Finite Group Databasebeta

← #2134×C251 #44546721

Subgroups — #2134×C251 #44546721

Order 32128.

262 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 84 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#535×C251C211yesmaximal, characteristic
2#413×C251C212yesmaximal, complemented
3#355×C251C214yesmaximal, complemented
4C2×#41×C251C2211yescharacteristic
5C22×D4×C251C411yescharacteristic
6C2×M16×C251—22no—
7#127×C251C412yes—
8#124×C251C2212yes—
9#121×C251—22no—
10#120×C251—22no—
11#117×C251C411yescharacteristic
12#101×C251C2214yescomplemented
13#100×C251—22no—
14C24×C251—21no—
15#41×C251—42no—
16C2×Q8×C251C2311yescharacteristic
17C2×D4×C251—24no—
18C2×D4×C251—21no—
19C2×D4×C251—42no—
20C2×D4×C251C2×C412yes—
21C2×D4×C251C2×C411yescharacteristic
22C2×D4×C251D412yes—
23C22×C4×C251C2×C411yescharacteristic
24C22×C4×C251D412yes—
25M16×C251—24no—
26M16×C251—42no—
27C2×C8×C251—42no—
28C4⋊C4×C251—21no—
29C22⋊C4×C251—21no—
30C22⋊C4×C251—24no—
31C42×C251C2×C412yes—
32C23×C251—22no—
33C23×C251—42no—
34C23×C251—82no—
35C23×C251C22⋊C411yescharacteristic
36C23×C251C22⋊C412yes—
37Q8×C251—41no—
38D4×C251—41no—
39D4×C251—42no—
40D4×C251—42no—
41D4×C251—42no—
42D4×C251—84no—
43C2×C4×C251—22no—
44C2×C4×C251—22no—
45C2×C4×C251—22no—
46C2×C4×C251—22no—
47C2×C4×C251C22⋊C411yescharacteristic
48C2×C4×C251C22×C411yescharacteristic
49C2×C4×C251C2×D412yes—
50C8×C251—44no—
51C22×C251—21no—
52C22×C251—22no—
53C22×C251—42no—
54C22×C251—41no—
55C22×C251—44no—
56C22×C251—42no—
57C22×C251—84no—
58C22×C251C2×C22⋊C411yescharacteristic
59C4×C251—22no—
60C4×C251—21no—
61C4×C251—42no—
62C4×C251#9912yes—
63C2×C251—21no—
64C2×C251—42no—
65C2×C251—41no—
66C2×C251—82no—
67C2×C251C2×#9911yescharacteristic
68C251#213411yescharacteristic, direct product factor, Sylow
69#2134C25111yesmaximal, characteristic, direct product factor, Sylow
70#535C2×C25111yescharacteristic
71#413C2×C25112yescomplemented
72#355C2×C25114yescomplemented
73C2×#41C22×C25111yescharacteristic
74C22×D4C4×C25111yescharacteristic
75C2×M16—22no—
76#127C4×C25112yes—
77#124C22×C25112yes—
78#121—22no—
79#120—22no—
80#117C4×C25111yescharacteristic
81#101C22×C25114yescomplemented
82#100—22no—
83C24—21no—
84#41—42no—
85C2×Q8C23×C25111yescharacteristic
86C2×D4—24no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4C2×C4×C25112yes—
90C2×D4C2×C4×C25111yescharacteristic
91C2×D4D4×C25112yes—
92C22×C4C2×C4×C25111yescharacteristic
93C22×C4D4×C25112yes—
94M16—24no—
95M16—42no—
96C2×C8—42no—
97C4⋊C4—21no—
98C22⋊C4—21no—
99C22⋊C4—24no—
100C42C2×C4×C25112yes—
101C23—22no—
102C23—42no—
103C23—82no—
104C23C22⋊C4×C25111yescharacteristic
105C23C22⋊C4×C25112yes—
106Q8—41no—
107D4—41no—
108D4—42no—
109D4—42no—
110D4—42no—
111D4—84no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4C22⋊C4×C25111yescharacteristic
117C2×C4C22×C4×C25111yescharacteristic
118C2×C4C2×D4×C25112yes—
119C8—44no—
120C22—21no—
121C22—22no—
122C22—42no—
123C22—41no—
124C22—44no—
125C22—42no—
126C22—84no—
127C22C2×C22⋊C4×C25111yescharacteristic
128C4—22no—
129C4—21no—
130C4—42no—
131C4#99×C25112yes—
132C2—21no—
133C2—41no—
134C2—42no—
135C2—82no—
136C2C2×#99×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C2×#99
commutatorC2×C4C22×C4×C251
Frattini, cosocleC2×Q8C23×C251
prime radical#535×C251C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C2×C4×C251 ◁ C2×#41×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #413×C251C2semidirect
3. #355×C251C2semidirect
12. #101×C251C22semidirect
68. C251#2134direct (N × H)
69. #2134C251direct (N × H)
71. #413C2×C251semidirect
72. #355C2×C251semidirect
81. #101C22×C251semidirect

Maximal subgroups

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

subgroupindex
#535×C2512
#413×C2512
#355×C2512
#2134251