Finite Group Databasebeta

← C5×#11500 #11934220

Subgroups — C5×#11500 #11934220

Order 1280.

706 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#381C211yesmaximal, characteristic
2C5×#1462C212yesmaximal
3C5×#1317C214yesmaximal, complemented
4C24×C4×C5C411yescharacteristic
5C22×C5×C22⋊C4C411yescharacteristic
6C22×C5×C22⋊C4C2211yescharacteristic
7C5×#407—22no—
8C5×#381—22no—
9C5×#381C412yes—
10C5×#381C2212yes—
11C2×C5×#95—22no—
12C5×#323—22no—
13C5×#323C2214yes—
14#11500C511yesmaximal, characteristic, direct product factor, Sylow
15C25×C5C2×C411yescharacteristic
16C23×C4×C5—21no—
17C23×C4×C5—21no—
18C23×C4×C5—24no—
19C23×C4×C5—21no—
20C23×C4×C5—42no—
21C23×C4×C5—42no—
22C23×C4×C5C2×C412yes—
23C23×C4×C5D412yes—
24C2×C5×M16—42no—
25C2×C5×C22⋊C4—21no—
26C2×C5×C22⋊C4—42no—
27C2×C5×C22⋊C4—42no—
28C2×C5×C22⋊C4C2×C412yes—
29C2×C5×C22⋊C4C2×C411yescharacteristic
30C2×C5×C22⋊C4D412yes—
31C2×C5×C22⋊C4C2311yescharacteristic
32C5×C22⋊C8—24no—
33C5×C22⋊C8—42no—
34C5×#95—24no—
35C5×#95—24no—
36C2×#381C2×C511yescharacteristic
37#1462C2×C512yes—
38#1317C2×C514yescomplemented
39C24×C5—22no—
40C24×C5—22no—
41C24×C5—22no—
42C24×C5—44no—
43C24×C5C22⋊C411yescharacteristic
44C24×C5C22⋊C411yescharacteristic
45C24×C5C22×C411yescharacteristic
46C22×C4×C5—24no—
47C22×C4×C5—24no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—22no—
53C22×C4×C5—416no—
54C22×C4×C5—42no—
55C22×C4×C5—42no—
56C22×C4×C5—42no—
57C22×C4×C5—42no—
58C22×C4×C5—42no—
59C22×C4×C5—42no—
60C22×C4×C5—44no—
61C22×C4×C5—42no—
62C22×C4×C5—44no—
63C22×C4×C5C22⋊C412yes—
64C22×C4×C5M1614yescomplemented
65C22×C4×C5C2×D412yes—
66C5×M16—84no—
67C2×C5×C8—44no—
68C5×C22⋊C4—41no—
69C5×C22⋊C4—42no—
70C5×C22⋊C4—41no—
71C5×C22⋊C4—42no—
72C5×C22⋊C4—42no—
73C24×C4C4×C511yescharacteristic
74C22×C22⋊C4C4×C511yescharacteristic
75C22×C22⋊C4C22×C511yescharacteristic
76#407—22no—
77#381—22no—
78#381C4×C512yes—
79#381C22×C512yes—
80C2×#95—22no—
81#323—22no—
82#323C22×C514yes—
83C23×C5—24no—
84C23×C5—21no—
85C23×C5—21no—
86C23×C5—28no—
87C23×C5—21no—
88C23×C5—21no—
89C23×C5—21no—
90C23×C5—21no—
91C23×C5—42no—
92C23×C5—42no—
93C23×C5—48no—
94C23×C5—48no—
95C23×C5—44no—
96C23×C5—44no—
97C23×C5#9912yes—
98C23×C5#10012yes—
99C23×C5C2×C22⋊C411yescharacteristic
100C23×C5C2×M1612yes—
101C2×C4×C5—24no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—48no—
105C2×C4×C5—48no—
106C2×C4×C5—44no—
107C2×C4×C5—44no—
108C2×C4×C5—42no—
109C2×C4×C5—42no—
110C2×C4×C5—44no—
111C2×C4×C5—42no—
112C2×C4×C5—42no—
113C2×C4×C5—44no—
114C2×C4×C5—42no—
115C2×C4×C5—82no—
116C2×C4×C5—84no—
117C5×C8—84no—
118C25C2×C4×C511yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4C2×C4×C512yes—
126C23×C4C5×D412yes—
127C2×M16—42no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4C2×C4×C512yes—
132C2×C22⋊C4C2×C4×C511yescharacteristic
133C2×C22⋊C4C5×D412yes—
134C2×C22⋊C4C23×C511yescharacteristic
135C22⋊C8—24no—
136C22⋊C8—42no—
137#95—24no—
138#95—24no—
139C22×C5—24no—
140C22×C5—24no—
141C22×C5—22no—
142C22×C5—22no—
143C22×C5—22no—
144C22×C5—22no—
145C22×C5—22no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5—416no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5#35214yes—
154C22×C5#40711yescharacteristic
155C22×C5C2×#9911yescharacteristic
156C22×C5C2×#10011yescharacteristic
157C4×C5—42no—
158C4×C5—42no—
159C4×C5—44no—
160C4×C5—82no—
161C24—22no—
162C24—22no—
163C24—22no—
164C24—44no—
165C24C5×C22⋊C411yescharacteristic
166C24C5×C22⋊C411yescharacteristic
167C24C22×C4×C511yescharacteristic
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—416no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4C5×C22⋊C412yes—
186C22×C4C5×M1614yescomplemented
187C22×C4C2×C5×D412yes—
188M16—84no—
189C2×C8—44no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—41no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C2×C5—21no—
196C2×C5—24no—
197C2×C5—21no—
198C2×C5—42no—
199C2×C5—42no—
200C2×C5#146211yescharacteristic
201C2×C5C2×#35212yes—
202C23—21no—
203C23—21no—
204C23—24no—
205C23—28no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—42no—
211C23—42no—
212C23—48no—
213C23—48no—
214C23—44no—
215C23—44no—
216C23C5×#9912yes—
217C23C5×#10012yes—
218C23C2×C5×C22⋊C411yescharacteristic
219C23C2×C5×M1612yes—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—48no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—84no—
236C8—84no—
237C5#1150011yescharacteristic, direct product factor, Sylow
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—416no—
250C22—42no—
251C22—42no—
252C22C5×#35214yes—
253C22C5×#40711yescharacteristic
254C22C2×C5×#9911yescharacteristic
255C22C2×C5×#10011yescharacteristic
256C4—42no—
257C4—42no—
258C4—44no—
259C4—82no—
260C2—21no—
261C2—21no—
262C2—24no—
263C2—42no—
264C2—42no—
265C2C5×#146211yescharacteristic
266C2C2×C5×#35212yes—
subgroupG/H
center, socleC22×C5C2×#99
commutatorC24C22×C4×C5
Frattini, cosocleC2×C22⋊C4C23×C5
prime radicalC25×C5C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#1317C2semidirect
14. #11500C5direct (N × H)
38. #1317C2×C5semidirect
64. C22×C4×C5M16semidirect
186. C22×C4C5×M16semidirect
237. C5#11500direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#3812
C5×#14622
C5×#13172
#115005