Finite Group Databasebeta

← C2×#560×C109 #28360779

Subgroups — C2×#560×C109 #28360779

Order 13952.

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×C109C2116yesmaximal, direct product factor
2C2×#127×C109C211yesmaximal, characteristic, complemented
3C2×#126×C109C212yesmaximal, complemented
4C2×#123×C109C213yesmaximal, complemented
5C2×#121×C109C216yesmaximal, complemented
6C2×#120×C109C213yesmaximal, complemented
7C25×C109C2211yescharacteristic
8C22×D4×C109C2216yes—
9C22×D4×C109C2213yes—
10C22×D4×C109C2213yescomplemented
11C23×C4×C109C2213yescomplemented
12#127×C109C2218yescomplemented
13#126×C109C22116yescomplemented
14#123×C109C22124yescomplemented
15#121×C109C22148yescomplemented
16#120×C109C22124yescomplemented
17C2×C4⋊C4×C109C2216yescomplemented
18C2×C22⋊C4×C109C2216yes—
19C2×C22⋊C4×C109C2213yes—
20C2×C22⋊C4×C109C2213yescomplemented
21C2×C42×C109C2211yescharacteristic, complemented
22C24×C109—46no—
23C24×C109C2312yes—
24C24×C109C2311yescharacteristic
25C24×C109C2313yes—
26C24×C109C2314yes—
27C2×D4×C109—412no—
28C2×D4×C109—46no—
29C2×D4×C109—46no—
30C2×D4×C109C23124yes—
31C2×D4×C109C23112yes—
32C2×D4×C109C23112yescomplemented
33C22×C4×C109—46no—
34C22×C4×C109C2313yes—
35C22×C4×C109C2316yes—
36C22×C4×C109C23112yescomplemented
37C4⋊C4×C109C23124yescomplemented
38C22⋊C4×C109C23124yes—
39C22⋊C4×C109C23112yes—
40C22⋊C4×C109C23112yescomplemented
41C42×C109C2314yescomplemented
42C23×C109—26no—
43C23×C109—212no—
44C23×C109—212no—
45C23×C109—26no—
46C23×C109—412no—
47C23×C109—416no—
48C23×C109C2414yes—
49C23×C109C2412yes—
50C23×C109C2416yes—
51C23×C109C2411yescharacteristic
52D4×C109—424no—
53D4×C109—412no—
54D4×C109—412no—
55C2×C4×C109—212no—
56C2×C4×C109—26no—
57C2×C4×C109—412no—
58C2×C4×C109C2416yes—
59C2×C4×C109C24112yes—
60C22×C109—26no—
61C22×C109—212no—
62C22×C109—212no—
63C22×C109—26no—
64C22×C109—416no—
65C22×C109—412no—
66C22×C109—48no—
67C22×C109—44no—
68C22×C109H(5,𝔽2)16yes—
69C22×C109C2511yescharacteristic
70C4×C109—212no—
71C4×C109—26no—
72C2×C109—46no—
73C2×C109—44no—
74C2×C109—42no—
75C2×C109#56014yesdirect product factor
76C2×C109C2×H(5,𝔽2)13yes—
77C2×#560C10911yesmaximal, characteristic, direct product factor, Sylow
78C109C2×#56011yescharacteristic, direct product factor, Sylow
79#560C2×C109116yesdirect product factor
80C2×#127C2×C10911yescharacteristic, complemented
81C2×#126C2×C10912yescomplemented
82C2×#123C2×C10913yescomplemented
83C2×#121C2×C10916yescomplemented
84C2×#120C2×C10913yescomplemented
85C25C22×C10911yescharacteristic
86C22×D4C22×C10916yes—
87C22×D4C22×C10913yes—
88C22×D4C22×C10913yescomplemented
89C23×C4C22×C10913yescomplemented
90#127C22×C10918yescomplemented
91#126C22×C109116yescomplemented
92#123C22×C109124yescomplemented
93#121C22×C109148yescomplemented
94#120C22×C109124yescomplemented
95C2×C4⋊C4C22×C10916yescomplemented
96C2×C22⋊C4C22×C10916yes—
97C2×C22⋊C4C22×C10913yes—
98C2×C22⋊C4C22×C10913yescomplemented
99C2×C42C22×C10911yescharacteristic, complemented
100C24—46no—
101C24C23×C10912yes—
102C24C23×C10911yescharacteristic
103C24C23×C10913yes—
104C24C23×C10914yes—
105C2×D4—412no—
106C2×D4—46no—
107C2×D4—46no—
108C2×D4C23×C109124yes—
109C2×D4C23×C109112yes—
110C2×D4C23×C109112yescomplemented
111C22×C4—46no—
112C22×C4C23×C10913yes—
113C22×C4C23×C10916yes—
114C22×C4C23×C109112yescomplemented
115C4⋊C4C23×C109124yescomplemented
116C22⋊C4C23×C109124yes—
117C22⋊C4C23×C109112yes—
118C22⋊C4C23×C109112yescomplemented
119C42C23×C10914yescomplemented
120C23—26no—
121C23—212no—
122C23—212no—
123C23—26no—
124C23—412no—
125C23—416no—
126C23C24×C10914yes—
127C23C24×C10912yes—
128C23C24×C10916yes—
129C23C24×C10911yescharacteristic
130D4—424no—
131D4—412no—
132D4—412no—
133C2×C4—212no—
134C2×C4—26no—
135C2×C4—412no—
136C2×C4C24×C10916yes—
137C2×C4C24×C109112yes—
138C22—26no—
139C22—212no—
140C22—212no—
141C22—26no—
142C22—416no—
143C22—48no—
144C22—44no—
145C22—412no—
146C22H(5,𝔽2)×C10916yes—
147C22C25×C10911yescharacteristic
148C4—212no—
149C4—26no—
150C2—44no—
151C2—42no—
152C2—46no—
153C2#560×C10914yesdirect product factor
154C2C2×H(5,𝔽2)×C10913yes—
subgroupG/H
center, socleC23×C109C24
commutator, Frattini, cosocleC22C25×C109
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. #560×C109C2direct (N × H)
2. C2×#127×C109C2semidirect
3. C2×#126×C109C2semidirect
4. C2×#123×C109C2semidirect
5. C2×#121×C109C2semidirect
6. C2×#120×C109C2semidirect
10. C22×D4×C109C22semidirect
11. C23×C4×C109C22semidirect
12. #127×C109C22semidirect
13. #126×C109C22semidirect
14. #123×C109C22semidirect
15. #121×C109C22semidirect
16. #120×C109C22semidirect
17. C2×C4⋊C4×C109C22semidirect
20. C2×C22⋊C4×C109C22semidirect
21. C2×C42×C109C22semidirect
32. C2×D4×C109C23semidirect
36. C22×C4×C109C23semidirect
37. C4⋊C4×C109C23semidirect
40. C22⋊C4×C109C23semidirect
41. C42×C109C23semidirect
75. C2×C109#560direct (N × H)
77. C2×#560C109direct (N × H)
78. C109C2×#560direct (N × H)
79. #560C2×C109direct (N × H)
80. C2×#127C2×C109semidirect
81. C2×#126C2×C109semidirect
82. C2×#123C2×C109semidirect
83. C2×#121C2×C109semidirect
84. C2×#120C2×C109semidirect
88. C22×D4C22×C109semidirect
89. C23×C4C22×C109semidirect
90. #127C22×C109semidirect
91. #126C22×C109semidirect
92. #123C22×C109semidirect
93. #121C22×C109semidirect
94. #120C22×C109semidirect
95. C2×C4⋊C4C22×C109semidirect
98. C2×C22⋊C4C22×C109semidirect
99. C2×C42C22×C109semidirect
110. C2×D4C23×C109semidirect
114. C22×C4C23×C109semidirect
115. C4⋊C4C23×C109semidirect
118. C22⋊C4C23×C109semidirect
119. C42C23×C109semidirect
153. C2#560×C109direct (N × H)

Maximal subgroups

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

subgroupindex
#560×C1092
C2×#127×C1092
C2×#126×C1092
C2×#123×C1092
C2×#121×C1092
C2×#120×C1092
C2×#560109