Finite Group Databasebeta

← #30636877

Subgroups — #30636877

Order 16256.

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

normal subgroups only

This group has 112 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#22550053C211yesmaximal, characteristic
2#22549616C211yesmaximal, characteristic, complemented
3#355×C127C211yesmaximal, characteristic, complemented
4#16861254C411yescharacteristic
5#16861240—21no—
6#16861137—21no—
7#16861126C411yescharacteristic
8#16861120—21no—
9#124×C127C2211yescharacteristic
10#101×C127—21no—
11#14523248—21no—
12#14523247—41no—
13C2×C22⋊D127—41no—
14D4×D127—41no—
15#14523238—41no—
16#14523235—41no—
17#14523232—21no—
18C2×C127⋊C8—41no—
19C2×Q8×C127D411yescharacteristic
20C2×D4×C127C2×C411yescharacteristic
21M16×C127—41no—
22C22⋊C4×C127—41no—
23C42×C127D411yescharacteristic
24C22×D127—41no—
25Dic254—21no—
26D508—21no—
27C22⋊D127—42no—
28C22⋊D127—41no—
29C2×Dic127—41no—
30C4×D127—41no—
31C127⋊C8—41no—
32C127⋊C8—42no—
33C23×C127—21no—
34Q8×C127—41no—
35D4×C127—41no—
36C2×C4×C127—21no—
37C2×C4×C127—21no—
38C2×C4×C127C22⋊C411yescharacteristic
39C8×C127—81no—
40C22×C127—41no—
41C22×C127—81no—
42C22×C127#9911yescharacteristic
43C2×D127—41no—
44C2×D127—161no—
45C4×C127—21no—
46C4×C127—41no—
47C4×C127—41no—
48Dic127—41no—
49C2×C127—21no—
50C2×C127—81no—
51C2×C127AΣL(1,𝔽16)11yescharacteristic
52D127—81no—
53#1403—1271nomaximal, Sylow
54C127#140311yescharacteristic, complemented, Sylow
55#455—1271no—
56#355D12711yescharacteristic, complemented
57#329—1271no—
58H(5,𝔽2)—1271no—
59#137—2541no—
60#124C2×D12711yescharacteristic
61#104—2541no—
62#101—21no—
63#100—1271no—
64#97—2541no—
65#41—5081no—
66#41—2541no—
67C2×Q8C22⋊D12711yescharacteristic
68C2×D4—5081no—
69C2×D4—5081no—
70C2×D4C4×D12711yescharacteristic
71Q16—5081no—
72SD16—5081no—
73M16—41no—
74M16—2541no—
75C2×C8—5081no—
76C22⋊C4—41no—
77C42D50811yescharacteristic
78C23—21no—
79C23—5081no—
80Q8—41no—
81Q8—2541no—
82D4—41no—
83D4—5082no—
84D4—5081no—
85D4—2541no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—5081no—
89C2×C4—5081no—
90C2×C4(C2×D127)⋊C411yescharacteristic
91C8—81no—
92C8—5081no—
93C8—5082no—
94C22—41no—
95C22—81no—
96C22—20321no—
97C22—5081no—
98C22#1686112311yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—5081no—
103C2—21no—
104C2—81no—
105C2—10161no—
106C2#2254966211yescharacteristic
subgroupG/H
centerC2#22549662
commutatorC2×D4×C127C2×C4
Frattini#124C2×D127
Fitting#355×C127C2
socleC2×C127AΣL(1,𝔽16)
prime radical#16861254C4
cosocle#124×C127C22
hypercenter#355D127
hypocenterC127#1403
LCSG ▷ C2×D4×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #22549616C2semidirect
3. #355×C127C2semidirect
54. C127#1403semidirect
56. #355D127semidirect

Maximal subgroups

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

subgroupindex
#225500532
#225496162
#355×C1272
#1403127