Finite Group Databasebeta

← C22×#347×C137 #46882713

Subgroups — C22×#347×C137 #46882713

Order 35072.

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×M32×C137C212yesmaximal
2C22×#97×C137C211yesmaximal, characteristic
3C2×#347×C137C2112yesmaximal, direct product factor
4C23×C8×C137C411yescharacteristic
5C23×C8×C137C2211yescharacteristic
6C22×C42×C137C411yescharacteristic
7C2×M32×C137C4112yescomplemented
8C2×M32×C137C22112yes—
9C22×C16×C137C414yescomplemented
10C2×#97×C137C416yes—
11C2×#97×C137C2216yes—
12#347×C137C22116yesdirect product factor
13C23×C4×C137—21no—
14C23×C4×C137C2×C411yescharacteristic
15C22×C8×C137—21no—
16C22×C8×C137—23no—
17C22×C8×C137C2×C416yes—
18C22×C8×C137C2×C413yes—
19C22×C8×C137C2×C416yes—
20C22×C8×C137C2×C412yes—
21C22×C8×C137C2313yes—
22C2×C42×C137—23no—
23C2×C42×C137C2×C416yes—
24M32×C137C2×C4124yescomplemented
25M32×C137C2318yes—
26C2×C16×C137C2×C4124yescomplemented
27#97×C137C2×C4112yes—
28#97×C137C2314yes—
29C24×C137M1611yescharacteristic
30C22×C4×C137—26no—
31C22×C4×C137—44no—
32C22×C4×C137C4213yes—
33C22×C4×C137C4211yescharacteristic
34C22×C4×C137M1613yes—
35C22×C4×C137M1611yescharacteristic
36C22×C4×C137M1613yes—
37C22×C4×C137C22×C413yes—
38C2×C8×C137—26no—
39C2×C8×C137—26no—
40C2×C8×C137C4216yes—
41C2×C8×C137C4216yes—
42C2×C8×C137C22×C414yes—
43C2×C8×C137C22×C413yes—
44C2×C8×C137C22×C416yes—
45C2×C8×C137C22×C416yes—
46C2×C8×C137C2411yescharacteristic
47C42×C137—26no—
48C42×C137C22×C414yes—
49C16×C137C22×C4116yescomplemented
50C23×C137—24no—
51C23×C137#9713yes—
52C23×C137#9711yescharacteristic
53C23×C137C2×M1613yes—
54C2×C4×C137—24no—
55C2×C4×C137—412no—
56C2×C4×C137#9716yes—
57C2×C4×C137#9713yes—
58C2×C4×C137#9713yes—
59C2×C4×C137C2×C4213yes—
60C2×C4×C137C2×C4213yes—
61C2×C4×C137C2×M1613yes—
62C2×C4×C137C2×M1613yes—
63C2×C4×C137C2×M1613yes—
64C2×C4×C137C23×C411yescharacteristic
65C8×C137—24no—
66C8×C137C2×C4216yes—
67C8×C137C23×C412yes—
68C22×C137—26no—
69C22×C137—26no—
70C22×C137#34714yesdirect product factor
71C22×C137C2×#9713yes—
72C22×C137C2×#9713yes—
73C22×C137C22×M1611yescharacteristic
74C4×C137—44no—
75C4×C137C2×#9713yes—
76C4×C137C2×#9713yes—
77C4×C137C22×C4211yescharacteristic
78C4×C137C22×M1611yescharacteristic
79C2×C137—21no—
80C2×C137—23no—
81C2×C137C2×#34716yesdirect product factor
82C2×C137C22×#9711yescharacteristic
83C22×#347C13711yesmaximal, characteristic, direct product factor, Sylow
84C137C22×#34711yescharacteristic, direct product factor, Sylow
85C22×M32C2×C13712yes—
86C22×#97C2×C13711yescharacteristic
87C2×#347C2×C137112yesdirect product factor
88C23×C8C4×C13711yescharacteristic
89C23×C8C22×C13711yescharacteristic
90C22×C42C4×C13711yescharacteristic
91C2×M32C4×C137112yescomplemented
92C2×M32C22×C137112yes—
93C22×C16C4×C13714yescomplemented
94C2×#97C4×C13716yes—
95C2×#97C22×C13716yes—
96#347C22×C137116yesdirect product factor
97C23×C4—21no—
98C23×C4C2×C4×C13711yescharacteristic
99C22×C8—23no—
100C22×C8—21no—
101C22×C8C2×C4×C13716yes—
102C22×C8C2×C4×C13713yes—
103C22×C8C2×C4×C13712yes—
104C22×C8C2×C4×C13716yes—
105C22×C8C23×C13713yes—
106C2×C42—23no—
107C2×C42C2×C4×C13716yes—
108M32C2×C4×C137124yescomplemented
109M32C23×C13718yes—
110C2×C16C2×C4×C137124yescomplemented
111#97C2×C4×C137112yes—
112#97C23×C13714yes—
113C24M16×C13711yescharacteristic
114C22×C4—26no—
115C22×C4—44no—
116C22×C4C42×C13713yes—
117C22×C4C42×C13711yescharacteristic
118C22×C4M16×C13713yes—
119C22×C4M16×C13713yes—
120C22×C4M16×C13711yescharacteristic
121C22×C4C22×C4×C13713yes—
122C2×C8—26no—
123C2×C8—26no—
124C2×C8C42×C13716yes—
125C2×C8C42×C13716yes—
126C2×C8C22×C4×C13714yes—
127C2×C8C22×C4×C13713yes—
128C2×C8C22×C4×C13716yes—
129C2×C8C22×C4×C13716yes—
130C2×C8C24×C13711yescharacteristic
131C42—26no—
132C42C22×C4×C13714yes—
133C16C22×C4×C137116yescomplemented
134C23—24no—
135C23#97×C13713yes—
136C23#97×C13711yescharacteristic
137C23C2×M16×C13713yes—
138C2×C4—24no—
139C2×C4—412no—
140C2×C4#97×C13716yes—
141C2×C4#97×C13713yes—
142C2×C4#97×C13713yes—
143C2×C4C2×C42×C13713yes—
144C2×C4C2×C42×C13713yes—
145C2×C4C2×M16×C13713yes—
146C2×C4C2×M16×C13713yes—
147C2×C4C2×M16×C13713yes—
148C2×C4C23×C4×C13711yescharacteristic
149C8—24no—
150C8C2×C42×C13716yes—
151C8C23×C4×C13712yes—
152C22—26no—
153C22—26no—
154C22#347×C13714yesdirect product factor
155C22C2×#97×C13713yes—
156C22C2×#97×C13713yes—
157C22C22×M16×C13711yescharacteristic
158C4—44no—
159C4C2×#97×C13713yes—
160C4C2×#97×C13713yes—
161C4C22×C42×C13711yescharacteristic
162C4C22×M16×C13711yescharacteristic
163C2—21no—
164C2—23no—
165C2C2×#347×C13716yesdirect product factor
166C2C22×#97×C13711yescharacteristic
subgroupG/H
centerC22×C4×C137C42
commutatorC4C22×C42×C137
Frattini, cosocleC2×C8C24×C137
socleC23×C137#97
prime radicalC24×C137M16
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C22×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#347×C137C2direct (N × H)
7. C2×M32×C137C4semidirect
9. C22×C16×C137C4semidirect
12. #347×C137C22direct (N × H)
24. M32×C137C2×C4semidirect
26. C2×C16×C137C2×C4semidirect
49. C16×C137C22×C4semidirect
70. C22×C137#347direct (N × H)
81. C2×C137C2×#347direct (N × H)
83. C22×#347C137direct (N × H)
84. C137C22×#347direct (N × H)
87. C2×#347C2×C137direct (N × H)
91. C2×M32C4×C137semidirect
93. C22×C16C4×C137semidirect
96. #347C22×C137direct (N × H)
108. M32C2×C4×C137semidirect
110. C2×C16C2×C4×C137semidirect
133. C16C22×C4×C137semidirect
154. C22#347×C137direct (N × H)
165. C2C2×#347×C137direct (N × H)

Maximal subgroups

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

subgroupindex
C22×M32×C1372
C22×#97×C1372
C2×#347×C1372
C22×#347137