Finite Group Databasebeta

← C2×#442×C251 #44547529

Subgroups — C2×#442×C251 #44547529

Order 32128.

608 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 296 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
1C22×D8×C251C211yesmaximal, characteristic
2C2×C4×D4×C251C212yesmaximal, complemented
3#442×C251C218yesmaximal, direct product factor
4C2×#106×C251C211yesmaximal, characteristic, complemented
5C2×#102×C251C212yesmaximal, complemented
6C2×#97×C251C211yesmaximal, characteristic, complemented
7C22×D4×C251C2212yes—
8C23×C4×C251—22no—
9C2×D8×C251C418yescomplemented
10C2×D8×C251C2214yes—
11C22×C8×C251C2211yescharacteristic
12C22×C8×C251C2211yescharacteristic
13C4×D4×C251—24no—
14C4×D4×C251C2218yescomplemented
15C2×C4⋊C4×C251C2212yes—
16C2×C22⋊C4×C251—22no—
17C2×C42×C251C2211yescharacteristic
18#106×C251C2214yescomplemented
19#102×C251C2218yescomplemented
20#97×C251C2214yescomplemented
21C24×C251—22no—
22C2×D4×C251—22no—
23C2×D4×C251—24no—
24C2×D4×C251C2×C418yescomplemented
25C2×D4×C251C2314yes—
26C22×C4×C251—22no—
27C22×C4×C251—28no—
28C22×C4×C251—48no—
29C22×C4×C251D412yes—
30C22×C4×C251C2311yescharacteristic
31D8×C251C2×C4116yescomplemented
32C2×C8×C251—22no—
33C2×C8×C251C2×C414yescomplemented
34C2×C8×C251C2312yes—
35C2×C8×C251C2312yes—
36C4⋊C4×C251—22no—
37C4⋊C4×C251C2314yes—
38C22⋊C4×C251—28no—
39C42×C251D412yes—
40C42×C251C2312yes—
41C23×C251—28no—
42C23×C251—24no—
43C23×C251—44no—
44C23×C251—44no—
45C23×C251C2×D411yescharacteristic
46D4×C251—28no—
47D4×C251—24no—
48D4×C251C22×C418yescomplemented
49C2×C4×C251—24no—
50C2×C4×C251—24no—
51C2×C4×C251—416no—
52C2×C4×C251—44no—
53C2×C4×C251C22×C412yes—
54C2×C4×C251C2×D414yes—
55C2×C4×C251C2×D411yescharacteristic
56C2×C4×C251#4112yes—
57C2×C4×C251C2411yescharacteristic
58C8×C251—22no—
59C8×C251C22×C414yescomplemented
60C22×C251—28no—
61C22×C251—44no—
62C22×C251—44no—
63C22×C251—416no—
64C22×C251C4×D412yes—
65C22×C251AGL(1,ℤ/8)14yes—
66C22×C251C22×D411yescharacteristic
67C4×C251—24no—
68C4×C251—44no—
69C4×C251C4×D412yes—
70C4×C251C23×C411yescharacteristic
71C4×C251C2×#4111yescharacteristic
72C2×C251—48no—
73C2×C251#44214yesdirect product factor
74C2×C251C2×C4×D411yescharacteristic
75C2×C251C2×AGL(1,ℤ/8)12yes—
76C251C2×#44211yescharacteristic, direct product factor, Sylow
77C2×#442C25111yesmaximal, characteristic, direct product factor, Sylow
78C22×D8C2×C25111yescharacteristic
79C2×C4×D4C2×C25112yescomplemented
80#442C2×C25118yesdirect product factor
81C2×#106C2×C25111yescharacteristic, complemented
82C2×#102C2×C25112yescomplemented
83C2×#97C2×C25111yescharacteristic, complemented
84C22×D4C22×C25112yes—
85C23×C4—22no—
86C2×D8C4×C25118yescomplemented
87C2×D8C22×C25114yes—
88C22×C8C22×C25111yescharacteristic
89C22×C8C22×C25111yescharacteristic
90C4×D4—24no—
91C4×D4C22×C25118yescomplemented
92C2×C4⋊C4C22×C25112yes—
93C2×C22⋊C4—22no—
94C2×C42C22×C25111yescharacteristic
95#106C22×C25114yescomplemented
96#102C22×C25118yescomplemented
97#97C22×C25114yescomplemented
98C24—22no—
99C2×D4—22no—
100C2×D4—24no—
101C2×D4C2×C4×C25118yescomplemented
102C2×D4C23×C25114yes—
103C22×C4—22no—
104C22×C4—28no—
105C22×C4—48no—
106C22×C4D4×C25112yes—
107C22×C4C23×C25111yescharacteristic
108D8C2×C4×C251116yescomplemented
109C2×C8—22no—
110C2×C8C2×C4×C25114yescomplemented
111C2×C8C23×C25112yes—
112C2×C8C23×C25112yes—
113C4⋊C4—22no—
114C4⋊C4C23×C25114yes—
115C22⋊C4—28no—
116C42D4×C25112yes—
117C42C23×C25112yes—
118C23—28no—
119C23—24no—
120C23—44no—
121C23—44no—
122C23C2×D4×C25111yescharacteristic
123D4—28no—
124D4—24no—
125D4C22×C4×C25118yescomplemented
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—416no—
129C2×C4—44no—
130C2×C4C22×C4×C25112yes—
131C2×C4C2×D4×C25114yes—
132C2×C4C2×D4×C25111yescharacteristic
133C2×C4#41×C25112yes—
134C2×C4C24×C25111yescharacteristic
135C8—22no—
136C8C22×C4×C25114yescomplemented
137C22—28no—
138C22—416no—
139C22—44no—
140C22—44no—
141C22C4×D4×C25112yes—
142C22AGL(1,ℤ/8)×C25114yes—
143C22C22×D4×C25111yescharacteristic
144C4—24no—
145C4—44no—
146C4C4×D4×C25112yes—
147C4C23×C4×C25111yescharacteristic
148C4C2×#41×C25111yescharacteristic
149C2—48no—
150C2#442×C25114yesdirect product factor
151C2C2×C4×D4×C25111yescharacteristic
152C2C2×AGL(1,ℤ/8)×C25112yes—
subgroupG/H
center, socleC23×C251C2×D4
commutatorC4C23×C4×C251
Frattini, cosocleC2×C4C24×C251
prime radicalC22×D8×C251C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C251 ◁ C2×C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×D4×C251C2semidirect
3. #442×C251C2direct (N × H)
4. C2×#106×C251C2semidirect
5. C2×#102×C251C2semidirect
6. C2×#97×C251C2semidirect
9. C2×D8×C251C4semidirect
14. C4×D4×C251C22semidirect
18. #106×C251C22semidirect
19. #102×C251C22semidirect
20. #97×C251C22semidirect
24. C2×D4×C251C2×C4semidirect
31. D8×C251C2×C4semidirect
33. C2×C8×C251C2×C4semidirect
48. D4×C251C22×C4semidirect
59. C8×C251C22×C4semidirect
73. C2×C251#442direct (N × H)
76. C251C2×#442direct (N × H)
77. C2×#442C251direct (N × H)
79. C2×C4×D4C2×C251semidirect
80. #442C2×C251direct (N × H)
81. C2×#106C2×C251semidirect
82. C2×#102C2×C251semidirect
83. C2×#97C2×C251semidirect
86. C2×D8C4×C251semidirect
91. C4×D4C22×C251semidirect
95. #106C22×C251semidirect
96. #102C22×C251semidirect
97. #97C22×C251semidirect
101. C2×D4C2×C4×C251semidirect
108. D8C2×C4×C251semidirect
110. C2×C8C2×C4×C251semidirect
125. D4C22×C4×C251semidirect
136. C8C22×C4×C251semidirect
150. C2#442×C251direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C2512
C2×C4×D4×C2512
#442×C2512
C2×#106×C2512
C2×#102×C2512
C2×#97×C2512
C2×#442251