Finite Group Databasebeta

← #13017097

Subgroups — #13017097

Order 1280.

516 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 132 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#10608212C211yesmaximal, characteristic
2C2×#64931C212yesmaximal
3#10604180C212yesmaximal
4#10598725C212yesmaximal
5C2×Q8×Dic5C411yescharacteristic
6C2×#4090C2211yescharacteristic
7C2×#4033C411yescharacteristic
8C2×#4004C2214yes—
9C2×#4002—22no—
10C5×#391C411yescharacteristic
11#65265C2211yescharacteristic
12C42×Dic5C2211yescharacteristic
13#65218C411yescharacteristic
14#64938—22no—
15#64936—22no—
16#64931C418yescomplemented
17#64918—22no—
18#8232—51nomaximal, Sylow
19C22×C5×Q8C2×C411yescharacteristic
20C22×Dic10C2×C411yescharacteristic
21C22×#170—24no—
22C2×C5×C4⋊C4C2×C411yescharacteristic
23C2×C42×C5C2×C411yescharacteristic
24Q8×Dic5—42no—
25#4090—22no—
26#4090C2×C414yes—
27C2×C4⋊Dic5—21no—
28C2×C4⋊Dic5C2×C411yescharacteristic
29C2×#683—22no—
30C2×#683—21no—
31C2×#683—22no—
32C2×C4×Dic5—21no—
33C2×C4×Dic5—21no—
34C2×C4×Dic5C2×C411yescharacteristic
35C2×C4×Dic5D412yes—
36C2×C4×Dic5D412yes—
37C2×C4×Dic5C2311yescharacteristic
38#4033—42no—
39#4004—24no—
40#4004C2×C418yes—
41#4002—44no—
42C5×#95—22no—
43#3963—22no—
44C4×C4⋊Q8—51no—
45#1767—52no—
46C2×#333—52no—
47#1296—52no—
48C2×C5×Q8—21no—
49C2×C5×Q8—41no—
50C2×C5×Q8—41no—
51C2×C5×Q8C4212yes—
52C22×C4×C5—21no—
53C22×C4×C5—21no—
54C22×C4×C5C22⋊C412yes—
55C22×C4×C5C22×C411yescharacteristic
56C22×Dic5—21no—
57C22×Dic5—21no—
58C22×Dic5C22⋊C411yescharacteristic
59C22×Dic5C22⋊C411yescharacteristic
60C22×Dic5C2×D412yes—
61C2×Dic10—21no—
62C2×Dic10—41no—
63C2×Dic10—41no—
64C2×Dic10C4212yes—
65C2×#170—28no—
66C2×#170—44no—
67C2×#170—44no—
68C5×C4⋊C4—41no—
69C42×C5—41no—
70C4⋊Dic5—42no—
71C4⋊Dic5—41no—
72#683—28no—
73#683—42no—
74#683—42no—
75#683—42no—
76C4×Dic5—22no—
77C4×Dic5—22no—
78C4×Dic5—24no—
79C4×Dic5—22no—
80C4×Dic5—42no—
81C4×Dic5—42no—
82C4×Dic5C22×C412yes—
83C4×Dic5#4112yes—
84C2×C4⋊Q8—51no—
85C2×C4×Q8—51no—
86C2×C4×Q8—51no—
87C2×C4⋊C8—54no—
88C2×#97—102no—
89#391—51no—
90#391AGL(1,𝔽5)11yescharacteristic
91#389—102no—
92#384—51no—
93C4×C4⋊C4—102no—
94C43—51no—
95#336—102no—
96#333—58no—
97C23×C5C2×C22⋊C411yescharacteristic
98C5×Q8—42no—
99C5×Q8—42no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—41no—
107C2×C4×C5—41no—
108C2×C4×C5—41no—
109C2×C4×C5—42no—
110C2×C4×C5C2×C4211yescharacteristic
111C2×C4×C5#11711yescharacteristic
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—21no—
115C2×Dic5—22no—
116C2×Dic5—41no—
117C2×Dic5—41no—
118C2×Dic5—41no—
119C2×Dic5—42no—
120C2×Dic5#10012yes—
121C2×Dic5C4×D414yes—
122C2×Dic5AGL(1,ℤ/8)12yes—
123C2×Dic5#13714yes—
124C2×Dic5#13712yes—
125Dic10—42no—
126Dic10—42no—
127#170—48no—
128C22×Q8—51no—
129C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
130C22×C8—104no—
131C4⋊Q8—54no—
132C4⋊Q8—102no—
133C4×Q8—202no—
134C4×Q8—202no—
135C2×C4⋊C4—51no—
136C2×C4⋊C4—102no—
137C2×C4⋊C4—101no—
138C2×C4⋊C4—101no—
139C2×C4⋊C4—102no—
140C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
141C2×C42—51no—
142C2×C42—52no—
143C2×C42—52no—
144C2×C42—51no—
145C2×C42—101no—
146C2×C42—101no—
147C2×C42C2×AGL(1,𝔽5)11yescharacteristic
148C4⋊C8—58no—
149C4⋊C8—104no—
150#97—204no—
151#95—22no—
152#95—102no—
153C22×C5C4×C22⋊C411yescharacteristic
154C22×C5C2×#10011yescharacteristic
155C22×C5#41311yescharacteristic
156C22×C5#41712yes—
157C22×C5#41912yes—
158C4×C5—21no—
159C4×C5—21no—
160C4×C5—42no—
161C4×C5—42no—
162C4×C5—81no—
163Dic5—22no—
164Dic5—41no—
165Dic5—42no—
166Dic5—81no—
167Dic5#44014yes—
168Dic5#44114yes—
169C2×Q8—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—52no—
173C2×Q8—101no—
174C2×Q8—201no—
175C2×Q8—201no—
176C2×Q8C4×AGL(1,𝔽5)12yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—52no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—101no—
183C22×C4—101no—
184C22×C4C22⋊AGL(1,𝔽5)12yes—
185C22×C4C22×AGL(1,𝔽5)11yescharacteristic
186C2×C8—108no—
187C2×C8—204no—
188C2×C8—204no—
189C4⋊C4—41no—
190C4⋊C4—108no—
191C4⋊C4—201no—
192C4⋊C4—202no—
193C4⋊C4—202no—
194C4⋊C4—202no—
195C4⋊C4—202no—
196C42—41no—
197C42—52no—
198C42—52no—
199C42—102no—
200C42—102no—
201C42—102no—
202C42—104no—
203C42—202no—
204C42—202no—
205C2×C5#155811yescharacteristic
206C2×C5#155912yes—
207C2×C5#156011yescharacteristic
208C2×C5C4×#10011yescharacteristic
209C2×C5#176312yes—
210C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
211Q8—42no—
212Q8—42no—
213Q8—202no—
214Q8—202no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—52no—
226C2×C4—52no—
227C2×C4—54no—
228C2×C4—54no—
229C2×C4—52no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—201no—
235C2×C4—201no—
236C2×C4—202no—
237C2×C4—201no—
238C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
239C2×C4#413011yescharacteristic
240C8—208no—
241C5#823211yescharacteristic, complemented, Sylow
242C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
243C22#6577711yescharacteristic
244C22D5⋊#10011yescharacteristic
245C22#6578212yes—
246C22#6579312yes—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—54no—
252C4—54no—
253C4—81no—
254C4—102no—
255C4—201no—
256C4—202no—
257C4—401no—
258C2#1060424511yescharacteristic
259C2#1060445412yes—
260C2#1060452012yes—
261C2#1060455811yescharacteristic
262C2#1060466211yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#391C4
socle, prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#391AGL(1,𝔽5)
hypocenterC5#8232
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
16. #64931C4semidirect
241. C5#8232semidirect

Maximal subgroups

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

subgroupindex
#106082122
C2×#649312
#106041802
#105987252
#82325