Finite Group Databasebeta

← C2×#560×C251 #44548112

Subgroups — C2×#560×C251 #44548112

Order 32128.

1344 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 776 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#560×C251C2116yesmaximal, direct product factor
2C2×#127×C251C211yesmaximal, characteristic, complemented
3C2×#126×C251C212yesmaximal, complemented
4C2×#123×C251C213yesmaximal, complemented
5C2×#121×C251C216yesmaximal, complemented
6C2×#120×C251C213yesmaximal, complemented
7C25×C251C2211yescharacteristic
8C22×D4×C251C2216yes—
9C22×D4×C251C2213yes—
10C22×D4×C251C2213yescomplemented
11C23×C4×C251C2213yescomplemented
12#127×C251C2218yescomplemented
13#126×C251C22116yescomplemented
14#123×C251C22124yescomplemented
15#121×C251C22148yescomplemented
16#120×C251C22124yescomplemented
17C2×C4⋊C4×C251C2216yescomplemented
18C2×C22⋊C4×C251C2216yes—
19C2×C22⋊C4×C251C2213yes—
20C2×C22⋊C4×C251C2213yescomplemented
21C2×C42×C251C2211yescharacteristic, complemented
22C24×C251—46no—
23C24×C251C2312yes—
24C24×C251C2311yescharacteristic
25C24×C251C2313yes—
26C24×C251C2314yes—
27C2×D4×C251—412no—
28C2×D4×C251—46no—
29C2×D4×C251—46no—
30C2×D4×C251C23124yes—
31C2×D4×C251C23112yes—
32C2×D4×C251C23112yescomplemented
33C22×C4×C251—46no—
34C22×C4×C251C2313yes—
35C22×C4×C251C2316yes—
36C22×C4×C251C23112yescomplemented
37C4⋊C4×C251C23124yescomplemented
38C22⋊C4×C251C23124yes—
39C22⋊C4×C251C23112yes—
40C22⋊C4×C251C23112yescomplemented
41C42×C251C2314yescomplemented
42C23×C251—26no—
43C23×C251—212no—
44C23×C251—212no—
45C23×C251—26no—
46C23×C251—412no—
47C23×C251—416no—
48C23×C251C2414yes—
49C23×C251C2412yes—
50C23×C251C2416yes—
51C23×C251C2411yescharacteristic
52D4×C251—424no—
53D4×C251—412no—
54D4×C251—412no—
55C2×C4×C251—212no—
56C2×C4×C251—26no—
57C2×C4×C251—412no—
58C2×C4×C251C2416yes—
59C2×C4×C251C24112yes—
60C22×C251—26no—
61C22×C251—212no—
62C22×C251—212no—
63C22×C251—26no—
64C22×C251—416no—
65C22×C251—412no—
66C22×C251—48no—
67C22×C251—44no—
68C22×C251H(5,𝔽2)16yes—
69C22×C251C2511yescharacteristic
70C4×C251—212no—
71C4×C251—26no—
72C2×C251—46no—
73C2×C251—44no—
74C2×C251—42no—
75C2×C251#56014yesdirect product factor
76C2×C251C2×H(5,𝔽2)13yes—
77C251C2×#56011yescharacteristic, direct product factor, Sylow
78C2×#560C25111yesmaximal, characteristic, direct product factor, Sylow
79#560C2×C251116yesdirect product factor
80C2×#127C2×C25111yescharacteristic, complemented
81C2×#126C2×C25112yescomplemented
82C2×#123C2×C25113yescomplemented
83C2×#121C2×C25116yescomplemented
84C2×#120C2×C25113yescomplemented
85C25C22×C25111yescharacteristic
86C22×D4C22×C25116yes—
87C22×D4C22×C25113yes—
88C22×D4C22×C25113yescomplemented
89C23×C4C22×C25113yescomplemented
90#127C22×C25118yescomplemented
91#126C22×C251116yescomplemented
92#123C22×C251124yescomplemented
93#121C22×C251148yescomplemented
94#120C22×C251124yescomplemented
95C2×C4⋊C4C22×C25116yescomplemented
96C2×C22⋊C4C22×C25116yes—
97C2×C22⋊C4C22×C25113yes—
98C2×C22⋊C4C22×C25113yescomplemented
99C2×C42C22×C25111yescharacteristic, complemented
100C24—46no—
101C24C23×C25112yes—
102C24C23×C25111yescharacteristic
103C24C23×C25113yes—
104C24C23×C25114yes—
105C2×D4—412no—
106C2×D4—46no—
107C2×D4—46no—
108C2×D4C23×C251124yes—
109C2×D4C23×C251112yes—
110C2×D4C23×C251112yescomplemented
111C22×C4—46no—
112C22×C4C23×C25113yes—
113C22×C4C23×C25116yes—
114C22×C4C23×C251112yescomplemented
115C4⋊C4C23×C251124yescomplemented
116C22⋊C4C23×C251124yes—
117C22⋊C4C23×C251112yes—
118C22⋊C4C23×C251112yescomplemented
119C42C23×C25114yescomplemented
120C23—26no—
121C23—212no—
122C23—212no—
123C23—26no—
124C23—412no—
125C23—416no—
126C23C24×C25114yes—
127C23C24×C25112yes—
128C23C24×C25116yes—
129C23C24×C25111yescharacteristic
130D4—424no—
131D4—412no—
132D4—412no—
133C2×C4—212no—
134C2×C4—26no—
135C2×C4—412no—
136C2×C4C24×C25116yes—
137C2×C4C24×C251112yes—
138C22—26no—
139C22—212no—
140C22—212no—
141C22—26no—
142C22—416no—
143C22—48no—
144C22—44no—
145C22—412no—
146C22H(5,𝔽2)×C25116yes—
147C22C25×C25111yescharacteristic
148C4—212no—
149C4—26no—
150C2—44no—
151C2—42no—
152C2—46no—
153C2#560×C25114yesdirect product factor
154C2C2×H(5,𝔽2)×C25113yes—
subgroupG/H
center, socleC23×C251C24
commutator, Frattini, cosocleC22C25×C251
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #560×C251C2direct (N × H)
2. C2×#127×C251C2semidirect
3. C2×#126×C251C2semidirect
4. C2×#123×C251C2semidirect
5. C2×#121×C251C2semidirect
6. C2×#120×C251C2semidirect
10. C22×D4×C251C22semidirect
11. C23×C4×C251C22semidirect
12. #127×C251C22semidirect
13. #126×C251C22semidirect
14. #123×C251C22semidirect
15. #121×C251C22semidirect
16. #120×C251C22semidirect
17. C2×C4⋊C4×C251C22semidirect
20. C2×C22⋊C4×C251C22semidirect
21. C2×C42×C251C22semidirect
32. C2×D4×C251C23semidirect
36. C22×C4×C251C23semidirect
37. C4⋊C4×C251C23semidirect
40. C22⋊C4×C251C23semidirect
41. C42×C251C23semidirect
75. C2×C251#560direct (N × H)
77. C251C2×#560direct (N × H)
78. C2×#560C251direct (N × H)
79. #560C2×C251direct (N × H)
80. C2×#127C2×C251semidirect
81. C2×#126C2×C251semidirect
82. C2×#123C2×C251semidirect
83. C2×#121C2×C251semidirect
84. C2×#120C2×C251semidirect
88. C22×D4C22×C251semidirect
89. C23×C4C22×C251semidirect
90. #127C22×C251semidirect
91. #126C22×C251semidirect
92. #123C22×C251semidirect
93. #121C22×C251semidirect
94. #120C22×C251semidirect
95. C2×C4⋊C4C22×C251semidirect
98. C2×C22⋊C4C22×C251semidirect
99. C2×C42C22×C251semidirect
110. C2×D4C23×C251semidirect
114. C22×C4C23×C251semidirect
115. C4⋊C4C23×C251semidirect
118. C22⋊C4C23×C251semidirect
119. C42C23×C251semidirect
153. C2#560×C251direct (N × H)

Maximal subgroups

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

subgroupindex
#560×C2512
C2×#127×C2512
C2×#126×C2512
C2×#123×C2512
C2×#121×C2512
C2×#120×C2512
C2×#560251