Finite Group Databasebeta

← #44548518

Subgroups — #44548518

Order 32128.

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#30631869C211yesmaximal, characteristic
2#30631436C211yesmaximal, characteristic, complemented
3#356×C251C211yesmaximal, characteristic, complemented
4#22548763C411yescharacteristic
5#22548746—21no—
6C4⋊(C251⋊C8)—21no—
7#22548643—21no—
8#22548634C411yescharacteristic
9C4⋊Q8×C251C2211yescharacteristic
10#101×C251—21no—
11#16860811—41no—
12#16860810—21no—
13#16860810—41no—
14Q8×D251—41no—
15#16860800—41no—
16#16860797—41no—
17#16860794—21no—
18C2×C251⋊C8—41no—
19C2×Q8×C251C2×C411yescharacteristic
20C2×Q8×C251D411yescharacteristic
21M16×C251—41no—
22C4⋊C4×C251—41no—
23C42×C251D411yescharacteristic
24Dic502—21no—
25Dic502—41no—
26D1004—21no—
27D1004—41no—
28C22⋊D251—41no—
29C4×D251—41no—
30C4×D251—42no—
31C251⋊C8—41no—
32C251⋊C8—81no—
33Q8×C251—41no—
34Q8×C251—41no—
35C2×C4×C251—21no—
36C2×C4×C251—21no—
37C2×C4×C251—21no—
38C2×C4×C251C22⋊C411yescharacteristic
39C8×C251—81no—
40C22×C251#9911yescharacteristic
41C2×D251—41no—
42C4×C251—21no—
43C4×C251—21no—
44C4×C251—21no—
45C4×C251—41no—
46C4×C251—41no—
47Dic251—41no—
48C2×C251—21no—
49C2×C251AΣL(1,𝔽16)11yescharacteristic
50D251—81no—
51C251#140511yescharacteristic, complemented, Sylow
52#1405—2511nomaximal, Sylow
53#456—2511no—
54#356D25111yescharacteristic, complemented
55#333—2511no—
562−1+4—2511no—
57#137—5021no—
58C4⋊Q8C2×D25111yescharacteristic
59C4⋊C8—5021no—
60#104—5021no—
61#101—21no—
62#101—2511no—
63#41—10041no—
64#41—10041no—
65#41—5021no—
66C2×Q8—10041no—
67C2×Q8C4×D25111yescharacteristic
68C2×Q8C22⋊D25111yescharacteristic
69Q16—10041no—
70SD16—10041no—
71M16—41no—
72M16—5021no—
73C2×C8—10041no—
74C4⋊C4—41no—
75C42D100411yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—10041no—
79Q8—5021no—
80D4—10041no—
81D4—10041no—
82D4—5021no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—10041no—
87C2×C4—10042no—
88C2×C4(C2×D251)⋊C411yescharacteristic
89C8—81no—
90C8—20081no—
91C8—10041no—
92C22—10041no—
93C22#2254862911yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—10041no—
100C2—21no—
101C2—20081no—
102C2#3063147311yescharacteristic
subgroupG/H
centerC2#30631473
commutatorC2×Q8×C251C2×C4
FrattiniC4⋊Q8C2×D251
Fitting#356×C251C2
socleC2×C251AΣL(1,𝔽16)
prime radical#22548763C4
cosocleC4⋊Q8×C251C22
hypercenter#356D251
hypocenterC251#1405
LCSG ▷ C2×Q8×C251 ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #30631436C2semidirect
3. #356×C251C2semidirect
51. C251#1405semidirect
54. #356D251semidirect

Maximal subgroups

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

subgroupindex
#306318692
#306314362
#356×C2512
#1405251