Finite Group Databasebeta

← #30636714

Subgroups — #30636714

Order 16256.

114 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 31 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#22550129C211yesmaximal, characteristic
2#22549658C211yesmaximal, characteristic, complemented
3H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
4C2×D1016C411yescharacteristic, complemented
5#16861183—21no—
6#16861149—21no—
7#16861141C411yescharacteristic, complemented
8#16861131—21no—
9C2×M16×C127C2211yescharacteristic
10#117×C127—21no—
11C2×D508—21no—
12C2×C22⋊D127—41no—
13D1016—41no—
14C4⋊Dic127—21no—
15(C2×D127)⋊C4—41no—
16C127⋊C16—41no—
17C22×C4×C127D411yescharacteristic
18M16×C127—41no—
19C2×C8×C127C2×C411yescharacteristic
20C2×C8×C127D411yescharacteristic
21C4⋊C4×C127—41no—
22C22⋊C4×C127—41no—
23C42×C127—22no—
24C22×D127—41no—
25D508—41no—
26D508—41no—
27C22⋊D127—82no—
28C2×Dic127—41no—
29C23×C127D811yescharacteristic
30C2×C4×C127—21no—
31C2×C4×C127—21no—
32C2×C4×C127—42no—
33C2×C4×C127C22⋊C411yescharacteristic
34C2×C4×C127SD1611yescharacteristic
35C8×C127—21no—
36C8×C127—21no—
37C22×C127—21no—
38C22×C127—41no—
39C22×C127#10211yescharacteristic
40C2×D127—81no—
41C2×D127—81no—
42C2×D127—81no—
43C4×C127—21no—
44C4×C127—82no—
45C4×C127#9911yescharacteristic
46C4×C127#10411yescharacteristic
47Dic127—81no—
48C2×C127—21no—
49C2×C127—41no—
50C2×C127#32711yescharacteristic
51D127—161no—
52#1339—1271nomaximal, Sylow
53C127#133911yescharacteristic, complemented, Sylow
54#469—1271no—
55#349—1271no—
56H(3,ℤ/4)D12711yescharacteristic, complemented
57C2×D8—1271no—
58C2×M16C2×D12711yescharacteristic
59#121—2541no—
60#117—21no—
61M32—2541no—
62#106—1271no—
63#102—2541no—
64C2×D4—5081no—
65C2×D4—2541no—
66C22×C4C22⋊D12711yescharacteristic
67D8—5081no—
68M16—41no—
69C2×C8C4×D12711yescharacteristic
70C2×C8D50811yescharacteristic
71C4⋊C4—41no—
72C4⋊C4—2541no—
73C22⋊C4—41no—
74C22⋊C4—5081no—
75C42—22no—
76C16—5081no—
77C23—5081no—
78C23#1452323311yescharacteristic
79D4—10162no—
80D4—5081no—
81D4—5081no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—5081no—
86C2×C4(C2×D127)⋊C411yescharacteristic
87C2×C4#1452323511yescharacteristic
88C8—21no—
89C8—21no—
90C22—21no—
91C22—41no—
92C22—10161no—
93C22—10161no—
94C22—10161no—
95C22#1686112911yescharacteristic
96C4—21no—
97C4—82no—
98C4—10161no—
99C4#1686112311yescharacteristic
100C4#1686113711yescharacteristic
101C2—21no—
102C2—41no—
103C2—20321no—
104C2#2254960911yescharacteristic
subgroupG/H
centerC2#22549609
commutatorC2×C8×C127C2×C4
FrattiniC2×M16C2×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#327
prime radical#22550129C2
cosocleC2×M16×C127C22
hypercenterH(3,ℤ/4)D127
hypocenterC127#1339
LCSG ▷ C2×C8×C127 ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #22549658C2semidirect
3. H(3,ℤ/4)×C127C2semidirect
4. C2×D1016C4semidirect
7. #16861141C4semidirect
53. C127#1339semidirect
56. H(3,ℤ/4)D127semidirect

Maximal subgroups

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

subgroupindex
#225501292
#225496582
H(3,ℤ/4)×C1272
#1339127