Finite Group Databasebeta

← C22×#102×C251 #44547477

Subgroups — C22×#102×C251 #44547477

Order 32128.

1072 conjugacy classes of subgroups; only the 440 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 440 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
1C23×D4×C251C211yesmaximal, characteristic
2C23×C8×C251C211yesmaximal, characteristic, complemented
3C22×C4⋊C4×C251C211yesmaximal, characteristic, complemented
4C2×#102×C251C2112yesmaximal, direct product factor
5C25×C251—21no—
6C22×D4×C251—23no—
7C22×D4×C251—24no—
8C22×D4×C251C418yescomplemented
9C22×D4×C251C2216yes—
10C23×C4×C251—21no—
11C23×C4×C251C2211yescharacteristic
12C22×C8×C251—24no—
13C22×C8×C251C2216yescomplemented
14C2×C4⋊C4×C251—23no—
15C2×C4⋊C4×C251C2216yescomplemented
16#102×C251C22116yesdirect product factor
17C24×C251—28no—
18C24×C251—26no—
19C24×C251—44no—
20C24×C251—44no—
21C24×C251D411yescharacteristic
22C2×D4×C251—224no—
23C2×D4×C251—26no—
24C2×D4×C251—212no—
25C2×D4×C251C2×C4124yescomplemented
26C2×D4×C251C2314yes—
27C22×C4×C251—26no—
28C22×C4×C251—44no—
29C22×C4×C251C2×C414yes—
30C22×C4×C251D413yes—
31C22×C4×C251D414yes—
32C22×C4×C251C2313yes—
33C2×C8×C251—212no—
34C2×C8×C251C2314yescomplemented
35C4⋊C4×C251—26no—
36C4⋊C4×C251C2314yescomplemented
37C23×C251—224no—
38C23×C251—24no—
39C23×C251—432no—
40C23×C251—412no—
41C23×C251—412no—
42C23×C251C22⋊C414yes—
43C23×C251D814yes—
44C23×C251SD1614yes—
45C23×C251C2×D413yes—
46D4×C251—224no—
47D4×C251—24no—
48D4×C251C22×C418yescomplemented
49C2×C4×C251—24no—
50C2×C4×C251—412no—
51C2×C4×C251C22⋊C4112yes—
52C2×C4×C251C22×C416yes—
53C2×C4×C251C2×D413yes—
54C2×C4×C251C2×D416yes—
55C2×C4×C251C2411yescharacteristic
56C8×C251—24no—
57C22×C251—28no—
58C22×C251—44no—
59C22×C251—44no—
60C22×C251—448no—
61C22×C251#102116yesdirect product factor
62C22×C251C2×C22⋊C416yes—
63C22×C251C2×D816yes—
64C22×C251C2×SD1616yes—
65C22×C251C22×D411yescharacteristic
66C4×C251—44no—
67C4×C251C2×C22⋊C416yes—
68C4×C251C23×C411yescharacteristic
69C4×C251C22×D411yescharacteristic
70C2×C251—48no—
71C2×C251C2×#102112yesdirect product factor
72C2×C251C22×C22⋊C411yescharacteristic
73C2×C251C22×D811yescharacteristic
74C2×C251C22×SD1611yescharacteristic
75C251C22×#10211yescharacteristic, direct product factor, Sylow
76C22×#102C25111yesmaximal, characteristic, direct product factor, Sylow
77C23×D4C2×C25111yescharacteristic
78C23×C8C2×C25111yescharacteristic, complemented
79C22×C4⋊C4C2×C25111yescharacteristic, complemented
80C2×#102C2×C251112yesdirect product factor
81C25—21no—
82C22×D4—23no—
83C22×D4—24no—
84C22×D4C4×C25118yescomplemented
85C22×D4C22×C25116yes—
86C23×C4—21no—
87C23×C4C22×C25111yescharacteristic
88C22×C8—24no—
89C22×C8C22×C25116yescomplemented
90C2×C4⋊C4—23no—
91C2×C4⋊C4C22×C25116yescomplemented
92#102C22×C251116yesdirect product factor
93C24—28no—
94C24—26no—
95C24—44no—
96C24—44no—
97C24D4×C25111yescharacteristic
98C2×D4—224no—
99C2×D4—26no—
100C2×D4—212no—
101C2×D4C2×C4×C251124yescomplemented
102C2×D4C23×C25114yes—
103C22×C4—26no—
104C22×C4—44no—
105C22×C4C2×C4×C25114yes—
106C22×C4D4×C25113yes—
107C22×C4D4×C25114yes—
108C22×C4C23×C25113yes—
109C2×C8—212no—
110C2×C8C23×C25114yescomplemented
111C4⋊C4—26no—
112C4⋊C4C23×C25114yescomplemented
113C23—224no—
114C23—24no—
115C23—432no—
116C23—412no—
117C23—412no—
118C23C22⋊C4×C25114yes—
119C23D8×C25114yes—
120C23SD16×C25114yes—
121C23C2×D4×C25113yes—
122D4—224no—
123D4—24no—
124D4C22×C4×C25118yescomplemented
125C2×C4—24no—
126C2×C4—412no—
127C2×C4C22⋊C4×C251112yes—
128C2×C4C22×C4×C25116yes—
129C2×C4C2×D4×C25113yes—
130C2×C4C2×D4×C25116yes—
131C2×C4C24×C25111yescharacteristic
132C8—24no—
133C22—28no—
134C22—448no—
135C22—44no—
136C22—44no—
137C22#102×C251116yesdirect product factor
138C22C2×C22⋊C4×C25116yes—
139C22C2×D8×C25116yes—
140C22C2×SD16×C25116yes—
141C22C22×D4×C25111yescharacteristic
142C4—44no—
143C4C2×C22⋊C4×C25116yes—
144C4C23×C4×C25111yescharacteristic
145C4C22×D4×C25111yescharacteristic
146C2—48no—
147C2C2×#102×C251112yesdirect product factor
148C2C22×C22⋊C4×C25111yescharacteristic
149C2C22×D8×C25111yescharacteristic
150C2C22×SD16×C25111yescharacteristic
subgroupG/H
center, socleC24×C251D4
commutatorC4C23×C4×C251
Frattini, cosocleC2×C4C24×C251
prime radicalC23×D4×C251C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C251 ◁ C23×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C8×C251C2semidirect
3. C22×C4⋊C4×C251C2semidirect
4. C2×#102×C251C2direct (N × H)
8. C22×D4×C251C4semidirect
13. C22×C8×C251C22semidirect
15. C2×C4⋊C4×C251C22semidirect
16. #102×C251C22direct (N × H)
25. C2×D4×C251C2×C4semidirect
34. C2×C8×C251C23semidirect
36. C4⋊C4×C251C23semidirect
48. D4×C251C22×C4semidirect
61. C22×C251#102direct (N × H)
71. C2×C251C2×#102direct (N × H)
75. C251C22×#102direct (N × H)
76. C22×#102C251direct (N × H)
78. C23×C8C2×C251semidirect
79. C22×C4⋊C4C2×C251semidirect
80. C2×#102C2×C251direct (N × H)
84. C22×D4C4×C251semidirect
89. C22×C8C22×C251semidirect
91. C2×C4⋊C4C22×C251semidirect
92. #102C22×C251direct (N × H)
101. C2×D4C2×C4×C251semidirect
110. C2×C8C23×C251semidirect
112. C4⋊C4C23×C251semidirect
124. D4C22×C4×C251semidirect
137. C22#102×C251direct (N × H)
147. C2C2×#102×C251direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4×C2512
C23×C8×C2512
C22×C4⋊C4×C2512
C2×#102×C2512
C22×#102251