Finite Group Databasebeta

← #11998683

Subgroups — #11998683

Order 1280.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#10610194C211yesmaximal, characteristic
2C5×#1450C211yesmaximal, characteristic, complemented
3#10601586C212yesmaximal, complemented
4#10601583C212yesmaximal
5#10600556C211yesmaximal, characteristic, complemented
6#65811C2211yescharacteristic
7#65804C2211yescharacteristic
8C5×#431C2212yes—
9C2×C4×C5×C8C2211yescharacteristic
10#65402—22no—
11#65225C2212yes—
12#64986—21no—
13#64986—21no—
14#64983C412yescomplemented
15#64970C412yescomplemented
16C5×#322—21no—
17C5×C82—21no—
18#64771—22no—
19#64690—24no—
20#64687—24no—
21#9757—51nomaximal, Sylow
22C2×#709—21no—
23C22×C5×C8D412yes—
24C2×C42×C5C2311yescharacteristic
25#4076—22no—
26#4072—21no—
27C2×C4⋊Dic5—22no—
28#4070—22no—
29#4021—21no—
30#4020C2×C411yescharacteristic
31C4×D20—22no—
32#4016—21no—
33#4015C2×C412yes—
34#4015C2×C411yescharacteristic
35C4×Dic10—22no—
36C5×C4⋊C8—22no—
37C5×C22⋊C8D414yes—
38C4×C5×C8—22no—
39C4×C5×C8C2×C412yes—
40C4×C5×C8D411yescharacteristic
41C4×C5×C8D411yescharacteristic
42#3963—22no—
43#3963—22no—
44#3953—22no—
45#3953—22no—
46#3948—22no—
47#3948—22no—
48#3131—51no—
49#1835—51no—
50#1518—52no—
51#1517—52no—
52#1450D511yescharacteristic, complemented
53C22×C4×C5C2×D412yes—
54C22×C4×C5C2×D411yescharacteristic
55C2×C22⋊D5—41no—
56C22×Dic5—42no—
57#709—44no—
58C2×D20—21no—
59C2×D20—22no—
60C2×C4×D5—41no—
61C2×Dic10—21no—
62C2×Dic10—22no—
63C2×C5×C8—24no—
64C2×C5×C8—24no—
65C2×C5×C8C22⋊C412yes—
66C2×C5×C8C22⋊C412yes—
67C42×C5C22×C411yescharacteristic
68C42×C5C2×D411yescharacteristic
69C42×C5#4112yes—
70C22⋊Dic5—41no—
71#685—42no—
72C4⋊Dic5—22no—
73C4⋊Dic5—22no—
74C4⋊Dic5—22no—
75C4⋊Dic5—44no—
76#683—42no—
77C4×Dic5—41no—
78#533—51no—
79#532—51no—
80#488—101no—
81#488—101no—
82#487—52no—
83#486—52no—
84#431C2×D512yes—
85#416—102no—
86C2×C4×C8C2×D511yescharacteristic
87#382—52no—
88#340—102no—
89#328—104no—
90#327—104no—
91#322—21no—
92C82—21no—
93C23×C5#12711yescharacteristic
94C22×D5—41no—
95C2×C4×C5—42no—
96C2×C4×C5C2×C22⋊C411yescharacteristic
97C2×C4×C5C4×D412yes—
98C2×C4×C5#12112yes—
99C2×C4×C5#12411yescharacteristic
100C2×C4×C5#13514yes—
101C22⋊D5—82no—
102C2×Dic5—42no—
103C2×Dic5—41no—
104C2×Dic5—84no—
105C2×Dic5—82no—
106D20—42no—
107D20—82no—
108C4×D5—82no—
109Dic10—42no—
110Dic10—82no—
111C5×C8—22no—
112C5×C8—22no—
113C5×C8—44no—
114C2×#41—101no—
115C22×C8D2012yes—
116C4⋊Q8—51no—
117C4⋊Q8—52no—
118#127—51no—
119#125—101no—
120#124—101no—
121C22⋊Q8—102no—
122#121—102no—
123C4×Q8—102no—
124C4×D4—102no—
125#117—101no—
126C2×C4⋊C4—102no—
127C2×C42C22×D511yescharacteristic
128C4⋊C8—22no—
129#103—102no—
130#103—102no—
131#102—102no—
132#102—102no—
133C22⋊C8D2014yes—
134C4×C8—22no—
135C4×C8C4×D512yes—
136C4×C8C22⋊D511yescharacteristic
137C4×C8C22⋊D511yescharacteristic
138#95—102no—
139#95—102no—
140C22×C5—22no—
141C22×C5#39011yescharacteristic
142C22×C5#49512yes—
143C2×D5—81no—
144C2×D5—82no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5#41612yes—
148C4×C5#48812yes—
149Dic5—81no—
150Dic5—82no—
151Dic5—162no—
152#41—204no—
153C2×Q8—102no—
154C2×Q8—101no—
155C2×D4—102no—
156C2×D4—101no—
157C2×D4—201no—
158C22×C4—201no—
159C22×C4—202no—
160C22×C4C2×D2012yes—
161C22×C4C2×D2011yescharacteristic
162C2×C8—24no—
163C2×C8—24no—
164C2×C8#68512yes—
165C2×C8#68512yes—
166C4⋊C4—102no—
167C4⋊C4—102no—
168C4⋊C4—102no—
169C4⋊C4—202no—
170C4⋊C4—204no—
171C22⋊C4—201no—
172C22⋊C4—202no—
173C42—201no—
174C42C2×C4×D511yescharacteristic
175C42#70912yes—
176C42C2×C22⋊D511yescharacteristic
177C2×C5—22no—
178C2×C5#195811yescharacteristic
179C2×C5#197612yes—
180D5—161no—
181C23—201no—
182C23#402011yescharacteristic
183Q8—202no—
184Q8—402no—
185D4—202no—
186D4—402no—
187D4—402no—
188C2×C4—42no—
189C2×C4—202no—
190C2×C4—201no—
191C2×C4—402no—
192C2×C4—404no—
193C2×C4—402no—
194C2×C4C4×D2012yes—
195C2×C4#402111yescharacteristic
196C2×C4#405014yes—
197C2×C4C2×#68511yescharacteristic
198C2×C4#407612yes—
199C8—22no—
200C8—22no—
201C8—44no—
202C5#975711yescharacteristic, complemented, Sylow
203C22—22no—
204C22—401no—
205C22—402no—
206C22#6498512yes—
207C22#6522811yescharacteristic
208C4—22no—
209C4—22no—
210C4—401no—
211C4—402no—
212C4—802no—
213C4#6498612yes—
214C4#6540212yes—
215C2—22no—
216C2—801no—
217C2#1060056611yescharacteristic
218C2#1060069012yes—
subgroupG/H
centerC22#65228
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#1450C2
socleC22×C5#390
prime radical#65811C22
cosocleC2×C42×C5C23
hypercenter#1450D5
hypocenterC5#9757
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. C5×#1450C2semidirect
3. #10601586C2semidirect
5. #10600556C2semidirect
14. #64983C4semidirect
15. #64970C4semidirect
52. #1450D5semidirect
202. C5#9757semidirect

Maximal subgroups

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

subgroupindex
#106101942
C5×#14502
#106015862
#106015832
#106005562
#97575