Finite Group Databasebeta

← #11998684

Subgroups — #11998684

Order 1280.

400 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 119 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×#3131C211yesmaximal, characteristic
2#10602631C212yesmaximal, complemented
3#10602607C212yesmaximal
4#10600556C211yesmaximal, characteristic, complemented
5#10600517C211yesmaximal, characteristic, complemented
6C5×#533C2211yescharacteristic
7C5×#532C2211yescharacteristic
8C5×#488—21no—
9C5×#488—21no—
10C5×#487C412yescomplemented
11C5×#486C412yescomplemented
12C5×#416—22no—
13C2×C4×C5×C8C2211yescharacteristic
14#65225C2212yes—
15#65214C2212yes—
16#64771—22no—
17#64758—24no—
18#64757—24no—
19#64675—21no—
20C8×#168—21no—
21#9757—51nomaximal, Sylow
22C2×C5×#41—21no—
23C22×C5×C8D412yes—
24C5×C4⋊Q8C2×C412yes—
25C5×C4⋊Q8C2×C411yescharacteristic
26C5×#127C2×C411yescharacteristic
27C5×#125—21no—
28C5×#124—21no—
29C5×C22⋊Q8—22no—
30C5×#121—22no—
31C4×C5×Q8—22no—
32C4×C5×D4—22no—
33C5×#117—21no—
34C2×C42×C5C2311yescharacteristic
35C2×C4⋊Dic5—22no—
36C5×#103—22no—
37C5×#103—22no—
38C5×#102—22no—
39C5×#102—22no—
40C4×C5×C8C2×C412yes—
41C4×C5×C8D411yescharacteristic
42C4×C5×C8D411yescharacteristic
43#3963—22no—
44#3963—22no—
45C22⋊#168D414yes—
46C4⋊#168—22no—
47C4×#168—22no—
48#3131D511yescharacteristic
49#1835—51no—
50#1518—52no—
51#1517—52no—
52#1450—51no—
53C5×#41—44no—
54C2×C5×Q8—21no—
55C2×C5×Q8—22no—
56C2×C5×D4—21no—
57C2×C5×D4—22no—
58C2×C5×D4—41no—
59C22×C4×C5—41no—
60C22×C4×C5C2×D412yes—
61C22×C4×C5C2×D411yescharacteristic
62C22×Dic5—42no—
63C2×C5×C8—24no—
64C2×C5×C8C22⋊C412yes—
65C2×C5×C8C22⋊C412yes—
66C5×C4⋊C4—22no—
67C5×C4⋊C4—22no—
68C5×C4⋊C4—22no—
69C5×C4⋊C4—42no—
70C5×C22⋊C4—41no—
71C5×C22⋊C4—42no—
72C42×C5—41no—
73C42×C5C22×C411yescharacteristic
74C42×C5C2×D411yescharacteristic
75C42×C5#4112yes—
76C4⋊Dic5—44no—
77C2×#168—24no—
78#533C2×D511yescharacteristic
79#532C2×D511yescharacteristic
80#488—21no—
81#488—21no—
82#487Dic512yescomplemented
83#486Dic512yescomplemented
84#431—52no—
85#416—22no—
86C2×C4×C8C2×D511yescharacteristic
87#382—52no—
88#340—102no—
89#328—104no—
90#327—104no—
91#322—101no—
92C82—101no—
93C23×C5—41no—
94C23×C5#12711yescharacteristic
95C5×Q8—42no—
96C5×Q8—82no—
97C5×D4—42no—
98C5×D4—82no—
99C5×D4—82no—
100C2×C4×C5—42no—
101C2×C4×C5—42no—
102C2×C4×C5—41no—
103C2×C4×C5—82no—
104C2×C4×C5C2×C22⋊C411yescharacteristic
105C2×C4×C5C4×D412yes—
106C2×C4×C5#12112yes—
107C2×C4×C5#12411yescharacteristic
108C2×C4×C5#13514yes—
109C2×Dic5—84no—
110C2×Dic5—82no—
111C5×C8—22no—
112C5×C8—22no—
113#168—44no—
114C2×#41—21no—
115C22×C8C22⋊D512yes—
116C4⋊Q8C2×Dic511yescharacteristic
117C4⋊Q8C2×Dic512yes—
118#127C2×Dic511yescharacteristic
119#125—21no—
120#124—21no—
121C22⋊Q8—22no—
122#121—22no—
123C4×Q8—22no—
124C4×D4—22no—
125#117—21no—
126C2×C4⋊C4—102no—
127C2×C42C22×D511yescharacteristic
128C4⋊C8—102no—
129#103—22no—
130#103—22no—
131#102—22no—
132#102—22no—
133C22⋊C8—54no—
134C4×C8—102no—
135C4×C8C2×Dic512yes—
136C4×C8C22⋊D511yescharacteristic
137C4×C8C22⋊D511yescharacteristic
138#95—102no—
139#95—102no—
140C22×C5—22no—
141C22×C5—81no—
142C22×C5—82no—
143C22×C5#39011yescharacteristic
144C22×C5#49512yes—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—81no—
148C4×C5—82no—
149C4×C5#41612yes—
150C4×C5#48812yes—
151Dic5—162no—
152#41—44no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×D4—21no—
156C2×D4—22no—
157C2×D4—41no—
158C22×C4—41no—
159C22×C4—202no—
160C22×C4D4×D512yes—
161C22×C4C2×C22⋊D511yescharacteristic
162C2×C8—24no—
163C2×C8—104no—
164C2×C8C22⋊Dic512yes—
165C2×C8C22⋊Dic512yes—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—204no—
171C22⋊C4—41no—
172C22⋊C4—42no—
173C42—41no—
174C42#71112yes—
175C42C22×Dic511yescharacteristic
176C42C2×C22⋊D511yescharacteristic
177C2×C5—22no—
178C2×C5—161no—
179C2×C5#195811yescharacteristic
180C2×C5#197612yes—
181C23—41no—
182C23#408611yescharacteristic
183Q8—42no—
184Q8—82no—
185D4—42no—
186D4—82no—
187D4—82no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—82no—
192C2×C4—404no—
193C2×C4—402no—
194C2×C4D4×Dic512yes—
195C2×C4#408211yescharacteristic
196C2×C4#408412yes—
197C2×C4#409614yes—
198C2×C4C2×C22⋊Dic511yescharacteristic
199C8—22no—
200C8—22no—
201C8—204no—
202C5#975711yescharacteristic, complemented, Sylow
203C22—22no—
204C22—81no—
205C22—82no—
206C22#6534812yes—
207C22#6551011yescharacteristic
208C4—22no—
209C4—22no—
210C4—81no—
211C4—82no—
212C4—802no—
213C4#6534512yes—
214C4#6552212yes—
215C2—22no—
216C2—161no—
217C2#1060251811yescharacteristic
218C2#1060287912yes—
subgroupG/H
centerC22#65510
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#3131C2
socleC22×C5#390
prime radicalC5×#532C22
cosocleC2×C42×C5C23
hypercenter#3131D5
hypocenterC5#9757
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
2. #10602631C2semidirect
4. #10600556C2semidirect
5. #10600517C2semidirect
10. C5×#487C4semidirect
11. C5×#486C4semidirect
82. #487Dic5semidirect
83. #486Dic5semidirect
202. C5#9757semidirect

Maximal subgroups

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

subgroupindex
C5×#31312
#106026312
#106026072
#106005562
#106005172
#97575