Finite Group Databasebeta

← C5×#12781 #11935501

Subgroups — C5×#12781 #11935501

Order 1280.

758 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3545C211yesmaximal, characteristic
2C5×#2133C212yesmaximal, complemented
3C2×C5×#355C211yesmaximal, characteristic, complemented
4C5×#1912C212yesmaximal, complemented
5C5×#1888C211yesmaximal, characteristic, complemented
6C5×#563C411yescharacteristic, complemented
7C5×#562—21no—
8C5×H(3,𝔽4)C411yescharacteristic, complemented
9C5×#555C412yescomplemented
10C5×#551—21no—
11C5×#551—22no—
12C5×#546—22no—
13C5×#542—22no—
14C5×#538—22no—
15C5×#536C2212yes—
16C5×#534—21no—
17C5×#529—21no—
18C2×C5×#124C2211yescharacteristic
19C5×#519C2211yescharacteristic, complemented
20C5×#421—22no—
21C5×#418—21no—
22C2×C5×#101C2212yescomplemented
23C2×C5×#101C2211yescharacteristic, complemented
24C5×#356—22no—
25C5×#355—21no—
26C5×#355—21no—
27C5×#355—22no—
28C5×#339—21no—
29C5×H(3,ℤ/4)—22no—
30#12781C511yesmaximal, characteristic, direct product factor, Sylow
31C2×C5×#41D412yes—
32C22×C5×Q8C2311yescharacteristic
33C22×C5×D4—21no—
34C22×C5×D4D411yescharacteristic
35C2×C5×M16—22no—
36C2×C5×M16—21no—
37C5×C4⋊Q8C2×C412yes—
38C5×C4⋊Q8D412yes—
39C5×#126—22no—
40C5×#126—22no—
41C5×#126—22no—
42C5×#125—21no—
43C5×#124—22no—
44C5×#124—22no—
45C5×#124—21no—
46C5×#124—22no—
47C5×#124C2×C411yescharacteristic
48C5×#124C2×C411yescharacteristic
49C5×#124C2×C412yes—
50C5×#124D411yescharacteristic
51C5×#124D411yescharacteristic
52C5×#124D412yes—
53C5×#123—21no—
54C5×#123—22no—
55C5×#123—22no—
56C5×#123—22no—
57C5×#123—21no—
58C5×#123—22no—
59C5×#123—22no—
60C5×C22⋊Q8—21no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8—22no—
63C5×C22⋊Q8—22no—
64C5×C22⋊Q8—22no—
65C5×#121—21no—
66C5×#121—22no—
67C5×#121—22no—
68C5×#121—22no—
69C5×#121—22no—
70C5×#120—21no—
71C5×#120—22no—
72C5×#120—21no—
73C5×#120—22no—
74C4×C5×Q8—21no—
75C4×C5×Q8—42no—
76C4×C5×D4—21no—
77C4×C5×D4—22no—
78C4×C5×D4—22no—
79C4×C5×D4—22no—
80C5×#117—22no—
81C5×#117—22no—
82C5×#117—22no—
83C5×#117—21no—
84C5×#117D412yes—
85C2×C5×C22⋊C4—21no—
86C2×C5×C22⋊C4—22no—
87C2×C5×C22⋊C4—22no—
88C2×C42×C5D411yescharacteristic
89C5×#104—44no—
90C5×#102—42no—
91C5×#101—22no—
92C5×#101—22no—
93C5×#101—22no—
94#3545C2×C511yescharacteristic
95#2133C2×C512yescomplemented
96C2×#355C2×C511yescharacteristic, complemented
97#1912C2×C512yescomplemented
98#1888C2×C511yescharacteristic, complemented
99C24×C5—21no—
100C24×C5—21no—
101C5×#41—42no—
102C5×#41—42no—
103C2×C5×Q8—21no—
104C2×C5×Q8—41no—
105C2×C5×Q8—41no—
106C2×C5×Q8C22⋊C412yes—
107C2×C5×Q8C22×C411yescharacteristic
108C2×C5×Q8C2×D411yescharacteristic
109C2×C5×Q8C2×D412yes—
110C2×C5×D4—22no—
111C2×C5×D4—22no—
112C2×C5×D4—22no—
113C2×C5×D4—21no—
114C2×C5×D4—22no—
115C2×C5×D4—21no—
116C2×C5×D4—41no—
117C2×C5×D4—41no—
118C2×C5×D4—81no—
119C2×C5×D4—81no—
120C2×C5×D4C22⋊C412yes—
121C2×C5×D4C22⋊C411yescharacteristic
122C2×C5×D4C22⋊C411yescharacteristic
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—22no—
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×D412yes—
130C5×M16—41no—
131C5×M16—42no—
132C5×M16—41no—
133C5×M16—42no—
134C2×C5×C8—42no—
135C2×C5×C8—41no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4—22no—
141C5×C4⋊C4—22no—
142C5×C4⋊C4—22no—
143C5×C4⋊C4—22no—
144C5×C4⋊C4—22no—
145C5×C4⋊C4—22no—
146C5×C4⋊C4—42no—
147C5×C22⋊C4—22no—
148C5×C22⋊C4—22no—
149C5×C22⋊C4—21no—
150C5×C22⋊C4—21no—
151C5×C22⋊C4—22no—
152C5×C22⋊C4—22no—
153C5×C22⋊C4—22no—
154C5×C22⋊C4—22no—
155C5×C22⋊C4—22no—
156C5×C22⋊C4—22no—
157C5×C22⋊C4—22no—
158C5×C22⋊C4—22no—
159C5×C22⋊C4—22no—
160C5×C22⋊C4—22no—
161C5×C22⋊C4—22no—
162C5×C22⋊C4—41no—
163C42×C5—22no—
164C42×C5—21no—
165C42×C5—24no—
166C42×C5—41no—
167C42×C5C22⋊C411yescharacteristic
168C42×C5C22⋊C411yescharacteristic
169C42×C5C22⋊C412yes—
170C42×C5C22⋊C412yes—
171#563C4×C511yescharacteristic, complemented
172#562—21no—
173H(3,𝔽4)C4×C511yescharacteristic, complemented
174#555C4×C512yescomplemented
175#551—21no—
176#551—22no—
177#546—22no—
178#542—22no—
179#538—22no—
180#536C22×C512yes—
181#534—21no—
182#529—21no—
183C2×#124C22×C511yescharacteristic
184#519C22×C511yescharacteristic, complemented
185#421—22no—
186#418—21no—
187C2×#101C22×C512yescomplemented
188C2×#101C22×C511yescharacteristic, complemented
189#356—22no—
190#355—21no—
191#355—21no—
192#355—22no—
193#339—21no—
194H(3,ℤ/4)—22no—
195C23×C5—21no—
196C23×C5—21no—
197C23×C5—22no—
198C23×C5—41no—
199C23×C5—41no—
200C23×C5—81no—
201C23×C5—82no—
202C23×C5—82no—
203C23×C5#12011yescharacteristic
204C5×Q8—41no—
205C5×Q8—41no—
206C5×Q8—42no—
207C5×Q8—42no—
208C5×Q8—81no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4—42no—
212C5×D4—42no—
213C5×D4—42no—
214C5×D4—81no—
215C5×D4—82no—
216C5×D4—82no—
217C5×D4—81no—
218C5×D4—82no—
219C2×C4×C5—22no—
220C2×C4×C5—21no—
221C2×C4×C5—22no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—22no—
225C2×C4×C5—21no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—42no—
229C2×C4×C5—82no—
230C2×C4×C5—82no—
231C2×C4×C5—81no—
232C2×C4×C5—82no—
233C2×C4×C5—81no—
234C2×C4×C5C2×C22⋊C412yes—
235C2×C4×C5C2×C22⋊C411yescharacteristic
236C2×C4×C5#12011yescharacteristic
237C2×C4×C5#12012yes—
238C5×C8—82no—
239C5×C8—81no—
240C2×#41C5×D412yes—
241C22×Q8C23×C511yescharacteristic
242C22×D4—21no—
243C22×D4C5×D411yescharacteristic
244C2×M16—22no—
245C2×M16—21no—
246C4⋊Q8C2×C4×C512yes—
247C4⋊Q8C5×D412yes—
248#126—22no—
249#126—22no—
250#126—22no—
251#125—21no—
252#124—22no—
253#124—22no—
254#124—21no—
255#124—22no—
256#124C2×C4×C511yescharacteristic
257#124C2×C4×C511yescharacteristic
258#124C2×C4×C512yes—
259#124C5×D411yescharacteristic
260#124C5×D411yescharacteristic
261#124C5×D412yes—
262#123—21no—
263#123—22no—
264#123—22no—
265#123—22no—
266#123—21no—
267#123—22no—
268#123—22no—
269C22⋊Q8—21no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274#121—21no—
275#121—22no—
276#121—22no—
277#121—22no—
278#121—22no—
279#120—21no—
280#120—22no—
281#120—21no—
282#120—22no—
283C4×Q8—21no—
284C4×Q8—42no—
285C4×D4—21no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—21no—
293#117C5×D412yes—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C42C5×D411yescharacteristic
298#104—44no—
299#102—42no—
300#101—22no—
301#101—22no—
302#101—22no—
303C22×C5—41no—
304C22×C5—42no—
305C22×C5—42no—
306C22×C5—42no—
307C22×C5—42no—
308C22×C5—42no—
309C22×C5—41no—
310C22×C5—41no—
311C22×C5—161no—
312C22×C5—161no—
313C22×C5—162no—
314C22×C5#37911yescharacteristic
315C22×C5B(4,𝔽2)11yescharacteristic
316C22×C5#45811yescharacteristic
317C4×C5—22no—
318C4×C5—21no—
319C4×C5—21no—
320C4×C5—22no—
321C4×C5—41no—
322C4×C5—42no—
323C4×C5—42no—
324C4×C5—41no—
325C4×C5—42no—
326C4×C5—82no—
327C24—21no—
328C24—21no—
329#41—42no—
330#41—42no—
331C2×Q8—21no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C5×C22⋊C412yes—
335C2×Q8C22×C4×C511yescharacteristic
336C2×Q8C2×C5×D411yescharacteristic
337C2×Q8C2×C5×D412yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—21no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C5×C22⋊C412yes—
349C2×D4C5×C22⋊C411yescharacteristic
350C2×D4C5×C22⋊C411yescharacteristic
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4C2×C5×D411yescharacteristic
357C22×C4C2×C5×D412yes—
358M16—42no—
359M16—41no—
360M16—41no—
361M16—42no—
362C2×C8—41no—
363C2×C8—42no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—41no—
391C42—22no—
392C42—21no—
393C42—24no—
394C42—41no—
395C42C5×C22⋊C411yescharacteristic
396C42C5×C22⋊C411yescharacteristic
397C42C5×C22⋊C412yes—
398C42C5×C22⋊C412yes—
399C2×C5—21no—
400C2×C5—21no—
401C2×C5—21no—
402C2×C5—81no—
403C2×C5—82no—
404C2×C5—81no—
405C2×C5#179011yescharacteristic
406C23—21no—
407C23—21no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23C5×#12011yescharacteristic
415Q8—41no—
416Q8—41no—
417Q8—42no—
418Q8—42no—
419Q8—81no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—81no—
426D4—82no—
427D4—82no—
428D4—82no—
429D4—81no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4C2×C5×C22⋊C412yes—
446C2×C4C2×C5×C22⋊C411yescharacteristic
447C2×C4C5×#12011yescharacteristic
448C2×C4C5×#12012yes—
449C8—82no—
450C8—81no—
451C5#1278111yescharacteristic, direct product factor, Sylow
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—41no—
459C22—41no—
460C22—161no—
461C22—161no—
462C22—162no—
463C22C5×#37911yescharacteristic
464C22C5×B(4,𝔽2)11yescharacteristic
465C22C5×#45811yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—81no—
480C2—81no—
481C2—82no—
482C2C5×#179011yescharacteristic
subgroupG/H
center, socleC2×C5#1790
commutatorC2×Q8C22×C4×C5
Frattini, cosocleC22×Q8C23×C5
prime radicalC5×#3545C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C5 ◁ C23×C5 ◁ C22×C5×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2133C2semidirect
3. C2×C5×#355C2semidirect
4. C5×#1912C2semidirect
5. C5×#1888C2semidirect
6. C5×#563C4semidirect
8. C5×H(3,𝔽4)C4semidirect
9. C5×#555C4semidirect
19. C5×#519C22semidirect
22. C2×C5×#101C22semidirect
23. C2×C5×#101C22semidirect
30. #12781C5direct (N × H)
95. #2133C2×C5semidirect
96. C2×#355C2×C5semidirect
97. #1912C2×C5semidirect
98. #1888C2×C5semidirect
171. #563C4×C5semidirect
173. H(3,𝔽4)C4×C5semidirect
174. #555C4×C5semidirect
184. #519C22×C5semidirect
187. C2×#101C22×C5semidirect
188. C2×#101C22×C5semidirect
451. C5#12781direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35452
C5×#21332
C2×C5×#3552
C5×#19122
C5×#18882
#127815