Finite Group Databasebeta

← #13102958

Subgroups — #13102958

Order 1600.

568 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 111 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#11717520C211yesmaximal, characteristic
2C52×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11717262C212yesmaximal, complemented
4#11716703C211yesmaximal, characteristic, complemented
5#11716672C212yesmaximal, complemented
6C2×C52×Q8C2211yescharacteristic
7C2×C52×Q8C2211yescharacteristic
8#87904—22no—
9Q8×C5⋊D5—22no—
10#87900—22no—
11#87900C2212yes—
12C52×C4⋊C4—22no—
13C42×C52C2211yescharacteristic
14#87812—22no—
15#87811—22no—
16#87804C2212yes—
17#65384—56nomaximal
18C52×Q8—22no—
19C52×Q8—22no—
20C2×C4×C52—21no—
21C2×C4×C52—21no—
22C2×C4×C52D412yes—
23C2×C4×C52C2311yescharacteristic
24C22⋊(C5⋊D5)—22no—
25C4⋊(C5⋊D5)—22no—
26C4⋊(C5⋊D5)D412yes—
27C4×C5⋊D5—24no—
28C4×C5⋊D5—22no—
29#6045—22no—
30#6045D412yes—
31#6029—22no—
32#4147—56no—
33C5×C4⋊Q8D516yescomplemented
34#4088—512no—
35#3968—56no—
36#3937—512no—
37C22×C52C2×D411yescharacteristic
38C2×C5⋊D5—22no—
39C4×C52—22no—
40C4×C52—22no—
41C4×C52—41no—
42C4×C52C2×D412yes—
43#1039—22no—
44C2×C5×Q8C2×D516yes—
45C2×C5×Q8C2×D516yes—
46#713—1012no—
47Q8×D5—1012no—
48#709—512no—
49#709—1012no—
50C5×C4⋊C4—212no—
51C42×C5C2×D516yes—
52#689—1012no—
53#688—1012no—
54#681—512no—
55#456—251noSylow
56C2×C52—21no—
57C2×C52#12011yescharacteristic
58C5⋊D5—42no—
59C5×Q8—212no—
60C5×Q8—212no—
61C2×C4×C5—26no—
62C2×C4×C5—26no—
63C2×C4×C5C22⋊D5112yes—
64C2×C4×C5C22×D516yes—
65C22⋊D5—1012no—
66D20—512no—
67D20—1012no—
68C4×D5—1024no—
69C4×D5—1012no—
70Dic10—512no—
71Dic10—1012no—
72#168—1012no—
732−1+4—251no—
74#137—252no—
75C4⋊Q8C5⋊D511yescharacteristic, complemented
76#104—252no—
77#101—251no—
78C52#45611yescharacteristic, complemented, Sylow
79C22×C5C2×C22⋊D516yes—
80C2×D5—1012no—
81C4×C5—212no—
82C4×C5—212no—
83C4×C5—46no—
84C4×C5D4×D5112yes—
85Dic5—1012no—
86#41—252no—
87#41—502no—
88#41—502no—
89C2×Q8—502no—
90C2×Q8C2×C5⋊D511yescharacteristic
91C2×Q8C2×C5⋊D511yescharacteristic
92Q16—502no—
93SD16—502no—
94M16—252no—
95C4⋊C4—22no—
96C42C2×C5⋊D511yescharacteristic
97C2×C5—26no—
98C2×C5#408316yes—
99D5—2012no—
100Q8—22no—
101Q8—22no—
102Q8—252no—
103Q8—502no—
104D4—252no—
105D4—502no—
106D4—502no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—504no—
110C2×C4—502no—
111C2×C4C22⋊(C5⋊D5)12yes—
112C2×C4C22×C5⋊D511yescharacteristic
113C8—502no—
114C5#6538416yescomplemented
115C22—502no—
116C22C2×C22⋊(C5⋊D5)11yescharacteristic
117C4—22no—
118C4—22no—
119C4—41no—
120C4—502no—
121C4D4×C5⋊D512yes—
122C2—21no—
123C2—1002no—
124C2#1171725711yescharacteristic
subgroupG/H
centerC2#11717257
commutator, cosocleC2×C4×C52C23
FrattiniC2×C4C22×C5⋊D5
FittingC52×C4⋊Q8C2
socleC2×C52#120
prime radical#11717520C2
hypercenterC4⋊Q8C5⋊D5
hypocenterC52#456
LCSG ▷ C2×C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C52×C4⋊Q8C2semidirect
3. #11717262C2semidirect
4. #11716703C2semidirect
5. #11716672C2semidirect
33. C5×C4⋊Q8D5semidirect
75. C4⋊Q8C5⋊D5semidirect
78. C52#456semidirect
114. C5#65384semidirect

Maximal subgroups

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

subgroupindex
#117175202
C52×C4⋊Q82
#117172622
#117167032
#117166722
#653845