Finite Group Databasebeta

← #22586276

Subgroups — #22586276

Order 9088.

128 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 37 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#17985781C212yesmaximal, complemented
2#407×C71C211yesmaximal, characteristic
3#14556068—22no—
4C23×C4×C71C411yescharacteristic
5C2×M16×C71C411yescharacteristic
6C2×M16×C71C2211yescharacteristic
7C22⋊C8×C71—22no—
8C71⋊C16—42no—
9C24×C71C811yescharacteristic
10C22×C4×C71—21no—
11C22×C4×C71—21no—
12C22×C4×C71—41no—
13C22×C4×C71—41no—
14C22×C4×C71C811yescharacteristic
15C22×C4×C71C2×C411yescharacteristic
16M16×C71—41no—
17M16×C71—41no—
18C2×C8×C71—21no—
19C2×C8×C71D412yes—
20C23×C71—21no—
21C23×C71—41no—
22C23×C71—42no—
23C23×C71C2×C811yescharacteristic
24C2×C4×C71—21no—
25C2×C4×C71—21no—
26C2×C4×C71—21no—
27C2×C4×C71—42no—
28C2×C4×C71—41no—
29C2×C4×C71—41no—
30C2×C4×C71—41no—
31C2×C4×C71C22⋊C411yescharacteristic
32C2×C4×C71M1611yescharacteristic
33C8×C71—22no—
34C8×C71—41no—
35C22×C71—22no—
36C22×C71—21no—
37C22×C71—41no—
38C22×C71—41no—
39C22×C71—42no—
40C22×C71—42no—
41C22×C71—41no—
42C22×C71C22⋊C811yescharacteristic
43C4×C71—21no—
44C4×C71—41no—
45C4×C71#9911yescharacteristic
46C4×C71#10011yescharacteristic
47C2×C71—21no—
48C2×C71—42no—
49C2×C71—41no—
50C2×C71#32311yescharacteristic
51#1320—711nomaximal, Sylow
52C71#132011yescharacteristic, complemented, Sylow
53#407D7111yescharacteristic
54#349—712no—
55C23×C4Dic7111yescharacteristic
56C2×M16Dic7111yescharacteristic
57C2×M16C2×D7111yescharacteristic
58M32—1422no—
59C22⋊C8—22no—
60C24C71⋊C811yescharacteristic
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4C71⋊C811yescharacteristic
66C22×C4C2×Dic7111yescharacteristic
67M16—41no—
68M16—41no—
69C2×C8—21no—
70C2×C8C22⋊D7112yes—
71C16—2842no—
72C23—21no—
73C23—41no—
74C23—42no—
75C23C2×C71⋊C811yescharacteristic
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—42no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4C22⋊Dic7111yescharacteristic
84C2×C4#1176584911yescharacteristic
85C8—22no—
86C8—41no—
87C22—22no—
88C22—21no—
89C22—41no—
90C22—41no—
91C22—42no—
92C22—42no—
93C22—41no—
94C22C22⋊(C71⋊C8)11yescharacteristic
95C4—21no—
96C4—41no—
97C4#1455604311yescharacteristic
98C4#1455604511yescharacteristic
99C2—21no—
100C2—42no—
101C2—41no—
102C2#1798572311yescharacteristic
subgroupG/H
centerC4#14556043
commutatorC23×C71C2×C8
FrattiniC2×M16C2×D71
Fitting#407×C71C2
socleC2×C71#323
hypercenter#407D71
hypocenterC71#1320
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #17985781C2semidirect
52. C71#1320semidirect