Finite Group Databasebeta

← C5×#12765 #11935485

Subgroups — C5×#12765 #11935485

Order 1280.

852 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3481C211yesmaximal, characteristic
2C5×#2132C211yesmaximal, characteristic, complemented
3C5×#2120C212yesmaximal, complemented
4C5×#1891C212yesmaximal, complemented
5C5×#1884C211yesmaximal, characteristic, complemented
6C5×#568C2211yescharacteristic, complemented
7C5×#559C411yescharacteristic, complemented
8C5×#554—21no—
9C5×#553—22no—
10C5×#551—21no—
11C5×#551C412yescomplemented
12C5×#546—22no—
13C5×D42C411yescharacteristic, complemented
14C5×#532C2211yescharacteristic
15C5×#529—22no—
16C5×#525—21no—
17C5×#525C2212yes—
18C4×C5×#41—21no—
19C2×C4×C5×D4—22no—
20C5×#421—21no—
21C5×#413C2212yescomplemented
22C5×#410C2211yescharacteristic, complemented
23C5×#407—22no—
24C5×#356—21no—
25C5×#355—21no—
26C5×#352—22no—
27C5×AΣL(1,𝔽16)—22no—
28C5×#344—22no—
29C5×#339—21no—
30#12765C511yesmaximal, characteristic, direct product factor, Sylow
31C2×C5×#41—21no—
32C2×C5×#41D411yescharacteristic
33C2×C5×#41C2311yescharacteristic
34C22×C5×D4—22no—
35C23×C4×C5—21no—
36C23×C4×C5D412yes—
37C5×#131—42no—
38C2×C5×M16—22no—
39C2×C5×M16—21no—
40C2×C5×M16—22no—
41C5×C4⋊Q8D411yescharacteristic
42C5×#127C2×C411yescharacteristic
43C5×#126—22no—
44C5×#126—22no—
45C5×#125—22no—
46C5×#124—22no—
47C5×#124C2×C411yescharacteristic
48C5×#124D411yescharacteristic
49C5×#123—21no—
50C5×#123—22no—
51C5×#123—21no—
52C5×#123—22no—
53C5×#123C2×C412yes—
54C5×#123D412yes—
55C5×C22⋊Q8—22no—
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8—22no—
58C5×#121—22no—
59C5×#121—22no—
60C5×#121—22no—
61C5×#121—22no—
62C5×#120—21no—
63C5×#120—21no—
64C5×#120C2×C412yes—
65C5×#120D412yes—
66C4×C5×Q8—21no—
67C4×C5×Q8—21no—
68C4×C5×D4—21no—
69C4×C5×D4—22no—
70C4×C5×D4—22no—
71C4×C5×D4—22no—
72C4×C5×D4—22no—
73C4×C5×D4—22no—
74C4×C5×D4—22no—
75C4×C5×D4—22no—
76C4×C5×D4—22no—
77C4×C5×D4—22no—
78C4×C5×D4—21no—
79C5×#117—21no—
80C5×#117—21no—
81C5×#117—22no—
82C5×#117—21no—
83C5×#117D412yes—
84C2×C5×C4⋊C4—22no—
85C2×C5×C22⋊C4—22no—
86C2×C5×C22⋊C4—22no—
87C2×C42×C5—22no—
88C2×C42×C5D411yescharacteristic
89C5×#104—42no—
90C5×#101—22no—
91C5×#100—22no—
92C5×#99—22no—
93C5×C22⋊C8—44no—
94#3481C2×C511yescharacteristic
95#2132C2×C511yescharacteristic, complemented
96#2120C2×C512yescomplemented
97#1891C2×C512yescomplemented
98#1884C2×C511yescharacteristic, complemented
99C24×C5—21no—
100C24×C5C22⋊C412yes—
101C5×#41—41no—
102C5×#41—41no—
103C5×#41—41no—
104C5×#41—41no—
105C5×#41—44no—
106C2×C5×Q8—21no—
107C2×C5×Q8C22⋊C411yescharacteristic
108C2×C5×Q8C2×D411yescharacteristic
109C2×C5×D4—21no—
110C2×C5×D4—22no—
111C2×C5×D4—22no—
112C2×C5×D4—22no—
113C2×C5×D4—22no—
114C2×C5×D4—21no—
115C2×C5×D4—48no—
116C2×C5×D4C22⋊C411yescharacteristic
117C2×C5×D4C22×C411yescharacteristic
118C2×C5×D4C2×D412yes—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—21no—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5—22no—
125C22×C4×C5—21no—
126C22×C4×C5—22no—
127C22×C4×C5—42no—
128C22×C4×C5—42no—
129C22×C4×C5—81no—
130C22×C4×C5—81no—
131C22×C4×C5C22⋊C412yes—
132C22×C4×C5C2×D411yescharacteristic
133C22×C4×C5C2×D412yes—
134C5×M16—22no—
135C5×M16—42no—
136C5×M16—41no—
137C5×M16—42no—
138C5×M16—42no—
139C2×C5×C8—24no—
140C2×C5×C8—41no—
141C2×C5×C8—42no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4—22no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4—22no—
146C5×C4⋊C4—22no—
147C5×C4⋊C4—22no—
148C5×C4⋊C4—22no—
149C5×C4⋊C4—22no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—41no—
152C5×C22⋊C4—22no—
153C5×C22⋊C4—22no—
154C5×C22⋊C4—22no—
155C5×C22⋊C4—21no—
156C5×C22⋊C4—21no—
157C5×C22⋊C4—22no—
158C5×C22⋊C4—22no—
159C5×C22⋊C4—22no—
160C5×C22⋊C4—22no—
161C5×C22⋊C4—22no—
162C5×C22⋊C4—22no—
163C5×C22⋊C4—41no—
164C5×C22⋊C4C22⋊C412yes—
165C5×C22⋊C4C22⋊C412yes—
166C42×C5—22no—
167C42×C5—22no—
168C42×C5—21no—
169C42×C5—22no—
170C42×C5—41no—
171C42×C5C22⋊C411yescharacteristic
172C42×C5C22⋊C411yescharacteristic
173#568C22×C511yescharacteristic, complemented
174#559C4×C511yescharacteristic, complemented
175#554—21no—
176#553—22no—
177#551—21no—
178#551C4×C512yescomplemented
179#546—22no—
180D42C4×C511yescharacteristic, complemented
181#532C22×C511yescharacteristic
182#529—22no—
183#525—21no—
184#525C22×C512yes—
185C4×#41—21no—
186C2×C4×D4—22no—
187#421—21no—
188#413C22×C512yescomplemented
189#410C22×C511yescharacteristic, complemented
190#407—22no—
191#356—21no—
192#355—21no—
193#352—22no—
194AΣL(1,𝔽16)—22no—
195#344—22no—
196#339—21no—
197C23×C5—22no—
198C23×C5—21no—
199C23×C5—42no—
200C23×C5—44no—
201C23×C5—44no—
202C23×C5—81no—
203C23×C5C2×C22⋊C412yes—
204C23×C5#12011yescharacteristic
205C5×Q8—41no—
206C5×Q8—41no—
207C5×Q8—42no—
208C5×D4—24no—
209C5×D4—41no—
210C5×D4—42no—
211C5×D4—41no—
212C5×D4—44no—
213C5×D4—44no—
214C5×D4—44no—
215C5×D4—44no—
216C5×D4—81no—
217C2×C4×C5—21no—
218C2×C4×C5—21no—
219C2×C4×C5—22no—
220C2×C4×C5—21no—
221C2×C4×C5—22no—
222C2×C4×C5—21no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—22no—
226C2×C4×C5—21no—
227C2×C4×C5—44no—
228C2×C4×C5—42no—
229C2×C4×C5—44no—
230C2×C4×C5—42no—
231C2×C4×C5—41no—
232C2×C4×C5—42no—
233C2×C4×C5—41no—
234C2×C4×C5—42no—
235C2×C4×C5—82no—
236C2×C4×C5—81no—
237C2×C4×C5—82no—
238C2×C4×C5—81no—
239C2×C4×C5C2×C22⋊C411yescharacteristic
240C2×C4×C5#12012yes—
241C2×C4×C5#12011yescharacteristic
242C5×C8—42no—
243C5×C8—82no—
244C2×#41—21no—
245C2×#41C5×D411yescharacteristic
246C2×#41C23×C511yescharacteristic
247C22×D4—22no—
248C23×C4—21no—
249C23×C4C5×D412yes—
250#131—42no—
251C2×M16—22no—
252C2×M16—21no—
253C2×M16—22no—
254C4⋊Q8C5×D411yescharacteristic
255#127C2×C4×C511yescharacteristic
256#126—22no—
257#126—22no—
258#125—22no—
259#124—22no—
260#124C2×C4×C511yescharacteristic
261#124C5×D411yescharacteristic
262#123—21no—
263#123—22no—
264#123—21no—
265#123—22no—
266#123C2×C4×C512yes—
267#123C5×D412yes—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271#121—22no—
272#121—22no—
273#121—22no—
274#121—22no—
275#120—21no—
276#120—21no—
277#120C2×C4×C512yes—
278#120C5×D412yes—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×D4—21no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—21no—
292#117—21no—
293#117—21no—
294#117—22no—
295#117—21no—
296#117C5×D412yes—
297C2×C4⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C42—22no—
301C2×C42C5×D411yescharacteristic
302#104—42no—
303#101—22no—
304#100—22no—
305#99—22no—
306C22⋊C8—44no—
307C22×C5—24no—
308C22×C5—22no—
309C22×C5—21no—
310C22×C5—41no—
311C22×C5—42no—
312C22×C5—42no—
313C22×C5—42no—
314C22×C5—44no—
315C22×C5—42no—
316C22×C5—41no—
317C22×C5—82no—
318C22×C5—84no—
319C22×C5#37911yescharacteristic
320C4×C5—22no—
321C4×C5—22no—
322C4×C5—21no—
323C4×C5—41no—
324C4×C5—42no—
325C4×C5—42no—
326C4×C5—42no—
327C4×C5—41no—
328C4×C5—161no—
329C4×C5B(4,𝔽2)11yescharacteristic
330C4×C5#45811yescharacteristic
331C24—21no—
332C24C5×C22⋊C412yes—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—44no—
338C2×Q8—21no—
339C2×Q8C5×C22⋊C411yescharacteristic
340C2×Q8C2×C5×D411yescharacteristic
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—48no—
348C2×D4C5×C22⋊C411yescharacteristic
349C2×D4C22×C4×C511yescharacteristic
350C2×D4C2×C5×D412yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4C5×C22⋊C412yes—
364C22×C4C2×C5×D411yescharacteristic
365C22×C4C2×C5×D412yes—
366M16—22no—
367M16—42no—
368M16—41no—
369M16—42no—
370M16—42no—
371C2×C8—24no—
372C2×C8—41no—
373C2×C8—42no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4C5×C22⋊C412yes—
397C22⋊C4C5×C22⋊C412yes—
398C42—22no—
399C42—22no—
400C42—21no—
401C42—22no—
402C42—41no—
403C42C5×C22⋊C411yescharacteristic
404C42C5×C22⋊C411yescharacteristic
405C2×C5—21no—
406C2×C5—41no—
407C2×C5—42no—
408C2×C5—44no—
409C2×C5—81no—
410C2×C5#179011yescharacteristic
411C23—22no—
412C23—21no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—81no—
417C23C2×C5×C22⋊C412yes—
418C23C5×#12011yescharacteristic
419Q8—41no—
420Q8—41no—
421Q8—42no—
422D4—24no—
423D4—41no—
424D4—42no—
425D4—41no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—81no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4C2×C5×C22⋊C411yescharacteristic
454C2×C4C5×#12012yes—
455C2×C4C5×#12011yescharacteristic
456C8—42no—
457C8—82no—
458C5#1276511yescharacteristic, direct product factor, Sylow
459C22—24no—
460C22—22no—
461C22—21no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—42no—
468C22—41no—
469C22—82no—
470C22—84no—
471C22C5×#37911yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—41no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—41no—
480C4—161no—
481C4C5×B(4,𝔽2)11yescharacteristic
482C4C5×#45811yescharacteristic
483C2—21no—
484C2—41no—
485C2—44no—
486C2—42no—
487C2—81no—
488C2C5×#179011yescharacteristic
subgroupG/H
centerC4×C5B(4,𝔽2)
commutatorC2×D4C22×C4×C5
Frattini, cosocleC2×#41C23×C5
socleC2×C5#1790
prime radicalC5×#3481C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2132C2semidirect
3. C5×#2120C2semidirect
4. C5×#1891C2semidirect
5. C5×#1884C2semidirect
6. C5×#568C22semidirect
7. C5×#559C4semidirect
11. C5×#551C4semidirect
13. C5×D42C4semidirect
21. C5×#413C22semidirect
22. C5×#410C22semidirect
30. #12765C5direct (N × H)
95. #2132C2×C5semidirect
96. #2120C2×C5semidirect
97. #1891C2×C5semidirect
98. #1884C2×C5semidirect
173. #568C22×C5semidirect
174. #559C4×C5semidirect
178. #551C4×C5semidirect
180. D42C4×C5semidirect
188. #413C22×C5semidirect
189. #410C22×C5semidirect
458. C5#12765direct (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×#21322
C5×#21202
C5×#18912
C5×#18842
#127655