Finite Group Databasebeta

← #13013667

Subgroups — #13013667

Order 1280.

286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#10604719C211yesmaximal, characteristic
2#10604705C211yesmaximal, characteristic
3#10599899C211yesmaximal, characteristic
4C22×C8×D5C2211yescharacteristic
5C4×C22⋊AGL(1,𝔽5)C411yescharacteristic
6D5⋊(C22⋊C8)C411yescharacteristic
7C2×#3998C412yescomplemented
8C2×#3992—21no—
9C2×#3930—21no—
10(C2×AGL(1,𝔽5))⋊C8—21no—
11#64887—21no—
12#64887—21no—
13C5×C22⋊C16C411yescharacteristic, complemented
14C22⋊#672C411yescharacteristic, complemented
15#64727—21no—
16#7201—51nomaximal, Sylow
17C22×C4×D5C2×C411yescharacteristic
18C2×C22⋊AGL(1,𝔽5)C812yes—
19C2×C4×AGL(1,𝔽5)—21no—
20C2×D5⋊C8—21no—
21C22×C5×C8C2×C411yescharacteristic
22C22×#168C2×C411yescharacteristic
23C2×C8×D5—42no—
24C2×C8×D5D411yescharacteristic
25C2×C8×D5D411yescharacteristic
26C2×C8×D5D411yescharacteristic
27C2×C8×D5Q811yescharacteristic
28C22⋊#170C812yes—
29(C2×AGL(1,𝔽5))⋊C4—21no—
30#4003—21no—
31#3998—42no—
32C2×#674—22no—
33#3992—42no—
34C2×C5×C16—21no—
35C2×#672—21no—
36#3930—42no—
37#2112—51no—
38#2112—51no—
39#1752—51no—
40C23×D5C2×C811yescharacteristic
41C22×AGL(1,𝔽5)—22no—
42C22×C4×C5C4211yescharacteristic
43C22×Dic5C2×C811yescharacteristic
44C2×C4×D5—41no—
45C2×C4×D5—41no—
46C2×C4×D5C22⋊C411yescharacteristic
47C2×C4×D5C4⋊C411yescharacteristic
48C2×C4×D5M1611yescharacteristic
49C2×C4×D5M1611yescharacteristic
50C22⋊AGL(1,𝔽5)—42no—
51C2×#170—22no—
52C4×AGL(1,𝔽5)—41no—
53C4×AGL(1,𝔽5)—41no—
54D5⋊C8—41no—
55D5⋊C8—41no—
56C2×C5×C8—41no—
57C2×C5×C8C22⋊C411yescharacteristic
58C2×C5×C8C4⋊C411yescharacteristic
59C2×#168—22no—
60C2×#168C22⋊C411yescharacteristic
61C2×#168C4⋊C411yescharacteristic
62C8×D5—22no—
63C8×D5—22no—
64C8×D5—41no—
65C8×D5—41no—
66#674—42no—
67C5×C16—41no—
68#672—41no—
69C23×C8—51no—
70C2×M32—52no—
71C2×M32—101no—
72C2×C22⋊C8—51no—
73C2×#97—101no—
74C4×C22⋊C4—51no—
75C22⋊C16—51no—
76C22⋊C16—101no—
77C22⋊C16—101no—
78C22⋊C16—101no—
79C22⋊C16AGL(1,𝔽5)11yescharacteristic, complemented
80#336—101no—
81C23×C5C4×C811yescharacteristic
82C22×D5—22no—
83C22×D5—41no—
84C22×D5—41no—
85C22×D5C22⋊C811yescharacteristic
86C22×D5C4⋊C811yescharacteristic
87C2×AGL(1,𝔽5)—42no—
88C2×AGL(1,𝔽5)—42no—
89C2×AGL(1,𝔽5)—42no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5#9511yescharacteristic
93C2×C4×C5#9711yescharacteristic
94C2×Dic5—41no—
95C2×Dic5C22⋊C811yescharacteristic
96C2×Dic5C4⋊C811yescharacteristic
97C4×D5—21no—
98C4×D5—21no—
99C4×D5—21no—
100C4×D5—21no—
101C4×D5—41no—
102C4×D5—41no—
103#170—42no—
104C5×C8—21no—
105C5×C8—21no—
106#168—21no—
107#168#9912yes—
108C23×C4—51no—
109C22×C8—51no—
110C22×C8—51no—
111C22×C8—51no—
112C22×C8—51no—
113C22×C8—51no—
114C22×C8—101no—
115C22×C8—202no—
116C22×C8C2×AGL(1,𝔽5)11yescharacteristic
117C2×C22⋊C4—52no—
118C2×C42—101no—
119M32—202no—
120M32—202no—
121C2×C16—21no—
122C2×C16—102no—
123C2×C16—101no—
124C22⋊C8—52no—
125C22⋊C8—101no—
126#97—202no—
127#95—101no—
128C22×C5—21no—
129C22×C5—21no—
130C22×C5—21no—
131C22×C5#33611yescharacteristic
132C2×D5—21no—
133C2×D5—21no—
134C2×D5—21no—
135C2×D5—21no—
136C2×D5—21no—
137C2×D5—44no—
138C2×D5#34912yes—
139AGL(1,𝔽5)—82no—
140C4×C5—21no—
141C4×C5#34211yescharacteristic
142C4×C5#34411yescharacteristic
143Dic5—21no—
144Dic5—21no—
145C24—51no—
146C22×C4—51no—
147C22×C4—51no—
148C22×C4—51no—
149C22×C4—51no—
150C22×C4—51no—
151C22×C4—102no—
152C22×C4—201no—
153C22×C4—201no—
154C22×C4C4×AGL(1,𝔽5)11yescharacteristic
155C2×C8—41no—
156C2×C8—51no—
157C2×C8—51no—
158C2×C8—102no—
159C2×C8—102no—
160C2×C8—102no—
161C2×C8—102no—
162C2×C8—201no—
163C2×C8—201no—
164C2×C8—201no—
165C2×C8—201no—
166C2×C8C4⋊AGL(1,𝔽5)11yescharacteristic
167C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
168C22⋊C4—202no—
169C42—201no—
170C42—201no—
171C16—41no—
172C16—201no—
173C16—202no—
174C2×C5—41no—
175C2×C5#128211yescharacteristic
176C2×C5#137711yescharacteristic
177C2×C5#137811yescharacteristic
178D5—22no—
179D5—41no—
180C23—51no—
181C23—51no—
182C23—102no—
183C23—201no—
184C23—201no—
185C23C8×AGL(1,𝔽5)11yescharacteristic
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—51no—
189C2×C4—51no—
190C2×C4—101no—
191C2×C4—101no—
192C2×C4—101no—
193C2×C4—101no—
194C2×C4—201no—
195C2×C4—201no—
196C2×C4—201no—
197C2×C4—202no—
198C2×C4—202no—
199C2×C4—202no—
200C2×C4#399211yescharacteristic
201C2×C4(C2×AGL(1,𝔽5))⋊C411yescharacteristic
202C8—21no—
203C8—21no—
204C8—52no—
205C8—101no—
206C8—202no—
207C5#720111yescharacteristic, complemented, Sylow
208C22—21no—
209C22—21no—
210C22—21no—
211C22—52no—
212C22—101no—
213C22—101no—
214C22—101no—
215C22—101no—
216C22—101no—
217C22—204no—
218C22(C2×AGL(1,𝔽5))⋊C811yescharacteristic
219C4—21no—
220C4—101no—
221C4—101no—
222C4—402no—
223C4#6486611yescharacteristic
224C4#6490211yescharacteristic
225C2—41no—
226C2—102no—
227C2—201no—
228C2#1059855911yescharacteristic
229C2#1059866111yescharacteristic
230C2#1059867111yescharacteristic
subgroupG/H
centerC2×C4(C2×AGL(1,𝔽5))⋊C4
commutatorC23×C5C4×C8
FrattiniC22×C8C2×AGL(1,𝔽5)
FittingC5×C22⋊C16C4
socleC22×C5#336
prime radicalC23×D5C2×C8
cosocleC22×C8×D5C22
hypercenterC22⋊C16AGL(1,𝔽5)
hypocenterC5#7201
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
7. C2×#3998C4semidirect
13. C5×C22⋊C16C4semidirect
14. C22⋊#672C4semidirect
79. C22⋊C16AGL(1,𝔽5)semidirect
207. C5#7201semidirect

Maximal subgroups

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

subgroupindex
#106047192
#106047052
#105998992
#72015