Finite Group Databasebeta

← #1405×C251 #44545992

Subgroups — #1405×C251 #44545992

Order 32128.

106 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 28 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#456×C251C211yesmaximal, characteristic
2#356×C251C211yesmaximal, characteristic, complemented
3#333×C251C211yesmaximal, characteristic, complemented
42−1+4×C251C411yescharacteristic
5#137×C251—21no—
6C4⋊Q8×C251C2211yescharacteristic
7C4⋊C8×C251—21no—
8#104×C251—21no—
9#101×C251—21no—
10#101×C251C411yescharacteristic
11#41×C251—21no—
12#41×C251—41no—
13#41×C251—41no—
14C2×Q8×C251—41no—
15C2×Q8×C251C2×C411yescharacteristic
16C2×Q8×C251D411yescharacteristic
17Q16×C251—41no—
18SD16×C251—41no—
19M16×C251—21no—
20M16×C251—41no—
21C2×C8×C251—41no—
22C4⋊C4×C251—41no—
23C42×C251D411yescharacteristic
24Q8×C251—21no—
25Q8×C251—41no—
26Q8×C251—41no—
27Q8×C251—41no—
28D4×C251—21no—
29D4×C251—41no—
30D4×C251—41no—
31C2×C4×C251—21no—
32C2×C4×C251—21no—
33C2×C4×C251—21no—
34C2×C4×C251—41no—
35C2×C4×C251—42no—
36C2×C4×C251C22⋊C411yescharacteristic
37C8×C251—41no—
38C8×C251—81no—
39C8×C251—81no—
40C22×C251—41no—
41C22×C251#9911yescharacteristic
42C4×C251—21no—
43C4×C251—21no—
44C4×C251—21no—
45C4×C251—41no—
46C4×C251—41no—
47C4×C251—41no—
48C2×C251—21no—
49C2×C251—81no—
50C2×C251AΣL(1,𝔽16)11yescharacteristic
51C251#140511yescharacteristic, direct product factor, Sylow
52#1405C25111yesmaximal, characteristic, direct product factor, Sylow
53#456C2×C25111yescharacteristic
54#356C2×C25111yescharacteristic, complemented
55#333C2×C25111yescharacteristic, complemented
562−1+4C4×C25111yescharacteristic
57#137—21no—
58C4⋊Q8C22×C25111yescharacteristic
59C4⋊C8—21no—
60#104—21no—
61#101—21no—
62#101C4×C25111yescharacteristic
63#41—21no—
64#41—41no—
65#41—41no—
66C2×Q8—41no—
67C2×Q8C2×C4×C25111yescharacteristic
68C2×Q8D4×C25111yescharacteristic
69Q16—41no—
70SD16—41no—
71M16—21no—
72M16—41no—
73C2×C8—41no—
74C4⋊C4—41no—
75C42D4×C25111yescharacteristic
76Q8—21no—
77Q8—41no—
78Q8—41no—
79Q8—41no—
80D4—21no—
81D4—41no—
82D4—41no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—41no—
87C2×C4—42no—
88C2×C4C22⋊C4×C25111yescharacteristic
89C8—41no—
90C8—81no—
91C8—81no—
92C22—41no—
93C22#99×C25111yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—41no—
100C2—21no—
101C2—81no—
102C2AΣL(1,𝔽16)×C25111yescharacteristic
subgroupG/H
center, socleC2×C251AΣL(1,𝔽16)
commutatorC2×Q8C2×C4×C251
Frattini, cosocleC4⋊Q8C22×C251
prime radical2−1+4×C251C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C22×C251 ◁ C2×C4×C251 ◁ C4⋊Q8×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #356×C251C2semidirect
3. #333×C251C2semidirect
51. C251#1405direct (N × H)
52. #1405C251direct (N × H)
54. #356C2×C251semidirect
55. #333C2×C251semidirect

Maximal subgroups

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

subgroupindex
#456×C2512
#356×C2512
#333×C2512
#1405251