Finite Group Databasebeta

← #44548511

Subgroups — #44548511

Order 32128.

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

normal subgroups only

This group has 122 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#30631854C211yesmaximal, characteristic
2#30631480C211yesmaximal, characteristic, complemented
3#331×C251C211yesmaximal, characteristic, complemented
4#22548760C411yescharacteristic, complemented
5#22548741—21no—
6#22548643—21no—
7#22548632C411yescharacteristic, complemented
8#22548630—21no—
9#127×C251C2211yescharacteristic
10C4⋊C8×C251—21no—
11#16860810—21no—
12#16860809—41no—
13C2×C22⋊D251—41no—
14D4×D251—41no—
15#16860798—41no—
16#16860795—41no—
17#16860794—21no—
18C22⋊Dic251—41no—
19C2×D4×C251—41no—
20C2×D4×C251C2×C411yescharacteristic
21C2×D4×C251D411yescharacteristic
22C2×C8×C251—41no—
23C42×C251D411yescharacteristic
24C22×D251—41no—
25Dic502—21no—
26D1004—21no—
27C22⋊D251—42no—
28C22⋊D251—41no—
29C2×Dic251—41no—
30C2×Dic251—81no—
31C4×D251—41no—
32C251⋊C8—41no—
33C23×C251—21no—
34C23×C251—21no—
35D4×C251—41no—
36D4×C251—41no—
37D4×C251—81no—
38D4×C251—81no—
39C2×C4×C251—21no—
40C2×C4×C251C22⋊C411yescharacteristic
41C8×C251—81no—
42C22×C251—41no—
43C22×C251—41no—
44C22×C251—81no—
45C22×C251—81no—
46C22×C251#9911yescharacteristic
47C2×D251—41no—
48C2×D251—161no—
49C4×C251—21no—
50C4×C251—21no—
51C4×C251—21no—
52Dic251—41no—
53Dic251—161no—
54C2×C251—21no—
55C2×C251—81no—
56C2×C251—81no—
57C2×C251AΣL(1,𝔽16)11yescharacteristic
58D251—81no—
59C251#140211yescharacteristic, complemented, Sylow
60#1402—2511nomaximal, Sylow
61#453—2511no—
62#353—2511no—
63#331D25111yescharacteristic, complemented
64H(5,𝔽2)—2511no—
65AGL(1,ℤ/8)—5021no—
66#127C2×D25111yescharacteristic
67C4⋊C8—21no—
68#104—5021no—
69#100—2511no—
70#99—5021no—
71#41—10041no—
72#41—5021no—
73C2×D4—41no—
74C2×D4—10041no—
75C2×D4—10041no—
76C2×D4C4×D25111yescharacteristic
77C2×D4D100411yescharacteristic
78SD16—10041no—
79D8—10041no—
80M16—5021no—
81C2×C8—41no—
82C22⋊C4—10041no—
83C42C22⋊D25111yescharacteristic
84C23—21no—
85C23—21no—
86C23—10041no—
87Q8—5021no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—10042no—
93D4—10041no—
94D4—5021no—
95C2×C4—21no—
96C2×C4—20081no—
97C2×C4—10041no—
98C2×C4—10041no—
99C2×C4(C2×D251)⋊C411yescharacteristic
100C8—81no—
101C8—10041no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—40161no—
107C22—10041no—
108C22#2254862911yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—40161no—
113C4—10041no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—20081no—
118C2#3063147511yescharacteristic
subgroupG/H
centerC2#30631475
commutatorC2×D4×C251C2×C4
Frattini#127C2×D251
Fitting#331×C251C2
socleC2×C251AΣL(1,𝔽16)
prime radical#30631854C2
cosocle#127×C251C22
hypercenter#331D251
hypocenterC251#1402
LCSG ▷ C2×D4×C251 ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30631480C2semidirect
3. #331×C251C2semidirect
4. #22548760C4semidirect
7. #22548632C4semidirect
59. C251#1402semidirect
63. #331D251semidirect

Maximal subgroups

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

subgroupindex
#306318542
#306314802
#331×C2512
#1402251