Finite Group Databasebeta

← C23×#131×C89 #36440921

Subgroups — C23×#131×C89 #36440921

Order 22784.

11422 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C89C211yesmaximal, characteristic
2C22×#131×C89C2156yesmaximal, direct product factor
3C23×M16×C89C213yesmaximal, complemented
4C24×C8×C89C213yesmaximal, complemented
5C22×#41×C89C4128yes—
6C22×#41×C89C22128yes—
7C23×Q8×C89C411yescharacteristic
8C23×D4×C89C413yes—
9C24×C4×C89C2213yes—
10C2×#131×C89C221448yesdirect product factor
11C22×M16×C89C22184yescomplemented
12C23×C8×C89C2213yes—
13C23×C8×C89C2211yescharacteristic
14C23×C8×C89C22184yescomplemented
15C25×C89C2×C413yes—
16C2×#41×C89C2×C41336yes—
17C2×#41×C89C231112yes—
18C22×Q8×C89C2×C4128yes—
19C22×D4×C89C2×C4184yes—
20C23×C4×C89C2×C413yes—
21C23×C4×C89C2×C4142yes—
22C23×C4×C89C2311yescharacteristic
23C23×C4×C89C23142yes—
24#131×C89C231512yesdirect product factor
25C2×M16×C89C231336yescomplemented
26C22×C8×C89C23142yes—
27C22×C8×C89C23114yes—
28C22×C8×C89C231336yescomplemented
29C24×C89—224no—
30C24×C89C22×C4142yes—
31C24×C89C22×C411yescharacteristic
32#41×C89C22×C41448yes—
33#41×C89C24164yes—
34C2×Q8×C89C22×C41112yes—
35C2×D4×C89C22×C41336yes—
36C22×C4×C89C22×C417yes—
37C22×C4×C89C22×C4142yes—
38C22×C4×C89C22×C41252yes—
39C22×C4×C89C2417yes—
40C22×C4×C89C24184yes—
41M16×C89C241192yescomplemented
42C2×C8×C89C24128yes—
43C2×C8×C89C24184yes—
44C2×C8×C89C241192yescomplemented
45C23×C89—2168no—
46C23×C89#13118yesdirect product factor
47C23×C89C23×C4184yes—
48C23×C89C23×C417yes—
49Q8×C89C23×C4164yes—
50D4×C89C23×C41192yes—
51C2×C4×C89C23×C4121yes—
52C2×C4×C89C23×C4184yes—
53C2×C4×C89C23×C41168yes—
54C2×C4×C89C2517yes—
55C2×C4×C89C25124yes—
56C8×C89C2518yes—
57C8×C89C25124yes—
58C22×C89—2168no—
59C22×C89C2×#131128yesdirect product factor
60C22×C89C24×C4124yes—
61C22×C89C24×C417yes—
62C4×C89C24×C417yes—
63C4×C89C24×C4124yes—
64C4×C89C2611yescharacteristic
65C23×#131C8911yesmaximal, characteristic, direct product factor, Sylow
66C2×C89—224no—
67C2×C89C22×#131114yesdirect product factor
68C2×C89C25×C411yescharacteristic
69C23×#41C2×C8911yescharacteristic
70C22×#131C2×C89156yesdirect product factor
71C23×M16C2×C8913yescomplemented
72C24×C8C2×C8913yescomplemented
73C89C23×#13111yescharacteristic, direct product factor, Sylow
74C22×#41C4×C89128yes—
75C22×#41C22×C89128yes—
76C23×Q8C4×C8911yescharacteristic
77C23×D4C4×C8913yes—
78C24×C4C22×C8913yes—
79C2×#131C22×C891448yesdirect product factor
80C22×M16C22×C89184yescomplemented
81C23×C8C22×C8913yes—
82C23×C8C22×C8911yescharacteristic
83C23×C8C22×C89184yescomplemented
84C25C2×C4×C8913yes—
85C2×#41C2×C4×C891336yes—
86C2×#41C23×C891112yes—
87C22×Q8C2×C4×C89128yes—
88C22×D4C2×C4×C89184yes—
89C23×C4C2×C4×C8913yes—
90C23×C4C2×C4×C89142yes—
91C23×C4C23×C8911yescharacteristic
92C23×C4C23×C89142yes—
93#131C23×C891512yesdirect product factor
94C2×M16C23×C891336yescomplemented
95C22×C8C23×C89114yes—
96C22×C8C23×C89142yes—
97C22×C8C23×C891336yescomplemented
98C24—224no—
99C24C22×C4×C89142yes—
100C24C22×C4×C8911yescharacteristic
101#41C22×C4×C891448yes—
102#41C24×C89164yes—
103C2×Q8C22×C4×C891112yes—
104C2×D4C22×C4×C891336yes—
105C22×C4C22×C4×C8917yes—
106C22×C4C22×C4×C89142yes—
107C22×C4C22×C4×C891252yes—
108C22×C4C24×C8917yes—
109C22×C4C24×C89184yes—
110M16C24×C891192yescomplemented
111C2×C8C24×C89128yes—
112C2×C8C24×C89184yes—
113C2×C8C24×C891192yescomplemented
114C23—2168no—
115C23#131×C8918yesdirect product factor
116C23C23×C4×C89184yes—
117C23C23×C4×C8917yes—
118Q8C23×C4×C89164yes—
119D4C23×C4×C891192yes—
120C2×C4C23×C4×C89121yes—
121C2×C4C23×C4×C89184yes—
122C2×C4C23×C4×C891168yes—
123C2×C4C25×C8917yes—
124C2×C4C25×C89124yes—
125C8C25×C8918yes—
126C8C25×C89124yes—
127C22—2168no—
128C22C2×#131×C89128yesdirect product factor
129C22C24×C4×C89124yes—
130C22C24×C4×C8917yes—
131C4C24×C4×C8917yes—
132C4C24×C4×C89124yes—
133C4C26×C8911yescharacteristic
134C2—224no—
135C2C22×#131×C89114yesdirect product factor
136C2C25×C4×C8911yescharacteristic
subgroupG/H
centerC23×C8×C89C22
commutatorC2C25×C4×C89
Frattini, cosocleC4C26×C89
socleC24×C89C22×C4
prime radicalC23×#41×C89C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C8×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#131×C89C2direct (N × H)
3. C23×M16×C89C2semidirect
4. C24×C8×C89C2semidirect
10. C2×#131×C89C22direct (N × H)
11. C22×M16×C89C22semidirect
14. C23×C8×C89C22semidirect
24. #131×C89C23direct (N × H)
25. C2×M16×C89C23semidirect
28. C22×C8×C89C23semidirect
41. M16×C89C24semidirect
44. C2×C8×C89C24semidirect
46. C23×C89#131direct (N × H)
59. C22×C89C2×#131direct (N × H)
65. C23×#131C89direct (N × H)
67. C2×C89C22×#131direct (N × H)
70. C22×#131C2×C89direct (N × H)
71. C23×M16C2×C89semidirect
72. C24×C8C2×C89semidirect
73. C89C23×#131direct (N × H)
79. C2×#131C22×C89direct (N × H)
80. C22×M16C22×C89semidirect
83. C23×C8C22×C89semidirect
93. #131C23×C89direct (N × H)
94. C2×M16C23×C89semidirect
97. C22×C8C23×C89semidirect
110. M16C24×C89semidirect
113. C2×C8C24×C89semidirect
115. C23#131×C89direct (N × H)
128. C22C2×#131×C89direct (N × H)
135. C2C22×#131×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C892
C22×#131×C892
C23×M16×C892
C24×C8×C892
C23×#13189