Finite Group Databasebeta

← #13103873

Subgroups — #13103873

Order 1600.

248 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 51 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#11717358C212yesmaximal, complemented
2#11716927C211yesmaximal, characteristic
3#11716889C211yesmaximal, characteristic, complemented
4#11716860C211yesmaximal, characteristic, complemented
5#11716770C212yesmaximal, complemented
6#87871C2212yes—
7#87863C2212yes—
8C4×#1043C2211yescharacteristic
9#87855C2211yescharacteristic
10#87834—22no—
11D5⋊#170—22no—
12C8×C5⋊D5C2211yescharacteristic
13C5×#677C412yescomplemented
14#87775C411yescharacteristic, complemented
15#87770C411yescharacteristic, complemented
16#65737—52nomaximal
17C2×#1043—21no—
18C4×C5⋊D5C2311yescharacteristic
19C5×D20C2×C412yes—
20C5×Dic10C2×C412yes—
21#6038—22no—
22D5×Dic5—22no—
23#6034—22no—
24#6033D412yes—
25C52×C8C2×C411yescharacteristic
26#6029C2×C411yescharacteristic
27#4133—52no—
28#4131—52no—
29#4062—52no—
30#4010—52no—
31#4008—52no—
32#3996—52no—
33#3992—52no—
34#3992—102no—
35C2×C5⋊D5#4111yescharacteristic
36C2×C5×D5—22no—
37#1043—41no—
38C4×C52C22×C411yescharacteristic
39#1039C2×D411yescharacteristic
40C5×Dic5—22no—
41#713—52no—
42#711—52no—
43#704—102no—
44C2×#170—102no—
45C4×AGL(1,𝔽5)—52no—
46C4×AGL(1,𝔽5)—102no—
47#700—52no—
48#700—52no—
49#700—102no—
50D5⋊C8—52no—
51D5⋊C8—102no—
52D5⋊C8—102no—
53C5×SD16—52no—
54#689—52no—
55#686—52no—
56#677AGL(1,𝔽5)12yescomplemented
57C8×D5—52no—
58C8×D5—102no—
59#444—251noSylow
60C2×C52C4×D411yescharacteristic
61C5⋊D5—21no—
62C5×D5—42no—
63C2×AGL(1,𝔽5)—102no—
64C2×AGL(1,𝔽5)—104no—
65C5×Q8—52no—
66C5×D4—52no—
67C22⋊D5—102no—
68C2×Dic5—102no—
69D20—102no—
70D20C2×AGL(1,𝔽5)12yes—
71C4×D5—52no—
72C4×D5—102no—
73C4×D5—102no—
74Dic10C2×AGL(1,𝔽5)12yes—
75#170—52no—
76#170—52no—
77#170—104no—
78#170—102no—
79#170—102no—
80C5×C8—22no—
81C5×C8C2×AGL(1,𝔽5)12yes—
82#168—52no—
83#168—102no—
84#135—251no—
85#131—252no—
86#108—251no—
87#104—252no—
88#97—251no—
89C52#44411yescharacteristic, complemented, Sylow
90C22×C5—102no—
91C2×D5—22no—
92C2×D5—52no—
93C2×D5—102no—
94AGL(1,𝔽5)—204no—
95AGL(1,𝔽5)—202no—
96C4×C5—22no—
97C4×C5—102no—
98C4×C5C22×AGL(1,𝔽5)12yes—
99Dic5—22no—
100Dic5—52no—
101Dic5—102no—
102#41—252no—
103Q16—251no—
104SD16—52no—
105D8—251no—
106M16—252no—
107M16—502no—
108C2×C8—251no—
109C2×C8—251no—
110C2×C8—502no—
111C42—251no—
112C2×C5—22no—
113C2×C5—202no—
114C2×C5D4×AGL(1,𝔽5)12yes—
115D5—42no—
116D5—104no—
117D5—102no—
118Q8—52no—
119D4—52no—
120D4—502no—
121C2×C4—251no—
122C2×C4—502no—
123C2×C4—501no—
124C8—252no—
125C8—251no—
126C8—502no—
127C8D5⋊AGL(1,𝔽5)11yescharacteristic
128C5—22no—
129C5#6573712yescomplemented
130C22—102no—
131C22—251no—
132C4—102no—
133C4—251no—
134C4—1001no—
135C4C2×D5⋊AGL(1,𝔽5)11yescharacteristic
136C2—202no—
137C2—501no—
138C2#1171736311yescharacteristic
subgroupG/H
centerC2#11717363
commutatorC4×C52C22×C4
FrattiniC4C2×D5⋊AGL(1,𝔽5)
FittingC52×C8C2×C4
socleC2×C52C4×D4
prime radical#11716927C2
cosocleC4×C5⋊D5C23
hypercenterC8D5⋊AGL(1,𝔽5)
hypocenterC52#444
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11717358C2semidirect
3. #11716889C2semidirect
4. #11716860C2semidirect
5. #11716770C2semidirect
13. C5×#677C4semidirect
14. #87775C4semidirect
15. #87770C4semidirect
56. #677AGL(1,𝔽5)semidirect
89. C52#444semidirect
129. C5#65737semidirect

Maximal subgroups

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

subgroupindex
#117173582
#117169272
#117168892
#117168602
#117167702
#657375