Finite Group Databasebeta

← #13022901

Subgroups — #13022901

Order 1280.

798 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 192 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
1D5×#382C211yesmaximal, characteristic
2C2×#64874C214yesmaximal
3#10604198C211yesmaximal, characteristic
4#10604171C211yesmaximal, characteristic
5C22×C4⋊AGL(1,𝔽5)C2212yes—
6C22×D5⋊C8C2212yes—
7C2×D5×C4⋊C4C2212yes—
8C2×C42×D5C2211yescharacteristic
9C2×(C2×AGL(1,𝔽5))⋊C4—22no—
10C5×#382C411yescharacteristic, complemented
11#65266C412yescomplemented
12#65225C411yescharacteristic, complemented
13D5×#95—22no—
14#64939—22no—
15#64918—22no—
16#64874—28no—
17#11358—51nomaximal, Sylow
18C23×AGL(1,𝔽5)—22no—
19C22×C4×D5—22no—
20C22×C4×D5D412yes—
21C22×C4×D5C2311yescharacteristic
22C22×#170—22no—
23C2×C4⋊AGL(1,𝔽5)—24no—
24C2×C4⋊AGL(1,𝔽5)—24no—
25C2×C4⋊AGL(1,𝔽5)—22no—
26C2×C4⋊AGL(1,𝔽5)D414yes—
27C2×D5⋊C8—24no—
28C2×D5⋊C8—24no—
29C2×D5⋊C8—22no—
30C2×D5⋊C8Q814yes—
31C2×C5×C4⋊C4C2×C412yes—
32C2×C42×C5C2×C411yescharacteristic
33C2×C4⋊Dic5C2×C412yes—
34C2×#683—22no—
35C2×C4×Dic5C2×C411yescharacteristic
36C2×C4×Dic5D412yes—
37D5×C4⋊C4—24no—
38D5×C4⋊C4—24no—
39C42×D5—24no—
40(C2×AGL(1,𝔽5))⋊C4—44no—
41C5×#95—22no—
42#3963—22no—
43#3963—24no—
44C2×#382—51no—
45C2×#340—54no—
46C2×#336—51no—
47C4⋊#95—51no—
48C23×D5C2×D411yescharacteristic
49C22×AGL(1,𝔽5)—24no—
50C22×AGL(1,𝔽5)—42no—
51C22×AGL(1,𝔽5)—42no—
52C22×AGL(1,𝔽5)—44no—
53C22×AGL(1,𝔽5)—44no—
54C22×C4×C5—22no—
55C22×C4×C5C22⋊C412yes—
56C22×C4×C5C22×C411yescharacteristic
57C22×Dic5—22no—
58C22×Dic5C22⋊C412yes—
59C22×Dic5C2×D411yescharacteristic
60C2×C4×D5—24no—
61C2×C4×D5—24no—
62C2×C4×D5—22no—
63C2×C4×D5—22no—
64C2×C4×D5—44no—
65C2×C4×D5—44no—
66C2×C4×D5—42no—
67C2×C4×D5—42no—
68C2×C4×D5C2×D412yes—
69C2×C4×D5C2×Q812yes—
70C2×C4×D5#4114yes—
71C2×C4×D5#4112yes—
72C2×C4×D5#4112yes—
73C2×#170—24no—
74C2×#170—42no—
75C2×#170—42no—
76C4⋊AGL(1,𝔽5)—28no—
77C4⋊AGL(1,𝔽5)—44no—
78D5⋊C8—28no—
79D5⋊C8—44no—
80C5×C4⋊C4—24no—
81C42×C5—22no—
82C4⋊Dic5—24no—
83#683—42no—
84#683—42no—
85C4×Dic5—22no—
86C4×Dic5—24no—
87C23×C8—52no—
88C22×C4⋊C4—52no—
89C22×C4⋊C4—52no—
90C22×C42—51no—
91#382—52no—
92#382—51no—
93#382—102no—
94#382AGL(1,𝔽5)11yescharacteristic, complemented
95C2×#95—102no—
96C2×#95—102no—
97#340—108no—
98#336—102no—
99C23×C5C2×C22⋊C411yescharacteristic
100C22×D5#12112yes—
101C22×D5#12312yes—
102C22×D5#12411yescharacteristic
103C22×D5C4⋊Q811yescharacteristic
104C22×D5#13514yes—
105C22×D5AGL(1,ℤ/8)12yes—
106C22×D5#13712yes—
107C2×AGL(1,𝔽5)—44no—
108C2×AGL(1,𝔽5)—48no—
109C2×AGL(1,𝔽5)—84no—
110C2×AGL(1,𝔽5)—82no—
111C2×AGL(1,𝔽5)—82no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—42no—
116C2×C4×C5—42no—
117C2×C4×C5—42no—
118C2×C4×C5#11712yes—
119C2×C4×C5C4×D412yes—
120C2×C4×C5C4×Q812yes—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—42no—
125C2×Dic5—42no—
126C2×Dic5—42no—
127C2×Dic5C22⋊Q812yes—
128C2×Dic5#12312yes—
129C2×Dic5#12411yescharacteristic
130C2×Dic5#12711yescharacteristic
131C4×D5—44no—
132C4×D5—44no—
133C4×D5—84no—
134C4×D5—84no—
135C4×D5C22⋊Q814yes—
136C4×D5#12314yes—
137C4×D5#12514yes—
138C4×D5#12614yes—
139#170—44no—
140C23×C4—51no—
141C23×C4—52no—
142C23×C4—102no—
143C23×C4—102no—
144C22×C8—54no—
145C22×C8—104no—
146C22×C8—102no—
147C22×C8—104no—
148C22×C8—102no—
149C2×C4⋊C4—54no—
150C2×C4⋊C4—52no—
151C2×C4⋊C4—104no—
152C2×C4⋊C4—104no—
153C2×C4⋊C4—104no—
154C2×C4⋊C4—104no—
155C2×C4⋊C4—102no—
156C2×C4⋊C4—102no—
157C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
158C2×C42—52no—
159C2×C42—51no—
160C2×C42—104no—
161C2×C42C2×AGL(1,𝔽5)11yescharacteristic
162#95—22no—
163#95—104no—
164#95—102no—
165#95—204no—
166C22×C5#38111yescharacteristic
167C22×C5#39011yescharacteristic
168C22×C5#39111yescharacteristic
169C22×C5#41612yes—
170C22×C5#41712yes—
171C2×D5#39712yes—
172C2×D5#39812yes—
173C2×D5#46314yes—
174C2×D5#46414yes—
175C2×D5#48214yes—
176C2×D5#48514yes—
177C2×D5#48812yes—
178C2×D5#49011yescharacteristic
179C2×D5#49111yescharacteristic
180C2×D5#49912yes—
181C2×D5#50112yes—
182AGL(1,𝔽5)—84no—
183C4×C5—42no—
184C4×C5—82no—
185C4×C5#38714yes—
186Dic5—42no—
187Dic5—82no—
188Dic5#39912yes—
189Dic5#40012yes—
190C24—51no—
191C22×C4—22no—
192C22×C4—51no—
193C22×C4—54no—
194C22×C4—52no—
195C22×C4—52no—
196C22×C4—52no—
197C22×C4—52no—
198C22×C4—52no—
199C22×C4—102no—
200C22×C4—104no—
201C22×C4—102no—
202C22×C4—104no—
203C22×C4—104no—
204C22×C4—102no—
205C22×C4—204no—
206C22×C4—202no—
207C22×C4—202no—
208C22×C4—204no—
209C22×C4—204no—
210C22×C4—202no—
211C22×C4—202no—
212C22×C4—204no—
213C22×C4C22⋊AGL(1,𝔽5)12yes—
214C22×C4C22×AGL(1,𝔽5)11yescharacteristic
215C2×C8—108no—
216C2×C8—104no—
217C2×C8—204no—
218C2×C8—202no—
219C2×C8—202no—
220C4⋊C4—24no—
221C4⋊C4—108no—
222C4⋊C4—104no—
223C4⋊C4—202no—
224C4⋊C4—202no—
225C4⋊C4—204no—
226C42—22no—
227C42—104no—
228C42—102no—
229C2×C5#144511yescharacteristic
230C2×C5#182512yes—
231C2×C5#198312yes—
232C2×C5#199111yescharacteristic
233C2×C5#199211yescharacteristic
234D5#207312yes—
235D5#207412yes—
236D5#208514yes—
237C23—52no—
238C23—52no—
239C23—54no—
240C23—52no—
241C23—52no—
242C23—51no—
243C23—51no—
244C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—52no—
252C2×C4—51no—
253C2×C4—51no—
254C2×C4—52no—
255C2×C4—54no—
256C2×C4—54no—
257C2×C4—54no—
258C2×C4—54no—
259C2×C4—102no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4—204no—
263C2×C4—204no—
264C2×C4—204no—
265C2×C4—208no—
266C2×C4—202no—
267C2×C4—202no—
268C2×C4—202no—
269C2×C4—404no—
270C2×C4—404no—
271C2×C4—402no—
272C2×C4—402no—
273C2×C4—404no—
274C2×C4#413012yes—
275C2×C4D4×AGL(1,𝔽5)12yes—
276C2×C4Q8×AGL(1,𝔽5)12yes—
277C8—204no—
278C5#1135811yescharacteristic, complemented, Sylow
279C22—54no—
280C22—54no—
281C22—52no—
282C22—52no—
283C22—51no—
284C22—51no—
285C22—52no—
286C22—52no—
287C22—52no—
288C22—54no—
289C22—54no—
290C22#6576411yescharacteristic
291C22#6578011yescharacteristic
292C22(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
293C22#6579312yes—
294C22#6579412yes—
295C4—42no—
296C4—52no—
297C4—52no—
298C4—82no—
299C4—202no—
300C4—404no—
301C4—402no—
302C4#6571114yes—
303C2—52no—
304C2—52no—
305C2—54no—
306C2#1060434511yescharacteristic
307C2#1060437412yes—
308C2#1060456112yes—
309C2#1060459011yescharacteristic
310C2#1060459111yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC5×#382C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C2×D4
cosocleC22×C4×D5C23
hypercenter#382AGL(1,𝔽5)
hypocenterC5#11358
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
10. C5×#382C4semidirect
11. #65266C4semidirect
12. #65225C4semidirect
94. #382AGL(1,𝔽5)semidirect
278. C5#11358semidirect

Maximal subgroups

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

subgroupindex
D5×#3822
C2×#648742
#106041982
#106041712
#113585