Finite Group Databasebeta

← #30636665

Subgroups — #30636665

Order 16256.

128 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 33 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#22549839C211yesmaximal, characteristic
2#22549658C211yesmaximal, characteristic, complemented
3#349×C127C211yesmaximal, characteristic, complemented
4C22×C4×D127C411yescharacteristic
5C2×#14523232C411yescharacteristic
6#16861149—21no—
7(C2×D127)⋊C8—21no—
8(C2×D127)⋊C8—21no—
9C2×M16×C127C2211yescharacteristic
10M32×C127—21no—
11C23×D127C811yescharacteristic
12C22×Dic127C811yescharacteristic
13C2×C4×D127—21no—
14C2×C4×D127—21no—
15C2×C4×D127—41no—
16C2×C4×D127—41no—
17#14523232—41no—
18C2×C127⋊C8—21no—
19C127⋊C16—41no—
20C22×C4×C127C2×C411yescharacteristic
21M16×C127—41no—
22C2×C8×C127D411yescharacteristic
23C2×C8×C127D411yescharacteristic
24C16×C127—41no—
25C22×D127—21no—
26C22×D127—42no—
27C22×D127—41no—
28C2×Dic127—21no—
29C2×Dic127—41no—
30C4×D127—42no—
31C4×D127—41no—
32C4×D127—41no—
33C127⋊C8—41no—
34C23×C127C2×C811yescharacteristic
35C2×C4×C127—21no—
36C2×C4×C127—21no—
37C2×C4×C127C22⋊C411yescharacteristic
38C2×C4×C127M1611yescharacteristic
39C8×C127—21no—
40C8×C127—21no—
41C22×C127—21no—
42C22×C127—41no—
43C22×C127C22⋊C811yescharacteristic
44C2×D127—22no—
45C2×D127—41no—
46C2×D127—42no—
47C2×D127—41no—
48C2×D127—42no—
49C4×C127—21no—
50C4×C127#9911yescharacteristic
51C4×C127#10011yescharacteristic
52Dic127—41no—
53C2×C127—21no—
54C2×C127—41no—
55C2×C127#32311yescharacteristic
56D127—42no—
57#1320—1271nomaximal, Sylow
58C127#132011yescharacteristic, complemented, Sylow
59#407—1271no—
60#349—1271no—
61#349D12711yescharacteristic, complemented
62C23×C4—1271no—
63C2×M16—1271no—
64C2×M16C2×D12711yescharacteristic
65M32—21no—
66M32—2541no—
67C22⋊C8—2541no—
68C22⋊C8—2541no—
69C24—1271no—
70C22×C4—1271no—
71C22×C4—5081no—
72C22×C4—5081no—
73C22×C4—2541no—
74C22×C4—2541no—
75C22×C4C4×D12711yescharacteristic
76M16—41no—
77M16—5081no—
78C2×C8—2541no—
79C2×C8C22⋊D12711yescharacteristic
80C2×C8D50811yescharacteristic
81C16—41no—
82C16—5081no—
83C23—5081no—
84C23—5082no—
85C23—2541no—
86C23C8×D12711yescharacteristic
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—5082no—
90C2×C4—5081no—
91C2×C4—5081no—
92C2×C4—5081no—
93C2×C4—2541no—
94C2×C4(C2×D127)⋊C411yescharacteristic
95C2×C4#1452323111yescharacteristic
96C8—21no—
97C8—21no—
98C8—5081no—
99C22—21no—
100C22—41no—
101C22—5081no—
102C22—5082no—
103C22—5082no—
104C22—5081no—
105C22—2542no—
106C22(C2×D127)⋊C811yescharacteristic
107C4—21no—
108C4—5081no—
109C4#1686112311yescharacteristic
110C4#1686112511yescharacteristic
111C2—21no—
112C2—41no—
113C2—5082no—
114C2#2254959911yescharacteristic
subgroupG/H
centerC4#16861123
commutatorC23×C127C2×C8
FrattiniC2×M16C2×D127
Fitting#349×C127C2
socleC2×C127#323
prime radicalC23×D127C8
cosocleC2×M16×C127C22
hypercenter#349D127
hypocenterC127#1320
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #22549658C2semidirect
3. #349×C127C2semidirect
58. C127#1320semidirect
61. #349D127semidirect

Maximal subgroups

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

subgroupindex
#225498392
#225496582
#349×C1272
#1320127