Finite Group Databasebeta

← #44548512

Subgroups — #44548512

Order 32128.

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#30631864C211yesmaximal, characteristic
2#30631427C211yesmaximal, characteristic, complemented
3#355×C251C211yesmaximal, characteristic, complemented
4#22548760C411yescharacteristic
5#22548746—21no—
6#22548643—21no—
7#22548632C411yescharacteristic
8#22548626—21no—
9#124×C251C2211yescharacteristic
10#101×C251—21no—
11#16860810—21no—
12#16860809—41no—
13C2×C22⋊D251—41no—
14D4×D251—41no—
15#16860800—41no—
16#16860797—41no—
17#16860794—21no—
18C2×C251⋊C8—41no—
19C2×Q8×C251D411yescharacteristic
20C2×D4×C251C2×C411yescharacteristic
21M16×C251—41no—
22C22⋊C4×C251—41no—
23C42×C251D411yescharacteristic
24C22×D251—41no—
25Dic502—21no—
26D1004—21no—
27C22⋊D251—42no—
28C22⋊D251—41no—
29C2×Dic251—41no—
30C4×D251—41no—
31C251⋊C8—41no—
32C251⋊C8—42no—
33C23×C251—21no—
34Q8×C251—41no—
35D4×C251—41no—
36C2×C4×C251—21no—
37C2×C4×C251—21no—
38C2×C4×C251C22⋊C411yescharacteristic
39C8×C251—81no—
40C22×C251—41no—
41C22×C251—81no—
42C22×C251#9911yescharacteristic
43C2×D251—41no—
44C2×D251—161no—
45C4×C251—21no—
46C4×C251—41no—
47C4×C251—41no—
48Dic251—41no—
49C2×C251—21no—
50C2×C251—81no—
51C2×C251AΣL(1,𝔽16)11yescharacteristic
52D251—81no—
53C251#140311yescharacteristic, complemented, Sylow
54#1403—2511nomaximal, Sylow
55#455—2511no—
56#355D25111yescharacteristic, complemented
57#329—2511no—
58H(5,𝔽2)—2511no—
59#137—5021no—
60#124C2×D25111yescharacteristic
61#104—5021no—
62#101—21no—
63#100—2511no—
64#97—5021no—
65#41—10041no—
66#41—5021no—
67C2×Q8C22⋊D25111yescharacteristic
68C2×D4—10041no—
69C2×D4—10041no—
70C2×D4C4×D25111yescharacteristic
71Q16—10041no—
72SD16—10041no—
73M16—41no—
74M16—5021no—
75C2×C8—10041no—
76C22⋊C4—41no—
77C42D100411yescharacteristic
78C23—21no—
79C23—10041no—
80Q8—41no—
81Q8—5021no—
82D4—41no—
83D4—10042no—
84D4—10041no—
85D4—5021no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—10041no—
89C2×C4—10041no—
90C2×C4(C2×D251)⋊C411yescharacteristic
91C8—81no—
92C8—10041no—
93C8—10042no—
94C22—41no—
95C22—81no—
96C22—40161no—
97C22—10041no—
98C22#2254862911yescharacteristic
99C4—21no—
100C4—41no—
101C4—41no—
102C4—10041no—
103C2—21no—
104C2—81no—
105C2—20081no—
106C2#3063147311yescharacteristic
subgroupG/H
centerC2#30631473
commutatorC2×D4×C251C2×C4
Frattini#124C2×D251
Fitting#355×C251C2
socleC2×C251AΣL(1,𝔽16)
prime radical#22548760C4
cosocle#124×C251C22
hypercenter#355D251
hypocenterC251#1403
LCSG ▷ C2×D4×C251 ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #30631427C2semidirect
3. #355×C251C2semidirect
53. C251#1403semidirect
56. #355D251semidirect

Maximal subgroups

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

subgroupindex
#306318642
#306314272
#355×C2512
#1403251