Finite Group Databasebeta

← #13021055

Subgroups — #13021055

Order 1280.

500 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 118 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#10606086C211yesmaximal, characteristic
2C2×#64870C214yesmaximal
3#10604181C211yesmaximal, characteristic
4C4×#4002C211yesmaximal, characteristic
5C2×#4036C2212yes—
6C2×#4002—21no—
7C2×#4002C2212yes—
8C2×#4002C2212yes—
9C2×C4⋊#170—21no—
10C2×C4×#170—21no—
11C5×#384C411yescharacteristic
12#65274—21no—
13#65274—21no—
14#65266C412yes—
15#65262—21no—
16#65262—21no—
17#65224C411yescharacteristic
18C42×Dic5C2211yescharacteristic
19#64918—21no—
20#64870—28no—
21#9686—51nomaximal, Sylow
22C22×#170—22no—
23C22×#170—22no—
24C2×C5×C4⋊C4—21no—
25C2×C5×C4⋊C4—21no—
26C2×C5×C4⋊C4C2×C412yes—
27C2×C42×C5C2×C411yescharacteristic
28C2×C4⋊Dic5—21no—
29C2×C4⋊Dic5—21no—
30C2×C4⋊Dic5—22no—
31C2×C4⋊Dic5C2×C412yes—
32C2×#683—22no—
33C2×C4×Dic5C2×C411yescharacteristic
34C2×C4×Dic5D411yescharacteristic
35C2×C4×Dic5D411yescharacteristic
36C2×C4×Dic5D411yescharacteristic
37C2×C4×Dic5D411yescharacteristic
38C2×C4×Dic5C2311yescharacteristic
39#4036—24no—
40#4036—24no—
41#4002—24no—
42#4002—24no—
43#4002—22no—
44#4002D414yes—
45C4⋊#170—24no—
46C4×#170—24no—
47#3963—22no—
48#3963—22no—
49#2859—51no—
50#1767—51no—
51C4×#97—51no—
52C2×#330—54no—
53C22×C4×C5—22no—
54C22×C4×C5C22⋊C411yescharacteristic
55C22×C4×C5C22⋊C411yescharacteristic
56C22×C4×C5C22×C411yescharacteristic
57C22×Dic5—22no—
58C22×Dic5C22⋊C411yescharacteristic
59C22×Dic5C22⋊C411yescharacteristic
60C22×Dic5C2×D411yescharacteristic
61C22×Dic5C2×D411yescharacteristic
62C2×#170—24no—
63C2×#170—24no—
64C2×#170—24no—
65C2×#170—24no—
66C2×#170—42no—
67C2×#170—42no—
68C5×C4⋊C4—24no—
69C5×C4⋊C4—42no—
70C5×C4⋊C4—42no—
71C42×C5—22no—
72C4⋊Dic5—24no—
73C4⋊Dic5—42no—
74C4⋊Dic5—42no—
75C4⋊Dic5—42no—
76C4⋊Dic5—42no—
77#683—44no—
78C4×Dic5—22no—
79C4×Dic5—22no—
80C4×Dic5—22no—
81C4×Dic5—22no—
82C4×Dic5—41no—
83C4×Dic5C2×D412yes—
84C4×Dic5#4112yes—
85C2×#125—52no—
86C2×C4⋊C8—101no—
87C2×#97—52no—
88C2×#97—52no—
89C2×#97—101no—
90C2×C4×C8—101no—
91#389—101no—
92#389—101no—
93#389—101no—
94#389—101no—
95#384—51no—
96#384AGL(1,𝔽5)11yescharacteristic
97#382—52no—
98C43—51no—
99#336—101no—
100#330—108no—
101C23×C5C2×C22⋊C411yescharacteristic
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5—42no—
107C2×C4×C5—42no—
108C2×C4×C5—42no—
109C2×C4×C5#10112yes—
110C2×C4×C5#10412yes—
111C2×C4×C5#10412yes—
112C2×C4×C5#10412yes—
113C2×C4×C5C4×D412yes—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—21no—
117C2×Dic5—21no—
118C2×Dic5—21no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5—21no—
122C2×Dic5—21no—
123C2×Dic5—42no—
124C2×Dic5—42no—
125C2×Dic5—42no—
126C2×Dic5#10112yes—
127C2×Dic5#12112yes—
128C2×Dic5#12411yescharacteristic
129C2×Dic5#12711yescharacteristic
130#170—28no—
131#170—44no—
132C22×C8—102no—
133C22×C8—102no—
134#125—104no—
135#125—104no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—102no—
140C2×C4⋊C4—102no—
141C2×C4⋊C4—101no—
142C2×C4⋊C4—101no—
143C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
144C2×C42—51no—
145C2×C42—51no—
146C2×C42—51no—
147C2×C42—51no—
148C2×C42—51no—
149C2×C42—51no—
150C2×C42C2×AGL(1,𝔽5)11yescharacteristic
151C4⋊C8—104no—
152#97—54no—
153#97—102no—
154#97—104no—
155#97—104no—
156C4×C8—104no—
157#95—102no—
158#95—102no—
159C22×C5#39011yescharacteristic
160C22×C5C2×#10111yescharacteristic
161C22×C5C2×#10111yescharacteristic
162C22×C5C2×#10411yescharacteristic
163C22×C5C2×#10411yescharacteristic
164C22×C5C2×#10411yescharacteristic
165C22×C5#42111yescharacteristic
166C4×C5—22no—
167C4×C5—22no—
168C4×C5—82no—
169C4×C5#33014yescomplemented
170Dic5—22no—
171Dic5—21no—
172Dic5—21no—
173Dic5—22no—
174Dic5—41no—
175Dic5—82no—
176C22×C4—22no—
177C22×C4—51no—
178C22×C4—51no—
179C22×C4—51no—
180C22×C4—51no—
181C22×C4—102no—
182C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
183C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
184C22×C4C22×AGL(1,𝔽5)11yescharacteristic
185C2×C8—104no—
186C2×C8—104no—
187C2×C8—104no—
188C2×C8—104no—
189C2×C8—202no—
190C2×C8—202no—
191C4⋊C4—24no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—104no—
195C4⋊C4—202no—
196C4⋊C4—204no—
197C4⋊C4—202no—
198C4⋊C4—202no—
199C4⋊C4—202no—
200C42—22no—
201C42—52no—
202C42—52no—
203C42—102no—
204C42—102no—
205C42—102no—
206C42—102no—
207C42—201no—
208C2×C5C2×#33011yescharacteristic
209C2×C5#152611yescharacteristic
210C2×C5#152811yescharacteristic
211C2×C5#153111yescharacteristic
212C2×C5#196211yescharacteristic
213C2×C5#196611yescharacteristic
214C2×C5#198011yescharacteristic
215C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—52no—
224C2×C4—51no—
225C2×C4—51no—
226C2×C4—52no—
227C2×C4—101no—
228C2×C4—101no—
229C2×C4—101no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—101no—
233C2×C4—101no—
234C2×C4—101no—
235C2×C4—101no—
236C2×C4—202no—
237C2×C4—202no—
238C2×C4—202no—
239C2×C4#400512yes—
240C2×C4#400812yes—
241C2×C4#400812yes—
242C2×C4#401012yes—
243C2×C4D4×AGL(1,𝔽5)12yes—
244C8—108no—
245C8—204no—
246C5#968611yescharacteristic, complemented, Sylow
247C22C2×#400511yescharacteristic
248C22C2×#400811yescharacteristic
249C22C2×#400811yescharacteristic
250C22#6578011yescharacteristic
251C22C2×#401011yescharacteristic
252C22#6578411yescharacteristic
253C22D5⋊#10111yescharacteristic
254C4—22no—
255C4—22no—
256C4—82no—
257C4—102no—
258C4—102no—
259C4—101no—
260C4—101no—
261C4—201no—
262C4—402no—
263C4#6487014yescomplemented
264C2C2×#6487011yescharacteristic
265C2#1060426511yescharacteristic
266C2#1060433911yescharacteristic
267C2#1060461111yescharacteristic
268C2#1060463411yescharacteristic
269C2#1060463611yescharacteristic
270C2#1060467811yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#384C4
socle, prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenter#384AGL(1,𝔽5)
hypocenterC5#9686
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
169. C4×C5#330semidirect
246. C5#9686semidirect
263. C4#64870semidirect

Maximal subgroups

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

subgroupindex
#106060862
C2×#648702
#106041812
C4×#40022
#96865