Finite Group Databasebeta

← C5×#10417 #11933137

Subgroups — C5×#10417 #11933137

Order 1280.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2547C211yesmaximal, characteristic
2C5×#1990C211yesmaximal, characteristic, complemented
3C5×#1948C211yesmaximal, characteristic, complemented
4C5×#1802C211yesmaximal, characteristic
5C2×C5×#332C211yesmaximal, characteristic, complemented
6C2×C5×D4⋊C8C211yesmaximal, characteristic
7C5×#1279C211yesmaximal, characteristic, complemented
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×C5×#124—21no—
10C2×C4×C5×D4C2211yescharacteristic
11C2×C5×#106—21no—
12C2×C5×C4⋊C8C2211yescharacteristic
13C2×C5×C4⋊C8C2211yescharacteristic
14C2×C5×#102—21no—
15C2×C4×C5×C8C2211yescharacteristic
16C5×#397—22no—
17C5×#391—21no—
18C5×#384C2211yescharacteristic
19C5×#380—21no—
20C5×#380—21no—
21C4×C5×C4⋊C4C2211yescharacteristic
22C5×#340—21no—
23C5×#340—21no—
24C5×#340—22no—
25C5×#332—22no—
26C5×D4⋊C8—22no—
27#10417C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8—21no—
29C22×C5×D4D411yescharacteristic
30C23×C4×C5—21no—
31C22×C5×C8—21no—
32C22×C5×C8—21no—
33C22×C5×C8—21no—
34C5×C4⋊Q8—21no—
35C5×C4⋊Q8—21no—
36C5×C4⋊Q8—21no—
37C5×C4⋊Q8—21no—
38C5×#124—42no—
39C5×#124—42no—
40C4×C5×D4—21no—
41C4×C5×D4—21no—
42C4×C5×D4—21no—
43C4×C5×D4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4—21no—
48C2×C5×C4⋊C4D411yescharacteristic
49C2×C5×C4⋊C4D411yescharacteristic
50C2×C5×C4⋊C4D411yescharacteristic
51C2×C5×C4⋊C4D412yes—
52C2×C5×C22⋊C4—22no—
53C2×C5×C22⋊C4—22no—
54C2×C5×C22⋊C4—21no—
55C2×C42×C5—21no—
56C2×C42×C5C2311yescharacteristic
57C5×#106—42no—
58C5×C4⋊C8—21no—
59C5×C4⋊C8—21no—
60C5×C4⋊C8—21no—
61C5×C4⋊C8D412yes—
62C5×#102—42no—
63C4×C5×C8—21no—
64C4×C5×C8—21no—
65C5×#95—22no—
66C5×#95—22no—
67C5×#95—21no—
68#2547C2×C511yescharacteristic
69#1990C2×C511yescharacteristic, complemented
70#1948C2×C511yescharacteristic, complemented
71#1802C2×C511yescharacteristic
72C2×#332C2×C511yescharacteristic, complemented
73C2×D4⋊C8C2×C511yescharacteristic
74#1279C2×C511yescharacteristic, complemented
75C24×C5—21no—
76C2×C5×Q8—41no—
77C2×C5×Q8—41no—
78C2×C5×Q8—41no—
79C2×C5×Q8—41no—
80C2×C5×Q8—41no—
81C2×C5×Q8—41no—
82C2×C5×D4—21no—
83C2×C5×D4—21no—
84C2×C5×D4—21no—
85C2×C5×D4—21no—
86C2×C5×D4—21no—
87C2×C5×D4SD1612yes—
88C22×C4×C5—21no—
89C22×C4×C5—21no—
90C22×C4×C5—22no—
91C22×C4×C5—21no—
92C22×C4×C5—41no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5—41no—
96C22×C4×C5—41no—
97C22×C4×C5—41no—
98C22×C4×C5—41no—
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C2×C5×C8—22no—
103C2×C5×C8—22no—
104C2×C5×C8—41no—
105C2×C5×C8—41no—
106C2×C5×C8—41no—
107C2×C5×C8—41no—
108C2×C5×C8—41no—
109C2×C5×C8—41no—
110C2×C5×C8—41no—
111C5×C4⋊C4—21no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—21no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—21no—
116C5×C4⋊C4—21no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—21no—
119C5×C4⋊C4—42no—
120C5×C4⋊C4—41no—
121C5×C4⋊C4—41no—
122C5×C4⋊C4—41no—
123C5×C4⋊C4—41no—
124C5×C4⋊C4—41no—
125C5×C4⋊C4—41no—
126C5×C22⋊C4—44no—
127C5×C22⋊C4—44no—
128C5×C22⋊C4—41no—
129C5×C22⋊C4—41no—
130C42×C5—41no—
131C42×C5—41no—
132C42×C5C2×D411yescharacteristic
133C42×C5#4111yescharacteristic
134C42×C5#4111yescharacteristic
135C42×C5#4111yescharacteristic
136C2×C4⋊Q8C22×C511yescharacteristic
137C2×#124—21no—
138C2×C4×D4C22×C511yescharacteristic
139C2×#106—21no—
140C2×C4⋊C8C22×C511yescharacteristic
141C2×C4⋊C8C22×C511yescharacteristic
142C2×#102—21no—
143C2×C4×C8C22×C511yescharacteristic
144#397—22no—
145#391—21no—
146#384C22×C511yescharacteristic
147#380—21no—
148#380—21no—
149C4×C4⋊C4C22×C511yescharacteristic
150#340—21no—
151#340—21no—
152#340—22no—
153#332—22no—
154D4⋊C8—22no—
155C23×C5—22no—
156C23×C5—41no—
157C23×C5—41no—
158C23×C5—41no—
159C23×C5—41no—
160C23×C5—41no—
161C23×C5—41no—
162C23×C5#12011yescharacteristic
163C5×Q8—81no—
164C5×Q8—81no—
165C5×Q8—81no—
166C5×Q8—81no—
167C5×D4—22no—
168C5×D4—22no—
169C5×D4—41no—
170C5×D4—41no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—41no—
174C2×C4×C5—42no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—42no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—42no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5—41no—
185C2×C4×C5—81no—
186C2×C4×C5—81no—
187C2×C4×C5—81no—
188C2×C4×C5—81no—
189C2×C4×C5—81no—
190C2×C4×C5—81no—
191C2×C4×C5—81no—
192C2×C4×C5#12111yescharacteristic
193C2×C4×C5#12111yescharacteristic
194C2×C4×C5#12111yescharacteristic
195C2×C4×C5#12311yescharacteristic
196C2×C4×C5#12311yescharacteristic
197C2×C4×C5#12411yescharacteristic
198C2×C4×C5C2×SD1611yescharacteristic
199C2×C4×C5#13511yescharacteristic
200C2×C4×C5#13511yescharacteristic
201C2×C4×C5#13511yescharacteristic
202C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C5#13711yescharacteristic
206C5×C8—42no—
207C5×C8—42no—
208C5×C8—81no—
209C22×Q8—21no—
210C22×D4C5×D411yescharacteristic
211C23×C4—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219#124—42no—
220#124—42no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C5×D411yescharacteristic
230C2×C4⋊C4C5×D412yes—
231C2×C4⋊C4C5×D411yescharacteristic
232C2×C4⋊C4C5×D411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42C23×C511yescharacteristic
238#106—42no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C5×D412yes—
243#102—42no—
244C4×C8—21no—
245C4×C8—21no—
246#95—22no—
247#95—22no—
248#95—21no—
249C22×C5—42no—
250C22×C5—42no—
251C22×C5—42no—
252C22×C5—81no—
253C22×C5—81no—
254C22×C5—81no—
255C22×C5—81no—
256C22×C5#39411yescharacteristic
257C22×C5#44911yescharacteristic
258C22×C5#44911yescharacteristic
259C22×C5#45011yescharacteristic
260C22×C5#45211yescharacteristic
261C22×C5#45411yescharacteristic
262C22×C5#45611yescharacteristic
263C4×C5—21no—
264C4×C5—21no—
265C4×C5—81no—
266C4×C5—81no—
267C4×C5—82no—
268C4×C5—81no—
269C4×C5—161no—
270C4×C5#46311yescharacteristic
271C4×C5#46411yescharacteristic
272C4×C5#46511yescharacteristic
273C4×C5#46911yescharacteristic
274C4×C5#48111yescharacteristic
275C4×C5#48211yescharacteristic
276C4×C5#48811yescharacteristic
277C4×C5#48911yescharacteristic
278C24—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4C5×SD1612yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C2×C5×D411yescharacteristic
303C22×C4C2×C5×D411yescharacteristic
304C22×C4C2×C5×D411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C42—41no—
334C42—41no—
335C42C2×C5×D411yescharacteristic
336C42C5×#4111yescharacteristic
337C42C5×#4111yescharacteristic
338C42C5×#4111yescharacteristic
339C2×C5—82no—
340C2×C5#162811yescharacteristic
341C2×C5#163511yescharacteristic
342C2×C5#165011yescharacteristic
343C2×C5#165411yescharacteristic
344C2×C5#203411yescharacteristic
345C2×C5#203811yescharacteristic
346C2×C5#204211yescharacteristic
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C5×#12011yescharacteristic
355Q8—81no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359D4—22no—
360D4—22no—
361D4—41no—
362D4—41no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4C5×#12111yescharacteristic
385C2×C4C5×#12111yescharacteristic
386C2×C4C5×#12111yescharacteristic
387C2×C4C5×#12311yescharacteristic
388C2×C4C5×#12311yescharacteristic
389C2×C4C5×#12411yescharacteristic
390C2×C4C2×C5×SD1611yescharacteristic
391C2×C4C5×#13511yescharacteristic
392C2×C4C5×#13511yescharacteristic
393C2×C4C5×#13511yescharacteristic
394C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
397C2×C4C5×#13711yescharacteristic
398C8—42no—
399C8—42no—
400C8—81no—
401C5#1041711yescharacteristic, direct product factor, Sylow
402C22—42no—
403C22—42no—
404C22—42no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C5×#39411yescharacteristic
410C22C5×#44911yescharacteristic
411C22C5×#44911yescharacteristic
412C22C5×#45011yescharacteristic
413C22C5×#45211yescharacteristic
414C22C5×#45411yescharacteristic
415C22C5×#45611yescharacteristic
416C4—21no—
417C4—21no—
418C4—82no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—161no—
423C4C5×#46311yescharacteristic
424C4C5×#46411yescharacteristic
425C4C5×#46511yescharacteristic
426C4C5×#46911yescharacteristic
427C4C5×#48111yescharacteristic
428C4C5×#48211yescharacteristic
429C4C5×#48811yescharacteristic
430C4C5×#48911yescharacteristic
431C2—82no—
432C2C5×#162811yescharacteristic
433C2C5×#163511yescharacteristic
434C2C5×#165011yescharacteristic
435C2C5×#165411yescharacteristic
436C2C5×#203411yescharacteristic
437C2C5×#203811yescharacteristic
438C2C5×#204211yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC22×C5×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1990C2semidirect
3. C5×#1948C2semidirect
5. C2×C5×#332C2semidirect
7. C5×#1279C2semidirect
27. #10417C5direct (N × H)
69. #1990C2×C5semidirect
70. #1948C2×C5semidirect
72. C2×#332C2×C5semidirect
74. #1279C2×C5semidirect
401. C5#10417direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25472
C5×#19902
C5×#19482
C5×#18022
C2×C5×#3322
C2×C5×D4⋊C82
C5×#12792
#104175