Finite Group Databasebeta

← #1352×C251 #44545939

Subgroups — #1352×C251 #44545939

Order 32128.

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

normal subgroups only

This group has 154 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#399×C251C211yesmaximal, characteristic
2#342×C251C211yesmaximal, characteristic
3#323×C251C211yesmaximal, characteristic
4C2×C4⋊C4×C251—21no—
5C2×C4⋊C4×C251C411yescharacteristic, complemented
6C2×C22⋊C4×C251—21no—
7C2×C22⋊C4×C251—21no—
8C2×C22⋊C4×C251C2211yescharacteristic
9C22⋊C8×C251—21no—
10#95×C251C411yescharacteristic, complemented
11C24×C251D411yescharacteristic
12C22×C4×C251—21no—
13C22×C4×C251—21no—
14C22×C4×C251—41no—
15C22×C4×C251C2×C411yescharacteristic
16C22×C4×C251D411yescharacteristic
17C2×C8×C251—41no—
18C4⋊C4×C251—41no—
19C4⋊C4×C251—42no—
20C22⋊C4×C251—22no—
21C22⋊C4×C251—41no—
22C22⋊C4×C251—42no—
23C22⋊C4×C251—41no—
24C23×C251—21no—
25C23×C251—21no—
26C23×C251—41no—
27C23×C251—41no—
28C23×C251C22⋊C411yescharacteristic
29C23×C251SD1611yescharacteristic
30C23×C251Q1611yescharacteristic
31C2×C4×C251—22no—
32C2×C4×C251—21no—
33C2×C4×C251—21no—
34C2×C4×C251—41no—
35C2×C4×C251—41no—
36C2×C4×C251—41no—
37C2×C4×C251—41no—
38C2×C4×C251—42no—
39C2×C4×C251—41no—
40C2×C4×C251—41no—
41C2×C4×C251—41no—
42C2×C4×C251—81no—
43C2×C4×C251—81no—
44C8×C251—81no—
45C22×C251—21no—
46C22×C251—21no—
47C22×C251—21no—
48C22×C251—21no—
49C22×C251—41no—
50C22×C251—41no—
51C22×C251—41no—
52C22×C251—41no—
53C22×C251—81no—
54C22×C251#9911yescharacteristic
55C22×C251#10311yescharacteristic
56C22×C251#10411yescharacteristic
57C4×C251—42no—
58C4×C251—41no—
59C4×C251—41no—
60C4×C251—82no—
61C4×C251—81no—
62C2×C251—21no—
63C2×C251—21no—
64C2×C251—41no—
65C2×C251—41no—
66C2×C251#32811yescharacteristic
67C2×C251#35211yescharacteristic
68C2×C251#35311yescharacteristic
69C251#135211yescharacteristic, direct product factor, Sylow
70#1352C25111yesmaximal, characteristic, direct product factor, Sylow
71#399C2×C25111yescharacteristic
72#342C2×C25111yescharacteristic
73#323C2×C25111yescharacteristic
74C2×C4⋊C4—21no—
75C2×C4⋊C4C4×C25111yescharacteristic, complemented
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4C22×C25111yescharacteristic
79C22⋊C8—21no—
80#95C4×C25111yescharacteristic, complemented
81C24D4×C25111yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4C2×C4×C25111yescharacteristic
86C22×C4D4×C25111yescharacteristic
87C2×C8—41no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C22⋊C4—22no—
91C22⋊C4—41no—
92C22⋊C4—42no—
93C22⋊C4—41no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C22⋊C4×C25111yescharacteristic
99C23SD16×C25111yescharacteristic
100C23Q16×C25111yescharacteristic
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—81no—
113C2×C4—81no—
114C8—81no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#99×C25111yescharacteristic
125C22#103×C25111yescharacteristic
126C22#104×C25111yescharacteristic
127C4—42no—
128C4—41no—
129C4—41no—
130C4—82no—
131C4—81no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#328×C25111yescharacteristic
137C2#352×C25111yescharacteristic
138C2#353×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#99
commutatorC22×C4C2×C4×C251
Frattini, cosocleC2×C22⋊C4C22×C251
prime radicalC24×C251D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
5. C2×C4⋊C4×C251C4semidirect
10. #95×C251C4semidirect
69. C251#1352direct (N × H)
70. #1352C251direct (N × H)
75. C2×C4⋊C4C4×C251semidirect
80. #95C4×C251semidirect

Maximal subgroups

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

subgroupindex
#399×C2512
#342×C2512
#323×C2512
#1352251