Finite Group Databasebeta

← C5×#7564 #11930284

Subgroups — C5×#7564 #11930284

Order 1280.

550 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C5×#2296C211yesmaximal, characteristic
2C4×C5×C22⋊C8C212yesmaximal
3C5×#1271C214yesmaximal, complemented
4C2×C5×#117C411yescharacteristic
5C2×C5×#117C411yescharacteristic
6C22×C42×C5C2211yescharacteristic
7C2×C5×C22⋊C8—22no—
8C2×C4×C5×C8—22no—
9C5×#389C412yes—
10C5×#389C2212yes—
11C5×#385—22no—
12C4×C5×C4⋊C4—22no—
13C5×#336C2214yes—
14#7564C511yesmaximal, characteristic, direct product factor, Sylow
15C23×C4×C5C2×C411yescharacteristic
16C23×C4×C5D412yes—
17C22×C5×C8—24no—
18C5×#117—22no—
19C5×#117—42no—
20C5×#117C814yescomplemented
21C2×C5×C4⋊C4—22no—
22C2×C5×C4⋊C4—22no—
23C2×C5×C4⋊C4C2×C412yes—
24C2×C5×C4⋊C4C2×C412yes—
25C2×C5×C22⋊C4—21no—
26C2×C5×C22⋊C4—21no—
27C2×C42×C5—21no—
28C2×C42×C5—21no—
29C2×C42×C5—42no—
30C2×C42×C5C2×C411yescharacteristic
31C2×C42×C5D412yes—
32C2×C42×C5C2311yescharacteristic
33C5×C22⋊C8—28no—
34C4×C5×C8—44no—
35C5×#95—22no—
36C5×#95—22no—
37#2296C2×C511yescharacteristic
38C4×C22⋊C8C2×C512yes—
39#1271C2×C514yescomplemented
40C24×C5C22⋊C411yescharacteristic
41C22×C4×C5—22no—
42C22×C4×C5—22no—
43C22×C4×C5—22no—
44C22×C4×C5—24no—
45C22×C4×C5—24no—
46C22×C4×C5—22no—
47C22×C4×C5—42no—
48C22×C4×C5C22⋊C412yes—
49C22×C4×C5C22⋊C411yescharacteristic
50C22×C4×C5C2×C812yes—
51C22×C4×C5M1612yes—
52C22×C4×C5C22×C411yescharacteristic
53C22×C4×C5C2×D412yes—
54C2×C5×C8—28no—
55C2×C5×C8—44no—
56C2×C5×C8—44no—
57C5×C4⋊C4—22no—
58C5×C4⋊C4—42no—
59C5×C4⋊C4—44no—
60C5×C4⋊C4—44no—
61C5×C4⋊C4C2×C814yes—
62C5×C22⋊C4—24no—
63C5×C22⋊C4—42no—
64C42×C5—24no—
65C42×C5—24no—
66C42×C5—41no—
67C42×C5—41no—
68C42×C5—42no—
69C2×#117C4×C511yescharacteristic
70C2×#117C4×C511yescharacteristic
71C22×C42C22×C511yescharacteristic
72C2×C22⋊C8—22no—
73C2×C4×C8—22no—
74#389C4×C512yes—
75#389C22×C512yes—
76#385—22no—
77C4×C4⋊C4—22no—
78#336C22×C514yes—
79C23×C5—21no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5C22⋊C812yes—
84C23×C5C2×C22⋊C411yescharacteristic
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—24no—
92C2×C4×C5—24no—
93C2×C4×C5—24no—
94C2×C4×C5—48no—
95C2×C4×C5—42no—
96C2×C4×C5—42no—
97C2×C4×C5—42no—
98C2×C4×C5—42no—
99C2×C4×C5—42no—
100C2×C4×C5C22⋊C812yes—
101C2×C4×C5#9912yes—
102C2×C4×C5#10112yes—
103C2×C4×C5#10414yes—
104C2×C4×C5C22×C811yescharacteristic
105C2×C4×C5C2×M1611yescharacteristic
106C2×C4×C5#13514yes—
107C5×C8—48no—
108C23×C4C2×C4×C511yescharacteristic
109C23×C4C5×D412yes—
110C22×C8—24no—
111#117—22no—
112#117—42no—
113#117C5×C814yescomplemented
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C2×C4×C512yes—
117C2×C4⋊C4C2×C4×C512yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—42no—
123C2×C42C2×C4×C511yescharacteristic
124C2×C42C5×D412yes—
125C2×C42C23×C511yescharacteristic
126C22⋊C8—28no—
127C4×C8—44no—
128#95—22no—
129#95—22no—
130C22×C5—22no—
131C22×C5—24no—
132C22×C5—41no—
133C22×C5—42no—
134C22×C5—41no—
135C22×C5C2×C22⋊C811yescharacteristic
136C22×C5C2×#9911yescharacteristic
137C22×C5C2×#10111yescharacteristic
138C22×C5#41612yes—
139C22×C5C2×#10412yes—
140C4×C5—21no—
141C4×C5—21no—
142C4×C5—24no—
143C4×C5—24no—
144C4×C5—44no—
145C4×C5—82no—
146C4×C5#32414yes—
147C24C5×C22⋊C411yescharacteristic
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4C5×C22⋊C412yes—
156C22×C4C5×C22⋊C411yescharacteristic
157C22×C4C2×C5×C812yes—
158C22×C4C5×M1612yes—
159C22×C4C22×C4×C511yescharacteristic
160C22×C4C2×C5×D412yes—
161C2×C8—28no—
162C2×C8—44no—
163C2×C8—44no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4C2×C5×C814yes—
169C22⋊C4—24no—
170C22⋊C4—42no—
171C42—24no—
172C42—24no—
173C42—41no—
174C42—41no—
175C42—42no—
176C2×C5—42no—
177C2×C5C2×#32411yescharacteristic
178C2×C5#147612yes—
179C2×C5#150112yes—
180C2×C5#152611yescharacteristic
181C2×C5#154411yescharacteristic
182C23—21no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23C5×C22⋊C812yes—
187C23C2×C5×C22⋊C411yescharacteristic
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4C5×C22⋊C812yes—
204C2×C4C5×#9912yes—
205C2×C4C5×#10112yes—
206C2×C4C5×#10414yes—
207C2×C4C22×C5×C811yescharacteristic
208C2×C4C2×C5×M1611yescharacteristic
209C2×C4C5×#13514yes—
210C8—48no—
211C5#756411yescharacteristic, direct product factor, Sylow
212C22—22no—
213C22—24no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22C2×C5×C22⋊C811yescharacteristic
218C22C2×C5×#9911yescharacteristic
219C22C2×C5×#10111yescharacteristic
220C22C5×#41612yes—
221C22C2×C5×#10412yes—
222C4—24no—
223C4—24no—
224C4—21no—
225C4—21no—
226C4—44no—
227C4—82no—
228C4C5×#32414yes—
229C2—42no—
230C2C2×C5×#32411yescharacteristic
231C2C5×#147612yes—
232C2C5×#150112yes—
233C2C5×#152611yescharacteristic
234C2C5×#154411yescharacteristic
subgroupG/H
centerC22×C4×C5C22⋊C4
commutatorC2×C4C22×C5×C8
Frattini, cosocleC2×C42C23×C5
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#1271C2semidirect
14. #7564C5direct (N × H)
20. C5×#117C8semidirect
39. #1271C2×C5semidirect
113. #117C5×C8semidirect
211. C5#7564direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#22962
C4×C5×C22⋊C82
C5×#12712
#75645