Finite Group Databasebeta

← #13022427

Subgroups — #13022427

Order 1280.

1318 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 184 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×#4034C211yesmaximal, characteristic
2C22×#4005C214yesmaximal
3#10604895C212yesmaximal
4C23×Dic10C2212yes—
5C22×#704C2214yes—
6C22×C5×C4⋊C4C411yescharacteristic
7C22×C4⋊Dic5C411yescharacteristic
8C22×#683C412yes—
9C22×C4×Dic5C2211yescharacteristic
10C2×#4034—24no—
11C2×#4034—28no—
12C2×#4005—28no—
13C2×#4005—216no—
14#64918—24no—
15#10983—51nomaximal, Sylow
16C23×C4×C5C2×C412yes—
17C23×C4×C5C2×C411yescharacteristic
18C23×Dic5C2×C411yescharacteristic
19C23×Dic5C2×C412yes—
20C23×Dic5C2311yescharacteristic
21C22×Dic10—24no—
22C22×Dic10—24no—
23C22×Dic10—28no—
24C22×Dic10—28no—
25C22×Dic10D418yes—
26C2×#704—24no—
27C2×#704—28no—
28C2×#704—28no—
29C2×#704D418yes—
30C22×#170—24no—
31C2×C5×C4⋊C4—24no—
32C2×C5×C4⋊C4—22no—
33C2×C4⋊Dic5—22no—
34C2×C4⋊Dic5—24no—
35C2×#683—24no—
36C2×#683—24no—
37C2×#683—24no—
38C2×C4×Dic5—24no—
39C2×C4×Dic5D414yes—
40#4034—232no—
41#4005—232no—
42C22×C4⋊Q8—51no—
43C22×#101—54no—
44#1728—52no—
45C24×C5C22×C411yescharacteristic
46C22×C4×C5—22no—
47C22×C4×C5—24no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—28no—
51C22×C4×C5C22⋊C414yes—
52C22×C4×C5C22⋊C412yes—
53C22×Dic5—22no—
54C22×Dic5—24no—
55C22×Dic5—22no—
56C22×Dic5—22no—
57C22×Dic5—28no—
58C22×Dic5C22⋊C414yes—
59C22×Dic5C22⋊C412yes—
60C22×Dic5C2×D414yes—
61C22×Dic5C2×D414yes—
62C22×Dic5C2×D412yes—
63C22×Dic5#4114yes—
64C2×Dic10—28no—
65C2×Dic10—216no—
66C2×Dic10—216no—
67C2×Dic10—28no—
68C2×Dic10—44no—
69C2×Dic10—48no—
70C2×Dic10—48no—
71C2×Dic10—44no—
72C2×Dic10—44no—
73C2×Dic10—44no—
74#704—216no—
75#704—48no—
76C2×#170—28no—
77C2×#170—44no—
78C2×#170—44no—
79C5×C4⋊C4—28no—
80C4⋊Dic5—28no—
81#683—216no—
82C4×Dic5—28no—
83C23×Q8—52no—
84C22×M16—54no—
85C2×C4⋊Q8—104no—
86C2×C4⋊Q8—108no—
87C22×C4⋊C4—52no—
88C22×C4⋊C4—51no—
89C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic
90C22×C42—51no—
91C2×#101—1016no—
92C2×#101—108no—
93#336—104no—
94C23×C5—22no—
95C23×C5—22no—
96C23×C5C2×C22⋊C412yes—
97C23×C5C2×C22⋊C411yescharacteristic
98C23×C5C4×D414yes—
99C2×C4×C5—28no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—24no—
103C2×C4×C5—28no—
104C2×C4×C5—44no—
105C2×C4×C5—44no—
106C2×C4×C5#10114yescomplemented
107C2×Dic5—28no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—24no—
111C2×Dic5—28no—
112C2×Dic5—44no—
113C2×Dic5—44no—
114C2×Dic5#10114yescomplemented
115C2×Dic5#12014yes—
116C2×Dic5#12014yes—
117C2×Dic5#12118yes—
118C2×Dic5#12114yes—
119C2×Dic5#12314yes—
120C2×Dic5#12412yes—
121C2×Dic5#12712yes—
122Dic10—416no—
123Dic10—416no—
124#170—48no—
125C22×Q8—58no—
126C22×Q8—104no—
127C22×Q8—104no—
128C22×Q8—108no—
129C22×Q8—108no—
130C23×C4—51no—
131C23×C4—52no—
132C23×C4—51no—
133C23×C4C2×AGL(1,𝔽5)12yes—
134C23×C4C2×AGL(1,𝔽5)11yescharacteristic
135C2×M16—58no—
136C2×M16—104no—
137C2×M16—108no—
138C2×M16—108no—
139C22×C8—104no—
140C4⋊Q8—1032no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—104no—
144C2×C4⋊C4—104no—
145C2×C4⋊C4—104no—
146C2×C4⋊C4—104no—
147C2×C4⋊C4—102no—
148C2×C42—54no—
149C2×C42—104no—
150#101—1032no—
151C22×C5—28no—
152C22×C5—22no—
153C22×C5—22no—
154C22×C5#37911yescharacteristic
155C22×C5#38614yes—
156C22×C5#39012yes—
157C22×C5C2×#10112yes—
158C22×C5C2×#10112yes—
159C4×C5—24no—
160C4×C5—44no—
161Dic5—24no—
162Dic5—44no—
163Dic5#39214yes—
164Dic5#39414yes—
165C24C22×AGL(1,𝔽5)11yescharacteristic
166C2×Q8—108no—
167C2×Q8—1016no—
168C2×Q8—1016no—
169C2×Q8—108no—
170C2×Q8—204no—
171C2×Q8—208no—
172C2×Q8—204no—
173C2×Q8—208no—
174C2×Q8—204no—
175C2×Q8—204no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—54no—
182C22×C4—54no—
183C22×C4—54no—
184C22×C4—52no—
185C22×C4—52no—
186C22×C4—54no—
187C22×C4—104no—
188C22×C4—102no—
189C22×C4—108no—
190C22×C4—102no—
191C22×C4—102no—
192C22×C4C22⋊AGL(1,𝔽5)14yes—
193C22×C4C22⋊AGL(1,𝔽5)12yes—
194M16—1016no—
195M16—208no—
196C2×C8—108no—
197C2×C8—204no—
198C2×C8—204no—
199C4⋊C4—28no—
200C4⋊C4—1016no—
201C4⋊C4—108no—
202C42—108no—
203C2×C5—24no—
204C2×C5#143811yescharacteristic
205C2×C5#178411yescharacteristic
206C2×C5#178411yescharacteristic
207C2×C5#196614yes—
208C23—22no—
209C23—22no—
210C23D4×AGL(1,𝔽5)14yes—
211C23C2×C22⋊AGL(1,𝔽5)12yes—
212C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
213Q8—2016no—
214Q8—2016no—
215C2×C4—28no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—28no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—54no—
223C2×C4—54no—
224C2×C4—58no—
225C2×C4—54no—
226C2×C4—54no—
227C2×C4—52no—
228C2×C4—52no—
229C2×C4—54no—
230C2×C4—108no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—104no—
234C2×C4—108no—
235C2×C4—204no—
236C2×C4—204no—
237C2×C4#400514yescomplemented
238C8—208no—
239C5#1098311yescharacteristic, complemented, Sylow
240C22—28no—
241C22—22no—
242C22—22no—
243C22C2×#400512yes—
244C22#6577914yes—
245C22#6578012yes—
246C22D5⋊#10112yes—
247C22#6580011yescharacteristic
248C4—24no—
249C4—44no—
250C4—54no—
251C4—54no—
252C4—104no—
253C4—204no—
254C2—24no—
255C2#1060433914yes—
256C2#1060490511yescharacteristic
257C2#1060495411yescharacteristic
258C2#1060498611yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutator, prime radicalC24×C5C22×C4
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
cosocleC23×Dic5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#10983
LCSG ▷ C24×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
106. C2×C4×C5#101semidirect
114. C2×Dic5#101semidirect
237. C2×C4#4005semidirect
239. C5#10983semidirect

Maximal subgroups

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

subgroupindex
C22×#40342
C22×#40052
#106048952
#109835