Finite Group Databasebeta

← #2883×C251 #44547470

Subgroups — #2883×C251 #44547470

Order 32128.

800 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 340 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
1C2×H(5,𝔽2)×C251C211yesmaximal, characteristic
2#518×C251C212yesmaximal, complemented
3#410×C251C214yesmaximal, complemented
4C2×#99×C251C218yesmaximal, complemented
5H(5,𝔽2)×C251—28no—
6C2×#41×C251C414yescomplemented
7C2×#41×C251C2212yes—
8C22×D4×C251C414yescomplemented
9C22×D4×C251C2214yes—
10C22×D4×C251C2211yescharacteristic
11C4×D4×C251—28no—
12#117×C251C2214yescomplemented
13C2×C22⋊C4×C251C2218yescomplemented
14#99×C251C22116yescomplemented
15C24×C251C2×C414yescomplemented
16C24×C251C2312yes—
17#41×C251—216no—
18#41×C251—28no—
19C2×Q8×C251C2×C412yescomplemented
20C2×D4×C251—216no—
21C2×D4×C251—216no—
22C2×D4×C251C2×C414yes—
23C2×D4×C251C2×C412yescomplemented
24C2×D4×C251C2×C418yescomplemented
25C2×D4×C251D418yes—
26C2×D4×C251C2314yes—
27C22×C4×C251—28no—
28C22×C4×C251C2×C414yescomplemented
29C22×C4×C251C2×C414yescomplemented
30C22×C4×C251C2311yescharacteristic
31C4⋊C4×C251—24no—
32C22⋊C4×C251—28no—
33C22⋊C4×C251C2318yes—
34C42×C251—24no—
35C23×C251—28no—
36C23×C251—82no—
37C23×C251—84no—
38C23×C251C22×C414yes—
39C23×C251C22×C414yes—
40C23×C251C2×D418yes—
41C23×C251C2411yescharacteristic
42Q8×C251—24no—
43D4×C251—216no—
44D4×C251—28no—
45D4×C251—24no—
46D4×C251C22⋊C4116yes—
47C2×C4×C251—28no—
48C2×C4×C251—28no—
49C2×C4×C251—28no—
50C2×C4×C251—88no—
51C2×C4×C251C22×C414yes—
52C2×C4×C251C22×C412yes—
53C2×C4×C251C2×D414yes—
54C22×C251—24no—
55C22×C251—48no—
56C22×C251—44no—
57C22×C251—41no—
58C22×C251—44no—
59C22×C251—88no—
60C22×C251C2×C22⋊C418yes—
61C22×C251C23×C411yescharacteristic
62C22×C251C22×D412yes—
63C4×C251—24no—
64C4×C251—48no—
65C4×C251C2×C22⋊C414yes—
66C2×C251—22no—
67C2×C251—21no—
68C2×C251—28no—
69C2×C251—44no—
70C2×C251C22×C22⋊C411yescharacteristic
71C251#288311yescharacteristic, direct product factor, Sylow
72#2883C25111yesmaximal, characteristic, direct product factor, Sylow
73C2×H(5,𝔽2)C2×C25111yescharacteristic
74#518C2×C25112yescomplemented
75#410C2×C25114yescomplemented
76C2×#99C2×C25118yescomplemented
77H(5,𝔽2)—28no—
78C2×#41C4×C25114yescomplemented
79C2×#41C22×C25112yes—
80C22×D4C4×C25114yescomplemented
81C22×D4C22×C25114yes—
82C22×D4C22×C25111yescharacteristic
83C4×D4—28no—
84#117C22×C25114yescomplemented
85C2×C22⋊C4C22×C25118yescomplemented
86#99C22×C251116yescomplemented
87C24C2×C4×C25114yescomplemented
88C24C23×C25112yes—
89#41—216no—
90#41—28no—
91C2×Q8C2×C4×C25112yescomplemented
92C2×D4—216no—
93C2×D4—216no—
94C2×D4C2×C4×C25114yes—
95C2×D4C2×C4×C25112yescomplemented
96C2×D4C2×C4×C25118yescomplemented
97C2×D4D4×C25118yes—
98C2×D4C23×C25114yes—
99C22×C4—28no—
100C22×C4C2×C4×C25114yescomplemented
101C22×C4C2×C4×C25114yescomplemented
102C22×C4C23×C25111yescharacteristic
103C4⋊C4—24no—
104C22⋊C4—28no—
105C22⋊C4C23×C25118yes—
106C42—24no—
107C23—28no—
108C23—82no—
109C23—84no—
110C23C22×C4×C25114yes—
111C23C22×C4×C25114yes—
112C23C2×D4×C25118yes—
113C23C24×C25111yescharacteristic
114Q8—24no—
115D4—216no—
116D4—28no—
117D4—24no—
118D4C22⋊C4×C251116yes—
119C2×C4—28no—
120C2×C4—28no—
121C2×C4—28no—
122C2×C4—88no—
123C2×C4C22×C4×C25114yes—
124C2×C4C22×C4×C25112yes—
125C2×C4C2×D4×C25114yes—
126C22—24no—
127C22—48no—
128C22—44no—
129C22—41no—
130C22—44no—
131C22—88no—
132C22C2×C22⋊C4×C25118yes—
133C22C23×C4×C25111yescharacteristic
134C22C22×D4×C25112yes—
135C4—24no—
136C4—48no—
137C4C2×C22⋊C4×C25114yes—
138C2—22no—
139C2—21no—
140C2—28no—
141C2—44no—
142C2C22×C22⋊C4×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C22×C22⋊C4
commutatorC22C23×C4×C251
Frattini, cosocleC23C24×C251
prime radicalC2×H(5,𝔽2)×C251C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C22×D4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. #518×C251C2semidirect
3. #410×C251C2semidirect
4. C2×#99×C251C2semidirect
6. C2×#41×C251C4semidirect
8. C22×D4×C251C4semidirect
12. #117×C251C22semidirect
13. C2×C22⋊C4×C251C22semidirect
14. #99×C251C22semidirect
15. C24×C251C2×C4semidirect
19. C2×Q8×C251C2×C4semidirect
23. C2×D4×C251C2×C4semidirect
24. C2×D4×C251C2×C4semidirect
28. C22×C4×C251C2×C4semidirect
29. C22×C4×C251C2×C4semidirect
71. C251#2883direct (N × H)
72. #2883C251direct (N × H)
74. #518C2×C251semidirect
75. #410C2×C251semidirect
76. C2×#99C2×C251semidirect
78. C2×#41C4×C251semidirect
80. C22×D4C4×C251semidirect
84. #117C22×C251semidirect
85. C2×C22⋊C4C22×C251semidirect
86. #99C22×C251semidirect
87. C24C2×C4×C251semidirect
91. C2×Q8C2×C4×C251semidirect
95. C2×D4C2×C4×C251semidirect
96. C2×D4C2×C4×C251semidirect
100. C22×C4C2×C4×C251semidirect
101. C22×C4C2×C4×C251semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C2512
#518×C2512
#410×C2512
C2×#99×C2512
#2883251