Finite Group Databasebeta

← #22558239

Subgroups — #22558239

Order 8576.

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

normal subgroups only

This group has 140 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#16863165C211yesmaximal, characteristic
2#16863122C211yesmaximal, characteristic
3#395×C67C211yesmaximal, characteristic
4C2×#11761612—21no—
5C22⋊(C67⋊C8)—21no—
6C22×Q8×C67C411yescharacteristic
7C2×C4⋊C4×C67—21no—
8C2×C4⋊C4×C67C2211yescharacteristic
9#95×C67—21no—
10#95×C67C411yescharacteristic
11#11761612—22no—
12#11761612—41no—
13C2×C67⋊C8—41no—
14C2×C67⋊C8—41no—
15C2×Q8×C67—42no—
16C2×Q8×C67—41no—
17C22×C4×C67—21no—
18C22×C4×C67—21no—
19C22×C4×C67C2×C411yescharacteristic
20C22×C4×C67D411yescharacteristic
21C22×C4×C67D411yescharacteristic
22C4⋊C4×C67—41no—
23C4⋊C4×C67—42no—
24C23×C67C22⋊C411yescharacteristic
25Q8×C67—42no—
26Q8×C67—42no—
27C2×C4×C67—21no—
28C2×C4×C67—21no—
29C2×C4×C67—22no—
30C2×C4×C67—22no—
31C2×C4×C67—21no—
32C2×C4×C67—21no—
33C2×C4×C67—41no—
34C2×C4×C67—41no—
35C2×C4×C67—41no—
36C2×C4×C67—41no—
37C2×C4×C67—41no—
38C2×C4×C67—41no—
39C2×C4×C67SD1611yescharacteristic
40C2×C4×C67Q1611yescharacteristic
41C67⋊C8—42no—
42C67⋊C8—81no—
43C22×C67—21no—
44C22×C67—21no—
45C22×C67#9911yescharacteristic
46C22×C67#10311yescharacteristic
47C22×C67#10411yescharacteristic
48C4×C67—21no—
49C4×C67—21no—
50C4×C67—42no—
51C4×C67—41no—
52C4×C67—42no—
53C4×C67—41no—
54C2×C67—21no—
55C2×C67—21no—
56C2×C67#32811yescharacteristic
57C2×C67AΣL(1,𝔽16)11yescharacteristic
58C2×C67#35611yescharacteristic
59#1354—671nomaximal, Sylow
60C67#135411yescharacteristic, complemented, Sylow
61#395D6711yescharacteristic
62#343—671no—
63#324—671no—
64C22×Q8Dic6711yescharacteristic
65C2×M16—1341no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×D6711yescharacteristic
68C22⋊C8—1341no—
69#95—21no—
70#95Dic6711yescharacteristic
71C2×Q8—42no—
72C2×Q8—41no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4C2×Dic6711yescharacteristic
76C22×C4C22⋊D6711yescharacteristic
77C22×C4C22⋊D6711yescharacteristic
78M16—2681no—
79M16—1342no—
80C2×C8—2681no—
81C2×C8—2681no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C23C22⋊Dic6711yescharacteristic
85Q8—42no—
86Q8—42no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—41no—
99C2×C4#1176161511yescharacteristic
100C2×C4#1176161811yescharacteristic
101C8—2682no—
102C8—5361no—
103C22—21no—
104C22—21no—
105C22#1452395811yescharacteristic
106C22#1452396711yescharacteristic
107C22#1452397011yescharacteristic
108C4—21no—
109C4—21no—
110C4—42no—
111C4—41no—
112C4—42no—
113C4—41no—
114C2—21no—
115C2—21no—
116C2#1686313311yescharacteristic
117C2#1686318311yescharacteristic
118C2#1686319511yescharacteristic
subgroupG/H
centerC22#14523958
commutatorC22×C4×C67C2×C4
FrattiniC2×C4⋊C4C2×D67
Fitting#395×C67C2
socleC22×C67#99
hypercenter#395D67
hypocenterC67#1354
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #395

Complements

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

NH ≅ G/Ntype
60. C67#1354semidirect