented
130#2975C2×C514yescomplemented
131#2969C2×C514yescomplemented
132#2965C2×C512yescomplemented
133D4×M16C2×C514yescomplemented
134#2931C2×C511yescharacteristic, complemented
135#2930C2×C514yescomplemented
136C24×C5C22×C414yes—
137C5×#41—42no—
138C5×#41—42no—
139C5×#41—44no—
140C2×C5×Q8C22×C412yes—
141C2×C5×Q8C22×C412yes—
142C2×C5×D4—48no—
143C2×C5×D4C22×C412yes—
144C2×C5×D4C22×C414yes—
145C2×C5×D4C22×C414yes—
146C2×C5×D4C22×C418yes—
147C2×C5×D4C22×C412yes—
148C2×C5×D4C22×C414yes—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5C22×C414yes—
152C22×C4×C5C22×C412yes—
153C22×C4×C5C22×C414yes—
154C22×C4×C5C22×C418yes—
155C22×C4×C5C22×C414yes—
156C22×C4×C5C2411yescharacteristic
157C22×C4×C5C2414yes—
158C22×C4×C5C2414yes—
159C5×M16—28no—
160C5×M16—48no—
161C5×M16—44no—
162C5×M16—44no—
163C5×M16—44no—
164C5×M16—42no—
165C2×C5×C8—28no—
166C2×C5×C8—22no—
167C2×C5×C8—41no—
168C2×C5×C8—44no—
169C2×C5×C8C2414yes—
170C2×C5×C8C2411yescharacteristic
171C2×C5×C8C2414yes—
172C2×C5×C8C2412yes—
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
197C22×M16C22×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×C512yescomplemented
210#446C22×C512yescomplemented
211C8×Q8C22×C514yescomplemented
212#436C22×C512yes—
213#436C22×C514yes—
214#436C22×C512yes—
215#436C22×C518yescomplemented
216#436C22×C518yescomplemented
217#436C22×C514yescomplemented
218#436C22×C518yescomplemented
219#436C22×C514yescomplemented
220#435C22×C514yescomplemented
221#435C22×C518yescomplemented
222#435C22×C518yescomplemented
223#435C22×C514yescomplemented
224C8×D4C22×C514yescomplemented
225C8×D4C22×C514yescomplemented
226#433C22×C518yescomplemented
227#431C22×C514yescomplemented
228#431C22×C512yescomplemented
229#431C22×C514yescomplemented
230#424C22×C514yescomplemented
231#423C22×C511yescharacteristic, complemented
232#423C22×C514yescomplemented
233#408C22×C514yescomplemented
234#408C22×C512yescomplemented
235#407C22×C514yescomplemented
236#407C22×C518yescomplemented
237#405C22×C514yescomplemented
238C4×M16C22×C514yescomplemented
239C2×C4×C8C22×C511yescharacteristic, complemented
240C23×C5—44no—
241C23×C5—48no—
242C23×C5C23×C414yes—
243C23×C5C23×C414yes—
244C23×C5C23×C411yescharacteristic
245C5×Q8—24no—
246C5×Q8—42no—
247C5×D4—24no—
248C5×D4—28no—
249C5×D4—48no—
250C5×D4—44no—
251C5×D4—42no—
252C5×D4—44no—
253C2×C4×C5—28no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—21no—
257C2×C4×C5—24no—
258C2×C4×C5—21no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—44no—
262C2×C4×C5—48no—
263C2×C4×C5—44no—
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—21no—
274C5×C8—24no—
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×M16—48no—
284C2×M16C23×C514yes—
285C2×M16C23×C518yes—
286C2×M16C23×C514yes—
287C2×M16C23×C512yes—
288C2×M16C23×C514yes—
289C2×M16C23×C514yes—
290C22×C8C23×C514yes—
291C22×C8C23×C511yescharacteristic
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×C514yes—
336C22⋊C8C23×C518yes—
337C22⋊C8C23×C518yes—
338C22⋊C8C23×C518yes—
339C22⋊C8C23×C514yes—
340#97C23×C514yes—
341#97C23×C514yes—
342C4×C8C23×C512yes—
343C4×C8C23×C512yes—
344C4×C8C23×C514yes—
345C4×C8C23×C518yes—
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—21no—
349C22×C5—22no—
350C22×C5—24no—
351C22×C5—24no—
352C22×C5—28no—
353C22×C5—88no—
354C22×C5C24×C411yescharacteristic
355C4×C5—24no—
356C4×C5—24no—
357C4×C5—21no—
358C4×C5—22no—
359C4×C5—24no—
360C4×C5—24no—
361C4×C5#56814yes—
362C4×C5C2×H(5,𝔽2)11yescharacteristic
363C4×C5#58511yescharacteristic
364C24C22×C4×C514yes—
365#41—42no—
366#41—42no—
367#41—44no—
368C2×Q8C22×C4×C512yes—
369C2×Q8C22×C4×C512yes—
370C2×D4—48no—
371C2×D4C22×C4×C512yes—
372C2×D4C22×C4×C514yes—
373C2×D4C22×C4×C514yes—
374C2×D4C22×C4×C518yes—
375C2×D4C22×C4×C512yes—
376C2×D4C22×C4×C514yes—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4C22×C4×C514yes—
380C22×C4C22×C4×C512yes—
381C22×C4C22×C4×C514yes—
382C22×C4C22×C4×C518yes—
383C22×C4C22×C4×C514yes—
384C22×C4C24×C511yescharacteristic
385C22×C4C24×C514yes—
386C22×C4C24×C514yes—
387M16—28no—
388M16—48no—
389M16—44no—
390M16—42no—
391M16—44no—
392M16—44no—
393C2×C8—28no—
394C2×C8—22no—
395C2×C8—44no—
396C2×C8—41no—
397C2×C8C24×C514yes—
398C2×C8C24×C512yes—
399C2×C8C24×C511yescharacteristic
400C2×C8C24×C514yes—
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—41no—
425C2×C5—44no—
426C2×C5—44no—
427C2×C5C4×H(5,𝔽2)11yescharacteristic
428C2×C5C2×#56812yes—
429C23—44no—
430C23—48no—
431C23C23×C4×C514yes—
432C23C23×C4×C514yes—
433C23C23×C4×C511yescharacteristic
434Q8—24no—
435Q8—42no—
436D4—24no—
437D4—28no—
438D4—48no—
439D4—44no—
440D4—42no—
441D4—44no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—44no—
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—21no—
463C8—24no—
464C8—22no—
465C8—24no—
466C8C5×H(5,𝔽2)12yes—
467C5#6031711yescharacteristic, direct product factor, Sylow
468C22—24no—
469C22—24no—
470C22—21no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—28no—
475C22—88no—
476C22C24×C4×C511yescharacteristic
477C4—24no—
478C4—24no—
479C4—21no—
480C4—22no—
481C4—24no—
482C4—24no—
483C4C5×#56814yes—
484C4C2×C5×H(5,𝔽2)11yescharacteristic
485C4C5×#58511yescharacteristic
486C2—44no—
487C2—44no—
488C2—41no—
489C2C4×C5×H(5,𝔽2)11yescharacteristic
490C2C2×C5×#56812yes—
subgroupG/H
centerC2×C4×C5C25
commutatorC22C24×C4×C5
Frattini, cosocleC2×C4C25×C5
socleC22×C5C24×C4
prime radicalC5×#3481C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2993C2semidirect
3. C5×#2991C2semidirect
4. C5×#2979C2semidirect
5. C5×#2978C2semidirect
6. C5×#2975C2semidirect
7. C5×#2969C2semidirect
8. C5×#2965C2semidirect
9. C5×D4×M16C2semidirect
10. C5×#2931C2semidirect
11. C5×#2930C2semidirect
12. C2×C5×#131C22semidirect
13. C22×C5×M16C22semidirect
25. C5×#446C22semidirect
26. C5×#446C22semidirect
27. C5×C8×Q8C22semidirect
31. C5×#436C22semidirect
32. C5×#436C22semidirect
33. C5×#436C22semidirect
34. C5×#436C22semidirect
35. C5×#436C22semidirect
36. C5×#435C22semidirect
37. C5×#435C22semidirect
38. C5×#435C22semidirect
39. C5×#435C22semidirect
40. C5×C8×D4C22semidirect
41. C5×C8×D4C22semidirect
42. C5×#433C22semidirect
43. C5×#431C22semidirect
44. C5×#431C22semidirect
45. C5×#431C22semidirect
46. C5×#424C22semidirect
47. C5×#423C22semidirect
48. C5×#423C22semidirect
49. C5×#408C22semidirect
50. C5×#408C22semidirect
51. C5×#407C22semidirect
52. C5×#407C22semidirect
53. C5×#405C22semidirect
54. C4×C5×M16C22semidirect
55. C2×C4×C5×C8C22semidirect
56. #60317C5direct (N × H)
126. #2993C2×C5semidirect
127. #2991C2×C5semidirect
128. #2979C2×C5semidirect
129. #2978C2×C5semidirect
130. #2975C2×C5semidirect
131. #2969C2×C5semidirect
132. #2965C2×C5semidirect
133. D4×M16C2×C5semidirect
134. #2931C2×C5semidirect
135. #2930C2×C5semidirect
196. C2×#131C22×C5semidirect
197. C22×M16C22×C5semidirect
209. #446C22×C5semidirect
210. #446C22×C5semidirect
211. C8×Q8C22×C5semidirect
215. #436C22×C5semidirect
216. #436C22×C5semidirect
217. #436C22×C5semidirect
218. #436C22×C5semidirect
219. #436C22×C5semidirect
220. #435C22×C5semidirect
221. #435C22×C5semidirect
222. #435C22×C5semidirect
223. #435C22×C5semidirect
224. C8×D4C22×C5semidirect
225. C8×D4C22×C5semidirect
226. #433C22×C5semidirect
227. #431C22×C5semidirect
228. #431C22×C5semidirect
229. #431C22×C5semidirect
230. #424C22×C5semidirect
231. #423C22×C5semidirect
232. #423C22×C5semidirect
233. #408C22×C5semidirect
234. #408C22×C5semidirect
235. #407C22×C5semidirect
236. #407C22×C5semidirect
237. #405C22×C5semidirect
238. C4×M16C22×C5semidirect
239. C2×C4×C8C22×C5semidirect
467. C5#60317direct (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×#29932
C5×#29912
C5×#29792
C5×#29782
C5×#29752
C5×#29692
C5×#29652
C5×D4×M162
C5×#29312
C5×#29302
#603175