Finite Group Databasebeta

← #30636878

Subgroups — #30636878

Order 16256.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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#22549673C211yesmaximal, characteristic, complemented
2#22549616C211yesmaximal, characteristic, complemented
3#455×C127C211yesmaximal, characteristic
4#16861128—21no—
5#16861120—21no—
6H(5,𝔽2)×C127C411yescharacteristic
7#137×C127—21no—
8#124×C127C2211yescharacteristic
9#104×C127—21no—
10#100×C127C411yescharacteristic
11#14523232—41no—
12C2×C127⋊C8—41no—
13#41×C127—21no—
14#41×C127—41no—
15C2×Q8×C127D411yescharacteristic
16C2×D4×C127—41no—
17C2×D4×C127—41no—
18C2×D4×C127C2×C411yescharacteristic
19Q16×C127—41no—
20SD16×C127—41no—
21M16×C127—21no—
22C22⋊C4×C127—41no—
23C42×C127D411yescharacteristic
24C127⋊C8—42no—
25C127⋊C8—81no—
26C23×C127—21no—
27C23×C127—41no—
28Q8×C127—21no—
29Q8×C127—41no—
30D4×C127—21no—
31D4×C127—41no—
32D4×C127—42no—
33D4×C127—41no—
34C2×C4×C127—21no—
35C2×C4×C127—21no—
36C2×C4×C127—41no—
37C2×C4×C127—41no—
38C2×C4×C127C22⋊C411yescharacteristic
39C8×C127—41no—
40C22×C127—41no—
41C22×C127—41no—
42C22×C127—81no—
43C22×C127—161no—
44C22×C127#9911yescharacteristic
45C4×C127—21no—
46C4×C127—41no—
47C4×C127—41no—
48C4×C127—41no—
49C2×C127—21no—
50C2×C127—81no—
51C2×C127—81no—
52C2×C127AΣL(1,𝔽16)11yescharacteristic
53#1403—1271nomaximal, Sylow
54C127#140311yescharacteristic, complemented, Sylow
55#455D12711yescharacteristic
56#355—1271no—
57#329—1271no—
58H(5,𝔽2)Dic12711yescharacteristic
59#137—21no—
60#124C2×D12711yescharacteristic
61#104—21no—
62#101—2541no—
63#100Dic12711yescharacteristic
64#97—2541no—
65#41—21no—
66#41—41no—
67C2×Q8C22⋊D12711yescharacteristic
68C2×D4—41no—
69C2×D4—41no—
70C2×D4C2×Dic12711yescharacteristic
71Q16—41no—
72SD16—41no—
73M16—21no—
74M16—5081no—
75C2×C8—5081no—
76C22⋊C4—41no—
77C42C22⋊D12711yescharacteristic
78C23—21no—
79C23—41no—
80Q8—21no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—42no—
85D4—41no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4C22⋊Dic12711yescharacteristic
91C8—41no—
92C8—10161no—
93C8—5082no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22—161no—
98C22#1686112411yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—41no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#2254966311yescharacteristic
subgroupG/H
centerC2#22549663
commutatorC2×D4×C127C2×C4
Frattini#124C2×D127
Fitting#455×C127C2
socleC2×C127AΣL(1,𝔽16)
prime radicalH(5,𝔽2)×C127C4
cosocle#124×C127C22
hypercenter#455D127
hypocenterC127#1403
LCSG ▷ C2×D4×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #22549673C2semidirect
2. #22549616C2semidirect
54. C127#1403semidirect

Maximal subgroups

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

subgroupindex
#225496732
#225496162
#455×C1272
#1403127