Finite Group Databasebeta

← #1345×C251 #44545932

Subgroups — #1345×C251 #44545932

Order 32128.

168 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 40 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#394×C251C211yesmaximal, characteristic
2#343×C251C211yesmaximal, characteristic, complemented
3#324×C251C211yesmaximal, characteristic, complemented
4C22×D4×C251C411yescharacteristic
5C2×M16×C251—21no—
6C2×C4⋊C4×C251C2211yescharacteristic
7C2×C22⋊C4×C251—21no—
8C2×C22⋊C4×C251—21no—
9C22⋊C8×C251—21no—
10#95×C251C411yescharacteristic
11C24×C251—21no—
12C2×D4×C251—41no—
13C2×D4×C251—42no—
14C22×C4×C251—21no—
15C22×C4×C251C2×C411yescharacteristic
16C22×C4×C251D411yescharacteristic
17C22×C4×C251D411yescharacteristic
18M16×C251—22no—
19M16×C251—41no—
20C2×C8×C251—41no—
21C2×C8×C251—41no—
22C4⋊C4×C251—41no—
23C22⋊C4×C251—42no—
24C22⋊C4×C251—41no—
25C22⋊C4×C251—41no—
26C23×C251—22no—
27C23×C251—41no—
28C23×C251—41no—
29C23×C251—41no—
30C23×C251—41no—
31C23×C251—41no—
32C23×C251—41no—
33C23×C251C22⋊C411yescharacteristic
34D4×C251—42no—
35D4×C251—42no—
36C2×C4×C251—21no—
37C2×C4×C251—21no—
38C2×C4×C251—21no—
39C2×C4×C251—21no—
40C2×C4×C251—41no—
41C2×C4×C251—41no—
42C2×C4×C251—41no—
43C2×C4×C251—41no—
44C2×C4×C251—41no—
45C2×C4×C251D811yescharacteristic
46C2×C4×C251SD1611yescharacteristic
47C8×C251—42no—
48C8×C251—81no—
49C22×C251—21no—
50C22×C251—21no—
51C22×C251—42no—
52C22×C251—42no—
53C22×C251—42no—
54C22×C251—81no—
55C22×C251—81no—
56C22×C251—81no—
57C22×C251—81no—
58C22×C251#9911yescharacteristic
59C22×C251#10211yescharacteristic
60C22×C251#10411yescharacteristic
61C4×C251—21no—
62C4×C251—21no—
63C4×C251—41no—
64C4×C251—41no—
65C4×C251—81no—
66C2×C251—21no—
67C2×C251—21no—
68C2×C251—82no—
69C2×C251#32711yescharacteristic
70C2×C251AΣL(1,𝔽16)11yescharacteristic
71C2×C251#35511yescharacteristic
72C251#134511yescharacteristic, direct product factor, Sylow
73#1345C25111yesmaximal, characteristic, direct product factor, Sylow
74#394C2×C25111yescharacteristic
75#343C2×C25111yescharacteristic, complemented
76#324C2×C25111yescharacteristic, complemented
77C22×D4C4×C25111yescharacteristic
78C2×M16—21no—
79C2×C4⋊C4C22×C25111yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C22⋊C8—21no—
83#95C4×C25111yescharacteristic
84C24—21no—
85C2×D4—41no—
86C2×D4—42no—
87C22×C4—21no—
88C22×C4C2×C4×C25111yescharacteristic
89C22×C4D4×C25111yescharacteristic
90C22×C4D4×C25111yescharacteristic
91M16—22no—
92M16—41no—
93C2×C8—41no—
94C2×C8—41no—
95C4⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C23—22no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C22⋊C4×C25111yescharacteristic
107D4—42no—
108D4—42no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4D8×C25111yescharacteristic
119C2×C4SD16×C25111yescharacteristic
120C8—42no—
121C8—81no—
122C22—21no—
123C22—21no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#99×C25111yescharacteristic
132C22#102×C25111yescharacteristic
133C22#104×C25111yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—81no—
139C2—21no—
140C2—21no—
141C2—82no—
142C2#327×C25111yescharacteristic
143C2AΣL(1,𝔽16)×C25111yescharacteristic
144C2#355×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#99
commutatorC22×C4C2×C4×C251
Frattini, cosocleC2×C4⋊C4C22×C251
prime radicalC22×D4×C251C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C251 ◁ C23×C251 ◁ C2×C4⋊C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #343×C251C2semidirect
3. #324×C251C2semidirect
72. C251#1345direct (N × H)
73. #1345C251direct (N × H)
75. #343C2×C251semidirect
76. #324C2×C251semidirect

Maximal subgroups

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

subgroupindex
#394×C2512
#343×C2512
#324×C2512
#1345251