Finite Group Databasebeta

← C22×C23×#347 #19180953

Subgroups — C22×C23×#347 #19180953

Order 5888.

782 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×M32C212yesmaximal
2C22×C23×#97C211yesmaximal, characteristic
3C2×C23×#347C2112yesmaximal, direct product factor
4C23×C8×C23C411yescharacteristic
5C23×C8×C23C2211yescharacteristic
6C22×C42×C23C411yescharacteristic
7C2×C23×M32C4112yescomplemented
8C2×C23×M32C22112yes—
9C22×C16×C23C414yescomplemented
10C2×C23×#97C416yes—
11C2×C23×#97C2216yes—
12C23×#347C22116yesdirect product factor
13C23×C4×C23—21no—
14C23×C4×C23C2×C411yescharacteristic
15C22×C8×C23—21no—
16C22×C8×C23—23no—
17C22×C8×C23C2×C416yes—
18C22×C8×C23C2×C413yes—
19C22×C8×C23C2×C416yes—
20C22×C8×C23C2×C412yes—
21C22×C8×C23C2313yes—
22C2×C42×C23—23no—
23C2×C42×C23C2×C416yes—
24C23×M32C2×C4124yescomplemented
25C23×M32C2318yes—
26C2×C16×C23C2×C4124yescomplemented
27C23×#97C2×C4112yes—
28C23×#97C2314yes—
29C24×C23M1611yescharacteristic
30C22×C4×C23—26no—
31C22×C4×C23—44no—
32C22×C4×C23C4213yes—
33C22×C4×C23C4211yescharacteristic
34C22×C4×C23M1613yes—
35C22×C4×C23M1611yescharacteristic
36C22×C4×C23M1613yes—
37C22×C4×C23C22×C413yes—
38C2×C8×C23—26no—
39C2×C8×C23—26no—
40C2×C8×C23C4216yes—
41C2×C8×C23C4216yes—
42C2×C8×C23C22×C414yes—
43C2×C8×C23C22×C413yes—
44C2×C8×C23C22×C416yes—
45C2×C8×C23C22×C416yes—
46C2×C8×C23C2411yescharacteristic
47C42×C23—26no—
48C42×C23C22×C414yes—
49C16×C23C22×C4116yescomplemented
50C22×#347C2311yesmaximal, characteristic, direct product factor, Sylow
51C23×C23—24no—
52C23×C23#9713yes—
53C23×C23#9711yescharacteristic
54C23×C23C2×M1613yes—
55C2×C4×C23—24no—
56C2×C4×C23—412no—
57C2×C4×C23#9716yes—
58C2×C4×C23#9713yes—
59C2×C4×C23#9713yes—
60C2×C4×C23C2×C4213yes—
61C2×C4×C23C2×C4213yes—
62C2×C4×C23C2×M1613yes—
63C2×C4×C23C2×M1613yes—
64C2×C4×C23C2×M1613yes—
65C2×C4×C23C23×C411yescharacteristic
66C8×C23—24no—
67C8×C23C2×C4216yes—
68C8×C23C23×C412yes—
69C22×M32C2×C2312yes—
70C22×#97C2×C2311yescharacteristic
71C2×#347C2×C23112yesdirect product factor
72C22×C23—26no—
73C22×C23—26no—
74C22×C23#34714yesdirect product factor
75C22×C23C2×#9713yes—
76C22×C23C2×#9713yes—
77C22×C23C22×M1611yescharacteristic
78C4×C23—44no—
79C4×C23C2×#9713yes—
80C4×C23C2×#9713yes—
81C4×C23C22×C4211yescharacteristic
82C4×C23C22×M1611yescharacteristic
83C23×C8C4×C2311yescharacteristic
84C23×C8C22×C2311yescharacteristic
85C22×C42C4×C2311yescharacteristic
86C2×M32C4×C23112yescomplemented
87C2×M32C22×C23112yes—
88C22×C16C4×C2314yescomplemented
89C2×#97C4×C2316yes—
90C2×#97C22×C2316yes—
91#347C22×C23116yesdirect product factor
92C2×C23—21no—
93C2×C23—23no—
94C2×C23C2×#34716yesdirect product factor
95C2×C23C22×#9711yescharacteristic
96C23×C4—21no—
97C23×C4C2×C4×C2311yescharacteristic
98C22×C8—23no—
99C22×C8—21no—
100C22×C8C2×C4×C2316yes—
101C22×C8C2×C4×C2313yes—
102C22×C8C2×C4×C2312yes—
103C22×C8C2×C4×C2316yes—
104C22×C8C23×C2313yes—
105C2×C42—23no—
106C2×C42C2×C4×C2316yes—
107M32C2×C4×C23124yescomplemented
108M32C23×C2318yes—
109C2×C16C2×C4×C23124yescomplemented
110#97C2×C4×C23112yes—
111#97C23×C2314yes—
112C23C22×#34711yescharacteristic, direct product factor, Sylow
113C24M16×C2311yescharacteristic
114C22×C4—26no—
115C22×C4—44no—
116C22×C4C42×C2313yes—
117C22×C4C42×C2311yescharacteristic
118C22×C4M16×C2313yes—
119C22×C4M16×C2311yescharacteristic
120C22×C4M16×C2313yes—
121C22×C4C22×C4×C2313yes—
122C2×C8—26no—
123C2×C8—26no—
124C2×C8C42×C2316yes—
125C2×C8C42×C2316yes—
126C2×C8C22×C4×C2314yes—
127C2×C8C22×C4×C2313yes—
128C2×C8C22×C4×C2316yes—
129C2×C8C22×C4×C2316yes—
130C2×C8C24×C2311yescharacteristic
131C42—26no—
132C42C22×C4×C2314yes—
133C16C22×C4×C23116yescomplemented
134C23—24no—
135C23C23×#9713yes—
136C23C23×#9711yescharacteristic
137C23C2×M16×C2313yes—
138C2×C4—24no—
139C2×C4—412no—
140C2×C4C23×#9716yes—
141C2×C4C23×#9713yes—
142C2×C4C23×#9713yes—
143C2×C4C2×C42×C2313yes—
144C2×C4C2×C42×C2313yes—
145C2×C4C2×M16×C2313yes—
146C2×C4C2×M16×C2313yes—
147C2×C4C2×M16×C2313yes—
148C2×C4C23×C4×C2311yescharacteristic
149C8—24no—
150C8C2×C42×C2316yes—
151C8C23×C4×C2312yes—
152C22—26no—
153C22—26no—
154C22C23×#34714yesdirect product factor
155C22C2×C23×#9713yes—
156C22C2×C23×#9713yes—
157C22C22×M16×C2311yescharacteristic
158C4—44no—
159C4C2×C23×#9713yes—
160C4C2×C23×#9713yes—
161C4C22×C42×C2311yescharacteristic
162C4C22×M16×C2311yescharacteristic
163C2—21no—
164C2—23no—
165C2C2×C23×#34716yesdirect product factor
166C2C22×C23×#9711yescharacteristic
subgroupG/H
centerC22×C4×C23C42
commutatorC4C22×C42×C23
Frattini, cosocleC2×C8C24×C23
socleC23×C23#97
prime radicalC24×C23M16
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#347C2direct (N × H)
7. C2×C23×M32C4semidirect
9. C22×C16×C23C4semidirect
12. C23×#347C22direct (N × H)
24. C23×M32C2×C4semidirect
26. C2×C16×C23C2×C4semidirect
49. C16×C23C22×C4semidirect
50. C22×#347C23direct (N × H)
71. C2×#347C2×C23direct (N × H)
74. C22×C23#347direct (N × H)
86. C2×M32C4×C23semidirect
88. C22×C16C4×C23semidirect
91. #347C22×C23direct (N × H)
94. C2×C23C2×#347direct (N × H)
107. M32C2×C4×C23semidirect
109. C2×C16C2×C4×C23semidirect
112. C23C22×#347direct (N × H)
133. C16C22×C4×C23semidirect
154. C22C23×#347direct (N × H)
165. C2C2×C23×#347direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×M322
C22×C23×#972
C2×C23×#3472
C22×#34723