Finite Group Databasebeta

← #20298801

Subgroups — #20298801

Order 7232.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C2×#14268635C211yesmaximal, characteristic
2#16813620C211yesmaximal, characteristic, complemented
3D4×Dic113C211yesmaximal, characteristic
4#16813551C211yesmaximal, characteristic, complemented
5#16813547C211yesmaximal, characteristic, complemented
6#16813546C211yesmaximal, characteristic, complemented
7C2×SD16×C113C211yesmaximal, characteristic
8C2×Dic226C2211yescharacteristic
9C22×Dic113—21no—
10#14268635—22no—
11C2×#11744751C2211yescharacteristic
12C4⋊Dic113C2211yescharacteristic
13#14268626—21no—
14#14268626—21no—
15C22⋊Dic113—21no—
16C4×Dic113C2211yescharacteristic
17C2×Q8×C113C2211yescharacteristic
18C2×D4×C113C2211yescharacteristic
19SD16×C113—22no—
20C2×C8×C113C2211yescharacteristic
21C23×C113—21no—
22Q8×C113—21no—
23Q8×C113—21no—
24D4×C113—21no—
25D4×C113D412yes—
26C2×C4×C113—21no—
27C2×C4×C113C2311yescharacteristic
28C2×Dic113—21no—
29C2×Dic113—21no—
30C2×Dic113—42no—
31C2×Dic113D411yescharacteristic
32C2×Dic113D411yescharacteristic
33Dic226—21no—
34Dic226—21no—
35C8×C113—21no—
36#11744751—21no—
37C22×C113—22no—
38C22×C113—41no—
39C22×C113—41no—
40C22×C113C2×D411yescharacteristic
41C4×C113—41no—
42C4×C113C2×D411yescharacteristic
43C4×C113#4111yescharacteristic
44Dic113—21no—
45Dic113—41no—
46Dic113—41no—
47Dic113SD1612yescomplemented
48C2×C113—42no—
49C2×C113#12111yescharacteristic
50C2×C113C2×SD1611yescharacteristic
51C2×C113#13711yescharacteristic
52C113#46111yescharacteristic, complemented, Sylow
53#461—1131nomaximal, Sylow
54C2×SD16—1131no—
55C2×SD16D11311yescharacteristic
56C4⋊Q8—1131no—
57C4×D4—1131no—
58C4⋊C8—1131no—
59#103—1131no—
60#103—1131no—
61C2×Q8—1131no—
62C2×Q8C2×D11311yescharacteristic
63C2×D4C2×D11311yescharacteristic
64C22×C4—2261no—
65SD16—22no—
66SD16—2262no—
67C2×C8—1131no—
68C2×C8C2×D11311yescharacteristic
69C4⋊C4—1131no—
70C4⋊C4—2261no—
71C4⋊C4—2261no—
72C22⋊C4—2261no—
73C42—1131no—
74C23—21no—
75Q8—21no—
76Q8—21no—
77Q8—2261no—
78Q8—2261no—
79D4—21no—
80D4C22⋊D11312yes—
81C2×C4—21no—
82C2×C4—1131no—
83C2×C4—1131no—
84C2×C4—4522no—
85C2×C4—2261no—
86C2×C4—2261no—
87C2×C4C22×D11311yescharacteristic
88C8—21no—
89C8—2261no—
90C22—22no—
91C22—41no—
92C22—41no—
93C22D4×D11311yescharacteristic
94C4—41no—
95C4—1132no—
96C4—4521no—
97C4—4521no—
98C4—2261no—
99C4C2×C22⋊D11311yescharacteristic
100C4#1426864611yescharacteristic
101C2—42no—
102C2#1681359411yescharacteristic
103C2SD16×D11311yescharacteristic
104C2#1681365111yescharacteristic
subgroupG/H
centerC22D4×D113
commutatorC2×C4×C113C23
FrattiniC2×C4C22×D113
FittingC2×SD16×C113C2
socleC22×C113C2×D4
hypercenterC2×SD16D113
hypocenterC113#461
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. #16813620C2semidirect
4. #16813551C2semidirect
5. #16813547C2semidirect
6. #16813546C2semidirect
47. Dic113SD16semidirect
52. C113#461semidirect