Finite Group Databasebeta

← #10604374

Subgroups — #10604374

Order 640.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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×#117C211yesmaximal, characteristic
2#65765C211yesmaximal, characteristic
3D5⋊(C22⋊C8)C211yesmaximal, characteristic
4#64874C212yesmaximal, complemented
5#64874C212yesmaximal, complemented
6C22×C4×D5C2211yescharacteristic
7C2×C22⋊AGL(1,𝔽5)—21no—
8C2×C4⋊AGL(1,𝔽5)C2212yes—
9C2×D5⋊C8C2212yes—
10C5×#117C411yescharacteristic, complemented
11#4072C411yescharacteristic, complemented
12#4038C411yescharacteristic, complemented
13#4038C411yescharacteristic, complemented
14D5×C4⋊C4C2211yescharacteristic
15D5×C4⋊C4C2211yescharacteristic
16D5×C22⋊C4—21no—
17#4023—21no—
18#4018—21no—
19C42×D5—21no—
20C22⋊#170—21no—
21(C2×AGL(1,𝔽5))⋊C4—21no—
22#4003—21no—
23#1825—51nomaximal, Sylow
24C23×D5D411yescharacteristic
25C22×AGL(1,𝔽5)—22no—
26C22×C4×C5C2×C411yescharacteristic
27C22×Dic5D411yescharacteristic
28C2×C4×D5—24no—
29C2×C4×D5—21no—
30C2×C4×D5—21no—
31C2×C4×D5C2×C411yescharacteristic
32C2×C4×D5D411yescharacteristic
33C2×C4×D5D411yescharacteristic
34C2×C4×D5C2311yescharacteristic
35C22⋊AGL(1,𝔽5)—42no—
36C2×#170—22no—
37C4⋊AGL(1,𝔽5)—24no—
38D5⋊C8—24no—
39C5×C4⋊C4C2×C411yescharacteristic
40C5×C4⋊C4C2×C411yescharacteristic
41C5×C22⋊C4—21no—
42C42×C5—21no—
43C22⋊Dic5—21no—
44#685—21no—
45#685—21no—
46C4⋊Dic5C2×C411yescharacteristic
47C4⋊Dic5C2×C411yescharacteristic
48#683—21no—
49#683—21no—
50C4×Dic5—21no—
51C4×Dic5—21no—
52C4×Dic5—21no—
53C2×#117—51no—
54C2×C22⋊C8—51no—
55#380—51no—
56#340—52no—
57#340—52no—
58C23×C5C22⋊C411yescharacteristic
59C22×D5—22no—
60C22×D5—22no—
61C22×D5—21no—
62C22×D5—21no—
63C22×D5C22⋊C411yescharacteristic
64C22×D5C2×D411yescharacteristic
65C2×AGL(1,𝔽5)—42no—
66C2×AGL(1,𝔽5)—42no—
67C2×AGL(1,𝔽5)—42no—
68C2×C4×C5—22no—
69C2×C4×C5—21no—
70C2×C4×C5—21no—
71C2×C4×C5C22⋊C411yescharacteristic
72C2×C4×C5C22×C411yescharacteristic
73C2×Dic5—22no—
74C2×Dic5—21no—
75C2×Dic5—21no—
76C2×Dic5C22⋊C411yescharacteristic
77C2×Dic5C2×D411yescharacteristic
78C4×D5—22no—
79C4×D5—22no—
80C4×D5—22no—
81C4×D5—42no—
82C4×D5—41no—
83C4×D5—41no—
84C4×D5#4114yes—
85#170—42no—
86C23×C4—51no—
87C22×C8—52no—
88#117—51no—
89#117—51no—
90#117—51no—
91#117—101no—
92#117—101no—
93#117AGL(1,𝔽5)11yescharacteristic, complemented
94C2×C4⋊C4—51no—
95C2×C4⋊C4—51no—
96C2×C4⋊C4—52no—
97C2×C22⋊C4—101no—
98C2×C22⋊C4—101no—
99C2×C42—101no—
100C22⋊C8—101no—
101C22⋊C8—101no—
102#95—101no—
103C22×C5—21no—
104C22×C5—21no—
105C22×C5—21no—
106C22×C5C2×C22⋊C411yescharacteristic
107C2×D5—21no—
108C2×D5—21no—
109C2×D5—21no—
110C2×D5—42no—
111C2×D5—42no—
112C2×D5#12312yes—
113C2×D5#13512yes—
114C2×D5AGL(1,ℤ/8)11yescharacteristic
115C2×D5#13711yescharacteristic
116AGL(1,𝔽5)—82no—
117C4×C5—21no—
118C4×C5—41no—
119C4×C5—41no—
120C4×C5#11712yes—
121Dic5—21no—
122Dic5—41no—
123Dic5—41no—
124Dic5#12312yes—
125C24—51no—
126C22×C4—51no—
127C22×C4—51no—
128C22×C4—51no—
129C22×C4—51no—
130C22×C4—51no—
131C22×C4—104no—
132C22×C4—101no—
133C22×C4—101no—
134C22×C4—102no—
135C22×C4C2×AGL(1,𝔽5)11yescharacteristic
136C2×C8—102no—
137C2×C8—104no—
138C4⋊C4—51no—
139C4⋊C4—51no—
140C4⋊C4—101no—
141C4⋊C4—101no—
142C4⋊C4—104no—
143C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
144C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—101no—
147C22⋊C4—101no—
148C22⋊C4—101no—
149C22⋊C4—202no—
150C42—21no—
151C42—101no—
152C42—101no—
153C42—101no—
154C2×C5—41no—
155C2×C5#38111yescharacteristic
156C2×C5#41611yescharacteristic
157C2×C5#41711yescharacteristic
158D5—41no—
159D5#48212yes—
160D5#48512yes—
161C23—51no—
162C23—51no—
163C23—101no—
164C23—102no—
165C23—101no—
166C23—102no—
167C23C22⋊AGL(1,𝔽5)11yescharacteristic
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—51no—
172C2×C4—54no—
173C2×C4—51no—
174C2×C4—102no—
175C2×C4—102no—
176C2×C4—102no—
177C2×C4—102no—
178C2×C4—101no—
179C2×C4—101no—
180C2×C4—201no—
181C2×C4—201no—
182C2×C4—202no—
183C2×C4—202no—
184C2×C4—202no—
185C2×C4—202no—
186C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
187C2×C4C22×AGL(1,𝔽5)11yescharacteristic
188C8—202no—
189C5#182511yescharacteristic, complemented, Sylow
190C22—21no—
191C22—21no—
192C22—21no—
193C22—51no—
194C22—51no—
195C22—52no—
196C22—52no—
197C22—101no—
198C22—101no—
199C22—101no—
200C22—202no—
201C22—202no—
202C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—52no—
207C4—101no—
208C4—201no—
209C4—201no—
210C4—402no—
211C4#413012yes—
212C2—41no—
213C2—52no—
214C2—52no—
215C2—201no—
216C2#6576411yescharacteristic
217C2#6579311yescharacteristic
218C2#6579411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#117C4
socleC22×C5C2×C22⋊C4
prime radicalC23×D5D4
cosocleC2×C4×D5C23
hypercenter#117AGL(1,𝔽5)
hypocenterC5#1825
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #64874C2semidirect
5. #64874C2semidirect
10. C5×#117C4semidirect
11. #4072C4semidirect
12. #4038C4semidirect
13. #4038C4semidirect
93. #117AGL(1,𝔽5)semidirect
189. C5#1825semidirect

Maximal subgroups

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

subgroupindex
D5×#1172
#657652
D5⋊(C22⋊C8)2
#648742
#18255