Finite Group Databasebeta

← C5×#10474 #11933194

Subgroups — C5×#10474 #11933194

Order 1280.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#460C211yesmaximal, characteristic
2C5×#1984C211yesmaximal, characteristic, complemented
3C5×#1957C211yesmaximal, characteristic
4C5×#1803C211yesmaximal, characteristic, complemented
5C2×C5×#335C211yesmaximal, characteristic, complemented
6C2×C5×#331C211yesmaximal, characteristic
7C5×#1296C211yesmaximal, characteristic, complemented
8C22×C5×SD16—21no—
9C2×C5×#127C2211yescharacteristic
10C2×C4×C5×Q8C2211yescharacteristic
11C5×#460—22no—
12C5×#460—22no—
13C2×C5×C4⋊C8C2211yescharacteristic
14C2×C5×C4⋊C8C2211yescharacteristic
15C2×C5×C4⋊C8C2211yescharacteristic
16C2×C5×#103—21no—
17C2×C5×#102—21no—
18C2×C5×#102—21no—
19C2×C5×#102—21no—
20C2×C4×C5×C8C2211yescharacteristic
21C5×#384C2211yescharacteristic
22C5×#340—21no—
23C5×#336—21no—
24C5×#336—21no—
25C5×#335—22no—
26C5×#331—22no—
27#10474C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8D411yescharacteristic
29C22×C5×D4—21no—
30C22×C5×D4—21no—
31C22×C5×D4—21no—
32C2×C5×SD16—24no—
33C2×C5×SD16—42no—
34C2×C5×SD16—42no—
35C22×C5×C8—21no—
36C22×C5×C8—21no—
37C22×C5×C8—21no—
38C22×C5×C8D411yescharacteristic
39C22×C5×C8D411yescharacteristic
40C5×#127—21no—
41C5×#127—21no—
42C5×#127—21no—
43C5×#127D412yes—
44C4×C5×Q8—21no—
45C4×C5×Q8—21no—
46C4×C5×Q8—21no—
47C4×C5×Q8—21no—
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4D411yescharacteristic
53C2×C42×C5—21no—
54C2×C42×C5C2311yescharacteristic
55C5×C4⋊C8—21no—
56C5×C4⋊C8—21no—
57C5×C4⋊C8—21no—
58C5×C4⋊C8—21no—
59C5×C4⋊C8—21no—
60C5×C4⋊C8D412yes—
61C5×#103—42no—
62C5×#102—24no—
63C5×#102—42no—
64C5×#102—42no—
65C4×C5×C8—21no—
66C4×C5×C8—21no—
67C2×#460C2×C511yescharacteristic
68#1984C2×C511yescharacteristic, complemented
69#1957C2×C511yescharacteristic
70#1803C2×C511yescharacteristic, complemented
71C2×#335C2×C511yescharacteristic, complemented
72C2×#331C2×C511yescharacteristic
73#1296C2×C511yescharacteristic, complemented
74C24×C5—41no—
75C2×C5×Q8—21no—
76C2×C5×Q8—21no—
77C2×C5×Q8—21no—
78C2×C5×Q8—21no—
79C2×C5×Q8—21no—
80C2×C5×Q8D812yes—
81C2×C5×D4—22no—
82C2×C5×D4—22no—
83C2×C5×D4—22no—
84C2×C5×D4—41no—
85C2×C5×D4—41no—
86C2×C5×D4—41no—
87C2×C5×D4—41no—
88C2×C5×D4—41no—
89C2×C5×D4—41no—
90C2×C5×D4—41no—
91C2×C5×D4—41no—
92C2×C5×D4—41no—
93C2×C5×D4—41no—
94C2×C5×D4—41no—
95C2×C5×D4—81no—
96C2×C5×D4—81no—
97C22×C4×C5—21no—
98C22×C4×C5—21no—
99C22×C4×C5—41no—
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5#4111yescharacteristic
103C5×SD16—48no—
104C2×C5×C8—21no—
105C2×C5×C8—21no—
106C2×C5×C8—22no—
107C2×C5×C8—21no—
108C2×C5×C8—21no—
109C2×C5×C8—21no—
110C2×C5×C8—41no—
111C2×C5×C8—41no—
112C2×C5×C8—41no—
113C2×C5×C8—41no—
114C2×C5×C8—41no—
115C2×C5×C8—41no—
116C2×C5×C8—41no—
117C2×C5×C8—41no—
118C2×C5×C8SD1612yescomplemented
119C5×C4⋊C4—21no—
120C5×C4⋊C4—21no—
121C5×C4⋊C4—41no—
122C5×C4⋊C4—41no—
123C5×C4⋊C4—41no—
124C5×C4⋊C4—41no—
125C5×C4⋊C4—41no—
126C5×C4⋊C4—41no—
127C5×C4⋊C4—81no—
128C42×C5—41no—
129C42×C5—41no—
130C42×C5C2×D411yescharacteristic
131C42×C5C2×D411yescharacteristic
132C42×C5#4111yescharacteristic
133C42×C5#4111yescharacteristic
134C22×SD16—21no—
135C2×#127C22×C511yescharacteristic
136C2×C4×Q8C22×C511yescharacteristic
137#460—22no—
138#460—22no—
139C2×C4⋊C8C22×C511yescharacteristic
140C2×C4⋊C8C22×C511yescharacteristic
141C2×C4⋊C8C22×C511yescharacteristic
142C2×#103—21no—
143C2×#102—21no—
144C2×#102—21no—
145C2×#102—21no—
146C2×C4×C8C22×C511yescharacteristic
147#384C22×C511yescharacteristic
148#340—21no—
149#336—21no—
150#336—21no—
151#335—22no—
152#331—22no—
153C23×C5—42no—
154C23×C5—81no—
155C23×C5—81no—
156C23×C5—81no—
157C23×C5—81no—
158C23×C5—81no—
159C23×C5—81no—
160C23×C5#12111yescharacteristic
161C5×Q8—22no—
162C5×Q8—22no—
163C5×Q8—41no—
164C5×Q8—41no—
165C5×D4—42no—
166C5×D4—42no—
167C5×D4—42no—
168C5×D4—42no—
169C5×D4—82no—
170C5×D4—82no—
171C5×D4—81no—
172C5×D4—81no—
173C5×D4—81no—
174C5×D4—81no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—22no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—81no—
184C2×C4×C5—81no—
185C2×C4×C5—81no—
186C2×C4×C5#12011yescharacteristic
187C2×C4×C5#12111yescharacteristic
188C2×C4×C5#12111yescharacteristic
189C2×C4×C5#12311yescharacteristic
190C2×C4×C5#12311yescharacteristic
191C2×C4×C5#12411yescharacteristic
192C2×C4×C5C2×D811yescharacteristic
193C2×C4×C5C2×SD1611yescharacteristic
194C2×C4×C5#13511yescharacteristic
195C2×C4×C5#13511yescharacteristic
196C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C5#13711yescharacteristic
199C2×C4×C5#13711yescharacteristic
200C5×C8—22no—
201C5×C8—42no—
202C5×C8—41no—
203C5×C8—81no—
204C5×C8—81no—
205C22×Q8C5×D411yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C2×SD16—24no—
210C2×SD16—42no—
211C2×SD16—42no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8C5×D411yescharacteristic
216C22×C8C5×D411yescharacteristic
217#127—21no—
218#127—21no—
219#127—21no—
220#127C5×D412yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C5×D411yescharacteristic
230C2×C42—21no—
231C2×C42C23×C511yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C5×D412yes—
238#103—42no—
239#102—24no—
240#102—42no—
241#102—42no—
242C4×C8—21no—
243C4×C8—21no—
244C22×C5—82no—
245C22×C5—82no—
246C22×C5—82no—
247C22×C5—161no—
248C22×C5—161no—
249C22×C5—161no—
250C22×C5—161no—
251C22×C5#39411yescharacteristic
252C22×C5#46011yescharacteristic
253C22×C5#46411yescharacteristic
254C22×C5#46611yescharacteristic
255C22×C5#46811yescharacteristic
256C22×C5#47111yescharacteristic
257C22×C5#47211yescharacteristic
258C4×C5—21no—
259C4×C5—21no—
260C4×C5—42no—
261C4×C5—81no—
262C4×C5—161no—
263C4×C5#44711yescharacteristic
264C4×C5#45211yescharacteristic
265C4×C5#46111yescharacteristic
266C4×C5#46311yescharacteristic
267C4×C5#46511yescharacteristic
268C4×C5#46811yescharacteristic
269C4×C5#48011yescharacteristic
270C4×C5#48211yescharacteristic
271C24—41no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8C5×D812yes—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—81no—
293C2×D4—81no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4C2×C5×D411yescharacteristic
298C22×C4C2×C5×D411yescharacteristic
299C22×C4C5×#4111yescharacteristic
300SD16—48no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8C5×SD1612yescomplemented
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C42—41no—
326C42—41no—
327C42C2×C5×D411yescharacteristic
328C42C2×C5×D411yescharacteristic
329C42C5×#4111yescharacteristic
330C42C5×#4111yescharacteristic
331C2×C5—162no—
332C2×C5#166811yescharacteristic
333C2×C5#168211yescharacteristic
334C2×C5#168611yescharacteristic
335C2×C5#169811yescharacteristic
336C2×C5#203311yescharacteristic
337C2×C5#205111yescharacteristic
338C2×C5#205511yescharacteristic
339C23—42no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23C5×#12111yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—41no—
350Q8—41no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—82no—
356D4—82no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4C5×#12011yescharacteristic
373C2×C4C5×#12111yescharacteristic
374C2×C4C5×#12111yescharacteristic
375C2×C4C5×#12311yescharacteristic
376C2×C4C5×#12311yescharacteristic
377C2×C4C5×#12411yescharacteristic
378C2×C4C2×C5×D811yescharacteristic
379C2×C4C2×C5×SD1611yescharacteristic
380C2×C4C5×#13511yescharacteristic
381C2×C4C5×#13511yescharacteristic
382C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
384C2×C4C5×#13711yescharacteristic
385C2×C4C5×#13711yescharacteristic
386C8—22no—
387C8—42no—
388C8—41no—
389C8—81no—
390C8—81no—
391C5#1047411yescharacteristic, direct product factor, Sylow
392C22—82no—
393C22—82no—
394C22—82no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22C5×#39411yescharacteristic
400C22C5×#46011yescharacteristic
401C22C5×#46411yescharacteristic
402C22C5×#46611yescharacteristic
403C22C5×#46811yescharacteristic
404C22C5×#47111yescharacteristic
405C22C5×#47211yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—161no—
411C4C5×#44711yescharacteristic
412C4C5×#45211yescharacteristic
413C4C5×#46111yescharacteristic
414C4C5×#46311yescharacteristic
415C4C5×#46511yescharacteristic
416C4C5×#46811yescharacteristic
417C4C5×#48011yescharacteristic
418C4C5×#48211yescharacteristic
419C2—162no—
420C2C5×#166811yescharacteristic
421C2C5×#168211yescharacteristic
422C2C5×#168611yescharacteristic
423C2C5×#169811yescharacteristic
424C2C5×#203311yescharacteristic
425C2C5×#205111yescharacteristic
426C2C5×#205511yescharacteristic
subgroupG/H
center, socleC23×C5#121
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC2×C5×#127C22
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×#1984C2semidirect
4. C5×#1803C2semidirect
5. C2×C5×#335C2semidirect
7. C5×#1296C2semidirect
27. #10474C5direct (N × H)
68. #1984C2×C5semidirect
70. #1803C2×C5semidirect
71. C2×#335C2×C5semidirect
73. #1296C2×C5semidirect
118. C2×C5×C8SD16semidirect
315. C2×C8C5×SD16semidirect
391. C5#10474direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4602
C5×#19842
C5×#19572
C5×#18032
C2×C5×#3352
C2×C5×#3312
C5×#12962
#104745