Finite Group Databasebeta

← #35203964

Subgroups — #35203964

Order 20288.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#23778777C211yesmaximal, characteristic, complemented
2#23778774C211yesmaximal, characteristic, complemented
3#23778765C213yesmaximal, complemented
4#23778755C213yesmaximal, complemented
5#23778753C213yesmaximal, complemented
6#23778750C213yesmaximal, complemented
7#126×C317C211yesmaximal, characteristic, complemented
8C23×D317C2211yescharacteristic
9C2×D1268C2213yes—
10C2×D1268C2213yes—
11C2×D1268C2213yescomplemented
12C2×C22⋊D317C2213yes—
13C2×C4×D317C2213yescomplemented
14#19099553C2213yescomplemented
15#19099551C2213yes—
16#19099551C2213yes—
17#19099551C2213yescomplemented
18C4⋊C4×C317C2213yescomplemented
19C22⋊C4×C317C2213yes—
20C42×C317C2211yescharacteristic, complemented
21C22×D317—43no—
22C22×D317C2311yescharacteristic
23C22×D317C2311yescharacteristic
24C22×D317C2313yes—
25D1268—43no—
26D1268—43no—
27D1268—43no—
28C22⋊D317—43no—
29C2×Dic317C2313yes—
30C4×D317—43no—
31C23×C317C2311yescharacteristic
32C2×C4×C317C2313yes—
33C2×C4×C317C2313yes—
34C22×C317—23no—
35C22×C317C2411yescharacteristic
36C2×D317—23no—
37C2×D317—23no—
38C2×D317—23no—
39C2×D317—23no—
40C2×D317—23no—
41C2×D317—44no—
42C4×C317—23no—
43C4×C317—23no—
44Dic317—23no—
45C2×C317—41no—
46C2×C317H(5,𝔽2)13yes—
47D317—43no—
48D317—41no—
49D317—41no—
50C317#56011yescharacteristic, complemented, Sylow
51#560—3171nomaximal, Sylow
52#127—3171no—
53#126—3171no—
54#126D31711yescharacteristic, complemented
55#123—3173no—
56#121—3173no—
57#121—3173no—
58#120—3173no—
59C24—3171no—
60C2×D4—3173no—
61C2×D4—3173no—
62C2×D4—3173no—
63C2×D4—3173no—
64C22×C4—3173no—
65C4⋊C4—3173no—
66C4⋊C4C2×D31713yescomplemented
67C22⋊C4—3173no—
68C22⋊C4—3173no—
69C22⋊C4—3173no—
70C22⋊C4C2×D31713yes—
71C42C2×D31711yescharacteristic, complemented
72C23—3171no—
73C23—3171no—
74C23—3173no—
75C23—12683no—
76C23C22×D31711yescharacteristic
77D4—12683no—
78D4—12683no—
79D4—12683no—
80D4—12683no—
81C2×C4—3173no—
82C2×C4—12683no—
83C2×C4C22×D31713yes—
84C2×C4C22×D31713yes—
85C22—23no—
86C22—6343no—
87C22—6343no—
88C22—6343no—
89C22—6343no—
90C22—6343no—
91C22—12684no—
92C22C23×D31711yescharacteristic
93C4—23no—
94C4—23no—
95C4—6343no—
96C2—41no—
97C2—12681no—
98C2—12681no—
99C2—12683no—
100C2#2377882613yes—
subgroupG/H
center, FrattiniC22C23×D317
commutator, socle, cosocleC22×C317C24
Fitting#126×C317C2
hypercenter#126D317
hypocenterC317#560
LCSG ▷ C22×C317 ▷ C317
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #23778777C2semidirect
2. #23778774C2semidirect
3. #23778765C2semidirect
4. #23778755C2semidirect
5. #23778753C2semidirect
6. #23778750C2semidirect
7. #126×C317C2semidirect
11. C2×D1268C22semidirect
13. C2×C4×D317C22semidirect
14. #19099553C22semidirect
17. #19099551C22semidirect
18. C4⋊C4×C317C22semidirect
20. C42×C317C22semidirect
50. C317#560semidirect
54. #126D317semidirect
66. C4⋊C4C2×D317semidirect
71. C42C2×D317semidirect

Maximal subgroups

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

subgroupindex
#237787772
#237787742
#237787652
#237787552
#237787532
#237787502
#126×C3172
#560317