Finite Group Databasebeta

← #13018081

Subgroups — #13018081

Order 1280.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C22×#4090C211yesmaximal, characteristic
2#10604905C212yesmaximal
3#10604882C214yesmaximal
4C23×C5×Q8C411yescharacteristic
5C23×Dic10C2211yescharacteristic
6C2×#4090—28no—
7C2×#4090—24no—
8C22×#683C412yes—
9C22×#683C2212yes—
10C22×C4×Dic5C411yescharacteristic
11#65798C2214yes—
12C2×#4005—24no—
13#64914—28no—
14#8551—51nomaximal, Sylow
15C22×C5×Q8—24no—
16C22×C5×Q8—22no—
17C22×C5×Q8—44no—
18C23×C4×C5C2×C411yescharacteristic
19C23×C4×C5C2×C412yes—
20C23×Dic5C2×C412yes—
21C23×Dic5C2×C411yescharacteristic
22C23×Dic5C2311yescharacteristic
23C22×Dic10—24no—
24C22×Dic10—44no—
25C22×Dic10D414yes—
26C2×#704—24no—
27#4090—232no—
28C2×#683—24no—
29C2×#683—24no—
30C2×#683—24no—
31C2×#683—24no—
32C2×#683—24no—
33C2×#683D418yes—
34C2×C4×Dic5—22no—
35C2×C4×Dic5—24no—
36C22⋊#170—28no—
37#4005—48no—
38C22×C4⋊Q8—51no—
39#1784—52no—
40#1463—54no—
41C24×C5C22×C411yescharacteristic
42C2×C5×Q8—28no—
43C2×C5×Q8—416no—
44C2×C5×Q8—44no—
45C2×C5×Q8—44no—
46C22×C4×C5—22no—
47C22×C4×C5—24no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—44no—
51C22×C4×C5C22⋊C412yes—
52C22×C4×C5C22⋊C414yes—
53C22×Dic5—22no—
54C22×Dic5—22no—
55C22×Dic5—28no—
56C22×Dic5—22no—
57C22×Dic5—42no—
58C22×Dic5—42no—
59C22×Dic5C22⋊C412yes—
60C22×Dic5C22⋊C414yes—
61C22×Dic5C2×D412yes—
62C22×Dic5C2×D414yes—
63C2×Dic10—28no—
64C2×Dic10—28no—
65C2×Dic10—44no—
66C2×Dic10—416no—
67#704—48no—
68C2×#170—44no—
69#683—216no—
70#683—216no—
71C4×Dic5—28no—
72C23×Q8—51no—
73C23×Q8AGL(1,𝔽5)11yescharacteristic
74C2×C4⋊Q8—108no—
75C2×C4⋊Q8—104no—
76C22×C4⋊C4—52no—
77C22×C4⋊C4—52no—
78C22×C42—51no—
79C2×#101—104no—
80#407—54no—
81#324—108no—
82C23×C5—22no—
83C23×C5—24no—
84C23×C5C2×C22⋊C411yescharacteristic
85C23×C5C2×C22⋊C412yes—
86C5×Q8—416no—
87C2×C4×C5—28no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—24no—
91C2×C4×C5—44no—
92C2×C4×C5—44no—
93C2×C4×C5—44no—
94C2×C4×C5#9914yes—
95C2×Dic5—28no—
96C2×Dic5—28no—
97C2×Dic5—24no—
98C2×Dic5—24no—
99C2×Dic5—24no—
100C2×Dic5—22no—
101C2×Dic5—24no—
102C2×Dic5—44no—
103C2×Dic5—44no—
104C2×Dic5#10118yescomplemented
105C2×Dic5#12014yes—
106Dic10—416no—
107#170—84no—
108C22×Q8—24no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8—54no—
112C22×Q8—104no—
113C22×Q8—204no—
114C23×C4—51no—
115C23×C4—52no—
116C23×C4—51no—
117C23×C4C2×AGL(1,𝔽5)11yescharacteristic
118C23×C4C2×AGL(1,𝔽5)12yes—
119C2×M16—104no—
120C4⋊Q8—1032no—
121C2×C4⋊C4—58no—
122C2×C4⋊C4—104no—
123C2×C4⋊C4—104no—
124C2×C4⋊C4—104no—
125C2×C4⋊C4—104no—
126C2×C4⋊C4—104no—
127C2×C42—104no—
128C2×C42—102no—
129#101—208no—
130C22⋊C8—108no—
131C22×C5—28no—
132C22×C5—22no—
133C22×C5—22no—
134C22×C5—22no—
135C22×C5#37911yescharacteristic
136C22×C5C2×#9912yes—
137C22×C5C2×#10114yes—
138C4×C5—24no—
139C4×C5—44no—
140Dic5—24no—
141Dic5—28no—
142Dic5—42no—
143C24C22×AGL(1,𝔽5)11yescharacteristic
144C2×Q8—28no—
145C2×Q8—416no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—108no—
149C2×Q8—108no—
150C2×Q8—204no—
151C2×Q8—2016no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—44no—
157C22×C4—54no—
158C22×C4—52no—
159C22×C4—54no—
160C22×C4—52no—
161C22×C4—108no—
162C22×C4—102no—
163C22×C4—102no—
164C22×C4—102no—
165C22×C4—202no—
166C22×C4—202no—
167C22×C4C22⋊AGL(1,𝔽5)12yes—
168C22×C4C22⋊AGL(1,𝔽5)14yes—
169M16—208no—
170C2×C8—204no—
171C4⋊C4—1016no—
172C4⋊C4—1016no—
173C42—108no—
174C2×C5—22no—
175C2×C5—24no—
176C2×C5#178111yescharacteristic
177C2×C5#178412yes—
178C23—22no—
179C23—24no—
180C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
181C23C2×C22⋊AGL(1,𝔽5)12yes—
182Q8—416no—
183Q8—2016no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—54no—
192C2×C4—58no—
193C2×C4—108no—
194C2×C4—108no—
195C2×C4—104no—
196C2×C4—104no—
197C2×C4—104no—
198C2×C4—104no—
199C2×C4—102no—
200C2×C4—204no—
201C2×C4—204no—
202C2×C4#401114yes—
203C8—404no—
204C5#855111yescharacteristic, complemented, Sylow
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22D5⋊#10114yes—
210C22C2×#401112yes—
211C22#6580011yescharacteristic
212C4—24no—
213C4—44no—
214C4—108no—
215C4—104no—
216C4—202no—
217C2—22no—
218C2—24no—
219C2#1060498612yes—
220C2#1060502411yescharacteristic
subgroupG/H
centerC22#65800
commutator, prime radicalC24×C5C22×C4
FrattiniC24C22×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC22×C5#379
cosocleC23×Dic5C23
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5#8551
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
104. C2×Dic5#101semidirect
204. C5#8551semidirect

Maximal subgroups

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

subgroupindex
C22×#40902
#106049052
#106048822
#85515