Finite Group Databasebeta

← #13021336

Subgroups — #13021336

Order 1280.

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

normal subgroups only

This group has 526 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×#494C211yesmaximal, characteristic
2#10604674C212yesmaximal
3#10604674C212yesmaximal
4#10603845C211yesmaximal, characteristic
5#10603806C211yesmaximal, characteristic
6C2×D5×Q16—22no—
7D5×C4⋊Q8C2212yes—
8#65688C2212yes—
9C4×D5⋊C8C2211yescharacteristic
10C4×D5⋊C8C2211yescharacteristic
11C5×#494C411yescharacteristic, complemented
12#65470—22no—
13#65381C412yescomplemented
14#64988C411yescharacteristic, complemented
15C4×C8×D5C2211yescharacteristic
16#64893—22no—
17#170⋊C8—22no—
18#64874—24no—
19#64874—24no—
20#9835—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—
27C2×C5×Q16—22no—
28C5×C4⋊Q8C2×C412yes—
29#4090—22no—
30C2×#689—24no—
31D5×Q16—48no—
32C2×Dic20—22no—
33C2×C8×D5D412yes—
34D5×C4⋊C4—22no—
35D5×C4⋊C4—22no—
36#4034—22no—
37#4034—22no—
38C42×D5C2311yescharacteristic
39#4015C2×C412yes—
40C4×#170—21no—
41C4×#170—21no—
42C4×#170—21no—
43C4×#170—21no—
44C4×C5×C8C2×C411yescharacteristic
45C8×Dic5D412yes—
46C4×#168C2×C411yescharacteristic
47C2×#494—51no—
48#1990—52no—
49#1990—52no—
50#1834—51no—
51C2×#322—51no—
52C22×AGL(1,𝔽5)—42no—
53C2×C5×Q8—22no—
54Q8×D5—48no—
55C2×C4×D5—42no—
56C2×C4×D5C2×D411yescharacteristic
57C2×C4×D5C2×D411yescharacteristic
58C2×C4×D5C2×D411yescharacteristic
59C2×Dic10—22no—
60C2×Dic10—42no—
61C2×#170—22no—
62C2×#170—22no—
63C4⋊AGL(1,𝔽5)—44no—
64C4⋊AGL(1,𝔽5)—44no—
65C4⋊AGL(1,𝔽5)—44no—
66D5⋊C8—22no—
67D5⋊C8—24no—
68D5⋊C8Q1614yescomplemented
69C5×Q16—44no—
70C2×C5×C8C22⋊C412yes—
71C5×C4⋊C4—22no—
72C5×C4⋊C4—22no—
73C42×C5C22×C411yescharacteristic
74#689—48no—
75C4⋊Dic5—22no—
76C4⋊Dic5—22no—
77#683—42no—
78#683—42no—
79C4×Dic5C2×D411yescharacteristic
80C4×Dic5#4111yescharacteristic
81C4×Dic5#4111yescharacteristic
82C2×#168C22⋊C412yes—
83Dic20—44no—
84C8×D5—24no—
85C22×Q16—102no—
86C2×C4⋊Q8—52no—
87#494—52no—
88#494—51no—
89#494—102no—
90#494AGL(1,𝔽5)11yescharacteristic, complemented
91C2×C4×C8—51no—
92C2×C4×C8—51no—
93C2×C4×C8—51no—
94#384—52no—
95#340—102no—
96#340—104no—
97#340—104no—
98#322—102no—
99C22×D5#12711yescharacteristic
100C2×AGL(1,𝔽5)—82no—
101C2×AGL(1,𝔽5)—82no—
102C2×AGL(1,𝔽5)—82no—
103C5×Q8—44no—
104C2×C4×C5—42no—
105C2×C4×C5C2×C22⋊C411yescharacteristic
106C2×C4×C5C4×D411yescharacteristic
107C2×C4×C5C4×D411yescharacteristic
108C2×Dic5—42no—
109C2×Dic5#12111yescharacteristic
110C2×Dic5#12111yescharacteristic
111C2×Dic5#12411yescharacteristic
112C4×D5—84no—
113C4×D5C2×Q1612yes—
114C4×D5#13512yes—
115C4×D5AGL(1,ℤ/8)14yes—
116C4×D5#13714yes—
117Dic10—44no—
118Dic10—84no—
119#170—24no—
120#170—42no—
121C5×C8—22no—
122#168—22no—
123C22×Q8—102no—
124C2×Q16—22no—
125C2×Q16—104no—
126C2×Q16—102no—
127C2×Q16—208no—
128C22×C8—52no—
129C22×C8—52no—
130C22×C8—52no—
131C4⋊Q8—52no—
132C4⋊Q8—102no—
133C4⋊Q8—102no—
134C4⋊Q8—102no—
135C4⋊Q8C2×AGL(1,𝔽5)12yes—
136C2×C4⋊C4—102no—
137C2×C4⋊C4—102no—
138C2×C4⋊C4—102no—
139C2×C4⋊C4—102no—
140C2×C4⋊C4—102no—
141C2×C42—51no—
142C4×C8—52no—
143C4×C8—51no—
144C4×C8—101no—
145C4×C8—101no—
146C4×C8—101no—
147C4×C8—101no—
148C4×C8C2×AGL(1,𝔽5)11yescharacteristic
149C22×C5#39011yescharacteristic
150C2×D5#49411yescharacteristic
151C2×D5#49511yescharacteristic
152C2×D5#49612yes—
153C2×D5#49712yes—
154AGL(1,𝔽5)—162no—
155C4×C5—82no—
156C4×C5#41812yes—
157C4×C5C4×Q1612yes—
158C4×C5#44112yes—
159Dic5—82no—
160Dic5#46712yes—
161Dic5#47012yes—
162Dic5#49012yes—
163C2×Q8—22no—
164C2×Q8—102no—
165C2×Q8—202no—
166C2×Q8—208no—
167C22×C4—51no—
168C22×C4—51no—
169C22×C4—51no—
170C22×C4—202no—
171C22×C4—202no—
172Q16—44no—
173Q16—204no—
174Q16—208no—
175C2×C8—54no—
176C2×C8—52no—
177C2×C8—104no—
178C2×C8—102no—
179C2×C8—102no—
180C2×C8—102no—
181C2×C8—104no—
182C2×C8C22⋊AGL(1,𝔽5)12yes—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—102no—
186C4⋊C4—102no—
187C4⋊C4—202no—
188C4⋊C4—202no—
189C4⋊C4—204no—
190C4⋊C4—204no—
191C4⋊C4—204no—
192C42—51no—
193C42—51no—
194C42—51no—
195C42C22×AGL(1,𝔽5)11yescharacteristic
196C2×C5#196111yescharacteristic
197C2×C5#196911yescharacteristic
198C2×C5#197111yescharacteristic
199D5#172012yes—
200D5#172312yes—
201C23—51no—
202Q8—44no—
203Q8—204no—
204Q8—404no—
205C2×C4—42no—
206C2×C4—51no—
207C2×C4—51no—
208C2×C4—51no—
209C2×C4—52no—
210C2×C4—52no—
211C2×C4—54no—
212C2×C4—54no—
213C2×C4—202no—
214C2×C4—404no—
215C2×C4—402no—
216C2×C4—402no—
217C2×C4—402no—
218C2×C4D4×AGL(1,𝔽5)11yescharacteristic
219C2×C4D4×AGL(1,𝔽5)11yescharacteristic
220C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
221C8—22no—
222C8—104no—
223C8—102no—
224C8—202no—
225C5#983511yescharacteristic, complemented, Sylow
226C22—51no—
227C22—51no—
228C22—52no—
229C22—52no—
230C22#6578011yescharacteristic
231C4—52no—
232C4—52no—
233C4—52no—
234C4—82no—
235C4—402no—
236C4—802no—
237C4Q16×AGL(1,𝔽5)12yes—
238C4#6573912yes—
239C4#6576712yes—
240C2—52no—
241C2—52no—
242C2#1060461711yescharacteristic
243C2#1060484811yescharacteristic
244C2#1060484911yescharacteristic
subgroupG/H
centerC22#65780
commutatorC42×C5C22×C4
FrattiniC42C22×AGL(1,𝔽5)
FittingC5×#494C4
socleC22×C5#390
prime radicalC22×D5#127
cosocleC42×D5C23
hypercenter#494AGL(1,𝔽5)
hypocenterC5#9835
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
11. C5×#494C4semidirect
13. #65381C4semidirect
14. #64988C4semidirect
68. D5⋊C8Q16semidirect
90. #494AGL(1,𝔽5)semidirect
225. C5#9835semidirect

Maximal subgroups

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

subgroupindex
D5×#4942
#106046742
#106038452
#106038062
#98355