Finite Group Databasebeta

← #13016159

Subgroups — #13016159

Order 1280.

638 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×#65757C212yesmaximal
2C2×#65266C211yesmaximal, characteristic
3#10598722C214yesmaximal, complemented
4C22×C5×C4⋊C4C411yescharacteristic
5C22×C4⋊Dic5C411yescharacteristic
6C22×C4×Dic5C2211yescharacteristic
7C2×C22⋊#170—22no—
8#65757—24no—
9#65757C418yescomplemented
10C2×#4004—22no—
11C2×#4002C2214yes—
12C2×#3963—22no—
13#65266—22no—
14#65266C412yes—
15#65266C2212yes—
16#8030—51nomaximal, Sylow
17C23×C4×C5—21no—
18C23×C4×C5C2×C411yescharacteristic
19C23×Dic5—21no—
20C23×Dic5D412yes—
21C22×#170—24no—
22C2×C5×C4⋊C4—21no—
23C2×C5×C4⋊C4—42no—
24C2×C5×C4⋊C4C2×C412yes—
25C2×C4⋊Dic5—21no—
26C2×C4⋊Dic5—42no—
27C2×C4⋊Dic5C2×C412yes—
28C2×C4×Dic5C2×C411yescharacteristic
29C2×C4×Dic5C2×C414yes—
30C2×C4×Dic5D412yes—
31C2×C4×Dic5D412yes—
32C2×C4×Dic5Q812yes—
33C2×C4×Dic5C2311yescharacteristic
34C22⋊#170—28no—
35#4004—28no—
36#4002—24no—
37#4002C2×C418yescomplemented
38#3963—24no—
39#3963—24no—
40C2×#432—52no—
41C2×#382—51no—
42#1295—54no—
43C24×C5C22⋊C411yescharacteristic
44C22×C4×C5—22no—
45C22×C4×C5—22no—
46C22×C4×C5—22no—
47C22×C4×C5—44no—
48C22×C4×C5—42no—
49C22×C4×C5C4212yes—
50C22×C4×C5C22⋊C411yescharacteristic
51C22×C4×C5C4⋊C412yes—
52C22×C4×C5C22×C411yescharacteristic
53C22×Dic5—22no—
54C22×Dic5—42no—
55C22×Dic5—42no—
56C22×Dic5—44no—
57C22×Dic5—42no—
58C22×Dic5C22⋊C412yes—
59C22×Dic5C22⋊C414yes—
60C22×Dic5C4⋊C414yes—
61C22×Dic5C2×D412yes—
62C2×#170—28no—
63C2×#170—44no—
64C2×#170—44no—
65C5×C4⋊C4—42no—
66C5×C4⋊C4—42no—
67C4⋊Dic5—42no—
68C4⋊Dic5—42no—
69C4×Dic5C4212yes—
70C4×Dic5C22⋊C414yes—
71C4×Dic5C4⋊C412yes—
72C4×Dic5C4⋊C414yes—
73C4×Dic5C22×C412yes—
74C4×Dic5C2×D411yescharacteristic
75C4×Dic5C2×Q811yescharacteristic
76C22×C4⋊C4—51no—
77C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
78C22×C42—51no—
79#432—58no—
80#432—104no—
81C2×C4⋊C8—102no—
82C2×C22⋊C8—102no—
83C2×#97—54no—
84#382—52no—
85#382—52no—
86#382—102no—
87C2×#95—102no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5#9512yes—
91C23×C5#10012yes—
92C23×C5#10112yes—
93C23×C5C2×C22⋊C411yescharacteristic
94C2×C4×C5—24no—
95C2×C4×C5—24no—
96C2×C4×C5—21no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—42no—
101C2×C4×C5—44no—
102C2×C4×C5—84no—
103C2×C4×C5#9512yes—
104C2×C4×C5C2×C4211yescharacteristic
105C2×C4×C5C2×C4⋊C411yescharacteristic
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—44no—
111C2×Dic5—44no—
112C2×Dic5—42no—
113C2×Dic5—44no—
114C2×Dic5—84no—
115C2×Dic5#9514yes—
116C2×Dic5C2×C22⋊C412yes—
117C2×Dic5C2×C4⋊C412yes—
118#170—48no—
119C23×C4—21no—
120C23×C4—52no—
121C23×C4—101no—
122C23×C4C2×AGL(1,𝔽5)11yescharacteristic
123C22×C8—104no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—101no—
128C2×C4⋊C4—202no—
129C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
130C2×C42—54no—
131C2×C42—52no—
132C2×C42—52no—
133C2×C42—52no—
134C2×C42—51no—
135C2×C42—51no—
136C4⋊C8—108no—
137C22⋊C8—108no—
138#97—58no—
139#97—104no—
140#95—104no—
141#95—104no—
142C22×C5—24no—
143C22×C5—22no—
144C22×C5—22no—
145C22×C5—24no—
146C22×C5#34314yes—
147C22×C5C2×#9511yescharacteristic
148C22×C5C2×#10011yescharacteristic
149C22×C5C2×#10111yescharacteristic
150C22×C5#42112yes—
151C22×C5#42112yes—
152C4×C5—22no—
153C4×C5—21no—
154C4×C5—21no—
155C4×C5—82no—
156Dic5—22no—
157Dic5—24no—
158Dic5—22no—
159Dic5—82no—
160C24C22⋊AGL(1,𝔽5)11yescharacteristic
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—52no—
167C22×C4—52no—
168C22×C4—54no—
169C22×C4—54no—
170C22×C4—102no—
171C22×C4—202no—
172C22×C4—202no—
173C22×C4—204no—
174C22×C4—202no—
175C22×C4C4×AGL(1,𝔽5)12yes—
176C22×C4C4⋊AGL(1,𝔽5)12yes—
177C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
178C22×C4C22×AGL(1,𝔽5)11yescharacteristic
179C2×C8—108no—
180C2×C8—204no—
181C2×C8—204no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—202no—
185C4⋊C4—202no—
186C42—52no—
187C42—52no—
188C42—51no—
189C42—52no—
190C42—51no—
191C42—54no—
192C42—54no—
193C2×C5—24no—
194C2×C5#153311yescharacteristic
195C2×C5#153311yescharacteristic
196C2×C5#153412yes—
197C2×C5#173312yes—
198C2×C5C2×#34311yescharacteristic
199C23—22no—
200C23—22no—
201C23#400512yes—
202C23(C2×AGL(1,𝔽5))⋊C412yes—
203C23#401312yes—
204C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—52no—
214C2×C4—54no—
215C2×C4—52no—
216C2×C4—84no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—204no—
222C2×C4—204no—
223C2×C4—202no—
224C2×C4—204no—
225C2×C4—404no—
226C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
227C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
228C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
229C8—208no—
230C5#803011yescharacteristic, complemented, Sylow
231C22—22no—
232C22—22no—
233C22—24no—
234C22—24no—
235C22#6491914yes—
236C22C2×#400511yescharacteristic
237C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
238C22#6576912yes—
239C22#6578412yes—
240C22C2×#401311yescharacteristic
241C4—21no—
242C4—21no—
243C4—22no—
244C4—82no—
245C4—102no—
246C4—104no—
247C4—102no—
248C4—402no—
249C2—24no—
250C2#1060426712yes—
251C2#1060428112yes—
252C2C2×#6491911yescharacteristic
253C2#1060491611yescharacteristic
254C2#1060496111yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#8030
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10598722C2semidirect
9. #65757C4semidirect
37. #4002C2×C4semidirect
230. C5#8030semidirect

Maximal subgroups

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

subgroupindex
C2×#657572
C2×#652662
#105987222
#80305