Finite Group Databasebeta

← C5×#35337 #11958057

Subgroups — C5×#35337 #11958057

Order 1280.

1554 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2862C211yesmaximal, characteristic, complemented
2C5×#2859C211yesmaximal, characteristic, complemented
3C5×#2851C212yesmaximal
4C5×#2647C212yesmaximal, complemented
5C5×#2628C212yesmaximal
6C5×#2589C218yesmaximal
7C5×#2588C218yesmaximal, complemented
8C5×#2331C212yesmaximal, complemented
9C5×#2314C212yesmaximal
10C4×C5×#125C212yesmaximal, complemented
11C4×C5×#117C211yesmaximal, characteristic
12C2×C5×C4⋊Q8C2212yes—
13C2×C5×#125C2212yes—
14C2×C5×#117C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16C22×C42×C5C2211yescharacteristic
17C5×#400C2214yes—
18C5×#400C2218yes—
19C5×#399C2214yes—
20C5×#398C2218yes—
21C5×#395C2214yes—
22C5×#393C2214yes—
23C5×#391C2218yes—
24C5×#391C2218yes—
25C5×#389C2214yes—
26C5×#389C2214yes—
27C5×#389C2218yes—
28C5×#388C2218yes—
29C5×#387C2218yes—
30C5×#387C2218yes—
31C5×#387C2218yes—
32C5×#387C2218yes—
33C5×#385C2218yes—
34C5×#384C2212yes—
35C5×#382C2212yes—
36C5×#382C2212yes—
37C5×#382C2212yes—
38C5×#380C2214yes—
39C4×C5×C4⋊C4C2214yes—
40C4×C5×C4⋊C4C2214yes—
41C4×C5×C4⋊C4C2218yes—
42C4×C5×C22⋊C4C2214yes—
43C5×#376C2211yescharacteristic
44C5×#376C2211yescharacteristic
45C43×C5C2211yescharacteristic
46C43×C5C2211yescharacteristic
47#35337C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8C2314yes—
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2312yes—
51C5×C4⋊Q8—28no—
52C5×#125—28no—
53C5×#117—24no—
54C5×#117Q818yescomplemented
55C2×C5×C4⋊C4C2312yes—
56C2×C5×C4⋊C4C2314yes—
57C2×C5×C4⋊C4C2314yes—
58C2×C5×C4⋊C4C2318yes—
59C2×C5×C4⋊C4C2314yes—
60C2×C5×C4⋊C4C2318yes—
61C2×C5×C4⋊C4C2318yes—
62C2×C5×C4⋊C4C2318yes—
63C2×C5×C4⋊C4C2312yes—
64C2×C5×C22⋊C4C2312yes—
65C2×C5×C22⋊C4C2318yes—
66C2×C5×C22⋊C4C2312yes—
67C2×C42×C5—24no—
68C2×C42×C5C2311yescharacteristic
69C2×C42×C5C2311yescharacteristic
70C2×C42×C5C2312yes—
71C2×C42×C5C2314yes—
72C2×C42×C5C2314yes—
73C2×C42×C5C2318yes—
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C5×#95C2314yes—
79C5×#95C2314yes—
80C5×#95C2318yes—
81C5×#95C2318yes—
82C5×#95C2318yes—
83C5×#95C2318yes—
84C5×#95C2314yes—
85C5×#95C2318yes—
86C5×#95C2314yes—
87C5×#95C2318yes—
88#2862C2×C511yescharacteristic, complemented
89#2859C2×C511yescharacteristic, complemented
90#2851C2×C512yes—
91#2647C2×C512yescomplemented
92#2628C2×C512yes—
93#2589C2×C518yes—
94#2588C2×C518yescomplemented
95#2331C2×C512yescomplemented
96#2314C2×C512yes—
97C4×#125C2×C512yescomplemented
98C4×#117C2×C511yescharacteristic
99C24×C5C2411yescharacteristic
100C2×C5×Q8—28no—
101C2×C5×Q8—48no—
102C22×C4×C5—22no—
103C22×C4×C5—22no—
104C22×C4×C5—24no—
105C22×C4×C5—44no—
106C22×C4×C5C2×Q814yes—
107C22×C4×C5C2412yes—
108C22×C4×C5C2412yes—
109C22×C4×C5C2418yes—
110C22×C4×C5C2418yes—
111C22×C4×C5C2412yes—
112C22×C4×C5C2412yes—
113C22×C4×C5C2411yescharacteristic
114C22×C4×C5C2412yes—
115C22×C4×C5C2412yes—
116C22×C4×C5C2411yescharacteristic
117C5×C4⋊C4—24no—
118C5×C4⋊C4—216no—
119C5×C4⋊C4—28no—
120C5×C4⋊C4—28no—
121C5×C4⋊C4—28no—
122C5×C4⋊C4—48no—
123C5×C4⋊C4—48no—
124C5×C4⋊C4—48no—
125C5×C4⋊C4C2×Q818yes—
126C5×C22⋊C4—24no—
127C5×C22⋊C4—48no—
128C5×C22⋊C4C2×Q818yes—
129C42×C5—24no—
130C42×C5—22no—
131C42×C5—22no—
132C42×C5—44no—
133C42×C5—48no—
134C42×C5—44no—
135C42×C5C2×Q814yes—
136C42×C5C2×Q814yes—
137C42×C5#4114yes—
138C42×C5#4114yes—
139C2×C4⋊Q8C22×C512yes—
140C2×#125C22×C512yes—
141C2×#117C22×C511yescharacteristic
142C2×#117C22×C511yescharacteristic
143C22×C42C22×C511yescharacteristic
144#400C22×C514yes—
145#400C22×C518yes—
146#399C22×C514yes—
147#398C22×C518yes—
148#395C22×C514yes—
149#393C22×C514yes—
150#391C22×C518yes—
151#391C22×C518yes—
152#389C22×C514yes—
153#389C22×C514yes—
154#389C22×C518yes—
155#388C22×C518yes—
156#387C22×C518yes—
157#387C22×C518yes—
158#387C22×C518yes—
159#387C22×C518yes—
160#385C22×C518yes—
161#384C22×C512yes—
162#382C22×C512yes—
163#382C22×C512yes—
164#382C22×C512yes—
165#380C22×C514yes—
166C4×C4⋊C4C22×C514yes—
167C4×C4⋊C4C22×C518yes—
168C4×C4⋊C4C22×C514yes—
169C4×C22⋊C4C22×C514yes—
170#376C22×C511yescharacteristic
171#376C22×C511yescharacteristic
172C43C22×C511yescharacteristic
173C43C22×C511yescharacteristic
174C23×C5—21no—
175C23×C5—21no—
176C23×C5—24no—
177C23×C5C22×Q812yes—
178C23×C5C2511yescharacteristic
179C5×Q8—88no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—24no—
183C2×C4×C5—24no—
184C2×C4×C5—24no—
185C2×C4×C5—24no—
186C2×C4×C5—28no—
187C2×C4×C5—22no—
188C2×C4×C5—22no—
189C2×C4×C5—28no—
190C2×C4×C5—28no—
191C2×C4×C5—28no—
192C2×C4×C5—28no—
193C2×C4×C5—24no—
194C2×C4×C5—28no—
195C2×C4×C5—24no—
196C2×C4×C5—28no—
197C2×C4×C5—48no—
198C2×C4×C5C22×Q814yes—
199C2×C4×C5C22×Q814yes—
200C2×C4×C5C22×Q812yes—
201C2×C4×C5C22×Q812yes—
202C2×C4×C5C2×#4114yes—
203C2×C4×C5C2×#4114yes—
204C2×C4×C5C2×#4112yes—
205C2×C4×C5C2×#4112yes—
206C22×Q8C23×C514yes—
207C23×C4C23×C511yescharacteristic
208C23×C4C23×C512yes—
209C4⋊Q8—28no—
210#125—28no—
211#117—24no—
212#117C5×Q818yescomplemented
213C2×C4⋊C4C23×C512yes—
214C2×C4⋊C4C23×C514yes—
215C2×C4⋊C4C23×C518yes—
216C2×C4⋊C4C23×C514yes—
217C2×C4⋊C4C23×C514yes—
218C2×C4⋊C4C23×C518yes—
219C2×C4⋊C4C23×C518yes—
220C2×C4⋊C4C23×C518yes—
221C2×C4⋊C4C23×C512yes—
222C2×C22⋊C4C23×C512yes—
223C2×C22⋊C4C23×C518yes—
224C2×C22⋊C4C23×C512yes—
225C2×C42—24no—
226C2×C42C23×C511yescharacteristic
227C2×C42C23×C511yescharacteristic
228C2×C42C23×C512yes—
229C2×C42C23×C514yes—
230C2×C42C23×C514yes—
231C2×C42C23×C518yes—
232C2×C42C23×C511yescharacteristic
233C2×C42C23×C511yescharacteristic
234C2×C42C23×C511yescharacteristic
235C2×C42C23×C511yescharacteristic
236#95C23×C514yes—
237#95C23×C514yes—
238#95C23×C518yes—
239#95C23×C518yes—
240#95C23×C518yes—
241#95C23×C518yes—
242#95C23×C514yes—
243#95C23×C518yes—
244#95C23×C514yes—
245#95C23×C518yes—
246C22×C5—22no—
247C22×C5—24no—
248C22×C5—42no—
249C22×C5—42no—
250C22×C5C23×Q811yescharacteristic
251C22×C5C22×#4111yescharacteristic
252C22×C5C22×#4111yescharacteristic
253C22×C5#58514yes—
254C4×C5—22no—
255C4×C5—24no—
256C4×C5—24no—
257C4×C5—24no—
258C4×C5—24no—
259C4×C5—48no—
260C4×C5—42no—
261C4×C5—42no—
262C4×C5—48no—
263C4×C5#53314yes—
264C24C24×C511yescharacteristic
265C2×Q8—28no—
266C2×Q8—48no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4C2×C5×Q814yes—
272C22×C4C24×C512yes—
273C22×C4C24×C518yes—
274C22×C4C24×C512yes—
275C22×C4C24×C518yes—
276C22×C4C24×C511yescharacteristic
277C22×C4C24×C512yes—
278C22×C4C24×C512yes—
279C22×C4C24×C512yes—
280C22×C4C24×C512yes—
281C22×C4C24×C511yescharacteristic
282C4⋊C4—24no—
283C4⋊C4—216no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4C2×C5×Q818yes—
291C22⋊C4—24no—
292C22⋊C4—48no—
293C22⋊C4C2×C5×Q818yes—
294C42—24no—
295C42—22no—
296C42—22no—
297C42—44no—
298C42—48no—
299C42—44no—
300C42C2×C5×Q814yes—
301C42C2×C5×Q814yes—
302C42C5×#4114yes—
303C42C5×#4114yes—
304C2×C5—42no—
305C2×C5C2×#53311yescharacteristic
306C2×C5#345512yes—
307C2×C5#350412yes—
308C2×C5#351012yes—
309C23—21no—
310C23—21no—
311C23—24no—
312C23C22×C5×Q812yes—
313C23C25×C511yescharacteristic
314Q8—88no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—48no—
333C2×C4C22×C5×Q814yes—
334C2×C4C22×C5×Q814yes—
335C2×C4C22×C5×Q812yes—
336C2×C4C22×C5×Q812yes—
337C2×C4C2×C5×#4114yes—
338C2×C4C2×C5×#4114yes—
339C2×C4C2×C5×#4112yes—
340C2×C4C2×C5×#4112yes—
341C5#3533711yescharacteristic, direct product factor, Sylow
342C22—22no—
343C22—24no—
344C22—42no—
345C22—42no—
346C22C23×C5×Q811yescharacteristic
347C22C22×C5×#4111yescharacteristic
348C22C22×C5×#4111yescharacteristic
349C22C5×#58514yes—
350C4—22no—
351C4—24no—
352C4—24no—
353C4—24no—
354C4—24no—
355C4—42no—
356C4—48no—
357C4—48no—
358C4—42no—
359C4C5×#53314yes—
360C2—42no—
361C2C2×C5×#53311yescharacteristic
362C2C5×#345512yes—
363C2C5×#350412yes—
364C2C5×#351012yes—
subgroupG/H
center, socleC23×C5C25
commutator, Frattini, cosocleC23C25×C5
prime radicalC24×C5C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2862C2semidirect
2. C5×#2859C2semidirect
4. C5×#2647C2semidirect
7. C5×#2588C2semidirect
8. C5×#2331C2semidirect
10. C4×C5×#125C2semidirect
47. #35337C5direct (N × H)
54. C5×#117Q8semidirect
88. #2862C2×C5semidirect
89. #2859C2×C5semidirect
91. #2647C2×C5semidirect
94. #2588C2×C5semidirect
95. #2331C2×C5semidirect
97. C4×#125C2×C5semidirect
212. #117C5×Q8semidirect
341. C5#35337direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28622
C5×#28592
C5×#28512
C5×#26472
C5×#26282
C5×#25892
C5×#25882
C5×#23312
C5×#23142
C4×C5×#1252
C4×C5×#1172
#353375