Finite Group Databasebeta

← #13023183

Subgroups — #13023183

Order 1280.

434 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 104 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#10613888C211yesmaximal, characteristic
2#10604877C212yesmaximal
3#10598710C214yesmaximal, complemented
4C22×C5×C4⋊C4C411yescharacteristic
5C2×#4072C411yescharacteristic
6C2×#4072C2211yescharacteristic
7#65748—22no—
8#65499—22no—
9#65266C2212yes—
10#65265C412yes—
11#65262—22no—
12#64933—22no—
13#64913C2214yes—
14#11449—51nomaximal, Sylow
15C23×C4×C5—21no—
16C23×C4×C5C2×C411yescharacteristic
17C2×C5×C4⋊C4—21no—
18C2×C5×C4⋊C4—42no—
19C2×C5×C4⋊C4C2×C412yes—
20C2×C22⋊Dic5—21no—
21C2×C22⋊Dic5D412yes—
22#4072—22no—
23#4072—42no—
24#4072C814yescomplemented
25C2×C4⋊Dic5—21no—
26C2×C4⋊Dic5C2×C411yescharacteristic
27C2×C4⋊Dic5C2311yescharacteristic
28C2×#683—24no—
29C2×C4×Dic5C2×C412yes—
30C2×C4×Dic5D412yes—
31C22⋊#170—24no—
32C4⋊#170—42no—
33C4×#170—42no—
34#3963—24no—
35#2317—51no—
36#1458—52no—
37#1324—54no—
38C24×C5C22⋊C411yescharacteristic
39C22×C4×C5—22no—
40C22×C4×C5—22no—
41C22×C4×C5—22no—
42C22×C4×C5—42no—
43C22×C4×C5—82no—
44C22×C4×C5C22⋊C411yescharacteristic
45C22×C4×C5C2×C812yes—
46C22×C4×C5M1612yes—
47C22×C4×C5C22×C411yescharacteristic
48C22×Dic5—22no—
49C22×Dic5C22⋊C412yes—
50C22×Dic5C2×D412yes—
51C2×#170—44no—
52C5×C4⋊C4—42no—
53C5×C4⋊C4—42no—
54C22⋊Dic5—24no—
55C22⋊Dic5—42no—
56C4⋊Dic5—24no—
57C4⋊Dic5—41no—
58C4⋊Dic5—41no—
59#683—84no—
60C4×Dic5—22no—
61C4×Dic5—22no—
62C4×Dic5—22no—
63C4×Dic5C2×C814yes—
64C2×#117—51no—
65C2×#117—51no—
66C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
67#431—102no—
68#389—102no—
69#385—102no—
70#384—52no—
71#382—52no—
72#324—54no—
73#323—102no—
74C23×C5—21no—
75C23×C5—21no—
76C23×C5—41no—
77C23×C5—41no—
78C23×C5C22⋊C812yes—
79C23×C5C2×C22⋊C411yescharacteristic
80C2×C4×C5—21no—
81C2×C4×C5—21no—
82C2×C4×C5—24no—
83C2×C4×C5—24no—
84C2×C4×C5—44no—
85C2×C4×C5—42no—
86C2×C4×C5—84no—
87C2×C4×C5C22⋊C812yes—
88C2×C4×C5#9912yes—
89C2×C4×C5#10112yes—
90C2×C4×C5C22×C811yescharacteristic
91C2×C4×C5C2×M1611yescharacteristic
92C2×Dic5—24no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—42no—
96C2×Dic5—42no—
97C2×Dic5—42no—
98C2×Dic5—42no—
99#170—84no—
100C23×C4—21no—
101C23×C4C2×AGL(1,𝔽5)11yescharacteristic
102#117—54no—
103#117—102no—
104#117—202no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—51no—
108C2×C4⋊C4—51no—
109C2×C4⋊C4—101no—
110C2×C4⋊C4—104no—
111C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—101no—
114C2×C42—52no—
115C2×C42—52no—
116C4⋊C8—202no—
117C22⋊C8—104no—
118C4×C8—202no—
119#95—104no—
120C22×C5—21no—
121C22×C5—21no—
122C22×C5—22no—
123C22×C5—42no—
124C22×C5—42no—
125C22×C5—41no—
126C22×C5—41no—
127C22×C5C2×C22⋊C811yescharacteristic
128C22×C5C2×#9911yescharacteristic
129C22×C5C2×#10111yescharacteristic
130C4×C5—21no—
131C4×C5—21no—
132C4×C5—82no—
133C4×C5#32414yes—
134Dic5—44no—
135Dic5—42no—
136Dic5—42no—
137C24C22⋊AGL(1,𝔽5)11yescharacteristic
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—52no—
143C22×C4—52no—
144C22×C4—82no—
145C22×C4—102no—
146C22×C4D5⋊C812yes—
147C22×C4#70012yes—
148C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
149C22×C4C22×AGL(1,𝔽5)11yescharacteristic
150C2×C8—204no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—104no—
154C4⋊C4—201no—
155C4⋊C4—201no—
156C4⋊C4—404no—
157C22⋊C4—104no—
158C22⋊C4—202no—
159C42—54no—
160C42—102no—
161C42—102no—
162C42—102no—
163C2×C5—21no—
164C2×C5—21no—
165C2×C5—42no—
166C2×C5C2×#32411yescharacteristic
167C2×C5#212911yescharacteristic
168C2×C5#213411yescharacteristic
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23#400312yes—
174C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—84no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—104no—
185C2×C4—202no—
186C2×C4—202no—
187C2×C4—202no—
188C2×C4—202no—
189C2×C4#400312yes—
190C2×C4#400512yes—
191C2×C4#401112yes—
192C2×C4C2×D5⋊C811yescharacteristic
193C2×C4C2×#70011yescharacteristic
194C8—404no—
195C5#1144911yescharacteristic, complemented, Sylow
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×#400311yescharacteristic
204C22C2×#400511yescharacteristic
205C22C2×#401111yescharacteristic
206C4—21no—
207C4—21no—
208C4—82no—
209C4—204no—
210C4—202no—
211C4—202no—
212C4#6491414yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#6491411yescharacteristic
217C2#1060492011yescharacteristic
218C2#1060496311yescharacteristic
subgroupG/H
centerC22C2×#4011
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C4⋊Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#11449
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #10598710C2semidirect
24. #4072C8semidirect
195. C5#11449semidirect

Maximal subgroups

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

subgroupindex
#106138882
#106048772
#105987102
#114495