Finite Group Databasebeta

← #30636851

Subgroups — #30636851

Order 16256.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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#22549850C212yesmaximal
2#408×C127C211yesmaximal, characteristic
3#16861168—22no—
4#16861124—22no—
5#16861124C414yescomplemented
6C2×#41×C127C2211yescharacteristic
7C2×M16×C127C411yescharacteristic, complemented
8C22×C8×C127C411yescharacteristic, complemented
9C22⋊C8×C127—22no—
10C4⋊Dic127—42no—
11C22⋊Dic127—24no—
12C4×Dic127—42no—
13#41×C127—41no—
14#41×C127—41no—
15C2×Q8×C127Q811yescharacteristic
16C2×D4×C127C2×C412yes—
17C2×D4×C127D411yescharacteristic
18C22×C4×C127C2×C411yescharacteristic
19C22×C4×C127D412yes—
20M16×C127—41no—
21C2×C8×C127—21no—
22C2×C8×C127—21no—
23C2×C8×C127—41no—
24C2×Dic127—44no—
25C23×C127C4211yescharacteristic
26C23×C127C22⋊C412yes—
27Q8×C127—41no—
28D4×C127—42no—
29D4×C127—41no—
30C2×C4×C127—21no—
31C2×C4×C127—21no—
32C2×C4×C127—22no—
33C2×C4×C127—22no—
34C2×C4×C127C22⋊C411yescharacteristic
35C2×C4×C127C4⋊C412yes—
36C2×C4×C127C4⋊C411yescharacteristic
37C8×C127—41no—
38C8×C127—41no—
39C22×C127—21no—
40C22×C127—22no—
41C22×C127—41no—
42C22×C127—42no—
43C22×C127#9511yescharacteristic
44C4×C127—22no—
45C4×C127—21no—
46C4×C127#9911yescharacteristic
47C4×C127#9911yescharacteristic
48Dic127—84no—
49C2×C127—21no—
50C2×C127—42no—
51C2×C127—41no—
52C2×C127#34211yescharacteristic
53#1391—1271nomaximal, Sylow
54C127#139111yescharacteristic, complemented, Sylow
55#410—1272no—
56#408D12711yescharacteristic
57C2×#41C2×D12711yescharacteristic
58C2×M16Dic12711yescharacteristic, complemented
59C22×C8Dic12711yescharacteristic, complemented
60#117—2542no—
61#99—1274no—
62#99—2542no—
63C22⋊C8—22no—
64#41—41no—
65#41—41no—
66C2×Q8Dic25411yescharacteristic
67C2×D4C4×D12712yes—
68C2×D4D50811yescharacteristic
69C22×C4C2×Dic12711yescharacteristic
70C22×C4C22⋊D12712yes—
71M16—41no—
72C2×C8—21no—
73C2×C8—21no—
74C2×C8—41no—
75C4⋊C4—5082no—
76C22⋊C4—2544no—
77C42—5082no—
78C23C4×Dic12711yescharacteristic
79C23(C2×D127)⋊C412yes—
80Q8—41no—
81D4—42no—
82D4—41no—
83C2×C4—22no—
84C2×C4—21no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—5084no—
88C2×C4C22⋊Dic12711yescharacteristic
89C2×C4Dic127⋊C412yes—
90C2×C4C4⋊Dic12711yescharacteristic
91C8—41no—
92C8—41no—
93C22—22no—
94C22—21no—
95C22—41no—
96C22—42no—
97C22C127⋊#9511yescharacteristic
98C4—22no—
99C4—21no—
100C4—10164no—
101C4#1686112411yescharacteristic
102C4#1686112411yescharacteristic
103C2—21no—
104C2—42no—
105C2—41no—
106C2#2254964311yescharacteristic
subgroupG/H
centerC4#16861124
commutatorC23×C127C42
FrattiniC2×#41C2×D127
Fitting#408×C127C2
socleC2×C127#342
prime radical, cosocleC2×#41×C127C22
hypercenter#408D127
hypocenterC127#1391
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
5. #16861124C4semidirect
7. C2×M16×C127C4semidirect
8. C22×C8×C127C4semidirect
54. C127#1391semidirect
58. C2×M16Dic127semidirect
59. C22×C8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498502
#408×C1272
#1391127