Finite Group Databasebeta

← C2×#10598711 #13023172

Subgroups — C2×#10598711 #13023172

Order 1280.

598 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 120 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×#65265C211yesmaximal, characteristic
2C2×#64913C212yesmaximal
3#10598711C214yesmaximal, direct product factor
4C22×C5×C4⋊C4C411yescharacteristic
5C22×C4⋊Dic5C2211yescharacteristic
6C22×#683—22no—
7C22×C4×Dic5C411yescharacteristic
8C2×C22⋊#170—22no—
9#65265—22no—
10#65265C412yes—
11#65265C2212yes—
12#64913—22no—
13#64913C2214yes—
14C2×#1325—51nomaximal, Sylow
15C23×C4×C5—21no—
16C23×C4×C5C2×C411yescharacteristic
17C23×Dic5—21no—
18C23×Dic5D412yes—
19C22×#170—42no—
20C2×C5×C4⋊C4—21no—
21C2×C5×C4⋊C4—41no—
22C2×C5×C4⋊C4—41no—
23C2×C5×C4⋊C4C2×C412yes—
24C2×C4⋊Dic5—41no—
25C2×C4⋊Dic5—41no—
26C2×C4⋊Dic5C2×C411yescharacteristic
27C2×C4⋊Dic5D412yes—
28C2×C4⋊Dic5C2311yescharacteristic
29C2×#683—24no—
30C2×#683—24no—
31C2×#683—48no—
32C2×C4×Dic5—21no—
33C2×C4×Dic5—22no—
34C2×C4×Dic5C814yescomplemented
35C2×C4×Dic5C2×C412yes—
36C22⋊#170—24no—
37C22⋊#170—42no—
38C2×#384—51no—
39C2×#324—52no—
40#1325—54no—
41C24×C5C22⋊C411yescharacteristic
42C22×C4×C5—21no—
43C22×C4×C5—21no—
44C22×C4×C5—22no—
45C22×C4×C5—22no—
46C22×C4×C5—44no—
47C22×C4×C5—41no—
48C22×C4×C5—41no—
49C22×C4×C5C22⋊C411yescharacteristic
50C22×C4×C5C2×C812yes—
51C22×C4×C5M1612yes—
52C22×C4×C5C22×C411yescharacteristic
53C22×Dic5—24no—
54C22×Dic5—24no—
55C22×Dic5—24no—
56C22×Dic5—22no—
57C22×Dic5—22no—
58C22×Dic5—22no—
59C22×Dic5—44no—
60C22×Dic5C22⋊C412yes—
61C22×Dic5C2×D412yes—
62C2×#170—44no—
63C2×#170—84no—
64C5×C4⋊C4—42no—
65C5×C4⋊C4—42no—
66C4⋊Dic5—41no—
67C4⋊Dic5—42no—
68C4⋊Dic5—41no—
69#683—48no—
70#683—48no—
71C4×Dic5—24no—
72C4×Dic5—22no—
73C4×Dic5C2×C814yescomplemented
74C22×C4⋊C4—51no—
75C22×C4⋊C4—102no—
76C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
77C22×C42—51no—
78C2×C22⋊C8—102no—
79#384—52no—
80#384—52no—
81#384—102no—
82#324—54no—
83#324—102no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5C22⋊C812yes—
87C23×C5#9911yescharacteristic
88C23×C5#9911yescharacteristic
89C23×C5#10112yes—
90C23×C5C2×C22⋊C411yescharacteristic
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—44no—
100C2×C4×C5—42no—
101C2×C4×C5—82no—
102C2×C4×C5—82no—
103C2×C4×C5C22⋊C812yes—
104C2×C4×C5C22×C811yescharacteristic
105C2×C4×C5C2×M1611yescharacteristic
106C2×Dic5—24no—
107C2×Dic5—216no—
108C2×Dic5—24no—
109C2×Dic5—24no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—44no—
113C2×Dic5—42no—
114C2×Dic5—42no—
115C2×Dic5—82no—
116#170—84no—
117C23×C4—21no—
118C23×C4—52no—
119C23×C4—101no—
120C23×C4C2×AGL(1,𝔽5)11yescharacteristic
121C22×C8—202no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—51no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—104no—
129C2×C4⋊C4—104no—
130C2×C4⋊C4—201no—
131C2×C4⋊C4—201no—
132C2×C4⋊C4—208no—
133C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
134C2×C42—54no—
135C2×C42—52no—
136C2×C42—102no—
137C2×C42—101no—
138C22⋊C8—104no—
139C22⋊C8—202no—
140C22×C5—24no—
141C22×C5—24no—
142C22×C5—21no—
143C22×C5—21no—
144C22×C5—21no—
145C22×C5—21no—
146C22×C5#32412yes—
147C22×C5#32412yes—
148C22×C5#35312yes—
149C22×C5#35612yes—
150C22×C5C2×C22⋊C811yescharacteristic
151C22×C5C2×#9911yescharacteristic
152C22×C5C2×#10111yescharacteristic
153C4×C5—22no—
154C4×C5—21no—
155C4×C5—21no—
156C4×C5—82no—
157Dic5—28no—
158Dic5—42no—
159Dic5—42no—
160C24C22⋊AGL(1,𝔽5)11yescharacteristic
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—44no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—52no—
169C22×C4—52no—
170C22×C4—104no—
171C22×C4—104no—
172C22×C4—102no—
173C22×C4—104no—
174C22×C4—102no—
175C22×C4—102no—
176C22×C4—204no—
177C22×C4D5⋊C812yes—
178C22×C4#70012yes—
179C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
180C22×C4C22×AGL(1,𝔽5)11yescharacteristic
181C2×C8—204no—
182C2×C8—404no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—202no—
186C4⋊C4—201no—
187C4⋊C4—201no—
188C4⋊C4—208no—
189C4⋊C4—208no—
190C42—54no—
191C42—102no—
192C42—104no—
193C2×C5—21no—
194C2×C5—22no—
195C2×C5—21no—
196C2×C5#132514yescomplemented
197C2×C5C2×#32411yescharacteristic
198C2×C5C2×#35311yescharacteristic
199C2×C5C2×#35611yescharacteristic
200C23—22no—
201C23—22no—
202C23#400312yes—
203C23#400512yes—
204C23#401111yescharacteristic
205C23#401111yescharacteristic
206C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—104no—
220C2×C4—1016no—
221C2×C4—104no—
222C2×C4—104no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—204no—
226C2×C4—202no—
227C2×C4—202no—
228C2×C4—402no—
229C2×C4#400312yes—
230C2×C4C2×D5⋊C811yescharacteristic
231C2×C4C2×#70011yescharacteristic
232C8—404no—
233C5C2×#132511yescharacteristic, complemented, Sylow
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#6491412yes—
241C22#6491412yes—
242C22#6492212yes—
243C22#6492912yes—
244C22C2×#400311yescharacteristic
245C22C2×#400511yescharacteristic
246C22C2×#401111yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—82no—
251C4—108no—
252C4—202no—
253C4—202no—
254C2—21no—
255C2—22no—
256C2—21no—
257C2#1059871114yesdirect product factor
258C2C2×#6491411yescharacteristic
259C2C2×#6492211yescharacteristic
260C2C2×#6492911yescharacteristic
subgroupG/H
centerC23#4011
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5#99
prime radicalC24×C5C22⋊C4
cosocleC2×C4⋊Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C2×#1325
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10598711C2direct (N × H)
34. C2×C4×Dic5C8semidirect
73. C4×Dic5C2×C8semidirect
196. C2×C5#1325semidirect
233. C5C2×#1325semidirect
257. C2#10598711direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#652652
C2×#649132
#105987112
C2×#13255