Finite Group Databasebeta

← #11994355

Subgroups — #11994355

Order 1280.

616 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 135 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×C5×#398C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic
3#10602346C212yesmaximal
4C2×#64748C211yesmaximal, characteristic
5#10598103C212yesmaximal
6C22×C5×C4⋊C4—21no—
7C22×C5×C4⋊C4C2211yescharacteristic
8C22×C5×C4⋊C4C2211yescharacteristic
9C5×#398—22no—
10C5×#398C414yescomplemented
11C2×C5×#95—21no—
12C2×C5×#95C2211yescharacteristic
13#65501C2212yes—
14#65500—21no—
15C4×C22⋊Dic5—21no—
16C2×C22⋊#168C2212yes—
17#65270—24no—
18#64749—24no—
19#64748—22no—
20#9026—51nomaximal, Sylow
21C23×C4×C5—21no—
22C23×C4×C5—21no—
23C23×C4×C5D411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5C2311yescharacteristic
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—24no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—44no—
33C2×C5×C4⋊C4C2×C412yes—
34C2×C5×C4⋊C4C2×C412yes—
35C2×C5×C4⋊C4D412yes—
36C2×C22⋊Dic5—22no—
37C2×C4⋊Dic5D414yes—
38C2×C4×Dic5—22no—
39C22×#168—22no—
40C5×#95—24no—
41C5×#95—21no—
42C5×#95C2×C412yes—
43#3963—22no—
44#3963—24no—
45#3963—22no—
46C22⋊#168—22no—
47C22⋊#168—22no—
48C2×#398D511yescharacteristic
49#2313—51no—
50#1823—52no—
51C2×#324—51no—
52#1294—52no—
53C24×C5C2×D411yescharacteristic
54C22×C4×C5—24no—
55C22×C4×C5—22no—
56C22×C4×C5—21no—
57C22×C4×C5—21no—
58C22×C4×C5—21no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—21no—
62C22×C4×C5—22no—
63C22×C4×C5—42no—
64C22×C4×C5—42no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—41no—
68C22×C4×C5—41no—
69C22×C4×C5—42no—
70C22×C4×C5—42no—
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22⋊C412yes—
73C22×C4×C5C22×C411yescharacteristic
74C22×C4×C5C2×D412yes—
75C22×C4×C5C2×D411yescharacteristic
76C22×C4×C5#4112yes—
77C22×Dic5—24no—
78C5×C4⋊C4—24no—
79C5×C4⋊C4—24no—
80C5×C4⋊C4—24no—
81C5×C4⋊C4—48no—
82C5×C4⋊C4—42no—
83C22⋊Dic5—42no—
84C22⋊Dic5—42no—
85C4⋊Dic5—44no—
86C4×Dic5—82no—
87C2×#168—44no—
88C2×#168—42no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2×D511yescharacteristic
91C22×C4⋊C4C2×D511yescharacteristic
92C2×C22⋊C8—52no—
93#398—22no—
94#398Dic514yescomplemented
95#387—104no—
96#381—101no—
97#380—52no—
98C4×C22⋊C4—101no—
99C2×#95—21no—
100C2×#95C2×D511yescharacteristic
101#340—104no—
102#324—102no—
103C23×C5—22no—
104C23×C5—22no—
105C23×C5C2×C22⋊C411yescharacteristic
106C23×C5#12011yescharacteristic
107C23×C5#12311yescharacteristic
108C23×C5#13512yes—
109C23×C5AGL(1,ℤ/8)12yes—
110C2×C4×C5—24no—
111C2×C4×C5—24no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—42no—
119C2×C4×C5—48no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—81no—
130C2×C4×C5—82no—
131C2×C4×C5—82no—
132C2×C4×C5C4×D412yes—
133C2×C4×C5#12112yes—
134C2×Dic5—44no—
135C2×Dic5—44no—
136C2×Dic5—44no—
137#168—82no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊D511yescharacteristic
141C23×C4C22⋊D511yescharacteristic
142C23×C4C22×D511yescharacteristic
143C22×C8—102no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—54no—
152C2×C4⋊C4C2×Dic512yes—
153C2×C4⋊C4C2×Dic512yes—
154C2×C4⋊C4C22⋊D512yes—
155C2×C22⋊C4—102no—
156C2×C42—102no—
157C22⋊C8—102no—
158C22⋊C8—102no—
159#95—24no—
160#95—21no—
161#95—102no—
162#95—104no—
163#95—102no—
164#95C2×Dic512yes—
165C22×C5—24no—
166C22×C5—24no—
167C22×C5—21no—
168C22×C5—21no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5#38611yescharacteristic
172C22×C5#41611yescharacteristic
173C22×C5#41811yescharacteristic
174C22×C5#44912yes—
175C22×C5#45612yes—
176C22×C5#45812yes—
177C22×C5#48212yes—
178C4×C5—22no—
179C4×C5—22no—
180C4×C5—44no—
181C4×C5—42no—
182C4×C5—42no—
183C4×C5—82no—
184Dic5—84no—
185C24C2×C22⋊D511yescharacteristic
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—104no—
204C22×C4C22⋊Dic512yes—
205C22×C4C22⋊Dic512yes—
206C22×C4D4×D512yes—
207C22×C4#71112yes—
208C22×C4C22×Dic511yescharacteristic
209C22×C4C2×C22⋊D511yescharacteristic
210C2×C8—204no—
211C2×C8—202no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—48no—
216C4⋊C4—42no—
217C4⋊C4—204no—
218C22⋊C4—202no—
219C22⋊C4—202no—
220C42—402no—
221C2×C5—21no—
222C2×C5—21no—
223C2×C5—22no—
224C2×C5#160914yes—
225C2×C5#187511yescharacteristic
226C2×C5#188611yescharacteristic
227C2×C5#189011yescharacteristic
228C23—22no—
229C23—22no—
230C23#407812yes—
231C23#408111yescharacteristic
232C23#408311yescharacteristic
233C23#409612yes—
234C23C2×C22⋊Dic511yescharacteristic
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—204no—
258C2×C4—204no—
259C2×C4—204no—
260C2×C4D4×Dic512yes—
261C2×C4#408512yes—
262C8—402no—
263C5#902611yescharacteristic, complemented, Sylow
264C22—21no—
265C22—21no—
266C22—24no—
267C22—24no—
268C22—21no—
269C22—21no—
270C22#6532412yes—
271C22#6532612yes—
272C22#6534212yes—
273C22#6538412yes—
274C22#6550411yescharacteristic
275C22#6550911yescharacteristic
276C22#6552211yescharacteristic
277C4—22no—
278C4—22no—
279C4—44no—
280C4—42no—
281C4—42no—
282C4—82no—
283C4—404no—
284C2—21no—
285C2—22no—
286C2—21no—
287C2#1060050414yes—
288C2#1060247111yescharacteristic
289C2#1060249711yescharacteristic
290C2#1060257711yescharacteristic
subgroupG/H
centerC23#4083
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC2×C5×#398C2
socleC23×C5#120
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenterC2×#398D5
hypocenterC5#9026
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
10. C5×#398C4semidirect
94. #398Dic5semidirect
263. C5#9026semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3982
#106138862
#106023462
C2×#647482
#105981032
#90265