Finite Group Databasebeta

← #12008711

Subgroups — #12008711

Order 1280.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2317C211yesmaximal, characteristic
2#10602365C212yesmaximal
3#10598123C214yesmaximal, complemented
4C2×C5×#117C411yescharacteristic
5C2×C5×#117C2211yescharacteristic
6C22×C5×C4⋊C4C411yescharacteristic
7C5×#389—22no—
8C5×#385—22no—
9C5×#384C412yes—
10C5×#382C2212yes—
11#65215—22no—
12#64748C2214yes—
13#64745—22no—
14#11515—51nomaximal, Sylow
15C23×C4×C5—21no—
16C23×C4×C5C2×C411yescharacteristic
17C5×#117—24no—
18C5×#117—42no—
19C2×C5×C4⋊C4—21no—
20C2×C5×C4⋊C4—21no—
21C2×C5×C4⋊C4—24no—
22C2×C5×C4⋊C4—42no—
23C2×C5×C4⋊C4C2×C412yes—
24C2×C5×C4⋊C4C2×C411yescharacteristic
25C2×C5×C4⋊C4C2311yescharacteristic
26C2×C5×C22⋊C4—21no—
27C2×C5×C22⋊C4D412yes—
28C2×C42×C5C2×C412yes—
29C2×C42×C5D412yes—
30C5×#95—24no—
31C22⋊#168—24no—
32C4⋊#168—42no—
33#3935—42no—
34#2317D511yescharacteristic
35#1464—52no—
36#1324—54no—
37C24×C5C22⋊C411yescharacteristic
38C22×C4×C5—22no—
39C22×C4×C5—22no—
40C22×C4×C5—22no—
41C22×C4×C5—22no—
42C22×C4×C5—22no—
43C22×C4×C5—42no—
44C22×C4×C5—82no—
45C22×C4×C5C22⋊C412yes—
46C22×C4×C5C22⋊C411yescharacteristic
47C22×C4×C5C22×C411yescharacteristic
48C22×C4×C5C2×D412yes—
49C5×C4⋊C4—24no—
50C5×C4⋊C4—41no—
51C5×C4⋊C4—41no—
52C5×C4⋊C4—42no—
53C5×C4⋊C4—42no—
54C5×C4⋊C4—84no—
55C5×C22⋊C4—24no—
56C5×C22⋊C4—42no—
57C42×C5—22no—
58C42×C5—22no—
59C42×C5—22no—
60C42×C5M1614yes—
61C2×#168—44no—
62C2×#117Dic511yescharacteristic
63C2×#117C2×D511yescharacteristic
64C22×C4⋊C4Dic511yescharacteristic
65#432—102no—
66#389—22no—
67#385—22no—
68#384Dic512yes—
69#382C2×D512yes—
70#324—54no—
71#323—102no—
72C23×C5—21no—
73C23×C5—21no—
74C23×C5—41no—
75C23×C5—41no—
76C23×C5#10112yes—
77C23×C5C2×C22⋊C411yescharacteristic
78C2×C4×C5—28no—
79C2×C4×C5—21no—
80C2×C4×C5—21no—
81C2×C4×C5—21no—
82C2×C4×C5—21no—
83C2×C4×C5—24no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—42no—
87C2×C4×C5—42no—
88C2×C4×C5—42no—
89C2×C4×C5—42no—
90C2×C4×C5—44no—
91C2×C4×C5—42no—
92C2×C4×C5—84no—
93C2×C4×C5#9912yes—
94C2×C4×C5C2×M1612yes—
95#168—84no—
96C23×C4—21no—
97C23×C4C2×Dic511yescharacteristic
98#117—24no—
99#117—42no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C2×Dic512yes—
105C2×C4⋊C4C2×Dic511yescharacteristic
106C2×C4⋊C4C22×D511yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4C22⋊D512yes—
109C2×C42C2×Dic512yes—
110C2×C42C22⋊D512yes—
111C4⋊C8—202no—
112C22⋊C8—104no—
113#97—202no—
114#95—24no—
115C22×C5—21no—
116C22×C5—21no—
117C22×C5—22no—
118C22×C5—42no—
119C22×C5—42no—
120C22×C5—41no—
121C22×C5—41no—
122C22×C5#40711yescharacteristic
123C22×C5C2×#9911yescharacteristic
124C22×C5C2×#10111yescharacteristic
125C4×C5—22no—
126C4×C5—22no—
127C4×C5—44no—
128C4×C5—42no—
129C4×C5—42no—
130C4×C5—82no—
131C24C22⋊Dic511yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—82no—
139C22×C4C22⋊Dic512yes—
140C22×C4C22⋊Dic511yescharacteristic
141C22×C4C22×Dic511yescharacteristic
142C22×C4C2×C22⋊D512yes—
143C2×C8—204no—
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—84no—
150C22⋊C4—24no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42#68114yes—
156C2×C5—21no—
157C2×C5—21no—
158C2×C5—42no—
159C2×C5#146311yescharacteristic
160C2×C5#212911yescharacteristic
161C2×C5#213411yescharacteristic
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23#396812yes—
167C23C2×C22⋊Dic511yescharacteristic
168C2×C4—28no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4#396612yes—
184C2×C4C2×#68112yes—
185C8—404no—
186C5#1151511yescharacteristic, complemented, Sylow
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#6549611yescharacteristic
195C22C2×#396611yescharacteristic
196C22C2×#396811yescharacteristic
197C4—22no—
198C4—22no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—82no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#1060234511yescharacteristic
207C2#1060265511yescharacteristic
208C2#1060266411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2317C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2317D5
hypocenterC5#11515
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #10598123C2semidirect
186. C5#11515semidirect

Maximal subgroups

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

subgroupindex
C5×#23172
#106023652
#105981232
#115155