Finite Group Databasebeta

← #10604596

Subgroups — #10604596

Order 640.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#66050C211yesmaximal, characteristic
2#65689C211yesmaximal, characteristic, complemented
3#65678C211yesmaximal, characteristic, complemented
4#64920C214yesmaximal, complemented
5C2×#711—21no—
6C2×C4⋊AGL(1,𝔽5)—21no—
7C5×#127C411yescharacteristic, complemented
8#4084—21no—
9#4084C2212yes—
10#4082—21no—
11D4×Dic5—21no—
12D4×Dic5C412yescomplemented
13C42×D5C2211yescharacteristic
14#4015C411yescharacteristic, complemented
15(C2×AGL(1,𝔽5))⋊C4—21no—
16(C2×AGL(1,𝔽5))⋊C4C2212yes—
17#4004—21no—
18#4003C2212yes—
19C4×#170—21no—
20#1517—51nomaximal, Sylow
21C22×AGL(1,𝔽5)—22no—
22C2×C5×D4—21no—
23C2×C5×D4—21no—
24C2×C5×D4C2×C412yes—
25C2×C22⋊D5—22no—
26C22×Dic5—22no—
27#711—42no—
28#711—42no—
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×C4×D5C2311yescharacteristic
32C2×Dic10—21no—
33C2×#170—22no—
34C4⋊AGL(1,𝔽5)—42no—
35C42×C5C2×C411yescharacteristic
36C22⋊Dic5—22no—
37C22⋊Dic5—22no—
38C4⋊Dic5—21no—
39C4⋊Dic5C2×C412yes—
40C4×Dic5C2×C411yescharacteristic
41C4×Dic5D411yescharacteristic
42C4×Dic5D411yescharacteristic
43#532—51no—
44#431—51no—
45#382—51no—
46#327—54no—
47C23×C5—22no—
48C22×D5C2×D411yescharacteristic
49C2×AGL(1,𝔽5)—42no—
50C2×AGL(1,𝔽5)—42no—
51C2×AGL(1,𝔽5)—42no—
52C5×D4—42no—
53C5×D4—42no—
54C5×D4—42no—
55C2×C4×C5C22⋊C411yescharacteristic
56C2×C4×C5C22⋊C411yescharacteristic
57C2×C4×C5C22×C411yescharacteristic
58C22⋊D5—42no—
59C22⋊D5—42no—
60C2×Dic5—22no—
61C2×Dic5—44no—
62C2×Dic5C22⋊C411yescharacteristic
63C2×Dic5C22⋊C411yescharacteristic
64C2×Dic5C2×D411yescharacteristic
65C4×D5—41no—
66C4×D5—41no—
67C4×D5D812yes—
68C4×D5SD1612yes—
69Dic10—42no—
70#170—42no—
71C2×#41—101no—
72C4⋊Q8—51no—
73#127AGL(1,𝔽5)11yescharacteristic, complemented
74#124—101no—
75#121—52no—
76#121—101no—
77C4×D4—52no—
78C4×D4—101no—
79C2×C4⋊C4—101no—
80C2×C42—51no—
81C4⋊C8—101no—
82C22⋊C8—52no—
83C4×C8—101no—
84#95—52no—
85#95—101no—
86C22×C5—42no—
87C22×C5—42no—
88C22×C5—42no—
89C22×C5C2×C22⋊C411yescharacteristic
90C2×D5—21no—
91C2×D5—21no—
92C2×D5C2×D811yescharacteristic
93C2×D5C2×SD1611yescharacteristic
94AGL(1,𝔽5)—82no—
95C4×C5—21no—
96C4×C5—21no—
97C4×C5#10212yes—
98Dic5—21no—
99Dic5—21no—
100Dic5—42no—
101Dic5#10212yes—
102#41—202no—
103#41—202no—
104C2×Q8—101no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—102no—
108C2×D4C2×AGL(1,𝔽5)12yes—
109C22×C4—51no—
110C22×C4—51no—
111C22×C4—51no—
112C22×C4—102no—
113C22×C4—102no—
114C2×C8—102no—
115C4⋊C4—52no—
116C4⋊C4—101no—
117C4⋊C4—202no—
118C22⋊C4—102no—
119C22⋊C4—102no—
120C42—51no—
121C42—51no—
122C42—51no—
123C42C2×AGL(1,𝔽5)11yescharacteristic
124C2×C5—82no—
125C2×C5#41011yescharacteristic
126C2×C5C2×#10211yescharacteristic
127C2×C5#42111yescharacteristic
128D5—21no—
129D5—21no—
130C23—22no—
131C23—51no—
132Q8—202no—
133D4—42no—
134D4—42no—
135D4—42no—
136D4—202no—
137D4—202no—
138C2×C4—51no—
139C2×C4—52no—
140C2×C4—52no—
141C2×C4—51no—
142C2×C4—51no—
143C2×C4—102no—
144C2×C4—201no—
145C2×C4—201no—
146C2×C4—204no—
147C2×C4—202no—
148C2×C4—202no—
149C2×C4—202no—
150C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
151C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
152C2×C4C22×AGL(1,𝔽5)11yescharacteristic
153C8—202no—
154C5#151711yescharacteristic, complemented, Sylow
155C22—42no—
156C22—42no—
157C22—42no—
158C22—51no—
159C22—51no—
160C22—101no—
161C22—101no—
162C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
163C4—21no—
164C4—21no—
165C4—52no—
166C4—101no—
167C4—101no—
168C4—202no—
169C4—402no—
170C4#400712yes—
171C2—82no—
172C2—101no—
173C2—101no—
174C2C2×#400711yescharacteristic
175C2#6576911yescharacteristic
176C2#6577111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#127C4
socleC22×C5C2×C22⋊C4
prime radical#66050C2
cosocleC2×C4×D5C23
hypercenter#127AGL(1,𝔽5)
hypocenterC5#1517
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #65689C2semidirect
3. #65678C2semidirect
4. #64920C2semidirect
7. C5×#127C4semidirect
12. D4×Dic5C4semidirect
14. #4015C4semidirect
73. #127AGL(1,𝔽5)semidirect
154. C5#1517semidirect

Maximal subgroups

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

subgroupindex
#660502
#656892
#656782
#649202
#15175