Finite Group Databasebeta

← #12012298

Subgroups — #12012298

Order 1280.

394 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C5×#2582C211yesmaximal, characteristic
2#10600506C212yesmaximal
3#10600366C211yesmaximal, characteristic
4#10598141C212yesmaximal
5#10598126C211yesmaximal, characteristic
6C2×C4×C5×Q8—21no—
7#65834C2211yescharacteristic
8C5×#398—21no—
9C5×#398C2212yes—
10C5×#391—22no—
11C5×#389—22no—
12C5×#384C2211yescharacteristic
13#65260—22no—
14#65253—21no—
15#64750C2211yescharacteristic
16#64748C2212yes—
17#64743—21no—
18#64672—22no—
19#12247—51nomaximal, Sylow
20C22×C5×Q8—21no—
21C4×C5×Q8—42no—
22C4×C5×Q8—42no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C4⋊C4—22no—
25C2×C5×C4⋊C4—22no—
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4D411yescharacteristic
28C2×C5×C4⋊C4D412yes—
29C2×C5×C4⋊C4C2311yescharacteristic
30C2×C42×C5—22no—
31C2×C42×C5D411yescharacteristic
32#4072—21no—
33#4070—22no—
34C2×#681—21no—
35#4036—41no—
36#4034—41no—
37C5×#95—22no—
38C5×#95—21no—
39C5×#95—22no—
40C5×#95D412yes—
41C22⋊#168—22no—
42#3942—42no—
43#3940—41no—
44#3940—42no—
45#3939—41no—
46#2582D511yescharacteristic
47#2060—51no—
48#1611—52no—
49#1346—52no—
50#1325—51no—
51C2×C5×Q8—24no—
52C2×C5×Q8—42no—
53C2×C5×Q8—42no—
54C22×C4×C5—21no—
55C22×C4×C5—22no—
56C22×C4×C5—22no—
57C22×C4×C5—21no—
58C22×C4×C5C2×D411yescharacteristic
59C22×C4×C5C2×D412yes—
60C2×Dic10—22no—
61C5×C4⋊C4—41no—
62C5×C4⋊C4—41no—
63C5×C4⋊C4—42no—
64C5×C4⋊C4—44no—
65C5×C4⋊C4—42no—
66C5×C4⋊C4—42no—
67C5×C4⋊C4—42no—
68C5×C4⋊C4—81no—
69C42×C5—21no—
70C42×C5—44no—
71C42×C5SD1612yes—
72C22⋊Dic5—41no—
73C4⋊Dic5—21no—
74C4⋊Dic5—22no—
75C4⋊Dic5—21no—
76#683—42no—
77C4×Dic5—41no—
78#681—22no—
79#681—41no—
80C2×#168—41no—
81C2×#168—42no—
82#544—51no—
83C2×C4×Q8—21no—
84#483—102no—
85#428—101no—
86#398—21no—
87#398C2×D512yes—
88#391—22no—
89#389—22no—
90#384C2×D511yescharacteristic
91#343—51no—
92#339—101no—
93#328—102no—
94#324—52no—
95C23×C5#12011yescharacteristic
96C5×Q8—48no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—24no—
103C2×C4×C5—22no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—42no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5—42no—
110C2×C4×C5—42no—
111C2×C4×C5—41no—
112C2×C4×C5C2×SD1611yescharacteristic
113C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
114C2×Dic5—41no—
115C2×Dic5—41no—
116Dic10—82no—
117#168—42no—
118#168—82no—
119C22×Q8—21no—
120C2×M16—101no—
121C4⋊Q8—201no—
122#125—201no—
123C22⋊Q8—102no—
124C4×Q8—42no—
125C4×Q8—42no—
126#117—101no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C22⋊D511yescharacteristic
132C2×C4⋊C4C22⋊D512yes—
133C2×C4⋊C4C22×D511yescharacteristic
134C2×C42—22no—
135C2×C42C22⋊D511yescharacteristic
136#107—201no—
137#106—201no—
138#106—202no—
139#103—202no—
140C22⋊C8—102no—
141#95—22no—
142#95—21no—
143#95—22no—
144#95C22⋊D512yes—
145C22×C5—21no—
146C22×C5—21no—
147C22×C5#45011yescharacteristic
148C22×C5#45611yescharacteristic
149C22×C5B(4,𝔽2)11yescharacteristic
150C4×C5—24no—
151C4×C5—21no—
152C4×C5—21no—
153C4×C5—44no—
154C4×C5—42no—
155C4×C5—82no—
156C4×C5—81no—
157Dic5—81no—
158Dic5—81no—
159C2×Q8—24no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—102no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4C2×C22⋊D511yescharacteristic
168C22×C4C2×C22⋊D512yes—
169M16—102no—
170M16—201no—
171C2×C8—201no—
172C2×C8—202no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—101no—
182C4⋊C4—102no—
183C4⋊C4—101no—
184C4⋊C4—202no—
185C22⋊C4—201no—
186C42—21no—
187C42—44no—
188C42—201no—
189C42#68712yes—
190C2×C5—21no—
191C2×C5—21no—
192C2×C5#160411yescharacteristic
193C2×C5#219811yescharacteristic
194C2×C5#220511yescharacteristic
195C23#409911yescharacteristic
196Q8—48no—
197Q8—402no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—201no—
214C2×C4—201no—
215C2×C4#407811yescharacteristic
216C2×C4C2×#68711yescharacteristic
217C8—202no—
218C8—402no—
219C5#1224711yescharacteristic, complemented, Sylow
220C22—21no—
221C22—21no—
222C22#6550711yescharacteristic
223C22#6553311yescharacteristic
224C22#6553511yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—401no—
233C4—401no—
234C2—21no—
235C2—21no—
236C2#1060267311yescharacteristic
237C2#1060273511yescharacteristic
238C2#1060273911yescharacteristic
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2582C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
hypercenter#2582D5
hypocenterC5#12247
LCSG ▷ C2×C5×C4⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
219. C5#12247semidirect

Maximal subgroups

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

subgroupindex
C5×#25822
#106005062
#106003662
#105981412
#105981262
#122475