Finite Group Databasebeta

← #12016131

Subgroups — #12016131

Order 1280.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#10614962C211yesmaximal, characteristic
2C2×C5×#356C211yesmaximal, characteristic
3#10602577C211yesmaximal, characteristic
4#10600895C212yesmaximal
5#10598190C212yesmaximal
6C2×C5×C4⋊Q8C2211yescharacteristic
7#66146C2211yescharacteristic
8C2×C5×#101C2212yes—
9#65519—21no—
10#65519C2212yes—
11C2×#3968C2211yescharacteristic
12#65514—21no—
13#65410—22no—
14#65270—22no—
15#65270—22no—
16#65226—21no—
17C5×#356C414yescomplemented
18#64775—22no—
19#64743—21no—
20#12971—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×C5×Q8C2311yescharacteristic
23C2×C5×M16—22no—
24C5×C4⋊Q8—22no—
25C5×C4⋊Q8C2×C412yes—
26C5×C4⋊Q8Q812yes—
27C2×C5×C4⋊C4—22no—
28C2×C42×C5D411yescharacteristic
29Q8×Dic5—42no—
30#4072—22no—
31#4072—21no—
32C2×C4⋊Dic5—21no—
33C2×C4⋊Dic5D412yes—
34C2×C4×Dic5—21no—
35C2×C4×Dic5D412yes—
36C2×#681—21no—
37C5×#101—22no—
38C5×#101C2×C414yes—
39#3968—22no—
40#3967—42no—
41#3963—21no—
42#3963—22no—
43#3963—21no—
44#3963—22no—
45#3963—22no—
46#3963—22no—
47#3950—42no—
48#3949—42no—
49#2381—51no—
50C2×#356D511yescharacteristic
51#1886—51no—
52#1857—52no—
53#1394—52no—
54C2×C5×Q8—21no—
55C2×C5×Q8—41no—
56C2×C5×Q8—41no—
57C2×C5×Q8—41no—
58C2×C5×Q8—41no—
59C2×C5×Q8C4⋊C412yes—
60C2×C5×Q8C22×C411yescharacteristic
61C2×C5×Q8C2×Q811yescharacteristic
62C2×C5×Q8#4112yes—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5C2×D411yescharacteristic
66C22×C4×C5C2×D412yes—
67C22×Dic5—22no—
68C22×Dic5—22no—
69C5×M16—24no—
70C5×M16—42no—
71C2×C5×C8—42no—
72C5×C4⋊C4—24no—
73C5×C4⋊C4—24no—
74C42×C5—21no—
75C42×C5C4⋊C412yes—
76C22⋊Dic5—41no—
77C4⋊Dic5—22no—
78C4⋊Dic5—22no—
79C4⋊Dic5—22no—
80C4⋊Dic5—42no—
81C4⋊Dic5—81no—
82C4×Dic5—41no—
83C4×Dic5—42no—
84C4×Dic5—42no—
85C4×Dic5—81no—
86#681—41no—
87#681—41no—
88C2×#168—41no—
89C2×C4⋊Q8C2×D511yescharacteristic
90#519—51no—
91#428—102no—
92#419—101no—
93C2×#101—51no—
94C2×#101C2×D512yes—
95#391—52no—
96#391—101no—
97#387—102no—
98#387—102no—
99#383—101no—
100#356Dic514yescomplemented
101#339—102no—
102#339—101no—
103C23×C5#12011yescharacteristic
104C5×Q8—41no—
105C5×Q8—41no—
106C5×Q8—42no—
107C5×Q8—42no—
108C5×Q8—81no—
109C2×C4×C5—21no—
110C2×C4×C5—22no—
111C2×C4×C5—21no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—42no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5C2×C4⋊C411yescharacteristic
119C2×C4×C5C4×D412yes—
120C2×C4×C5C22⋊Q812yes—
121C2×C4×C5#12311yescharacteristic
122C2×Dic5—44no—
123C2×Dic5—44no—
124C2×Dic5—42no—
125C2×Dic5—42no—
126C2×Dic5—42no—
127C5×C8—44no—
128#168—81no—
129C22×Q8D2011yescharacteristic
130C22×Q8C22×D511yescharacteristic
131C2×M16—22no—
132C2×M16—101no—
133C4⋊Q8—22no—
134C4⋊Q8Dic1012yes—
135C4⋊Q8C2×Dic512yes—
136C4×Q8—202no—
137#117—101no—
138#117—102no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—52no—
141C2×C4⋊C4—101no—
142C2×C42—52no—
143C2×C42—101no—
144C2×C42D2011yescharacteristic
145#107—202no—
146#106—202no—
147#103—202no—
148#101—22no—
149#101—102no—
150#101C2×Dic514yes—
151#95—101no—
152#95—102no—
153#95—101no—
154#95—102no—
155#95—102no—
156#95—102no—
157C22×C5—22no—
158C22×C5#38511yescharacteristic
159C22×C5B(4,𝔽2)11yescharacteristic
160C22×C5#45811yescharacteristic
161C4×C5—21no—
162C4×C5—22no—
163C4×C5—21no—
164C4×C5—22no—
165C4×C5—42no—
166C4×C5—42no—
167Dic5—82no—
168Dic5—82no—
169Dic5—82no—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8C4⋊Dic512yes—
176C2×Q8C2×Dic1011yescharacteristic
177C2×Q8#71112yes—
178C2×Q8C22×Dic511yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—102no—
182C22×C4—102no—
183C22×C4C2×D2011yescharacteristic
184C22×C4D4×D512yes—
185M16—24no—
186M16—42no—
187M16—201no—
188M16—201no—
189C2×C8—42no—
190C2×C8—201no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—102no—
194C4⋊C4—102no—
195C4⋊C4—102no—
196C4⋊C4—202no—
197C4⋊C4—401no—
198C22⋊C4—201no—
199C42—21no—
200C42—202no—
201C42—202no—
202C42—201no—
203C42—401no—
204C42C4⋊Dic512yes—
205C2×C5—21no—
206C2×C5—21no—
207C2×C5#186711yescharacteristic
208C2×C5#220312yes—
209C23#402811yescharacteristic
210Q8—41no—
211Q8—42no—
212Q8—41no—
213Q8—42no—
214Q8—81no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—204no—
225C2×C4—204no—
226C2×C4—202no—
227C2×C4—202no—
228C2×C4—202no—
229C2×C4#402412yes—
230C2×C4#403211yescharacteristic
231C2×C4C2×C4⋊Dic511yescharacteristic
232C2×C4D4×Dic512yes—
233C8—44no—
234C8—401no—
235C5#1297111yescharacteristic, complemented, Sylow
236C22—22no—
237C22#6503011yescharacteristic
238C22#6503111yescharacteristic
239C22#6523911yescharacteristic
240C4—22no—
241C4—21no—
242C4—22no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—402no—
247C4—402no—
248C4—402no—
249C2—21no—
250C2—21no—
251C2#1059996012yes—
252C2#1060081411yescharacteristic
subgroupG/H
centerC22#65030
commutatorC2×C5×Q8C22×C4
FrattiniC22×Q8C22×D5
FittingC2×C5×#356C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
cosocleC22×C5×Q8C23
hypercenterC2×#356D5
hypocenterC5#12971
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
17. C5×#356C4semidirect
100. #356Dic5semidirect
235. C5#12971semidirect

Maximal subgroups

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

subgroupindex
#106149622
C2×C5×#3562
#106025772
#106008952
#105981902
#129715