Finite Group Databasebeta

← C4×#64855 #13025843

Subgroups — C4×#64855 #13025843

Order 1280.

500 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C4×#4015C211yesmaximal, characteristic
2C4×#4005C212yesmaximal
3C2×#64855C212yesmaximal
4#10598694C212yesmaximal
5C2×#4015C2211yescharacteristic
6C2×C4×Dic10C411yescharacteristic
7C2×C4×Dic10C2211yescharacteristic
8C2×#4005C2214yes—
9C4×#704—22no—
10C43×C5C411yescharacteristic
11#65224C411yescharacteristic
12C4×C4⋊Dic5—22no—
13#65218C411yescharacteristic
14#65218C2211yescharacteristic
15#65217—22no—
16#64919—22no—
17#64855C418yesdirect product factor
18C4×#356—51nomaximal, Sylow
19C22×Dic10C2×C411yescharacteristic
20C22×Dic10C2311yescharacteristic
21C2×#704—24no—
22C2×C42×C5—21no—
23C2×C42×C5—21no—
24C2×C42×C5C2×C411yescharacteristic
25C2×C42×C5C2×C411yescharacteristic
26C2×C42×C5C2×C411yescharacteristic
27C2×C4⋊Dic5—22no—
28C2×C4⋊Dic5—22no—
29C2×C4⋊Dic5C2×C411yescharacteristic
30C2×C4⋊Dic5C2×C411yescharacteristic
31C2×#683—21no—
32C2×#683D412yes—
33C2×C4×Dic5—21no—
34C2×C4×Dic5D412yes—
35#4015—22no—
36#4015C2×C414yes—
37C4×Dic10—41no—
38C4×Dic10—41no—
39C4×Dic10—41no—
40C4×Dic10—41no—
41#4005—24no—
42#4005C2×C418yes—
43#4002—42no—
44C4×#170—42no—
45#3963—22no—
46#3963—22no—
47C4×C4⋊Q8—51no—
48C2×#356—52no—
49C4×#101—52no—
50#1394—52no—
51C22×C4×C5—21no—
52C22×C4×C5—21no—
53C22×C4×C5C22⋊C411yescharacteristic
54C22×C4×C5C22⋊C411yescharacteristic
55C22×C4×C5C22×C411yescharacteristic
56C22×Dic5—21no—
57C22×Dic5—21no—
58C22×Dic5C22⋊C412yes—
59C22×Dic5C2×D412yes—
60C2×Dic10—21no—
61C2×Dic10—41no—
62C2×Dic10—41no—
63C2×Dic10—41no—
64C2×Dic10—41no—
65C2×Dic10C4212yes—
66C2×Dic10C22×C412yes—
67C2×Dic10#4112yes—
68#704—28no—
69#704—44no—
70C2×#170—44no—
71C42×C5—24no—
72C42×C5—21no—
73C42×C5—21no—
74C42×C5—21no—
75C42×C5—41no—
76C42×C5—42no—
77C42×C5C4212yes—
78C4⋊Dic5—28no—
79C4⋊Dic5—41no—
80C4⋊Dic5—44no—
81C4⋊Dic5—41no—
82#683—42no—
83#683—42no—
84C4×Dic5—22no—
85C4×Dic5—22no—
86C4×Dic5—41no—
87C4×Dic5—41no—
88C2×C4⋊Q8—51no—
89C2×C4×Q8—51no—
90C2×C4×Q8—51no—
91C2×#101—54no—
92C4×M16—102no—
93#391—51no—
94#391—51no—
95#389—102no—
96#384—51no—
97C4×C4⋊C4—102no—
98C43AGL(1,𝔽5)11yescharacteristic
99#356—58no—
100#343—102no—
101C23×C5C2×C22⋊C411yescharacteristic
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—21no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—21no—
110C2×C4×C5—41no—
111C2×C4×C5—41no—
112C2×C4×C5—41no—
113C2×C4×C5#9911yescharacteristic
114C2×C4×C5#9911yescharacteristic
115C2×C4×C5C2×C4211yescharacteristic
116C2×C4×C5#11711yescharacteristic
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—41no—
122C2×Dic5—41no—
123C2×Dic5—42no—
124C2×Dic5—42no—
125C2×Dic5—41no—
126C2×Dic5C4×D414yes—
127Dic10—42no—
128Dic10—42no—
129Dic10—42no—
130Dic10—42no—
131#170—48no—
132C22×Q8—51no—
133C22×Q8—51no—
134C2×M16—104no—
135C4⋊Q8—54no—
136C4⋊Q8—102no—
137C4×Q8—201no—
138C4×Q8—201no—
139C4×Q8—201no—
140C4×Q8—201no—
141C2×C4⋊C4—52no—
142C2×C4⋊C4—51no—
143C2×C4⋊C4—51no—
144C2×C4⋊C4—102no—
145C2×C4⋊C4—102no—
146C2×C4⋊C4—101no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—52no—
150C2×C42—101no—
151C2×C42C2×AGL(1,𝔽5)11yescharacteristic
152C2×C42C2×AGL(1,𝔽5)11yescharacteristic
153C2×C42C2×AGL(1,𝔽5)11yescharacteristic
154#101—58no—
155#101—104no—
156#97—202no—
157C4×C8—202no—
158#95—102no—
159#95—102no—
160C22×C5—21no—
161C22×C5—21no—
162C22×C5C4×C22⋊C411yescharacteristic
163C22×C5C2×#9911yescharacteristic
164C22×C5#41011yescharacteristic
165C4×C5—24no—
166C4×C5—21no—
167C4×C5—21no—
168C4×C5—21no—
169C4×C5—42no—
170C4×C5—41no—
171C4×C5#35612yescomplemented
172Dic5—24no—
173Dic5—42no—
174Dic5—42no—
175Dic5—81no—
176C2×Q8—52no—
177C2×Q8—52no—
178C2×Q8—52no—
179C2×Q8—101no—
180C2×Q8—201no—
181C2×Q8—201no—
182C2×Q8—201no—
183C2×Q8—201no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—52no—
187C22×C4—52no—
188C22×C4—101no—
189C22×C4—101no—
190C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
191C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
192C22×C4C22×AGL(1,𝔽5)11yescharacteristic
193M16—108no—
194M16—204no—
195C2×C8—204no—
196C4⋊C4—108no—
197C4⋊C4—202no—
198C4⋊C4—201no—
199C4⋊C4—201no—
200C4⋊C4—202no—
201C4⋊C4—204no—
202C42—24no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—41no—
207C42—42no—
208C42—102no—
209C42—102no—
210C42—201no—
211C42—201no—
212C42C4×AGL(1,𝔽5)12yes—
213C2×C5—21no—
214C2×C5—21no—
215C2×C5C4×#9911yescharacteristic
216C2×C5C2×#35611yescharacteristic
217C2×C5#213211yescharacteristic
218C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
219Q8—202no—
220Q8—202no—
221Q8—202no—
222Q8—202no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—54no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—201no—
240C2×C4—201no—
241C2×C4—202no—
242C2×C4—202no—
243C2×C4—201no—
244C2×C4#399911yescharacteristic
245C2×C4#399911yescharacteristic
246C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
247C2×C4#413011yescharacteristic
248C8—208no—
249C5C4×#35611yescharacteristic, complemented, Sylow
250C22—21no—
251C22—21no—
252C22C2×#399911yescharacteristic
253C22#6574511yescharacteristic
254C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
255C4—24no—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—41no—
261C4—104no—
262C4—202no—
263C4—202no—
264C4—401no—
265C4#6485512yesdirect product factor
266C2—21no—
267C2—21no—
268C2C2×#6485511yescharacteristic
269C2#1060414511yescharacteristic
270C2C4×#399911yescharacteristic
subgroupG/H
centerC2×C4#3999
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC43×C5C4
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC22×Dic10C23
hypercenterC43AGL(1,𝔽5)
hypocenterC5C4×#356
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
17. #64855C4direct (N × H)
171. C4×C5#356semidirect
249. C5C4×#356semidirect
265. C4#64855direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#40152
C4×#40052
C2×#648552
#105986942
C4×#3565