Finite Group Databasebeta

← #13023111

Subgroups — #13023111

Order 1280.

686 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 112 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#10615775C211yesmaximal, characteristic
2#10604877C212yesmaximal, complemented
3#10598837C214yesmaximal, complemented
4C23×C5×D4C411yescharacteristic
5C2×#4072C411yescharacteristic
6C2×#4072C2211yescharacteristic
7#65748—22no—
8#65536C412yes—
9#65536C2212yes—
10#65510—22no—
11#65510—22no—
12#64933C2214yes—
13#64913—22no—
14#11427—51nomaximal, Sylow
15C25×C5C2×C412yes—
16C22×C5×D4—21no—
17C22×C5×D4—21no—
18C22×C5×D4—22no—
19C22×C5×D4—42no—
20C22×C5×D4—42no—
21C22×C5×D4C814yescomplemented
22C23×C4×C5C2×C411yescharacteristic
23C2×C22⋊Dic5—24no—
24C2×C22⋊Dic5—24no—
25C2×C22⋊Dic5C2×C412yes—
26C2×C22⋊Dic5C2×C411yescharacteristic
27C2×C22⋊Dic5C2311yescharacteristic
28#4072—42no—
29#4072—42no—
30C2×C4⋊Dic5—21no—
31C2×C4⋊Dic5D412yes—
32C2×C4×Dic5—21no—
33C2×C4×Dic5D412yes—
34C22⋊#170—24no—
35C4⋊#170—42no—
36C4×#170—42no—
37#2309—51no—
38#1458—52no—
39#1316—54no—
40C24×C5—22no—
41C24×C5—22no—
42C24×C5—42no—
43C24×C5—42no—
44C24×C5—84no—
45C24×C5C2×C814yes—
46C24×C5C22×C411yescharacteristic
47C2×C5×D4—24no—
48C2×C5×D4—24no—
49C2×C5×D4—42no—
50C2×C5×D4—42no—
51C2×C5×D4—48no—
52C2×C5×D4—48no—
53C2×C5×D4—42no—
54C2×C5×D4—42no—
55C22×C4×C5—22no—
56C22×C4×C5—22no—
57C22×C4×C5C22⋊C411yescharacteristic
58C22×C4×C5C22⋊C411yescharacteristic
59C22×C4×C5C2×C812yes—
60C22×C4×C5M1612yes—
61C22×Dic5—22no—
62C22×Dic5C22⋊C412yes—
63C22×Dic5C2×D412yes—
64C2×#170—44no—
65C22⋊Dic5—41no—
66C22⋊Dic5—41no—
67C22⋊Dic5—42no—
68C22⋊Dic5—84no—
69C22⋊Dic5—84no—
70C4⋊Dic5—42no—
71C4⋊Dic5—42no—
72C4×Dic5—22no—
73C4×Dic5—22no—
74C4×Dic5—42no—
75C23×D4AGL(1,𝔽5)11yescharacteristic
76C2×#117—51no—
77C2×#117—51no—
78#431—102no—
79#390—102no—
80#390—102no—
81#379—52no—
82#379—52no—
83#324—102no—
84#323—54no—
85C23×C5—21no—
86C23×C5—21no—
87C23×C5—24no—
88C23×C5—42no—
89C23×C5—44no—
90C23×C5—44no—
91C23×C5—44no—
92C23×C5—44no—
93C23×C5—42no—
94C23×C5—42no—
95C23×C5—84no—
96C23×C5—84no—
97C23×C5—88no—
98C23×C5—88no—
99C23×C5C2×C22⋊C411yescharacteristic
100C23×C5C22×C811yescharacteristic
101C23×C5C2×M1611yescharacteristic
102C5×D4—48no—
103C5×D4—48no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5—24no—
107C2×C4×C5—24no—
108C2×C4×C5C22⋊C812yes—
109C2×C4×C5C22⋊C812yes—
110C2×C4×C5#9912yes—
111C2×C4×C5#10012yes—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—42no—
115C2×Dic5—42no—
116C2×Dic5—44no—
117C2×Dic5—42no—
118#170—84no—
119C25C2×AGL(1,𝔽5)12yes—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4#17014yescomplemented
126C23×C4C2×AGL(1,𝔽5)11yescharacteristic
127#117—202no—
128#117—202no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—101no—
131C2×C22⋊C4—51no—
132C2×C22⋊C4—51no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—104no—
135C2×C22⋊C4—104no—
136C2×C42—52no—
137C2×C42—101no—
138C4⋊C8—202no—
139C22⋊C8—104no—
140C4×C8—202no—
141C22×C5—22no—
142C22×C5—21no—
143C22×C5—21no—
144C22×C5—41no—
145C22×C5—41no—
146C22×C5—42no—
147C22×C5—44no—
148C22×C5—44no—
149C22×C5—42no—
150C22×C5—44no—
151C22×C5—84no—
152C22×C5—88no—
153C22×C5—88no—
154C22×C5—84no—
155C22×C5C2×C22⋊C811yescharacteristic
156C22×C5C2×#9911yescharacteristic
157C22×C5C2×#10011yescharacteristic
158C4×C5—22no—
159C4×C5#32314yes—
160Dic5—42no—
161Dic5—42no—
162Dic5—82no—
163C24—22no—
164C24—22no—
165C24—42no—
166C24—42no—
167C24—84no—
168C24C2×#17014yes—
169C24C22×AGL(1,𝔽5)11yescharacteristic
170C2×D4—24no—
171C2×D4—24no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—48no—
175C2×D4—48no—
176C2×D4—42no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—52no—
181C22×C4—52no—
182C22×C4—102no—
183C22×C4C2×#17012yes—
184C22×C4#70412yes—
185C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
186C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
187C2×C8—204no—
188C4⋊C4—202no—
189C4⋊C4—202no—
190C22⋊C4—201no—
191C22⋊C4—201no—
192C22⋊C4—202no—
193C22⋊C4—404no—
194C22⋊C4—404no—
195C42—102no—
196C42—102no—
197C42—202no—
198C2×C5—21no—
199C2×C5—21no—
200C2×C5—41no—
201C2×C5—41no—
202C2×C5—84no—
203C2×C5C2×#32311yescharacteristic
204C2×C5#212212yes—
205C23—21no—
206C23—21no—
207C23—24no—
208C23—42no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—42no—
214C23—42no—
215C23—84no—
216C23—84no—
217C23—88no—
218C23—88no—
219C23C22×#17011yescharacteristic
220C23C2×#70411yescharacteristic
221C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
222D4—48no—
223D4—48no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—202no—
231C2×C4—202no—
232C2×C4—204no—
233C2×C4—202no—
234C2×C4#401112yes—
235C2×C4C22⋊#17012yes—
236C2×C4C22⋊#17012yes—
237C2×C4#401312yes—
238C8—404no—
239C5#1142711yescharacteristic, complemented, Sylow
240C22—22no—
241C22—21no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—44no—
247C22—44no—
248C22—42no—
249C22—44no—
250C22—84no—
251C22—88no—
252C22—88no—
253C22—84no—
254C22C2×#401111yescharacteristic
255C22C2×C22⋊#17011yescharacteristic
256C22C2×#401311yescharacteristic
257C4—22no—
258C4—202no—
259C4—202no—
260C4—402no—
261C4#6493314yes—
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2—84no—
267C2#1060494012yes—
268C2C2×#6493311yescharacteristic
subgroupG/H
centerC22C2×#4011
commutatorC23×C5C22×C8
FrattiniC24C22×AGL(1,𝔽5)
Fitting, prime radicalC23×C5×D4C4
socleC22×C5C2×#99
cosocleC2×C22⋊Dic5C23
hypercenterC23×D4AGL(1,𝔽5)
hypocenterC5#11427
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
2. #10604877C2semidirect
3. #10598837C2semidirect
21. C22×C5×D4C8semidirect
125. C22×D4#170semidirect
239. C5#11427semidirect

Maximal subgroups

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

subgroupindex
#106157752
#106048772
#105988372
#114275