Finite Group Databasebeta

← #3416×C251 #44548003

Subgroups — #3416×C251 #44548003

Order 32128.

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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#578×C251C211yesmaximal, characteristic, complemented
2#576×C251C211yesmaximal, characteristic, complemented
3#510×C251C213yesmaximal, complemented
4#509×C251C213yesmaximal, complemented
5#508×C251C213yesmaximal, complemented
6C2×SD32×C251C213yesmaximal, complemented
7#504×C251C211yesmaximal, characteristic, complemented
82−1+4×C251—21no—
9H(5,𝔽2)×C251—21no—
10#137×C251—23no—
11AGL(1,ℤ/8)×C251—23no—
12#135×C251C2213yes—
13#135×C251C2213yescomplemented
14C2×Q16×C251C2213yescomplemented
15C2×D8×C251C2213yes—
16#131×C251C2211yescharacteristic
17Q32×C251C2213yescomplemented
18SD32×C251C2213yescomplemented
19SD32×C251C2216yescomplemented
20SD32×C251C2211yescharacteristic, complemented
21D16×C251C2213yescomplemented
22M32×C251C2213yescomplemented
23C2×C16×C251C2213yescomplemented
24#41×C251—23no—
25#41×C251—23no—
26#41×C251—41no—
27#41×C251—43no—
28#41×C251D411yescharacteristic
29C2×Q8×C251—23no—
30C2×Q8×C251—41no—
31C2×D4×C251—23no—
32C2×D4×C251—43no—
33Q16×C251—23no—
34Q16×C251C2313yes—
35Q16×C251C2311yescharacteristic
36SD16×C251—23no—
37SD16×C251—23no—
38D8×C251—23no—
39D8×C251C2311yescharacteristic
40D8×C251C2313yes—
41M16×C251D413yes—
42C2×C8×C251C2313yes—
43C16×C251C2311yescharacteristic
44C16×C251C2313yes—
45C23×C251—43no—
46Q8×C251—21no—
47Q8×C251—23no—
48Q8×C251—43no—
49Q8×C251D811yescharacteristic
50D4×C251—23no—
51D4×C251—21no—
52D4×C251—43no—
53D4×C251—43no—
54D4×C251—43no—
55D4×C251D813yes—
56C2×C4×C251—43no—
57C2×C4×C251—43no—
58C2×C4×C251—43no—
59C2×C4×C251C2×D413yes—
60C8×C251C2×D413yes—
61C8×C251C2411yescharacteristic
62C22×C251—41no—
63C22×C251—43no—
64C22×C251—163no—
65C22×C251C2×D813yes—
66C4×C251—41no—
67C4×C251—43no—
68C4×C251C2×D813yes—
69C4×C251C22×D411yescharacteristic
70C2×C251—23no—
71C2×C251—81no—
72C2×C251—83no—
73C2×C251C22×D811yescharacteristic
74C251#341611yescharacteristic, direct product factor, Sylow
75#3416C25111yesmaximal, characteristic, direct product factor, Sylow
76#578C2×C25111yescharacteristic, complemented
77#576C2×C25111yescharacteristic, complemented
78#510C2×C25113yescomplemented
79#509C2×C25113yescomplemented
80#508C2×C25113yescomplemented
81C2×SD32C2×C25113yescomplemented
82#504C2×C25111yescharacteristic, complemented
832−1+4—21no—
84H(5,𝔽2)—21no—
85#137—23no—
86AGL(1,ℤ/8)—23no—
87#135C22×C25113yes—
88#135C22×C25113yescomplemented
89C2×Q16C22×C25113yescomplemented
90C2×D8C22×C25113yes—
91#131C22×C25111yescharacteristic
92Q32C22×C25113yescomplemented
93SD32C22×C25113yescomplemented
94SD32C22×C25116yescomplemented
95SD32C22×C25111yescharacteristic, complemented
96D16C22×C25113yescomplemented
97M32C22×C25113yescomplemented
98C2×C16C22×C25113yescomplemented
99#41—23no—
100#41—23no—
101#41—41no—
102#41—43no—
103#41D4×C25111yescharacteristic
104C2×Q8—23no—
105C2×Q8—41no—
106C2×D4—23no—
107C2×D4—43no—
108Q16—23no—
109Q16C23×C25113yes—
110Q16C23×C25111yescharacteristic
111SD16—23no—
112SD16—23no—
113D8—23no—
114D8C23×C25111yescharacteristic
115D8C23×C25113yes—
116M16D4×C25113yes—
117C2×C8C23×C25113yes—
118C16C23×C25111yescharacteristic
119C16C23×C25113yes—
120C23—43no—
121Q8—21no—
122Q8—23no—
123Q8—43no—
124Q8D8×C25111yescharacteristic
125D4—23no—
126D4—21no—
127D4—43no—
128D4—43no—
129D4—43no—
130D4D8×C25113yes—
131C2×C4—43no—
132C2×C4—43no—
133C2×C4—43no—
134C2×C4C2×D4×C25113yes—
135C8C2×D4×C25113yes—
136C8C24×C25111yescharacteristic
137C22—41no—
138C22—43no—
139C22—163no—
140C22C2×D8×C25113yes—
141C4—41no—
142C4—43no—
143C4C2×D8×C25113yes—
144C4C22×D4×C25111yescharacteristic
145C2—23no—
146C2—83no—
147C2—81no—
148C2C22×D8×C25111yescharacteristic
subgroupG/H
center, socleC2×C251C22×D8
commutator, Frattini, cosocleC8C24×C251
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ #41×C251 ◁ #131×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #578×C251C2semidirect
2. #576×C251C2semidirect
3. #510×C251C2semidirect
4. #509×C251C2semidirect
5. #508×C251C2semidirect
6. C2×SD32×C251C2semidirect
7. #504×C251C2semidirect
13. #135×C251C22semidirect
14. C2×Q16×C251C22semidirect
17. Q32×C251C22semidirect
18. SD32×C251C22semidirect
19. SD32×C251C22semidirect
20. SD32×C251C22semidirect
21. D16×C251C22semidirect
22. M32×C251C22semidirect
23. C2×C16×C251C22semidirect
74. C251#3416direct (N × H)
75. #3416C251direct (N × H)
76. #578C2×C251semidirect
77. #576C2×C251semidirect
78. #510C2×C251semidirect
79. #509C2×C251semidirect
80. #508C2×C251semidirect
81. C2×SD32C2×C251semidirect
82. #504C2×C251semidirect
88. #135C22×C251semidirect
89. C2×Q16C22×C251semidirect
92. Q32C22×C251semidirect
93. SD32C22×C251semidirect
94. SD32C22×C251semidirect
95. SD32C22×C251semidirect
96. D16C22×C251semidirect
97. M32C22×C251semidirect
98. C2×C16C22×C251semidirect

Maximal subgroups

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

subgroupindex
#578×C2512
#576×C2512
#510×C2512
#509×C2512
#508×C2512
C2×SD32×C2512
#504×C2512
#3416251