Finite Group Databasebeta

← #12012092

Subgroups — #12012092

Order 1280.

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

normal subgroups only

This group has 400 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×#2607C211yesmaximal, characteristic
2#10600494C212yesmaximal
3#10600329C211yesmaximal, characteristic
4#10598137C212yesmaximal
5#10598118C211yesmaximal, characteristic
6#65817C2211yescharacteristic
7C5×#399—21no—
8C5×#393C2212yes—
9C5×#391—22no—
10C5×#387—22no—
11C5×#381C2211yescharacteristic
12C4×C5×C22⋊C4—21no—
13#65517—22no—
14#65500—21no—
15#64747—21no—
16#64746C2211yescharacteristic
17#64745C2212yes—
18#64672—22no—
19#12215—51nomaximal, Sylow
20C22×C5×Q8D412yes—
21C23×C4×C5D411yescharacteristic
22C2×C5×C4⋊C4—22no—
23C2×C5×C4⋊C4—21no—
24C2×C5×C22⋊C4—22no—
25C2×C5×C22⋊C4D411yescharacteristic
26C2×C5×C22⋊C4C2311yescharacteristic
27C2×C42×C5—21no—
28C2×C42×C5—21no—
29C2×C22⋊Dic5—21no—
30C2×C22⋊Dic5—21no—
31C2×#689—42no—
32#4070—22no—
33#4024—42no—
34C5×#95—22no—
35C5×#95—21no—
36C5×#95—22no—
37C5×#95—22no—
38C5×#95D412yes—
39#3967—42no—
40#3963—42no—
41C22⋊#168—22no—
42#2607D511yescharacteristic
43#2030—51no—
44#1602—52no—
45#1351—52no—
46#1317—51no—
47C24×C5C2×D411yescharacteristic
48C2×C5×Q8—44no—
49C2×C5×Q8—42no—
50C22×C4×C5—21no—
51C22×C4×C5—22no—
52C22×C4×C5—21no—
53C22×C4×C5—22no—
54C22×C4×C5—21no—
55C22×C4×C5—41no—
56C22×C4×C5—41no—
57C22×C4×C5Q1612yescomplemented
58C22×C4×C5C2×D412yes—
59C22×Dic5—22no—
60C22×Dic5—41no—
61C2×Dic10—22no—
62C5×C4⋊C4—44no—
63C5×C4⋊C4—81no—
64C5×C22⋊C4—41no—
65C5×C22⋊C4—44no—
66C5×C22⋊C4—41no—
67C42×C5—81no—
68C42×C5—81no—
69C22⋊Dic5—41no—
70C22⋊Dic5—41no—
71C22⋊Dic5—41no—
72C22⋊Dic5—41no—
73#689—84no—
74C4⋊Dic5—22no—
75#683—42no—
76C2×#168—42no—
77#543—51no—
78#451—102no—
79#399—21no—
80#393C2×D512yes—
81#391—22no—
82#387—22no—
83#381—101no—
84#381C2×D511yescharacteristic
85C4×C22⋊C4—21no—
86#342—51no—
87#342—101no—
88#328—102no—
89#323—52no—
90C23×C5—21no—
91C23×C5—21no—
92C23×C5—42no—
93C23×C5#12011yescharacteristic
94C23×C5C2×Q1611yescharacteristic
95C23×C5#13711yescharacteristic
96C5×Q8—44no—
97C5×Q8—82no—
98C2×C4×C5—24no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—42no—
102C2×C4×C5—41no—
103C2×C4×C5—41no—
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×C5—81no—
113C2×Dic5—42no—
114C2×Dic5—82no—
115C2×Dic5—81no—
116C2×Dic5—82no—
117Dic10—82no—
118#168—82no—
119C22×Q8C22⋊D512yes—
120C23×C4C22⋊D511yescharacteristic
121C2×Q16—202no—
122C22⋊Q8—102no—
123C22⋊Q8—202no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—101no—
128C2×C22⋊C4—101no—
129C2×C22⋊C4C22⋊D511yescharacteristic
130C2×C22⋊C4C22×D511yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133#103—202no—
134C22⋊C8—102no—
135#95—22no—
136#95—21no—
137#95—22no—
138#95—22no—
139#95—202no—
140#95C22⋊D512yes—
141C22×C5—22no—
142C22×C5—21no—
143C22×C5—21no—
144C22×C5—41no—
145C22×C5—41no—
146C22×C5—42no—
147C22×C5—81no—
148C22×C5#45111yescharacteristic
149C22×C5#45511yescharacteristic
150C22×C5B(4,𝔽2)11yescharacteristic
151C4×C5—42no—
152C4×C5—42no—
153C4×C5—44no—
154C4×C5—82no—
155C4×C5—81no—
156Dic5—82no—
157Dic5—161no—
158C24C2×C22⋊D511yescharacteristic
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—102no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—102no—
170C22×C4—201no—
171C22×C4#68912yescomplemented
172C22×C4C2×C22⋊D512yes—
173Q16—404no—
174C2×C8—202no—
175C4⋊C4—44no—
176C4⋊C4—81no—
177C4⋊C4—102no—
178C4⋊C4—202no—
179C22⋊C4—41no—
180C22⋊C4—44no—
181C22⋊C4—41no—
182C22⋊C4—201no—
183C22⋊C4—201no—
184C22⋊C4—201no—
185C22⋊C4—201no—
186C42—81no—
187C42—81no—
188C2×C5—21no—
189C2×C5—21no—
190C2×C5—41no—
191C2×C5—41no—
192C2×C5#160211yescharacteristic
193C2×C5#219712yes—
194C23—21no—
195C23—21no—
196C23—42no—
197C23#408811yescharacteristic
198C23C2×#68911yescharacteristic
199C23#409911yescharacteristic
200Q8—44no—
201Q8—82no—
202Q8—402no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—202no—
219C2×C4—402no—
220C2×C4—401no—
221C2×C4—402no—
222C8—402no—
223C5#1221511yescharacteristic, complemented, Sylow
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#6551711yescharacteristic
232C22#6553111yescharacteristic
233C22#6553311yescharacteristic
234C4—42no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—402no—
240C4—801no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#1060268611yescharacteristic
246C2#1060271512yes—
subgroupG/H
centerC22#65533
commutator, cosocleC2×C5×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2607C2
socleC22×C5B(4,𝔽2)
prime radicalC24×C5C2×D4
hypercenter#2607D5
hypocenterC5#12215
LCSG ▷ C2×C5×C22⋊C4 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
57. C22×C4×C5Q16semidirect
171. C22×C4#689semidirect
223. C5#12215semidirect

Maximal subgroups

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

subgroupindex
C5×#26072
#106004942
#106003292
#105981372
#105981182
#122155