Finite Group Databasebeta

← #10604627

Subgroups — #10604627

Order 640.

252 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 56 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#66059C211yesmaximal, characteristic
2#65690C211yesmaximal, characteristic, complemented
3#65684C211yesmaximal, characteristic, complemented
4#64868C214yesmaximal, complemented
5C2×#713—21no—
6C5×C4⋊Q8C411yescharacteristic, complemented
7#4093—21no—
8#4041—21no—
9#4041C2212yes—
10#4039—21no—
11#4039C412yescomplemented
12#4020C411yescharacteristic, complemented
13C42×D5C2211yescharacteristic
14(C2×AGL(1,𝔽5))⋊C4—21no—
15(C2×AGL(1,𝔽5))⋊C4—21no—
16(C2×AGL(1,𝔽5))⋊C4C2212yes—
17#4004—21no—
18#4003C2212yes—
19#4002—21no—
20#1514—51nomaximal, Sylow
21C22×AGL(1,𝔽5)—22no—
22C2×C5×Q8—21no—
23#713—44no—
24C2×D20—22no—
25C2×D20—21no—
26C2×D20—21no—
27C2×D20C2×C412yes—
28C2×C4×D5—22no—
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×C4×D5C2311yescharacteristic
32C2×#170—22no—
33C5×C4⋊C4—21no—
34C5×C4⋊C4C2×C412yes—
35C42×C5C2×C411yescharacteristic
36#685—22no—
37#685—22no—
38C4×Dic5C2×C411yescharacteristic
39C4×Dic5D411yescharacteristic
40C4×Dic5D411yescharacteristic
41#532—51no—
42#432—51no—
43#383—51no—
44#327—54no—
45C22×D5—22no—
46C22×D5C2×D411yescharacteristic
47C2×AGL(1,𝔽5)—42no—
48C2×AGL(1,𝔽5)—42no—
49C2×AGL(1,𝔽5)—42no—
50C5×Q8—42no—
51C2×C4×C5—22no—
52C2×C4×C5C22⋊C411yescharacteristic
53C2×C4×C5C22⋊C411yescharacteristic
54C2×C4×C5C22×C411yescharacteristic
55C2×Dic5C22⋊C411yescharacteristic
56C2×Dic5C22⋊C411yescharacteristic
57C2×Dic5C2×D411yescharacteristic
58D20—42no—
59D20—42no—
60D20—42no—
61D20—44no—
62C4×D5—22no—
63C4×D5—41no—
64C4×D5—41no—
65C4×D5—44no—
66#170—42no—
67C2×#41—101no—
68C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
69#127—51no—
70#124—101no—
71#121—52no—
72#121—101no—
73C4×D4—52no—
74C4×D4—101no—
75C2×C42—51no—
76C4⋊C8—101no—
77C22⋊C8—52no—
78#97—101no—
79#95—52no—
80#95—101no—
81#95—101no—
82C22×C5C2×C22⋊C411yescharacteristic
83C2×D5—21no—
84C2×D5—21no—
85C2×D5—42no—
86C2×D5—42no—
87C2×D5—42no—
88C2×D5AGL(1,ℤ/8)12yes—
89AGL(1,𝔽5)—82no—
90C4×C5—21no—
91C4×C5—21no—
92C4×C5—21no—
93C4×C5—42no—
94Dic5—21no—
95Dic5—21no—
96Dic5—21no—
97#41—204no—
98C2×Q8—21no—
99C2×D4—52no—
100C2×D4—102no—
101C2×D4—101no—
102C2×D4—101no—
103C22×C4—51no—
104C22×C4—51no—
105C22×C4—51no—
106C22×C4—102no—
107C22×C4—102no—
108C2×C8—102no—
109C4⋊C4—21no—
110C4⋊C4C2×AGL(1,𝔽5)12yes—
111C22⋊C4—102no—
112C22⋊C4—102no—
113C42—51no—
114C42—51no—
115C42—51no—
116C42C2×AGL(1,𝔽5)11yescharacteristic
117C2×C5#41011yescharacteristic
118C2×C5#41811yescharacteristic
119C2×C5#42111yescharacteristic
120D5—22no—
121D5—82no—
122C23—51no—
123C23—102no—
124Q8—42no—
125D4—202no—
126D4—202no—
127D4—202no—
128D4—204no—
129C2×C4—22no—
130C2×C4—51no—
131C2×C4—51no—
132C2×C4—51no—
133C2×C4—102no—
134C2×C4—201no—
135C2×C4—201no—
136C2×C4—204no—
137C2×C4—202no—
138C2×C4—202no—
139C2×C4—202no—
140C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
141C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
142C2×C4C22×AGL(1,𝔽5)11yescharacteristic
143C8—202no—
144C5#151411yescharacteristic, complemented, Sylow
145C22—52no—
146C22—101no—
147C22—101no—
148C22—202no—
149C22—202no—
150C22—202no—
151C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—101no—
157C4—101no—
158C4—101no—
159C4—402no—
160C2—102no—
161C2—402no—
162C2#6576711yescharacteristic
163C2#6578411yescharacteristic
164C2#6578511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C4⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radical#66059C2
cosocleC2×C4×D5C23
hypercenterC4⋊Q8AGL(1,𝔽5)
hypocenterC5#1514
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #65690C2semidirect
3. #65684C2semidirect
4. #64868C2semidirect
6. C5×C4⋊Q8C4semidirect
11. #4039C4semidirect
12. #4020C4semidirect
68. C4⋊Q8AGL(1,𝔽5)semidirect
144. C5#1514semidirect

Maximal subgroups

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

subgroupindex
#660592
#656902
#656842
#648682
#15145