Finite Group Databasebeta

← C22×#19153870 #37550272

Subgroups — C22×#19153870 #37550272

Order 23168.

680 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 338 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
1C22×D181⋊C8C211yesmaximal, characteristic
2C22×C4×#10624382C211yesmaximal, characteristic
3C2×#19153870C2112yesmaximal, direct product factor
4C22×C8×D181C211yesmaximal, characteristic
5C23×#10624382C412yes—
6C2×D181⋊C8C416yescomplemented
7C2×D181⋊C8C2216yes—
8C22×#13089685C412yescomplemented
9C2×C4×#10624382C416yes—
10C2×C4×#10624382C2216yes—
11#19153870C22116yesdirect product factor
12C22×C4×D181C2211yescharacteristic
13C22×#13089678C411yescharacteristic, complemented
14C2×C8×D181C416yescomplemented
15C2×C8×D181C2216yes—
16C22×C8×C181C411yescharacteristic, complemented
17C22×#10624382—28no—
18C22×#10624382C2×C4112yes—
19D181⋊C8C2×C4112yescomplemented
20D181⋊C8C2314yes—
21C2×#13089685C2×C4112yescomplemented
22C4×#10624382C2×C4112yes—
23C4×#10624382C2314yes—
24C23×D181C2×C411yescharacteristic
25C22×Dic181C2×C411yescharacteristic
26C2×C4×D181C2×C413yes—
27C2×C4×D181C2×C413yes—
28C2×C4×D181C2×C413yes—
29C2×C4×D181C2313yes—
30C2×#13089678C2×C416yescomplemented
31C8×D181C2×C4112yescomplemented
32C8×D181C2314yes—
33C22×C4×C181C2×C411yescharacteristic
34C2×C8×C181C2×C416yescomplemented
35C2×#10624382—224no—
36C2×#10624382C22×C418yes—
37#13089685C22×C418yescomplemented
38C22×D181C4213yes—
39C22×D181M1618yes—
40C22×D181C22×C413yes—
41C2×Dic181C4213yes—
42C2×Dic181C22×C413yes—
43C4×D181C4216yes—
44C4×D181C22×C413yes—
45C4×D181C22×C413yes—
46C4×D181C22×C413yes—
47C4×D181C2411yescharacteristic
48#13089678C22×C414yescomplemented
49C23×C181C4211yescharacteristic
50C2×C4×C181C4213yes—
51C2×C4×C181C22×C413yes—
52C8×C181C22×C414yescomplemented
53C22×C181#9714yescomplemented
54C22×C181C2×C4213yes—
55C2×D181#97112yescomplemented
56C2×D181C2×C4213yes—
57C2×D181C2×M16112yes—
58C2×D181C23×C411yescharacteristic
59#10624382—28no—
60C4×C181C2×C4213yes—
61C4×C181C23×C411yescharacteristic
62Dic181C2×C4213yes—
63Dic181C23×C411yescharacteristic
64C2×C181C2×#9716yescomplemented
65C2×C181C22×C4211yescharacteristic
66D181C2×#9716yescomplemented
67D181C22×M1612yes—
68C181C22×#9711yescharacteristic, complemented, Sylow
69C22×#97—1811nomaximal, Sylow
70C23×C8—1811no—
71C23×C8—1811no—
72C22×C42—1811no—
73C2×#97—18112no—
74C23×C4—1812no—
75C23×C4—1811no—
76C22×C8—1816no—
77C22×C8—1812no—
78C22×C8—1816no—
79C22×C8—1811no—
80C22×C8—1816no—
81C22×C8—1816no—
82C22×C8#1062438211yescharacteristic, complemented
83C2×C42—1816no—
84C2×C42—1816no—
85#97—18116no—
86C24—1811no—
87C22×C4—3628no—
88C22×C4—18112no—
89C22×C4—1811no—
90C22×C4—1813no—
91C22×C4—1813no—
92C22×C4—1813no—
93C22×C4—1813no—
94C22×C4C2×#1062438211yescharacteristic
95C2×C8—1814no—
96C2×C8—18112no—
97C2×C8—18112no—
98C2×C8—1816no—
99C2×C8—1814no—
100C2×C8—18112no—
101C2×C8C2×#1062438216yescomplemented
102C42—1814no—
103C42—18112no—
104C23—1818no—
105C23—1813no—
106C23—1813no—
107C23C4×#1062438211yescharacteristic
108C2×C4—36224no—
109C2×C4—1818no—
110C2×C4—1813no—
111C2×C4—1813no—
112C2×C4—1813no—
113C2×C4—1813no—
114C2×C4—1816no—
115C2×C4—1811no—
116C2×C4—1813no—
117C2×C4C4×#1062438213yes—
118C2×C4C22×#1062438213yes—
119C8—1818no—
120C8—1814no—
121C8C22×#1062438214yescomplemented
122C22—18112no—
123C22—18112no—
124C22—1811no—
125C22—1813no—
126C22#1915387014yesdirect product factor
127C22C2×C4×#1062438213yes—
128C4—3628no—
129C4—1811no—
130C4—1813no—
131C4C2×C4×#1062438213yes—
132C4C23×#1062438211yescharacteristic
133C2—1812no—
134C2—1816no—
135C2C2×#1915387016yesdirect product factor
136C2C22×C4×#1062438211yescharacteristic
subgroupG/H
centerC22×C4C2×#10624382
commutatorC2×C181C22×C42
FrattiniC4C23×#10624382
FittingC22×C8×C181C4
socleC23×C181C42
prime radicalC23×D181C2×C4
cosocleC4×D181C24
hypercenterC22×C8#10624382
hypocenterC181C22×#97
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×#19153870C2direct (N × H)
6. C2×D181⋊C8C4semidirect
8. C22×#13089685C4semidirect
11. #19153870C22direct (N × H)
13. C22×#13089678C4semidirect
14. C2×C8×D181C4semidirect
16. C22×C8×C181C4semidirect
19. D181⋊C8C2×C4semidirect
21. C2×#13089685C2×C4semidirect
30. C2×#13089678C2×C4semidirect
31. C8×D181C2×C4semidirect
34. C2×C8×C181C2×C4semidirect
37. #13089685C22×C4semidirect
48. #13089678C22×C4semidirect
52. C8×C181C22×C4semidirect
53. C22×C181#97semidirect
55. C2×D181#97semidirect
64. C2×C181C2×#97semidirect
66. D181C2×#97semidirect
68. C181C22×#97semidirect
82. C22×C8#10624382semidirect
101. C2×C8C2×#10624382semidirect
121. C8C22×#10624382semidirect
126. C22#19153870direct (N × H)
135. C2C2×#19153870direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D181⋊C82
C22×C4×#106243822
C2×#191538702
C22×C8×D1812
C22×#97181