Finite Group Databasebeta

← #13017544

Subgroups — #13017544

Order 1280.

802 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C2×#65851C211yesmaximal, characteristic
2#10604900C212yesmaximal
3C2×#64873C214yesmaximal, complemented
4C2×C5×#117C411yescharacteristic
5C2×#4072C411yescharacteristic
6C22×C4×Dic5C2211yescharacteristic
7#65851—28no—
8C2×#4036C412yes—
9C2×#4034C2212yes—
10C2×#4024—22no—
11C2×C4×Dic10—22no—
12C2×C22⋊#170—22no—
13C2×#4004C2214yes—
14#64918—22no—
15#64873—28no—
16#8345—51nomaximal, Sylow
17C23×C4×C5C2×C411yescharacteristic
18C23×Dic5D412yes—
19C22×Dic10—22no—
20C22×#170—24no—
21C5×#117—24no—
22C2×C5×C4⋊C4C2×C412yes—
23C2×C5×C22⋊C4—21no—
24C2×C42×C5—21no—
25C2×C22⋊Dic5—21no—
26#4072—24no—
27C2×C4⋊Dic5—22no—
28C2×C4⋊Dic5C2×C412yes—
29C2×#683—22no—
30C2×#683—22no—
31C2×C4×Dic5—21no—
32C2×C4×Dic5—24no—
33C2×C4×Dic5C2×C411yescharacteristic
34C2×C4×Dic5D412yes—
35C2×C4×Dic5C2311yescharacteristic
36#4036—24no—
37#4036—24no—
38#4034—24no—
39#4034D418yes—
40#4024—216no—
41C4×Dic10—216no—
42C22⋊#170—44no—
43#4004—28no—
44C2×#533—51no—
45#1797—52no—
46C2×#332—54no—
47C24×C5C22⋊C411yescharacteristic
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5—22no—
52C22×C4×C5C22⋊C411yescharacteristic
53C22×C4×C5C22×C411yescharacteristic
54C22×Dic5—22no—
55C22×Dic5—24no—
56C22×Dic5—24no—
57C22×Dic5—22no—
58C22×Dic5—22no—
59C22×Dic5C22⋊C412yes—
60C22×Dic5C2×D412yes—
61C2×Dic10—28no—
62C2×Dic10—44no—
63C2×Dic10—42no—
64C2×Dic10—42no—
65C2×#170—44no—
66C2×#170—44no—
67C2×#170—44no—
68C5×C4⋊C4—22no—
69C5×C4⋊C4C22⋊C414yes—
70C5×C22⋊C4—24no—
71C42×C5—24no—
72C22⋊Dic5—24no—
73C4⋊Dic5—28no—
74C4⋊Dic5—22no—
75C4⋊Dic5C22⋊C414yes—
76#683—28no—
77#683—28no—
78C4×Dic5—24no—
79C4×Dic5—22no—
80C4×Dic5—24no—
81C4×Dic5C2×D414yes—
82#533—108no—
83C2×C4⋊Q8—52no—
84C2×#125—52no—
85C2×C22⋊Q8—102no—
86C2×C4×Q8—102no—
87C2×#117—51no—
88C2×#117AGL(1,𝔽5)11yescharacteristic
89C22×C42—51no—
90C2×C4⋊C8—54no—
91C2×C22⋊C8—102no—
92#336—102no—
93#332—108no—
94C23×C5—21no—
95C23×C5—21no—
96C23×C5—22no—
97C23×C5—22no—
98C23×C5#10112yes—
99C23×C5C2×C22⋊C411yescharacteristic
100C2×C4×C5—28no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—24no—
106C2×C4×C5—42no—
107C2×C4×C5—42no—
108C2×C4×C5#10112yes—
109C2×C4×C5C2×C22⋊C412yes—
110C2×Dic5—22no—
111C2×Dic5—24no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—42no—
115C2×Dic5—42no—
116C2×Dic5—44no—
117C2×Dic5—44no—
118C2×Dic5#12014yes—
119C2×Dic5#13514yes—
120C2×Dic5AGL(1,ℤ/8)12yes—
121C2×Dic5#13712yes—
122Dic10—48no—
123Dic10—48no—
124#170—84no—
125C22×Q8—102no—
126C23×C4—52no—
127C23×C4C2×AGL(1,𝔽5)11yescharacteristic
128C22×C8—104no—
129C4⋊Q8—58no—
130C4⋊Q8—104no—
131#125—104no—
132#125—104no—
133C22⋊Q8—1016no—
134C4×Q8—1016no—
135#117—24no—
136#117—104no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4—102no—
139C2×C4⋊C4—102no—
140C2×C4⋊C4—102no—
141C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—101no—
144C2×C42—21no—
145C2×C42—51no—
146C2×C42—52no—
147C2×C42—51no—
148C2×C42—104no—
149C2×C42—101no—
150C4⋊C8—108no—
151C22⋊C8—204no—
152C22×C5—24no—
153C22×C5—22no—
154C22×C5—41no—
155C22×C5—41no—
156C22×C5—42no—
157C22×C5#37911yescharacteristic
158C22×C5C2×#10111yescharacteristic
159C22×C5C2×#10111yescharacteristic
160C22×C5#41612yes—
161C22×C5#41712yes—
162C4×C5—22no—
163C4×C5—22no—
164C4×C5—44no—
165Dic5—44no—
166Dic5—42no—
167Dic5#44914yes—
168Dic5#45214yes—
169C24C22⋊AGL(1,𝔽5)11yescharacteristic
170C2×Q8—108no—
171C2×Q8—202no—
172C2×Q8—202no—
173C2×Q8—204no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—52no—
179C22×C4—52no—
180C22×C4—102no—
181C22×C4—104no—
182C22×C4—102no—
183C22×C4—104no—
184C22×C4—102no—
185C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
186C22×C4C22×AGL(1,𝔽5)11yescharacteristic
187C2×C8—204no—
188C2×C8—204no—
189C2×C8—204no—
190C4⋊C4—22no—
191C4⋊C4—54no—
192C4⋊C4—108no—
193C4⋊C4—108no—
194C4⋊C4—108no—
195C4⋊C4—102no—
196C4⋊C4C22⋊AGL(1,𝔽5)14yes—
197C22⋊C4—24no—
198C22⋊C4—104no—
199C42—24no—
200C42—54no—
201C42—104no—
202C42—104no—
203C42—102no—
204C2×C5—42no—
205C2×C5#154411yescharacteristic
206C2×C5#155212yes—
207C2×C5#156111yescharacteristic
208C2×C5#178411yescharacteristic
209C2×C5#178812yes—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23#400512yes—
215C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
216Q8—208no—
217Q8—208no—
218C2×C4—28no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—52no—
227C2×C4—52no—
228C2×C4—54no—
229C2×C4—54no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—104no—
234C2×C4—204no—
235C2×C4—204no—
236C2×C4—202no—
237C2×C4—202no—
238C2×C4#400512yes—
239C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
240C8—404no—
241C5#834511yescharacteristic, complemented, Sylow
242C22—24no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22C2×#400511yescharacteristic
248C22C2×#400511yescharacteristic
249C22#6579312yes—
250C22#6579412yes—
251C22#6580011yescharacteristic
252C4—22no—
253C4—22no—
254C4—44no—
255C4—54no—
256C4—54no—
257C4—202no—
258C4—204no—
259C2—42no—
260C2#1060436311yescharacteristic
261C2#1060436412yes—
262C2#1060436511yescharacteristic
263C2#1060490511yescharacteristic
264C2#1060501112yes—
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5#8345
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. C2×#64873C2semidirect
241. C5#8345semidirect

Maximal subgroups

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

subgroupindex
C2×#658512
#106049002
C2×#648732
#83455