Finite Group Databasebeta

← #1316×C251 #44545903

Subgroups — #1316×C251 #44545903

Order 32128.

294 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 44 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#379×C251C211yesmaximal, characteristic
2#323×C251C212yesmaximal, complemented
3C25×C251C411yescharacteristic
4C2×C22⋊C4×C251—22no—
5C2×C22⋊C4×C251C411yescharacteristic
6C2×C22⋊C4×C251C2211yescharacteristic
7C22⋊C8×C251—22no—
8C24×C251—21no—
9C24×C251—21no—
10C24×C251—44no—
11C24×C251—41no—
12C24×C251—41no—
13C24×C251C812yescomplemented
14C24×C251C2×C411yescharacteristic
15C22×C4×C251—21no—
16C22×C4×C251D412yes—
17C2×C8×C251—42no—
18C22⋊C4×C251—41no—
19C22⋊C4×C251—44no—
20C22⋊C4×C251—41no—
21C23×C251—24no—
22C23×C251—22no—
23C23×C251—21no—
24C23×C251—21no—
25C23×C251—44no—
26C23×C251—42no—
27C23×C251—42no—
28C23×C251—42no—
29C23×C251—44no—
30C23×C251—42no—
31C23×C251—44no—
32C23×C251—48no—
33C23×C251—44no—
34C23×C251—41no—
35C23×C251—41no—
36C23×C251C22⋊C411yescharacteristic
37C23×C251C2×C811yescharacteristic
38C23×C251M1611yescharacteristic
39C2×C4×C251—22no—
40C2×C4×C251—42no—
41C2×C4×C251—42no—
42C2×C4×C251—41no—
43C8×C251—82no—
44C22×C251—22no—
45C22×C251—24no—
46C22×C251—21no—
47C22×C251—21no—
48C22×C251—41no—
49C22×C251—41no—
50C22×C251—44no—
51C22×C251—42no—
52C22×C251—42no—
53C22×C251—44no—
54C22×C251—44no—
55C22×C251—48no—
56C22×C251—42no—
57C22×C251—44no—
58C22×C251—42no—
59C22×C251C22⋊C811yescharacteristic
60C22×C251#9911yescharacteristic
61C22×C251#10011yescharacteristic
62C4×C251—42no—
63C4×C251—81no—
64C2×C251—21no—
65C2×C251—21no—
66C2×C251—44no—
67C2×C251—41no—
68C2×C251—41no—
69C2×C251#32311yescharacteristic
70C2×C251AΣL(1,𝔽16)12yes—
71C251#131611yescharacteristic, direct product factor, Sylow
72#1316C25111yesmaximal, characteristic, direct product factor, Sylow
73#379C2×C25111yescharacteristic
74#323C2×C25112yescomplemented
75C25C4×C25111yescharacteristic
76C2×C22⋊C4—22no—
77C2×C22⋊C4C4×C25111yescharacteristic
78C2×C22⋊C4C22×C25111yescharacteristic
79C22⋊C8—22no—
80C24—21no—
81C24—21no—
82C24—44no—
83C24—41no—
84C24—41no—
85C24C8×C25112yescomplemented
86C24C2×C4×C25111yescharacteristic
87C22×C4—21no—
88C22×C4D4×C25112yes—
89C2×C8—42no—
90C22⋊C4—41no—
91C22⋊C4—44no—
92C22⋊C4—41no—
93C23—24no—
94C23—22no—
95C23—21no—
96C23—21no—
97C23—44no—
98C23—42no—
99C23—42no—
100C23—42no—
101C23—44no—
102C23—42no—
103C23—44no—
104C23—48no—
105C23—44no—
106C23—41no—
107C23—41no—
108C23C22⋊C4×C25111yescharacteristic
109C23C2×C8×C25111yescharacteristic
110C23M16×C25111yescharacteristic
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C8—82no—
116C22—22no—
117C22—24no—
118C22—21no—
119C22—21no—
120C22—41no—
121C22—41no—
122C22—44no—
123C22—42no—
124C22—42no—
125C22—44no—
126C22—44no—
127C22—48no—
128C22—44no—
129C22—42no—
130C22—42no—
131C22C22⋊C8×C25111yescharacteristic
132C22#99×C25111yescharacteristic
133C22#100×C25111yescharacteristic
134C4—42no—
135C4—81no—
136C2—21no—
137C2—21no—
138C2—44no—
139C2—41no—
140C2—41no—
141C2#323×C25111yescharacteristic
142C2AΣL(1,𝔽16)×C25112yes—
subgroupG/H
center, socleC22×C251#99
commutatorC23C2×C8×C251
Frattini, cosocleC2×C22⋊C4C22×C251
prime radicalC25×C251C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C251 ◁ C23×C251 ◁ C2×C22⋊C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #323×C251C2semidirect
13. C24×C251C8semidirect
71. C251#1316direct (N × H)
72. #1316C251direct (N × H)
74. #323C2×C251semidirect
85. C24C8×C251semidirect

Maximal subgroups

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

subgroupindex
#379×C2512
#323×C2512
#1316251