Finite Group Databasebeta

← C5×#7726 #11930446

Subgroups — C5×#7726 #11930446

Order 1280.

400 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2860C211yesmaximal, characteristic
2C5×#1839C211yesmaximal, characteristic, complemented
3C5×#1839C211yesmaximal, characteristic, complemented
4C5×#1271C212yesmaximal, complemented
5C5×#1270C212yesmaximal
6C2×C5×#126—21no—
7C5×#390C411yescharacteristic
8C5×#390C411yescharacteristic
9C5×#390C2211yescharacteristic
10C5×#389C411yescharacteristic
11C5×#389C411yescharacteristic
12C5×#389C2211yescharacteristic
13C5×#388—21no—
14C5×#388—21no—
15C5×#388—21no—
16C43×C5C2211yescharacteristic
17C5×#336—21no—
18C5×#336—21no—
19C5×#336—21no—
20C5×#336—21no—
21C5×#336C2212yes—
22C5×#336C2212yes—
23#7726C511yesmaximal, characteristic, direct product factor, Sylow
24C22×C5×D4C2×C411yescharacteristic
25C22×C5×C8—22no—
26C22×C5×C8—22no—
27C5×#126—44no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—21no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C42×C5C2×C411yescharacteristic
41C2×C42×C5C2×C411yescharacteristic
42C2×C42×C5D411yescharacteristic
43C2×C42×C5D411yescharacteristic
44C2×C42×C5D411yescharacteristic
45C2×C42×C5D411yescharacteristic
46C2×C42×C5C2311yescharacteristic
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50#2860C2×C511yescharacteristic
51#1839C2×C511yescharacteristic, complemented
52#1839C2×C511yescharacteristic, complemented
53#1271C2×C512yescomplemented
54#1270C2×C512yes—
55C24×C5—21no—
56C2×C5×D4—22no—
57C2×C5×D4—41no—
58C2×C5×D4—41no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5C22⋊C411yescharacteristic
63C22×C4×C5C22⋊C411yescharacteristic
64C22×C4×C5C22⋊C411yescharacteristic
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C22×C411yescharacteristic
67C22×C4×C5C2×D411yescharacteristic
68C22×C4×C5C2×D411yescharacteristic
69C2×C5×C8—42no—
70C2×C5×C8—42no—
71C2×C5×C8—42no—
72C2×C5×C8—42no—
73C2×C5×C8—42no—
74C2×C5×C8—42no—
75C5×C4⋊C4—22no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—42no—
79C5×C4⋊C4—42no—
80C5×C4⋊C4—42no—
81C5×C22⋊C4—42no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—42no—
85C5×C22⋊C4—42no—
86C5×C22⋊C4—42no—
87C42×C5—22no—
88C42×C5—22no—
89C42×C5—22no—
90C42×C5—41no—
91C42×C5—41no—
92C42×C5—41no—
93C42×C5—41no—
94C2×#126—21no—
95#390C4×C511yescharacteristic
96#390C4×C511yescharacteristic
97#390C22×C511yescharacteristic
98#389C4×C511yescharacteristic
99#389C4×C511yescharacteristic
100#389C22×C511yescharacteristic
101#388—21no—
102#388—21no—
103#388—21no—
104C43C22×C511yescharacteristic
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336C22×C512yes—
110#336C22×C512yes—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5C2×C22⋊C411yescharacteristic
119C5×D4—82no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5#13112yes—
148C2×C4×C5#13512yes—
149C2×C4×C5AGL(1,ℤ/8)12yes—
150C5×C8—82no—
151C5×C8—82no—
152C22×D4C2×C4×C511yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155#126—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×C4×C511yescharacteristic
160C2×C4⋊C4C2×C4×C511yescharacteristic
161C2×C4⋊C4C2×C4×C511yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C42C2×C4×C511yescharacteristic
169C2×C42C2×C4×C511yescharacteristic
170C2×C42C5×D411yescharacteristic
171C2×C42C5×D411yescharacteristic
172C2×C42C5×D411yescharacteristic
173C2×C42C5×D411yescharacteristic
174C2×C42C23×C511yescharacteristic
175#95—21no—
176#95—21no—
177#95—21no—
178C22×C5—81no—
179C22×C5—81no—
180C22×C5—81no—
181C22×C5—81no—
182C22×C5—81no—
183C22×C5—81no—
184C22×C5—81no—
185C22×C5#40811yescharacteristic
186C22×C5#41011yescharacteristic
187C22×C5#41311yescharacteristic
188C22×C5#41611yescharacteristic
189C22×C5#41811yescharacteristic
190C22×C5#42111yescharacteristic
191C22×C5#42111yescharacteristic
192C4×C5—22no—
193C4×C5—22no—
194C4×C5—22no—
195C4×C5—41no—
196C4×C5—41no—
197C4×C5—41no—
198C4×C5—41no—
199C4×C5—81no—
200C4×C5—81no—
201C4×C5—81no—
202C24—21no—
203C2×D4—22no—
204C2×D4—41no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4C5×C22⋊C411yescharacteristic
210C22×C4C5×C22⋊C411yescharacteristic
211C22×C4C5×C22⋊C411yescharacteristic
212C22×C4C5×C22⋊C411yescharacteristic
213C22×C4C22×C4×C511yescharacteristic
214C22×C4C2×C5×D411yescharacteristic
215C22×C4C2×C5×D411yescharacteristic
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C2×C5—161no—
242C2×C5#147311yescharacteristic
243C2×C5#148211yescharacteristic
244C2×C5#149611yescharacteristic
245C2×C5#151411yescharacteristic
246C2×C5#152111yescharacteristic
247C2×C5#153511yescharacteristic
248C2×C5#155311yescharacteristic
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×C5×C22⋊C411yescharacteristic
257D4—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4C5×#13112yes—
286C2×C4C5×#13512yes—
287C2×C4C5×AGL(1,ℤ/8)12yes—
288C8—82no—
289C8—82no—
290C5#772611yescharacteristic, direct product factor, Sylow
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C5×#40811yescharacteristic
299C22C5×#41011yescharacteristic
300C22C5×#41311yescharacteristic
301C22C5×#41611yescharacteristic
302C22C5×#41811yescharacteristic
303C22C5×#42111yescharacteristic
304C22C5×#42111yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—81no—
315C2—161no—
316C2C5×#147311yescharacteristic
317C2C5×#148211yescharacteristic
318C2C5×#149611yescharacteristic
319C2C5×#151411yescharacteristic
320C2C5×#152111yescharacteristic
321C2C5×#153511yescharacteristic
322C2C5×#155311yescharacteristic
subgroupG/H
center, socleC23×C5C2×C22⋊C4
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C42C23×C5
prime radicalC22×C5×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ C43×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1839C2semidirect
3. C5×#1839C2semidirect
4. C5×#1271C2semidirect
23. #7726C5direct (N × H)
51. #1839C2×C5semidirect
52. #1839C2×C5semidirect
53. #1271C2×C5semidirect
290. C5#7726direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28602
C5×#18392
C5×#12712
C5×#12702
#77265