Finite Group Databasebeta

← #72275813

Subgroups — #72275813

Order 8000.

340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 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#72275737C211yesmaximal, characteristic
2M16×D25⋊C5C211yesmaximal, characteristic
3C2×C4×AGL(1,ℤ/25)C211yesmaximal, characteristic
4C22×AGL(1,ℤ/25)—21no—
5C2×C4×D25⋊C5C2211yescharacteristic
6C22⋊AGL(1,ℤ/25)—21no—
7C4⋊AGL(1,ℤ/25)C412yescomplemented
8C4×AGL(1,ℤ/25)—21no—
9C4×AGL(1,ℤ/25)—21no—
10C4×AGL(1,ℤ/25)C412yescomplemented
11M16×5−1+2C411yescharacteristic, complemented
12#87716⋊C5C411yescharacteristic, complemented
13#87712⋊C5—21no—
14C8×D25⋊C5—21no—
15C5×#64900—51nomaximal
16#13097887C511yesmaximal, characteristic, complemented
17C22×D25⋊C5D411yescharacteristic
18C2×AGL(1,ℤ/25)—24no—
19C2×AGL(1,ℤ/25)—22no—
20C2×AGL(1,ℤ/25)—22no—
21C2×C4×5−1+2C2×C411yescharacteristic
22C2×Dic25⋊C5Q811yescharacteristic
23C4×D25⋊C5C2×C411yescharacteristic
24C4×D25⋊C5C2×C411yescharacteristic
25C4×D25⋊C5D411yescharacteristic
26C4×D25⋊C5D411yescharacteristic
27C8×5−1+2—21no—
28#6014⋊C5—21no—
29C5×#4130—51no—
30C2×C4×C5×AGL(1,𝔽5)—51no—
31C5×D5×M16—51no—
32#11716549C2×C511yescharacteristic
33C2×C4×#1035C2×C511yescharacteristic
34M16×D25C2×C511yescharacteristic
35C22×5−1+2C4⋊C411yescharacteristic
36C2×D25⋊C5—22no—
37C2×D25⋊C5C22⋊C411yescharacteristic
38C2×D25⋊C5C22⋊C411yescharacteristic
39AGL(1,ℤ/25)—24no—
40AGL(1,ℤ/25)—42no—
41C4×5−1+2C4211yescharacteristic
42C4×5−1+2C22⋊C411yescharacteristic
43Dic25⋊C5C4⋊C411yescharacteristic
44Dic25⋊C5C4⋊C411yescharacteristic
45C22×C5×AGL(1,𝔽5)—101no—
46C2×C4×C5×D5—51no—
47C5×C22⋊AGL(1,𝔽5)—101no—
48C5×C4⋊AGL(1,𝔽5)—52no—
49C4×C5×AGL(1,𝔽5)—52no—
50C4×C5×AGL(1,𝔽5)—101no—
51C4×C5×AGL(1,𝔽5)—101no—
52C52×M16AGL(1,𝔽5)11yescharacteristic
53C5×#681—51no—
54C5×#676—101no—
55C5×C8×D5—101no—
56C22×#1035—21no—
57C2×C4×D25C22×C511yescharacteristic
58C22⋊#1035—21no—
59C4⋊#1035C4×C512yescomplemented
60C4×#1035—21no—
61C4×#1035—21no—
62C4×#1035C4×C512yescomplemented
63M16×C25—41no—
64M16×C25C4×C511yescharacteristic, complemented
65#87716C4×C511yescharacteristic, complemented
66#87712—21no—
67C8×D25—21no—
68#64900—51no—
69C5×#339—251no—
70C2×5−1+2—21no—
71C2×5−1+2#9511yescharacteristic
72D25⋊C5—21no—
73D25⋊C5#10412yes—
74C22×C5×D5—51no—
75C2×C5×AGL(1,𝔽5)—102no—
76C2×C5×AGL(1,𝔽5)—104no—
77C2×C5×AGL(1,𝔽5)—102no—
78C2×C4×C52C2×AGL(1,𝔽5)11yescharacteristic
79C2×C5×Dic5—51no—
80C4×C5×D5—51no—
81C4×C5×D5—51no—
82C4×C5×D5—51no—
83C4×C5×D5—51no—
84C52×C8—21no—
85C5×#168—101no—
86C22×D25C5×D411yescharacteristic
87C2×#1035—24no—
88C2×#1035—22no—
89C2×#1035—22no—
90C2×C4×C25—41no—
91C2×C4×C25C2×C4×C511yescharacteristic
92C2×Dic25C5×Q811yescharacteristic
93C4×D25C2×C4×C511yescharacteristic
94C4×D25C2×C4×C511yescharacteristic
95C4×D25C5×D411yescharacteristic
96C4×D25C5×D411yescharacteristic
97C8×C25—21no—
98C8×C25—42no—
99#6014—21no—
100#4130—51no—
101C2×C4×AGL(1,𝔽5)—51no—
102C2×C5×M16—251no—
103C5×#117—251no—
104C2×C42×C5—251no—
105D5×M16—51no—
1065−1+2#33911yescharacteristic, complemented, Sylow
107C22×C52C4⋊AGL(1,𝔽5)11yescharacteristic
108C2×C5×D5—51no—
109C2×C5×D5—51no—
110C2×C5×D5—102no—
111C5×AGL(1,𝔽5)—104no—
112C5×AGL(1,𝔽5)—202no—
113C4×C52C4×AGL(1,𝔽5)11yescharacteristic
114C4×C52C22⋊AGL(1,𝔽5)11yescharacteristic
115C5×Dic5—51no—
116C5×Dic5—51no—
117C22×C25—41no—
118C22×C25C5×C4⋊C411yescharacteristic
119C2×D25—22no—
120C2×D25C5×C22⋊C411yescharacteristic
121C2×D25C5×C22⋊C411yescharacteristic
122#1035—24no—
123#1035—42no—
124C4×C25—41no—
125C4×C25—41no—
126C4×C25C42×C511yescharacteristic
127C4×C25C5×C22⋊C411yescharacteristic
128Dic25C5×C4⋊C411yescharacteristic
129Dic25C5×C4⋊C411yescharacteristic
130C22×AGL(1,𝔽5)—101no—
131C22×C4×C5—251no—
132C22×C4×C5—501no—
133C2×C4×D5—51no—
134C22⋊AGL(1,𝔽5)—101no—
135C4⋊AGL(1,𝔽5)—52no—
136C4×AGL(1,𝔽5)—52no—
137C4×AGL(1,𝔽5)—101no—
138C4×AGL(1,𝔽5)—101no—
139C5×M16—51no—
140C5×M16—251no—
141C5×M16—501no—
142C5×M16C5×AGL(1,𝔽5)11yescharacteristic
143C2×C5×C8—501no—
144C5×C4⋊C4—252no—
145C5×C22⋊C4—501no—
146C42×C5—252no—
147C42×C5—501no—
148C42×C5—501no—
149#681—51no—
150#676—101no—
151C8×D5—101no—
152#339—251noSylow
153C2×C52—21no—
154C2×C52(C2×AGL(1,𝔽5))⋊C411yescharacteristic
155C5×D5—52no—
156C5×D5—101no—
157C2×C25—21no—
158C2×C25—41no—
159C2×C25—81no—
160C2×C25C5×#9511yescharacteristic
161D25—21no—
162D25C5×#10412yes—
163C23×C5—251no—
164C22×D5—51no—
165C2×AGL(1,𝔽5)—102no—
166C2×AGL(1,𝔽5)—104no—
167C2×AGL(1,𝔽5)—102no—
168C2×C4×C5—51no—
169C2×C4×C5—251no—
170C2×C4×C5—251no—
171C2×C4×C5—251no—
172C2×C4×C5—251no—
173C2×C4×C5—251no—
174C2×C4×C5—502no—
175C2×C4×C5—504no—
176C2×C4×C5—502no—
177C2×C4×C5C2×C5×AGL(1,𝔽5)11yescharacteristic
178C2×Dic5—51no—
179C4×D5—51no—
180C4×D5—51no—
181C4×D5—51no—
182C4×D5—51no—
183C5×C8—21no—
184C5×C8—101no—
185C5×C8—501no—
186#168—101no—
187C2×M16—251no—
188#117—251no—
189C2×C42—251no—
190C52#6490011yescharacteristic
191C25—41no—
192C25C5×#33911yescharacteristic, complemented
193C22×C5—51no—
194C22×C5—251no—
195C22×C5—251no—
196C22×C5—502no—
197C22×C5C5×C4⋊AGL(1,𝔽5)11yescharacteristic
198C2×D5—51no—
199C2×D5—51no—
200C2×D5—102no—
201AGL(1,𝔽5)—104no—
202AGL(1,𝔽5)—202no—
203C4×C5—51no—
204C4×C5—51no—
205C4×C5—251no—
206C4×C5—251no—
207C4×C5—504no—
208C4×C5—1002no—
209C4×C5C4×C5×AGL(1,𝔽5)11yescharacteristic
210C4×C5C5×C22⋊AGL(1,𝔽5)11yescharacteristic
211Dic5—51no—
212Dic5—51no—
213C22×C4—251no—
214C22×C4—501no—
215M16—251no—
216M16—501no—
217M16AGL(1,ℤ/25)11yescharacteristic, complemented
218C2×C8—501no—
219C4⋊C4—252no—
220C22⋊C4—501no—
221C42—252no—
222C42—501no—
223C42—501no—
224C2×C5—21no—
225C2×C5—51no—
226C2×C5—101no—
227C2×C5—252no—
228C2×C5—501no—
229C2×C5C5×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
230D5—52no—
231D5—101no—
232C23—251no—
233C2×C4—251no—
234C2×C4—251no—
235C2×C4—251no—
236C2×C4—251no—
237C2×C4—251no—
238C2×C4—502no—
239C2×C4—504no—
240C2×C4—502no—
241C2×C4C2×AGL(1,ℤ/25)11yescharacteristic
242C8—21no—
243C8—501no—
244C5—51no—
245C5C5×#6490011yescharacteristic
246C22—251no—
247C22—251no—
248C22—502no—
249C22C4⋊AGL(1,ℤ/25)11yescharacteristic
250C4—251no—
251C4—251no—
252C4—504no—
253C4—1002no—
254C4C4×AGL(1,ℤ/25)11yescharacteristic
255C4C22⋊AGL(1,ℤ/25)11yescharacteristic
256C2—21no—
257C2—252no—
258C2—501no—
259C2#7227561011yescharacteristic
subgroupG/H
centerC4C22⋊AGL(1,ℤ/25)
commutatorC4×C25C42×C5
FrattiniC2×C4×C5C2×C5×AGL(1,𝔽5)
FittingM16×5−1+2C4
socleC2×C5C5×(C2×AGL(1,𝔽5))⋊C4
prime radicalC22×D25⋊C5D4
cosocleC2×C4×D25C22×C5
hypercenterM16AGL(1,ℤ/25)
hypocenterC25C5×#339
LCSG ▷ C4×C25 ▷ C2×C25 ▷ C25
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
7. C4⋊AGL(1,ℤ/25)C4semidirect
10. C4×AGL(1,ℤ/25)C4semidirect
11. M16×5−1+2C4semidirect
12. #87716⋊C5C4semidirect
16. #13097887C5semidirect
59. C4⋊#1035C4×C5semidirect
62. C4×#1035C4×C5semidirect
64. M16×C25C4×C5semidirect
65. #87716C4×C5semidirect
106. 5−1+2#339semidirect
192. C25C5×#339semidirect
217. M16AGL(1,ℤ/25)semidirect

Maximal subgroups

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

subgroupindex
#722757372
M16×D25⋊C52
C2×C4×AGL(1,ℤ/25)2
C5×#649005
#130978875