Finite Group Databasebeta

← #13013865

Subgroups — #13013865

Order 1280.

402 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 84 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#10604723C211yesmaximal, characteristic
2C2×#64893C211yesmaximal, characteristic
3D5×C22⋊C16C211yesmaximal, characteristic
4C22×C4⋊AGL(1,𝔽5)—21no—
5C22×C8×D5C2211yescharacteristic
6#65765—21no—
7C2×#3994C412yescomplemented
8C2×D5×C16—21no—
9#64893—21no—
10#64893—21no—
11#64893C412yescomplemented
12C5×C22⋊C16C411yescharacteristic, complemented
13C22⋊#672C411yescharacteristic, complemented
14#64727—21no—
15#7258—51nomaximal, Sylow
16C23×AGL(1,𝔽5)—41no—
17C22×C4×D5D411yescharacteristic
18C2×C22⋊AGL(1,𝔽5)—41no—
19C2×C4⋊AGL(1,𝔽5)—24no—
20C2×C4⋊AGL(1,𝔽5)—22no—
21C2×C4⋊AGL(1,𝔽5)—22no—
22C2×C4⋊AGL(1,𝔽5)—42no—
23C2×C4⋊AGL(1,𝔽5)—41no—
24C22×C5×C8C2×C411yescharacteristic
25C22×#168Q811yescharacteristic
26C2×C8×D5—42no—
27C2×C8×D5C2×C411yescharacteristic
28C2×C8×D5C2×C411yescharacteristic
29C2×C8×D5D411yescharacteristic
30C2×C8×D5D411yescharacteristic
31(C2×AGL(1,𝔽5))⋊C4—41no—
32#3994—42no—
33C2×C5×C16—21no—
34C2×#672—21no—
35D5×C16—42no—
36C2×C22⋊C16—51no—
37#1808—51no—
38C2×#340—51no—
39C23×D5D811yescharacteristic
40C22×AGL(1,𝔽5)—44no—
41C22×AGL(1,𝔽5)—42no—
42C22×AGL(1,𝔽5)—42no—
43C22×AGL(1,𝔽5)—82no—
44C22×AGL(1,𝔽5)—81no—
45C22×AGL(1,𝔽5)—81no—
46C22×C4×C5C4⋊C411yescharacteristic
47C22×Dic5Q1611yescharacteristic
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C2×C4×D5C22⋊C411yescharacteristic
51C2×C4×D5C22⋊C411yescharacteristic
52C2×C4×D5SD1611yescharacteristic
53C2×C4×D5SD1611yescharacteristic
54C22⋊AGL(1,𝔽5)—81no—
55C22⋊AGL(1,𝔽5)—81no—
56C4⋊AGL(1,𝔽5)—24no—
57C4⋊AGL(1,𝔽5)—42no—
58C4⋊AGL(1,𝔽5)—44no—
59C4⋊AGL(1,𝔽5)—42no—
60C4⋊AGL(1,𝔽5)—42no—
61C2×C5×C8—41no—
62C2×C5×C8C4211yescharacteristic
63C2×C5×C8C22⋊C411yescharacteristic
64C2×#168—41no—
65C2×#168C4⋊C411yescharacteristic
66C2×#168C4⋊C411yescharacteristic
67C8×D5—22no—
68C8×D5—22no—
69C8×D5—41no—
70C8×D5—41no—
71C5×C16—41no—
72#672—41no—
73C23×C8—51no—
74C22×C4⋊C4—101no—
75C22×C16—101no—
76C2×#107—52no—
77#380—101no—
78C22⋊C16—51no—
79C22⋊C16—101no—
80C22⋊C16AGL(1,𝔽5)11yescharacteristic, complemented
81#340—52no—
82#340—101no—
83#340—101no—
84C23×C5#10711yescharacteristic
85C22×D5—22no—
86C22×D5—22no—
87C22×D5#10211yescharacteristic
88C22×D5#10211yescharacteristic
89C22×D5D1611yescharacteristic
90C22×D5SD3211yescharacteristic
91C22×D5SD3211yescharacteristic
92C22×D5Q3211yescharacteristic
93C2×AGL(1,𝔽5)—44no—
94C2×AGL(1,𝔽5)—82no—
95C2×AGL(1,𝔽5)—84no—
96C2×AGL(1,𝔽5)—84no—
97C2×AGL(1,𝔽5)—82no—
98C2×AGL(1,𝔽5)—82no—
99C2×AGL(1,𝔽5)—82no—
100C2×AGL(1,𝔽5)—82no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5#9511yescharacteristic
104C2×C4×C5#10611yescharacteristic
105C2×Dic5—22no—
106C2×Dic5#10311yescharacteristic
107C2×Dic5#10311yescharacteristic
108C4×D5—22no—
109C4×D5—21no—
110C4×D5—21no—
111C4×D5—22no—
112C4×D5#9912yes—
113C4×D5#10412yes—
114C5×C8—21no—
115C5×C8—21no—
116#168—21no—
117#168—21no—
118C23×C4—51no—
119C23×C4—201no—
120C22×C8—51no—
121C22×C8—51no—
122C22×C8—51no—
123C22×C8—51no—
124C22×C8—51no—
125C22×C8—202no—
126C22×C8C2×AGL(1,𝔽5)11yescharacteristic
127C2×C4⋊C4—102no—
128C2×C4⋊C4—104no—
129C2×C4⋊C4—102no—
130C2×C4⋊C4—201no—
131C2×C4⋊C4—202no—
132C2×C22⋊C4—201no—
133C2×C16—21no—
134C2×C16—101no—
135C2×C16—202no—
136#107—202no—
137#95—201no—
138C22×C5—21no—
139C22×C5—21no—
140C22×C5#34011yescharacteristic
141C22×C5#36611yescharacteristic
142C22×C5#36711yescharacteristic
143C2×D5—21no—
144C2×D5—21no—
145C2×D5—22no—
146C2×D5—22no—
147C2×D5—22no—
148C2×D5—22no—
149C2×D5#32712yes—
150C2×D5#35711yescharacteristic
151C2×D5#35711yescharacteristic
152C2×D5#35811yescharacteristic
153C2×D5#35811yescharacteristic
154C2×D5#36012yes—
155AGL(1,𝔽5)—84no—
156AGL(1,𝔽5)—162no—
157C4×C5—21no—
158C4×C5#33911yescharacteristic
159C4×C5#34211yescharacteristic
160Dic5—21no—
161Dic5#32812yes—
162C24—51no—
163C22×C4—51no—
164C22×C4—51no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—51no—
168C22×C4—102no—
169C22×C4—102no—
170C22×C4—202no—
171C22×C4—204no—
172C22×C4—202no—
173C22×C4—401no—
174C22×C4—401no—
175C22×C4—402no—
176C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
177C2×C8—41no—
178C2×C8—51no—
179C2×C8—51no—
180C2×C8—102no—
181C2×C8—102no—
182C2×C8—201no—
183C2×C8—201no—
184C2×C8—201no—
185C2×C8C4×AGL(1,𝔽5)11yescharacteristic
186C2×C8C22⋊AGL(1,𝔽5)11yescharacteristic
187C4⋊C4—104no—
188C4⋊C4—202no—
189C4⋊C4—202no—
190C4⋊C4—204no—
191C4⋊C4—202no—
192C22⋊C4—401no—
193C22⋊C4—401no—
194C16—41no—
195C16—201no—
196C2×C5—21no—
197C2×C5—21no—
198C2×C5#129411yescharacteristic
199C2×C5#138011yescharacteristic
200C2×C5#138611yescharacteristic
201D5—21no—
202D5—21no—
203D5#134712yes—
204D5#134812yes—
205C23—51no—
206C23—51no—
207C23—51no—
208C23—51no—
209C23—51no—
210C23—51no—
211C23—102no—
212C23—102no—
213C23#399411yescharacteristic
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—51no—
217C2×C4—52no—
218C2×C4—52no—
219C2×C4—51no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—101no—
224C2×C4—101no—
225C2×C4—204no—
226C2×C4—402no—
227C2×C4—404no—
228C2×C4—404no—
229C2×C4—402no—
230C2×C4—402no—
231C2×C4—402no—
232C2×C4—402no—
233C2×C4#399311yescharacteristic
234C2×C4(C2×AGL(1,𝔽5))⋊C411yescharacteristic
235C8—21no—
236C8—21no—
237C8—101no—
238C8—101no—
239C5#725811yescharacteristic, complemented, Sylow
240C22—21no—
241C22—21no—
242C22—51no—
243C22—51no—
244C22—52no—
245C22—52no—
246C22—51no—
247C22—51no—
248C22—102no—
249C22—102no—
250C22—102no—
251C22—102no—
252C22—101no—
253C22—101no—
254C22#6484911yescharacteristic
255C22#6485011yescharacteristic
256C22#6489311yescharacteristic
257C4—21no—
258C4—52no—
259C4—101no—
260C4—404no—
261C4—802no—
262C4#6486611yescharacteristic
263C4#6490011yescharacteristic
264C2—21no—
265C2—21no—
266C2—52no—
267C2—52no—
268C2—101no—
269C2—101no—
270C2#1059855211yescharacteristic
271C2#1059866411yescharacteristic
272C2#1059867511yescharacteristic
subgroupG/H
centerC22#64893
commutatorC2×C5×C8C42
FrattiniC22×C8C2×AGL(1,𝔽5)
FittingC5×C22⋊C16C4
socleC22×C5#340
prime radicalC23×D5D8
cosocleC22×C8×D5C22
hypercenterC22⋊C16AGL(1,𝔽5)
hypocenterC5#7258
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×#3994C4semidirect
11. #64893C4semidirect
12. C5×C22⋊C16C4semidirect
13. C22⋊#672C4semidirect
80. C22⋊C16AGL(1,𝔽5)semidirect
239. C5#7258semidirect

Maximal subgroups

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

subgroupindex
#106047232
C2×#648932
D5×C22⋊C162
#72585