Finite Group Databasebeta

← #13013871

Subgroups — #13013871

Order 1280.

336 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 68 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#10604732C211yesmaximal, characteristic
2#10604721C211yesmaximal, characteristic
3D5×C22⋊C16C211yesmaximal, characteristic
4C2×#4130—21no—
5C22×C8×D5C2211yescharacteristic
6#65756—21no—
7C2×#3996C412yescomplemented
8C2×D5×C16—21no—
9#64896—21no—
10#64893—21no—
11#64893C412yes—
12C5×C22⋊C16C411yescharacteristic, complemented
13C22⋊#672C411yescharacteristic, complemented
14#64727—21no—
15#7260—51nomaximal, Sylow
16C22×C4×D5D411yescharacteristic
17C2×C22⋊AGL(1,𝔽5)—41no—
18#4130—24no—
19#4130—42no—
20C2×C4⋊AGL(1,𝔽5)—22no—
21C2×C4×AGL(1,𝔽5)—41no—
22C2×#700—22no—
23C22×C5×C8C2×C411yescharacteristic
24C22×#168Q811yescharacteristic
25C2×C8×D5—42no—
26C2×C8×D5C2×C411yescharacteristic
27C2×C8×D5C2×C411yescharacteristic
28C2×C8×D5D411yescharacteristic
29C2×C8×D5D411yescharacteristic
30C22⋊#170—41no—
31#4003—41no—
32#3996—42no—
33C2×C5×C16—21no—
34C2×#672—21no—
35D5×C16—42no—
36C2×C22⋊C16—51no—
37#1810—51no—
38#1735—51no—
39C23×D5SD1611yescharacteristic
40C22×AGL(1,𝔽5)—42no—
41C22×C4×C5C4⋊C411yescharacteristic
42C22×Dic5SD1611yescharacteristic
43C2×C4×D5—22no—
44C2×C4×D5—22no—
45C2×C4×D5C22⋊C411yescharacteristic
46C2×C4×D5C22⋊C411yescharacteristic
47C2×C4×D5D811yescharacteristic
48C2×C4×D5Q1611yescharacteristic
49C22⋊AGL(1,𝔽5)—44no—
50C2×#170—42no—
51C4⋊AGL(1,𝔽5)—24no—
52C4⋊AGL(1,𝔽5)—42no—
53C4×AGL(1,𝔽5)—44no—
54#700—42no—
55#700—42no—
56C2×C5×C8—41no—
57C2×C5×C8C4211yescharacteristic
58C2×C5×C8C22⋊C411yescharacteristic
59C2×#168—41no—
60C2×#168C4⋊C411yescharacteristic
61C2×#168C4⋊C411yescharacteristic
62C8×D5—22no—
63C8×D5—22no—
64C8×D5—41no—
65C8×D5—41no—
66C5×C16—41no—
67#672—41no—
68C23×C8—51no—
69C2×#117—101no—
70C22×C16—101no—
71C2×#108—52no—
72#407—101no—
73C22⋊C16—51no—
74C22⋊C16—101no—
75C22⋊C16AGL(1,𝔽5)11yescharacteristic, complemented
76#341—101no—
77#340—52no—
78#340—101no—
79C23×C5#10611yescharacteristic
80C22×D5—22no—
81C22×D5—22no—
82C22×D5—41no—
83C22×D5#10211yescharacteristic
84C22×D5#10311yescharacteristic
85C2×AGL(1,𝔽5)—44no—
86C2×AGL(1,𝔽5)—82no—
87C2×AGL(1,𝔽5)—82no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5#9511yescharacteristic
91C2×C4×C5#10711yescharacteristic
92C2×Dic5—22no—
93C2×Dic5#10211yescharacteristic
94C2×Dic5#10311yescharacteristic
95C4×D5—22no—
96C4×D5—21no—
97C4×D5—21no—
98C4×D5—22no—
99C4×D5#9912yes—
100C4×D5#10412yes—
101#170—82no—
102C5×C8—21no—
103C5×C8—21no—
104#168—21no—
105#168—21no—
106C23×C4—51no—
107C2×M16—102no—
108C22×C8—51no—
109C22×C8—51no—
110C22×C8—51no—
111C22×C8—51no—
112C22×C8—51no—
113C22×C8—202no—
114C22×C8C2×AGL(1,𝔽5)11yescharacteristic
115#117—104no—
116#117—202no—
117C2×C4⋊C4—102no—
118C2×C22⋊C4—201no—
119C2×C42—201no—
120C2×C16—21no—
121C2×C16—101no—
122C2×C16—202no—
123#108—202no—
124C22⋊C8—201no—
125C22⋊C8—201no—
126C22×C5—21no—
127C22×C5—21no—
128C22×C5—21no—
129C22×C5#34011yescharacteristic
130C2×D5—21no—
131C2×D5—21no—
132C2×D5—21no—
133C2×D5—21no—
134C2×D5—44no—
135C2×D5#32812yes—
136C2×D5#35912yes—
137AGL(1,𝔽5)—84no—
138C4×C5—21no—
139C4×C5#33911yescharacteristic
140C4×C5#34211yescharacteristic
141Dic5—21no—
142Dic5#32712yes—
143C24—51no—
144C22×C4—51no—
145C22×C4—51no—
146C22×C4—51no—
147C22×C4—51no—
148C22×C4—51no—
149C22×C4—102no—
150C22×C4—102no—
151C22×C4—202no—
152C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
153M16—202no—
154M16—202no—
155C2×C8—41no—
156C2×C8—51no—
157C2×C8—51no—
158C2×C8—102no—
159C2×C8—102no—
160C2×C8—201no—
161C2×C8—201no—
162C2×C8—201no—
163C2×C8—202no—
164C2×C8C4×AGL(1,𝔽5)11yescharacteristic
165C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
166C4⋊C4—104no—
167C4⋊C4—202no—
168C22⋊C4—204no—
169C42—204no—
170C16—41no—
171C16—201no—
172C2×C5—41no—
173C2×C5#129411yescharacteristic
174C2×C5#138211yescharacteristic
175C2×C5#138911yescharacteristic
176D5—22no—
177D5—22no—
178C23—51no—
179C23—51no—
180C23—102no—
181C23—102no—
182C23—201no—
183C23#399311yescharacteristic
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—51no—
187C2×C4—52no—
188C2×C4—52no—
189C2×C4—51no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—101no—
194C2×C4—101no—
195C2×C4—204no—
196C2×C4—402no—
197C2×C4—402no—
198C2×C4#399411yescharacteristic
199C2×C4(C2×AGL(1,𝔽5))⋊C411yescharacteristic
200C8—21no—
201C8—21no—
202C8—101no—
203C8—101no—
204C8—402no—
205C5#726011yescharacteristic, complemented, Sylow
206C22—21no—
207C22—21no—
208C22—21no—
209C22—52no—
210C22—52no—
211C22—101no—
212C22—101no—
213C22—101no—
214C22—101no—
215C22—204no—
216C22#6489311yescharacteristic
217C4—21no—
218C4—52no—
219C4—101no—
220C4—404no—
221C4#6486611yescharacteristic
222C4#6490011yescharacteristic
223C2—41no—
224C2—102no—
225C2—102no—
226C2#1059855211yescharacteristic
227C2#1059866511yescharacteristic
228C2#1059867711yescharacteristic
subgroupG/H
centerC22#64893
commutatorC2×C5×C8C42
FrattiniC22×C8C2×AGL(1,𝔽5)
FittingC5×C22⋊C16C4
socleC22×C5#340
prime radicalC23×D5SD16
cosocleC22×C8×D5C22
hypercenterC22⋊C16AGL(1,𝔽5)
hypocenterC5#7260
LCSG ▷ C2×C5×C8 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
7. C2×#3996C4semidirect
12. C5×C22⋊C16C4semidirect
13. C22⋊#672C4semidirect
75. C22⋊C16AGL(1,𝔽5)semidirect
205. C5#7260semidirect

Maximal subgroups

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

subgroupindex
#106047322
#106047212
D5×C22⋊C162
#72605