Finite Group Databasebeta

← #30636883

Subgroups — #30636883

Order 16256.

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

normal subgroups only

This group has 106 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#22550058C211yesmaximal, characteristic
2#22549625C211yesmaximal, characteristic, complemented
3#356×C127C211yesmaximal, characteristic, complemented
4#16861257C411yescharacteristic
5#16861240—21no—
6C4⋊(C127⋊C8)—21no—
7#16861137—21no—
8#16861128C411yescharacteristic
9C4⋊Q8×C127C2211yescharacteristic
10#101×C127—21no—
11#14523249—41no—
12#14523248—21no—
13#14523248—41no—
14Q8×D127—41no—
15#14523238—41no—
16#14523235—41no—
17#14523232—21no—
18C2×C127⋊C8—41no—
19C2×Q8×C127C2×C411yescharacteristic
20C2×Q8×C127D411yescharacteristic
21M16×C127—41no—
22C4⋊C4×C127—41no—
23C42×C127D411yescharacteristic
24Dic254—21no—
25Dic254—41no—
26D508—21no—
27D508—41no—
28C22⋊D127—41no—
29C4×D127—41no—
30C4×D127—42no—
31C127⋊C8—41no—
32C127⋊C8—81no—
33Q8×C127—41no—
34Q8×C127—41no—
35C2×C4×C127—21no—
36C2×C4×C127—21no—
37C2×C4×C127—21no—
38C2×C4×C127C22⋊C411yescharacteristic
39C8×C127—81no—
40C22×C127#9911yescharacteristic
41C2×D127—41no—
42C4×C127—21no—
43C4×C127—21no—
44C4×C127—21no—
45C4×C127—41no—
46C4×C127—41no—
47Dic127—41no—
48C2×C127—21no—
49C2×C127AΣL(1,𝔽16)11yescharacteristic
50D127—81no—
51#1405—1271nomaximal, Sylow
52C127#140511yescharacteristic, complemented, Sylow
53#456—1271no—
54#356D12711yescharacteristic, complemented
55#333—1271no—
562−1+4—1271no—
57#137—2541no—
58C4⋊Q8C2×D12711yescharacteristic
59C4⋊C8—2541no—
60#104—2541no—
61#101—21no—
62#101—1271no—
63#41—5081no—
64#41—5081no—
65#41—2541no—
66C2×Q8—5081no—
67C2×Q8C4×D12711yescharacteristic
68C2×Q8C22⋊D12711yescharacteristic
69Q16—5081no—
70SD16—5081no—
71M16—41no—
72M16—2541no—
73C2×C8—5081no—
74C4⋊C4—41no—
75C42D50811yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—5081no—
79Q8—2541no—
80D4—5081no—
81D4—5081no—
82D4—2541no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—5081no—
87C2×C4—5082no—
88C2×C4(C2×D127)⋊C411yescharacteristic
89C8—81no—
90C8—10161no—
91C8—5081no—
92C22—5081no—
93C22#1686112311yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—5081no—
100C2—21no—
101C2—10161no—
102C2#2254966211yescharacteristic
subgroupG/H
centerC2#22549662
commutatorC2×Q8×C127C2×C4
FrattiniC4⋊Q8C2×D127
Fitting#356×C127C2
socleC2×C127AΣL(1,𝔽16)
prime radical#16861257C4
cosocleC4⋊Q8×C127C22
hypercenter#356D127
hypocenterC127#1405
LCSG ▷ C2×Q8×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #22549625C2semidirect
3. #356×C127C2semidirect
52. C127#1405semidirect
54. #356D127semidirect

Maximal subgroups

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

subgroupindex
#225500582
#225496252
#356×C1272
#1405127