Finite Group Databasebeta

← #13013864

Subgroups — #13013864

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
2D5⋊(C22⋊C16)C211yesmaximal, characteristic
3D5×#340C211yesmaximal, characteristic
4C22×C8×D5C2211yescharacteristic
5C2×D5×C4⋊C4—21no—
6#65765—21no—
7C2×#3994C412yescomplemented
8C2×D5⋊C16—21no—
9C22⋊#674—21no—
10#64893—21no—
11#64887C412yescomplemented
12C5×#340C411yescharacteristic, complemented
13#64771C411yescharacteristic, complemented
14#64749—21no—
15#7258—51nomaximal, Sylow
16C22×C4×D5—41no—
17C22×C4×D5D411yescharacteristic
18C2×C22⋊AGL(1,𝔽5)—41no—
19C2×C4⋊AGL(1,𝔽5)—22no—
20C22×C5×C8C2×C411yescharacteristic
21C2×C5×C4⋊C4—21no—
22C2×C4⋊Dic5—21no—
23C2×#683—41no—
24C22×#168D411yescharacteristic
25C2×C8×D5—42no—
26C2×C8×D5C2×C411yescharacteristic
27C2×C8×D5C2×C411yescharacteristic
28C2×C8×D5D411yescharacteristic
29C2×C8×D5Q811yescharacteristic
30D5×C4⋊C4—24no—
31D5×C4⋊C4—42no—
32(C2×AGL(1,𝔽5))⋊C4—41no—
33C2×#674—22no—
34#3994—42no—
35D5⋊C16—42no—
36C2×C22⋊C16—51no—
37#1808—51no—
38C2×#340—51no—
39C23×D5D811yescharacteristic
40C22×AGL(1,𝔽5)—42no—
41C22×C4×C5—41no—
42C22×C4×C5C22⋊C411yescharacteristic
43C22×Dic5—41no—
44C22×Dic5SD1611yescharacteristic
45C2×C4×D5—22no—
46C2×C4×D5—22no—
47C2×C4×D5—44no—
48C2×C4×D5—82no—
49C2×C4×D5—81no—
50C2×C4×D5—81no—
51C2×C4×D5C22⋊C411yescharacteristic
52C2×C4×D5C4⋊C411yescharacteristic
53C2×C4×D5SD1611yescharacteristic
54C2×C4×D5Q1611yescharacteristic
55C22⋊AGL(1,𝔽5)—81no—
56C22⋊AGL(1,𝔽5)—81no—
57C4⋊AGL(1,𝔽5)—42no—
58C4⋊AGL(1,𝔽5)—42no—
59C2×C5×C8—41no—
60C2×C5×C8C4211yescharacteristic
61C2×C5×C8C4⋊C411yescharacteristic
62C5×C4⋊C4—22no—
63C5×C4⋊C4—41no—
64C4⋊Dic5—22no—
65C4⋊Dic5—41no—
66#683—44no—
67C2×#168—41no—
68C2×#168C22⋊C411yescharacteristic
69C2×#168C4⋊C411yescharacteristic
70C8×D5—22no—
71C8×D5—22no—
72C8×D5—41no—
73C8×D5—41no—
74#674—42no—
75C23×C8—51no—
76C22×C4⋊C4—101no—
77C22×C16—101no—
78C2×#107—52no—
79#380—101no—
80C22⋊C16—52no—
81C22⋊C16—101no—
82#340—51no—
83#340—101no—
84#340—101no—
85#340AGL(1,𝔽5)11yescharacteristic, complemented
86C23×C5#10211yescharacteristic
87C22×D5—22no—
88C22×D5—22no—
89C22×D5#10211yescharacteristic
90C22×D5#10711yescharacteristic
91C22×D5D1611yescharacteristic
92C22×D5SD3211yescharacteristic
93C22×D5SD3211yescharacteristic
94C22×D5Q3211yescharacteristic
95C2×AGL(1,𝔽5)—82no—
96C2×AGL(1,𝔽5)—82no—
97C2×AGL(1,𝔽5)—82no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—42no—
101C2×C4×C5—81no—
102C2×C4×C5—81no—
103C2×C4×C5#9511yescharacteristic
104C2×C4×C5#10311yescharacteristic
105C2×Dic5—22no—
106C2×Dic5—42no—
107C2×Dic5—81no—
108C2×Dic5—81no—
109C2×Dic5#10311yescharacteristic
110C2×Dic5#10611yescharacteristic
111C4×D5—22no—
112C4×D5—21no—
113C4×D5—21no—
114C4×D5—22no—
115C4×D5—84no—
116C4×D5—84no—
117C4×D5#9912yes—
118C4×D5#10412yes—
119C5×C8—21no—
120C5×C8—21no—
121#168—21no—
122#168—21no—
123C23×C4—51no—
124C23×C4—201no—
125C22×C8—51no—
126C22×C8—51no—
127C22×C8—51no—
128C22×C8—51no—
129C22×C8—51no—
130C22×C8—202no—
131C22×C8C2×AGL(1,𝔽5)11yescharacteristic
132C2×C4⋊C4—21no—
133C2×C4⋊C4—104no—
134C2×C4⋊C4—101no—
135C2×C4⋊C4—102no—
136C2×C4⋊C4—202no—
137C2×C4⋊C4—201no—
138C2×C22⋊C4—201no—
139C2×C16—102no—
140C2×C16—202no—
141#107—202no—
142#95—201no—
143C22×C5—21no—
144C22×C5—21no—
145C22×C5#34011yescharacteristic
146C22×C5#35711yescharacteristic
147C22×C5#35811yescharacteristic
148C2×D5—21no—
149C2×D5—21no—
150C2×D5—22no—
151C2×D5—22no—
152C2×D5—22no—
153C2×D5—22no—
154C2×D5#32712yes—
155C2×D5#35711yescharacteristic
156C2×D5#35811yescharacteristic
157C2×D5#36012yes—
158C2×D5#36611yescharacteristic
159C2×D5#36711yescharacteristic
160AGL(1,𝔽5)—162no—
161C4×C5—21no—
162C4×C5—82no—
163C4×C5#33911yescharacteristic
164C4×C5#34211yescharacteristic
165Dic5—21no—
166Dic5—82no—
167Dic5#32812yes—
168C24—51no—
169C22×C4—41no—
170C22×C4—51no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—51no—
174C22×C4—51no—
175C22×C4—102no—
176C22×C4—102no—
177C22×C4—204no—
178C22×C4—201no—
179C22×C4—202no—
180C22×C4—401no—
181C22×C4—401no—
182C22×C4—402no—
183C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
184C2×C8—41no—
185C2×C8—51no—
186C2×C8—51no—
187C2×C8—102no—
188C2×C8—102no—
189C2×C8—201no—
190C2×C8—201no—
191C2×C8—201no—
192C2×C8C4×AGL(1,𝔽5)11yescharacteristic
193C2×C8C4⋊AGL(1,𝔽5)11yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—102no—
197C4⋊C4—201no—
198C4⋊C4—204no—
199C4⋊C4—202no—
200C4⋊C4—202no—
201C22⋊C4—401no—
202C22⋊C4—401no—
203C16—202no—
204C2×C5—21no—
205C2×C5—21no—
206C2×C5#129411yescharacteristic
207C2×C5#138011yescharacteristic
208C2×C5#138611yescharacteristic
209D5—21no—
210D5—21no—
211D5#134712yes—
212D5#134812yes—
213C23—51no—
214C23—51no—
215C23—51no—
216C23—51no—
217C23—51no—
218C23—51no—
219C23—102no—
220C23—102no—
221C23#400711yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—51no—
226C2×C4—52no—
227C2×C4—52no—
228C2×C4—51no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—101no—
235C2×C4—101no—
236C2×C4—202no—
237C2×C4—404no—
238C2×C4—404no—
239C2×C4—401no—
240C2×C4—401no—
241C2×C4—402no—
242C2×C4—402no—
243C2×C4—402no—
244C2×C4(C2×AGL(1,𝔽5))⋊C411yescharacteristic
245C2×C4#400911yescharacteristic
246C8—21no—
247C8—21no—
248C8—101no—
249C8—101no—
250C5#725811yescharacteristic, complemented, Sylow
251C22—21no—
252C22—21no—
253C22—52no—
254C22—52no—
255C22—51no—
256C22—51no—
257C22—51no—
258C22—51no—
259C22—102no—
260C22—102no—
261C22—102no—
262C22—102no—
263C22—101no—
264C22—101no—
265C22#6487411yescharacteristic
266C22#6490411yescharacteristic
267C22#6490811yescharacteristic
268C4—21no—
269C4—52no—
270C4—82no—
271C4—101no—
272C4—402no—
273C4—802no—
274C4#6486211yescharacteristic
275C4#6486611yescharacteristic
276C2—21no—
277C2—21no—
278C2—52no—
279C2—52no—
280C2—101no—
281C2—101no—
282C2#1059850411yescharacteristic
283C2#1059860511yescharacteristic
284C2#1059861911yescharacteristic
subgroupG/H
centerC22#64874
commutatorC2×C5×C8C42
FrattiniC22×C8C2×AGL(1,𝔽5)
FittingC5×#340C4
socleC22×C5#340
prime radicalC23×D5D8
cosocleC22×C8×D5C22
hypercenter#340AGL(1,𝔽5)
hypocenterC5#7258
LCSG ▷ C2×C5×C8 ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #340

Complements

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

NH ≅ G/Ntype
7. C2×#3994C4semidirect
11. #64887C4semidirect
12. C5×#340C4semidirect
13. #64771C4semidirect
85. #340AGL(1,𝔽5)semidirect
250. C5#7258semidirect

Maximal subgroups

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

subgroupindex
#106047232
D5⋊(C22⋊C16)2
D5×#3402
#72585