Finite Group Databasebeta

← #10604624

Subgroups — #10604624

Order 640.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#65686C211yesmaximal, characteristic, complemented
3#65683C211yesmaximal, characteristic, complemented
4#64868C214yesmaximal, complemented
5C2×#713—21no—
6C2×C4×AGL(1,𝔽5)—21no—
7C5×C4⋊Q8C411yescharacteristic, complemented
8#4093—21no—
9#4041—21no—
10#4041C2212yes—
11#4039—21no—
12#4039C412yescomplemented
13#4020C411yescharacteristic, complemented
14C42×D5C2211yescharacteristic
15(C2×AGL(1,𝔽5))⋊C4—21no—
16(C2×AGL(1,𝔽5))⋊C4C2212yes—
17#4003C2212yes—
18#4002—21no—
19C4⋊#170—21no—
20#1505—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—
33C4×AGL(1,𝔽5)—24no—
34C5×C4⋊C4—21no—
35C5×C4⋊C4C2×C412yes—
36C42×C5C2×C411yescharacteristic
37#685—22no—
38#685—22no—
39C4×Dic5C2×C411yescharacteristic
40C4×Dic5D411yescharacteristic
41C4×Dic5D411yescharacteristic
42#532—51no—
43#432—51no—
44#376—51no—
45#327—54no—
46C22×D5—22no—
47C22×D5C2×D411yescharacteristic
48C2×AGL(1,𝔽5)—24no—
49C2×AGL(1,𝔽5)—42no—
50C2×AGL(1,𝔽5)—42no—
51C5×Q8—42no—
52C2×C4×C5—22no—
53C2×C4×C5C22⋊C411yescharacteristic
54C2×C4×C5C22⋊C411yescharacteristic
55C2×C4×C5C22×C411yescharacteristic
56C2×Dic5C22⋊C411yescharacteristic
57C2×Dic5C22⋊C411yescharacteristic
58C2×Dic5C2×D411yescharacteristic
59D20—42no—
60D20—42no—
61D20—44no—
62D20—42no—
63C4×D5—22no—
64C4×D5—22no—
65C4×D5—41no—
66C4×D5—44no—
67#170—42no—
68C2×#41—101no—
69C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
70#127—51no—
71#124—101no—
72#121—52no—
73#121—101no—
74C4×D4—52no—
75C4×D4—101no—
76C2×C42—51no—
77C2×C42—101no—
78C4⋊C8—101no—
79C22⋊C8—52no—
80#97—101no—
81#95—52no—
82#95—101no—
83C22×C5C2×C22⋊C411yescharacteristic
84C2×D5—21no—
85C2×D5—21no—
86C2×D5—42no—
87C2×D5—42no—
88C2×D5—42no—
89C2×D5AGL(1,ℤ/8)12yes—
90AGL(1,𝔽5)—44no—
91C4×C5—21no—
92C4×C5—21no—
93C4×C5—42no—
94C4×C5#10412yes—
95Dic5—21no—
96Dic5—21no—
97Dic5—21no—
98#41—204no—
99C2×Q8—21no—
100C2×D4—52no—
101C2×D4—102no—
102C2×D4—101no—
103C2×D4—101no—
104C22×C4—51no—
105C22×C4—51no—
106C22×C4—51no—
107C22×C4—102no—
108C22×C4—102no—
109C2×C8—102no—
110C4⋊C4—21no—
111C4⋊C4C2×AGL(1,𝔽5)12yes—
112C22⋊C4—102no—
113C22⋊C4—102no—
114C42—51no—
115C42—51no—
116C42—51no—
117C42—104no—
118C42C2×AGL(1,𝔽5)11yescharacteristic
119C2×C5#41011yescharacteristic
120C2×C5#41811yescharacteristic
121C2×C5C2×#10411yescharacteristic
122D5—22no—
123D5—82no—
124C23—51no—
125C23—102no—
126Q8—42no—
127D4—202no—
128D4—202no—
129D4—204no—
130D4—202no—
131C2×C4—22no—
132C2×C4—51no—
133C2×C4—51no—
134C2×C4—51no—
135C2×C4—102no—
136C2×C4—102no—
137C2×C4—104no—
138C2×C4—201no—
139C2×C4—204no—
140C2×C4—202no—
141C2×C4—202no—
142C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
143C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
144C2×C4C22×AGL(1,𝔽5)11yescharacteristic
145C8—202no—
146C5#150511yescharacteristic, complemented, Sylow
147C22—52no—
148C22—101no—
149C22—101no—
150C22—202no—
151C22—202no—
152C22—202no—
153C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—101no—
158C4—101no—
159C4—101no—
160C4—204no—
161C4#401012yes—
162C2—102no—
163C2—402no—
164C2#6576711yescharacteristic
165C2C2×#401011yescharacteristic
166C2#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#1505
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. #65686C2semidirect
3. #65683C2semidirect
4. #64868C2semidirect
7. C5×C4⋊Q8C4semidirect
12. #4039C4semidirect
13. #4020C4semidirect
69. C4⋊Q8AGL(1,𝔽5)semidirect
146. C5#1505semidirect

Maximal subgroups

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

subgroupindex
#660592
#656862
#656832
#648682
#15055