Finite Group Databasebeta

← #30636892

Subgroups — #30636892

Order 16256.

120 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 23 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#22550252C211yesmaximal, characteristic
2#22549669C211yesmaximal, characteristic, complemented
3#353×C127C211yesmaximal, characteristic, complemented
4C2×#14523233—21no—
5C2×#14523233—21no—
6#16861206C411yescharacteristic, complemented
7#16861124—21no—
8C4×C127⋊C8C411yescharacteristic, complemented
9#127×C127C2211yescharacteristic
10#99×C127—21no—
11C2×D508—21no—
12C2×D508—41no—
13#14523233—42no—
14#14523233—42no—
15C2×C127⋊C8—21no—
16C22⋊Dic127—41no—
17C2×D4×C127—41no—
18C2×D4×C127D411yescharacteristic
19C2×D4×C127D411yescharacteristic
20C22⋊C4×C127—41no—
21C42×C127C2×C411yescharacteristic
22C22×D127—41no—
23D508—41no—
24D508—41no—
25D508—82no—
26C2×Dic127—81no—
27C127⋊C8—22no—
28C23×C127—21no—
29C23×C127—21no—
30D4×C127—41no—
31D4×C127—41no—
32D4×C127—82no—
33C2×C4×C127—21no—
34C2×C4×C127—81no—
35C2×C4×C127C22⋊C411yescharacteristic
36C22×C127—41no—
37C22×C127—41no—
38C22×C127—81no—
39C22×C127—81no—
40C22×C127#9911yescharacteristic
41C2×D127—82no—
42C2×D127—81no—
43C4×C127—22no—
44C4×C127—21no—
45C4×C127—161no—
46Dic127—161no—
47C2×C127—21no—
48C2×C127—81no—
49C2×C127—81no—
50C2×C127#35311yescharacteristic
51D127—161no—
52#1408—1271nomaximal, Sylow
53C127#140811yescharacteristic, complemented, Sylow
54#493—1271no—
55#353—1271no—
56#353D12711yescharacteristic, complemented
57C2×D8—2541no—
58C2×D8—2541no—
59#127—1271no—
60#127C2×D12711yescharacteristic
61#99—21no—
62#99—2541no—
63C4×C8—1271no—
64C2×D4—41no—
65C2×D4—5081no—
66C2×D4—2541no—
67C2×D4C22⋊D12711yescharacteristic
68C2×D4D50811yescharacteristic
69D8—5082no—
70D8—5082no—
71C2×C8—2541no—
72C22⋊C4—41no—
73C22⋊C4—5081no—
74C42C4×D12711yescharacteristic
75C23—21no—
76C23—21no—
77C23—5081no—
78D4—41no—
79D4—41no—
80D4—82no—
81D4—10162no—
82D4—5081no—
83D4—5081no—
84C2×C4—21no—
85C2×C4—81no—
86C2×C4—10161no—
87C2×C4(C2×D127)⋊C411yescharacteristic
88C8—2542no—
89C22—41no—
90C22—41no—
91C22—81no—
92C22—81no—
93C22—10162no—
94C22—10161no—
95C22#1686112311yescharacteristic
96C4—22no—
97C4—21no—
98C4—161no—
99C4—20321no—
100C2—21no—
101C2—81no—
102C2—81no—
103C2—20321no—
104C2#2254966811yescharacteristic
subgroupG/H
centerC2#22549668
commutatorC42×C127C2×C4
Frattini#127C2×D127
Fitting#353×C127C2
socleC2×C127#353
prime radical#22550252C2
cosocle#127×C127C22
hypercenter#353D127
hypocenterC127#1408
LCSG ▷ C42×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #22549669C2semidirect
3. #353×C127C2semidirect
6. #16861206C4semidirect
8. C4×C127⋊C8C4semidirect
53. C127#1408semidirect
56. #353D127semidirect

Maximal subgroups

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

subgroupindex
#225502522
#225496692
#353×C1272
#1408127