Finite Group Databasebeta

← C5×#11450 #11934170

Subgroups — C5×#11450 #11934170

Order 1280.

434 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2317C211yesmaximal, characteristic
2C5×#1458C212yesmaximal
3C5×#1325C214yesmaximal, complemented
4C2×C5×#117C411yescharacteristic
5C2×C5×#117C2211yescharacteristic
6C22×C5×C4⋊C4C411yescharacteristic
7C5×#431—22no—
8C5×#389—22no—
9C5×#385—22no—
10C5×#384C2212yes—
11C5×#382C412yes—
12C5×#324C2214yes—
13C5×#323—22no—
14#11450C511yesmaximal, characteristic, direct product factor, Sylow
15C23×C4×C5—21no—
16C23×C4×C5C2×C411yescharacteristic
17C5×#117—22no—
18C5×#117—42no—
19C5×#117C814yescomplemented
20C2×C5×C4⋊C4—21no—
21C2×C5×C4⋊C4—21no—
22C2×C5×C4⋊C4—24no—
23C2×C5×C4⋊C4—42no—
24C2×C5×C4⋊C4C2×C412yes—
25C2×C5×C4⋊C4C2×C411yescharacteristic
26C2×C5×C4⋊C4C2311yescharacteristic
27C2×C5×C22⋊C4—21no—
28C2×C5×C22⋊C4D412yes—
29C2×C42×C5C2×C412yes—
30C2×C42×C5D412yes—
31C5×C4⋊C8—42no—
32C5×C22⋊C8—24no—
33C4×C5×C8—42no—
34C5×#95—24no—
35#2317C2×C511yescharacteristic
36#1458C2×C512yes—
37#1325C2×C514yescomplemented
38C24×C5C22⋊C411yescharacteristic
39C22×C4×C5—22no—
40C22×C4×C5—22no—
41C22×C4×C5—22no—
42C22×C4×C5—22no—
43C22×C4×C5—42no—
44C22×C4×C5—82no—
45C22×C4×C5C22⋊C412yes—
46C22×C4×C5C22⋊C411yescharacteristic
47C22×C4×C5C2×C812yes—
48C22×C4×C5M1612yes—
49C22×C4×C5C22×C411yescharacteristic
50C22×C4×C5C2×D412yes—
51C2×C5×C8—44no—
52C5×C4⋊C4—24no—
53C5×C4⋊C4—41no—
54C5×C4⋊C4—41no—
55C5×C4⋊C4—42no—
56C5×C4⋊C4—42no—
57C5×C4⋊C4—84no—
58C5×C22⋊C4—24no—
59C5×C22⋊C4—42no—
60C42×C5—22no—
61C42×C5—22no—
62C42×C5—22no—
63C42×C5C2×C814yes—
64C2×#117C4×C511yescharacteristic
65C2×#117C22×C511yescharacteristic
66C22×C4⋊C4C4×C511yescharacteristic
67#431—22no—
68#389—22no—
69#385—22no—
70#384C22×C512yes—
71#382C4×C512yes—
72#324C22×C514yes—
73#323—22no—
74C23×C5—21no—
75C23×C5—21no—
76C23×C5—41no—
77C23×C5—41no—
78C23×C5C22⋊C812yes—
79C23×C5C2×C22⋊C411yescharacteristic
80C2×C4×C5—21no—
81C2×C4×C5—21no—
82C2×C4×C5—24no—
83C2×C4×C5—24no—
84C2×C4×C5—24no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5—42no—
88C2×C4×C5—42no—
89C2×C4×C5—42no—
90C2×C4×C5—42no—
91C2×C4×C5—44no—
92C2×C4×C5—42no—
93C2×C4×C5—84no—
94C2×C4×C5C22⋊C812yes—
95C2×C4×C5#9912yes—
96C2×C4×C5#10112yes—
97C2×C4×C5C22×C811yescharacteristic
98C2×C4×C5C2×M1611yescharacteristic
99C5×C8—84no—
100C23×C4—21no—
101C23×C4C2×C4×C511yescharacteristic
102#117—22no—
103#117—42no—
104#117C5×C814yescomplemented
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C2×C4×C512yes—
110C2×C4⋊C4C2×C4×C511yescharacteristic
111C2×C4⋊C4C23×C511yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4C5×D412yes—
114C2×C42C2×C4×C512yes—
115C2×C42C5×D412yes—
116C4⋊C8—42no—
117C22⋊C8—24no—
118C4×C8—42no—
119#95—24no—
120C22×C5—21no—
121C22×C5—21no—
122C22×C5—22no—
123C22×C5—42no—
124C22×C5—42no—
125C22×C5—41no—
126C22×C5—41no—
127C22×C5C2×C22⋊C811yescharacteristic
128C22×C5C2×#9911yescharacteristic
129C22×C5C2×#10111yescharacteristic
130C4×C5—21no—
131C4×C5—21no—
132C4×C5—44no—
133C4×C5—42no—
134C4×C5—42no—
135C4×C5—82no—
136C4×C5#32414yes—
137C24C5×C22⋊C411yescharacteristic
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—82no—
144C22×C4C5×C22⋊C412yes—
145C22×C4C5×C22⋊C411yescharacteristic
146C22×C4C2×C5×C812yes—
147C22×C4C5×M1612yes—
148C22×C4C22×C4×C511yescharacteristic
149C22×C4C2×C5×D412yes—
150C2×C8—44no—
151C4⋊C4—24no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42C2×C5×C814yes—
163C2×C5—21no—
164C2×C5—21no—
165C2×C5—42no—
166C2×C5C2×#32411yescharacteristic
167C2×C5#212811yescharacteristic
168C2×C5#213511yescharacteristic
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23C5×C22⋊C812yes—
174C23C2×C5×C22⋊C411yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—84no—
189C2×C4C5×C22⋊C812yes—
190C2×C4C5×#9912yes—
191C2×C4C5×#10112yes—
192C2×C4C22×C5×C811yescharacteristic
193C2×C4C2×C5×M1611yescharacteristic
194C8—84no—
195C5#1145011yescharacteristic, direct product factor, Sylow
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×C5×C22⋊C811yescharacteristic
204C22C2×C5×#9911yescharacteristic
205C22C2×C5×#10111yescharacteristic
206C4—21no—
207C4—21no—
208C4—44no—
209C4—42no—
210C4—42no—
211C4—82no—
212C4C5×#32414yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×C5×#32411yescharacteristic
217C2C5×#212811yescharacteristic
218C2C5×#213511yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC2×C4C22×C5×C8
Frattini, cosocleC2×C4⋊C4C23×C5
prime radicalC24×C5C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#1325C2semidirect
14. #11450C5direct (N × H)
19. C5×#117C8semidirect
37. #1325C2×C5semidirect
104. #117C5×C8semidirect
195. C5#11450direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#23172
C5×#14582
C5×#13252
#114505