Finite Group Databasebeta

← #13016163

Subgroups — #13016163

Order 1280.

610 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 160 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×#65684C212yesmaximal
2#10606049C211yesmaximal, characteristic
3#10598722C214yesmaximal, complemented
4C2×#4038C412yes—
5C2×C42×D5C2211yescharacteristic
6C2×#4004—22no—
7C2×#4003—22no—
8C2×#4002C2214yes—
9#65684—24no—
10#65684C418yescomplemented
11C5×#384C411yescharacteristic
12#65276—22no—
13#65266C2212yes—
14C4⋊C4×Dic5—22no—
15#65224C411yescharacteristic
16#8031—51nomaximal, Sylow
17C22×C4×D5C2×C411yescharacteristic
18C22×C4×D5D412yes—
19C22×#170—24no—
20C2×C5×C4⋊C4—22no—
21C2×C5×C4⋊C4C2×C412yes—
22C2×C42×C5C2×C411yescharacteristic
23C2×#685—22no—
24C2×C4⋊Dic5—22no—
25C2×C4⋊Dic5C2×C412yes—
26C2×C4×Dic5—22no—
27C2×C4×Dic5D412yes—
28C2×C4×Dic5C2311yescharacteristic
29#4038—44no—
30C42×D5C2×C414yes—
31C42×D5D412yes—
32C42×D5Q812yes—
33#4004—28no—
34#4003—28no—
35#4002—24no—
36#4002C2×C418yes—
37#3963—24no—
38C2×#432—52no—
39#2285—51no—
40#1295—54no—
41C23×D5C22⋊C411yescharacteristic
42C22×C4×C5—22no—
43C22×C4×C5C22⋊C412yes—
44C22×C4×C5C22×C411yescharacteristic
45C22×Dic5—22no—
46C22×Dic5C22⋊C411yescharacteristic
47C22×Dic5C2×D412yes—
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C2×C4×D5—42no—
51C2×C4×D5—42no—
52C2×C4×D5C4212yes—
53C2×C4×D5C22⋊C414yes—
54C2×C4×D5C4⋊C412yes—
55C2×C4×D5C4⋊C414yes—
56C2×#170—28no—
57C2×#170—44no—
58C2×#170—44no—
59C5×C4⋊C4—42no—
60C5×C4⋊C4—44no—
61C42×C5C4212yes—
62C42×C5C4⋊C412yes—
63#685—44no—
64C4⋊Dic5—44no—
65C4⋊Dic5—42no—
66C4×Dic5—44no—
67C4×Dic5C22⋊C414yes—
68C4×Dic5C4⋊C414yes—
69C4×Dic5C22×C412yes—
70C4×Dic5C2×D411yescharacteristic
71C4×Dic5C2×Q811yescharacteristic
72C2×#117—52no—
73C22×C42—51no—
74#432—58no—
75#432—104no—
76C2×C4⋊C8—102no—
77C2×C22⋊C8—102no—
78C2×#97—54no—
79#384—51no—
80#384AGL(1,𝔽5)11yescharacteristic
81#382—52no—
82#380—102no—
83C4×C4⋊C4—102no—
84C23×C5C2×C22⋊C411yescharacteristic
85C22×D5—21no—
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—22no—
89C22×D5#9512yes—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—42no—
95C2×C4×C5—44no—
96C2×C4×C5#9514yes—
97C2×C4×C5C2×C4211yescharacteristic
98C2×C4×C5C2×C4⋊C411yescharacteristic
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—42no—
102C2×Dic5—44no—
103C2×Dic5#9512yes—
104C2×Dic5#10012yes—
105C2×Dic5#10112yes—
106C2×Dic5C2×C22⋊C412yes—
107C2×Dic5C2×C4⋊C412yes—
108C4×D5—24no—
109C4×D5—24no—
110C4×D5—44no—
111C4×D5—44no—
112#170—48no—
113C23×C4—51no—
114C23×C4—52no—
115C22×C8—104no—
116#117—204no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—102no—
120C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
121C2×C22⋊C4—102no—
122C2×C42—52no—
123C2×C42—51no—
124C2×C42—54no—
125C2×C42—52no—
126C2×C42—52no—
127C2×C42—102no—
128C2×C42C2×AGL(1,𝔽5)11yescharacteristic
129C4⋊C8—108no—
130C22⋊C8—108no—
131#97—58no—
132#97—104no—
133#95—104no—
134C22×C5C2×#9511yescharacteristic
135C22×C5C2×#10011yescharacteristic
136C22×C5C2×#10111yescharacteristic
137C22×C5#42112yes—
138C22×C5#42112yes—
139C2×D5—24no—
140C2×D5—22no—
141C2×D5—41no—
142C2×D5—41no—
143C2×D5—42no—
144C4×C5—24no—
145C4×C5—21no—
146C4×C5—21no—
147C4×C5—82no—
148Dic5—22no—
149Dic5—22no—
150Dic5—82no—
151Dic5#34314yes—
152C24—51no—
153C22×C4—22no—
154C22×C4—51no—
155C22×C4—52no—
156C22×C4—52no—
157C22×C4—52no—
158C22×C4—54no—
159C22×C4—54no—
160C22×C4—102no—
161C22×C4—102no—
162C22×C4—102no—
163C22×C4—202no—
164C22×C4—202no—
165C22×C4C22⋊AGL(1,𝔽5)12yes—
166C22×C4C22×AGL(1,𝔽5)11yescharacteristic
167C2×C8—108no—
168C2×C8—204no—
169C2×C8—204no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—202no—
173C4⋊C4—204no—
174C22⋊C4—204no—
175C42—52no—
176C42—51no—
177C42—51no—
178C42—54no—
179C42—54no—
180C42—204no—
181C42C4×AGL(1,𝔽5)12yes—
182C42C4⋊AGL(1,𝔽5)12yes—
183C2×C5#153611yescharacteristic
184C2×C5#153611yescharacteristic
185C2×C5#153712yes—
186C2×C5#173312yes—
187C2×C5C2×#34311yescharacteristic
188D5—42no—
189C23—52no—
190C23—102no—
191C23—101no—
192C23—102no—
193C23—101no—
194C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—52no—
202C2×C4—52no—
203C2×C4—52no—
204C2×C4—52no—
205C2×C4—52no—
206C2×C4—104no—
207C2×C4—104no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4—204no—
211C2×C4—204no—
212C2×C4—202no—
213C2×C4—204no—
214C2×C4(C2×AGL(1,𝔽5))⋊C414yes—
215C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
216C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
217C8—208no—
218C5#803111yescharacteristic, complemented, Sylow
219C22—102no—
220C22—104no—
221C22—201no—
222C22—201no—
223C22—202no—
224C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
225C22#6576912yes—
226C22D5⋊#10011yescharacteristic
227C22#6578412yes—
228C22D5⋊#10111yescharacteristic
229C4—21no—
230C4—21no—
231C4—24no—
232C4—54no—
233C4—82no—
234C4—102no—
235C4—102no—
236C4—402no—
237C2—202no—
238C2#1060428112yes—
239C2#1060433411yescharacteristic
240C2#1060459911yescharacteristic
241C2#1060463811yescharacteristic
242C2#1060463912yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#384C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#384AGL(1,𝔽5)
hypocenterC5#8031
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #10598722C2semidirect
10. #65684C4semidirect
218. C5#8031semidirect

Maximal subgroups

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

subgroupindex
C2×#656842
#106060492
#105987222
#80315