Finite Group Databasebeta

← #13021332

Subgroups — #13021332

Order 1280.

494 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 116 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×#500C211yesmaximal, characteristic
2#10604674C212yesmaximal
3#10604672C212yesmaximal
4#10603845C211yesmaximal, characteristic
5#10603806C211yesmaximal, characteristic
6D5×C4⋊Q8C2212yes—
7C2×#4009—24no—
8#65688C2212yes—
9C4×D5⋊C8C2211yescharacteristic
10C4×D5⋊C8C2211yescharacteristic
11C5×#500C411yescharacteristic, complemented
12#65367C412yescomplemented
13D5×#107—22no—
14#65163—22no—
15C4×C8×D5C2211yescharacteristic
16#64971C411yescharacteristic, complemented
17#64893—22no—
18#170⋊C8—22no—
19#64874—24no—
20#9831—51nomaximal, Sylow
21C2×Q8×D5—22no—
22C2×C4⋊AGL(1,𝔽5)—22no—
23C2×C4⋊AGL(1,𝔽5)—22no—
24C2×C4⋊AGL(1,𝔽5)—22no—
25C2×D5⋊C8D412yes—
26C2×D5⋊C8D412yes—
27C5×C4⋊Q8C2×C412yes—
28#4090—22no—
29C2×C8×D5D412yes—
30D5×C4⋊C4—22no—
31D5×C4⋊C4—22no—
32#4034—22no—
33#4034—22no—
34C42×D5C2311yescharacteristic
35#4015C2×C412yes—
36#4009—48no—
37C4×#170—21no—
38C4×#170—21no—
39C4×#170—21no—
40C4×#170—21no—
41C5×#107—22no—
42C4×C5×C8C2×C411yescharacteristic
43#3950—22no—
44C8×Dic5D412yes—
45#3939—24no—
46C4×#168C2×C411yescharacteristic
47C2×#500—51no—
48#1990—52no—
49#1956—52no—
50#1834—51no—
51C2×#322—51no—
52C22×AGL(1,𝔽5)—42no—
53C2×C5×Q8—22no—
54Q8×D5—44no—
55Q8×D5—44no—
56C2×C4×D5—42no—
57C2×C4×D5C2×D411yescharacteristic
58C2×C4×D5C2×D411yescharacteristic
59C2×C4×D5C2×D411yescharacteristic
60C2×Dic10—22no—
61C2×Dic10—42no—
62C2×#170—22no—
63C2×#170—22no—
64C4⋊AGL(1,𝔽5)—44no—
65C4⋊AGL(1,𝔽5)—44no—
66C4⋊AGL(1,𝔽5)—44no—
67D5⋊C8—22no—
68D5⋊C8—24no—
69D5⋊C8D814yes—
70C2×C5×C8C22⋊C412yes—
71C5×C4⋊C4—22no—
72C5×C4⋊C4—22no—
73C42×C5C22×C411yescharacteristic
74C4⋊Dic5—22no—
75C4⋊Dic5—22no—
76#683—42no—
77#683—42no—
78C4×Dic5C2×D411yescharacteristic
79C4×Dic5#4111yescharacteristic
80C4×Dic5#4111yescharacteristic
81C2×#168C22⋊C412yes—
82C8×D5—24no—
83C2×C4⋊Q8—52no—
84#500—52no—
85#500—51no—
86#500—102no—
87#500AGL(1,𝔽5)11yescharacteristic, complemented
88C2×#107—102no—
89C2×#103—104no—
90C2×C4×C8—51no—
91C2×C4×C8—51no—
92C2×C4×C8—51no—
93#384—52no—
94#340—102no—
95#340—104no—
96#322—102no—
97C22×D5#12711yescharacteristic
98C2×AGL(1,𝔽5)—82no—
99C2×AGL(1,𝔽5)—82no—
100C2×AGL(1,𝔽5)—82no—
101C5×Q8—44no—
102C2×C4×C5—42no—
103C2×C4×C5C2×C22⋊C411yescharacteristic
104C2×C4×C5C4×D411yescharacteristic
105C2×C4×C5C4×D411yescharacteristic
106C2×Dic5—42no—
107C2×Dic5#12111yescharacteristic
108C2×Dic5#12111yescharacteristic
109C2×Dic5#12411yescharacteristic
110C4×D5—82no—
111C4×D5—82no—
112C4×D5C2×D812yes—
113C4×D5#13512yes—
114C4×D5#13714yes—
115C4×D5#13714yes—
116Dic10—44no—
117Dic10—84no—
118#170—24no—
119#170—42no—
120C5×C8—22no—
121#168—22no—
122C22×Q8—102no—
123C22×C8—52no—
124C22×C8—52no—
125C22×C8—52no—
126C4⋊Q8—52no—
127C4⋊Q8—102no—
128C4⋊Q8—102no—
129C4⋊Q8—102no—
130C4⋊Q8C2×AGL(1,𝔽5)12yes—
131C2×C4⋊C4—102no—
132C2×C4⋊C4—102no—
133C2×C4⋊C4—102no—
134C2×C4⋊C4—102no—
135C2×C4⋊C4—102no—
136C2×C42—51no—
137#107—22no—
138#107—104no—
139#107—102no—
140#103—208no—
141C4×C8—52no—
142C4×C8—51no—
143C4×C8—101no—
144C4×C8—101no—
145C4×C8—101no—
146C4×C8—101no—
147C4×C8C2×AGL(1,𝔽5)11yescharacteristic
148C22×C5#39011yescharacteristic
149C2×D5#49311yescharacteristic
150C2×D5#49511yescharacteristic
151C2×D5#49712yes—
152C2×D5#49712yes—
153AGL(1,𝔽5)—162no—
154C4×C5—82no—
155C4×C5#41912yes—
156C4×C5C4×D812yes—
157C4×C5#44112yes—
158Dic5—82no—
159Dic5#46612yes—
160Dic5#47012yes—
161Dic5#49112yes—
162C2×Q8—22no—
163C2×Q8—102no—
164C2×Q8—202no—
165C2×Q8—204no—
166C2×Q8—204no—
167C22×C4—51no—
168C22×C4—51no—
169C22×C4—51no—
170C22×C4—202no—
171C22×C4—202no—
172C2×C8—54no—
173C2×C8—52no—
174C2×C8—104no—
175C2×C8—102no—
176C2×C8—102no—
177C2×C8—102no—
178C2×C8—104no—
179C2×C8C22⋊AGL(1,𝔽5)12yes—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—202no—
185C4⋊C4—202no—
186C4⋊C4—204no—
187C4⋊C4—204no—
188C4⋊C4—204no—
189C42—51no—
190C42—51no—
191C42—51no—
192C42C22×AGL(1,𝔽5)11yescharacteristic
193C2×C5#196011yescharacteristic
194C2×C5#197011yescharacteristic
195C2×C5#197111yescharacteristic
196D5#171812yes—
197D5#172212yes—
198C23—51no—
199Q8—44no—
200Q8—204no—
201Q8—404no—
202C2×C4—42no—
203C2×C4—51no—
204C2×C4—51no—
205C2×C4—51no—
206C2×C4—52no—
207C2×C4—52no—
208C2×C4—54no—
209C2×C4—54no—
210C2×C4—202no—
211C2×C4—402no—
212C2×C4—402no—
213C2×C4—402no—
214C2×C4—402no—
215C2×C4—402no—
216C2×C4D4×AGL(1,𝔽5)11yescharacteristic
217C2×C4D4×AGL(1,𝔽5)11yescharacteristic
218C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
219C8—22no—
220C8—104no—
221C8—102no—
222C8—202no—
223C5#983111yescharacteristic, complemented, Sylow
224C22—51no—
225C22—51no—
226C22—52no—
227C22—52no—
228C22#6578011yescharacteristic
229C4—52no—
230C4—52no—
231C4—52no—
232C4—82no—
233C4—402no—
234C4—802no—
235C4D8×AGL(1,𝔽5)12yes—
236C4#6573912yes—
237C4#6578212yes—
238C2—52no—
239C2—52no—
240C2#1060467311yescharacteristic
241C2#1060480211yescharacteristic
242C2#1060484911yescharacteristic
subgroupG/H
centerC22#65780
commutatorC42×C5C22×C4
FrattiniC42C22×AGL(1,𝔽5)
FittingC5×#500C4
socleC22×C5#390
prime radicalC22×D5#127
cosocleC42×D5C23
hypercenter#500AGL(1,𝔽5)
hypocenterC5#9831
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
11. C5×#500C4semidirect
12. #65367C4semidirect
16. #64971C4semidirect
87. #500AGL(1,𝔽5)semidirect
223. C5#9831semidirect

Maximal subgroups

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

subgroupindex
D5×#5002
#106046742
#106046722
#106038452
#106038062
#98315