Finite Group Databasebeta

← C2×C32×#686 #13087861

Subgroups — C2×C32×#686 #13087861

Order 1440.

456 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C2×C32×C5×D4C211yesmaximal, characteristic, complemented
2C2×C32×D20C211yesmaximal, characteristic, complemented
3C32×#686C214yesmaximal, direct product factor
4C2×C32×#168C211yesmaximal, characteristic, complemented
5C2×C3×#686C314yesmaximal, direct product factor
6C23×C32×C5—21no—
7C22×C32×D5—21no—
8C32×C5×D4—21no—
9C32×C5×D4C2212yescomplemented
10C2×C32×C4×C5C2211yescharacteristic
11C32×D20—21no—
12C32×D20C2212yescomplemented
13C32×#168C2212yescomplemented
14C2×C32×D8—51nomaximal, Hall
15C2×C3×C5×D4C2×C314yescomplemented
16C2×C3×D20C2×C314yescomplemented
17C3×#686C2×C3116yesdirect product factor
18C2×C3×#168C2×C314yescomplemented
19C22×C32×C5—22no—
20C22×C32×C5—42no—
21C22×C32×C5D411yescharacteristic
22C2×C32×D5—22no—
23C2×C32×D5—42no—
24C32×C4×C5D411yescharacteristic
25C32×C4×C5C2311yescharacteristic
26C2×#686C3211yescharacteristic, direct product factor, Hall
27C2×C32×D4—51no—
28C2×C32×D4D511yescharacteristic, complemented
29C32×D8—54no—
30C2×C32×C8—51no—
31C23×C3×C5—24no—
32C22×C3×D5—24no—
33C3×C5×D4—24no—
34C3×C5×D4C22×C318yescomplemented
35C2×C3×C4×C5C22×C314yes—
36C3×D20—24no—
37C3×D20C22×C318yescomplemented
38C3×#168C22×C318yescomplemented
39C2×C3×D8—54no—
40C2×C32×C5—42no—
41C2×C32×C5D812yescomplemented
42C2×C32×C5C2×D411yescharacteristic
43C32×D5—42no—
44C2×C5×D4C2×C3211yescharacteristic, complemented
45C2×D20C2×C3211yescharacteristic, complemented
46#686C2×C3214yesdirect product factor
47C2×#168C2×C3211yescharacteristic, complemented
48C23×C32—21no—
49C23×C32—101no—
50C32×D4—21no—
51C32×D4—52no—
52C32×D4—101no—
53C32×D4C2×D512yescomplemented
54C2×C32×C4C2×D511yescharacteristic
55C32×C8—52no—
56C22×C3×C5—28no—
57C22×C3×C5—48no—
58C22×C3×C5C3×D414yes—
59C2×C3×D5—28no—
60C2×C3×D5—48no—
61C3×C4×C5C3×D414yes—
62C3×C4×C5C23×C314yes—
63C2×C3×D4—54no—
64C2×C3×D4C3×D514yescomplemented
65C3×D8—516no—
66C2×C3×C8—54no—
67C32×C5C2×D811yescharacteristic, complemented, Hall
68C23×C5—21no—
69C22×D5—21no—
70C5×D4—21no—
71C5×D4C22×C3212yescomplemented
72C2×C4×C5C22×C3211yescharacteristic
73D20—21no—
74D20C22×C3212yescomplemented
75#168C22×C3212yescomplemented
76C22×C32—22no—
77C22×C32—42no—
78C22×C32—102no—
79C22×C32—202no—
80C22×C32C22⋊D511yescharacteristic
81C32×C4C22⋊D511yescharacteristic
82C32×C4C22×D511yescharacteristic
83C2×D8—51noSylow
84C2×C3×C5—48no—
85C2×C3×C5C3×D818yescomplemented
86C2×C3×C5C2×C3×D414yes—
87C3×D5—48no—
88C23×C3—24no—
89C23×C3—104no—
90C3×D4—24no—
91C3×D4—58no—
92C3×D4—104no—
93C3×D4C2×C3×D518yescomplemented
94C2×C3×C4C2×C3×D514yes—
95C3×C8—58no—
96C22×C5—22no—
97C22×C5—42no—
98C22×C5C32×D411yescharacteristic
99C2×D5—22no—
100C2×D5—42no—
101C4×C5C32×D411yescharacteristic
102C4×C5C23×C3211yescharacteristic
103C2×C32—42no—
104C2×C32—202no—
105C2×C32#68612yesdirect product factor
106C2×C32C2×C22⋊D511yescharacteristic
107C2×D4—51no—
108C2×D4C32×D511yescharacteristic, complemented
109D8—54no—
110C2×C8—51no—
111C3×C5C2×C3×D814yescomplemented
112C22×C3—28no—
113C22×C3—48no—
114C22×C3—108no—
115C22×C3—208no—
116C22×C3C3×C22⋊D514yes—
117C3×C4C3×C22⋊D514yes—
118C3×C4C22×C3×D514yes—
119C2×C5—42no—
120C2×C5C32×D812yescomplemented
121C2×C5C2×C32×D411yescharacteristic
122D5—42no—
123C32C2×#68611yescharacteristic, direct product factor, Sylow
124C23—21no—
125C23—101no—
126D4—21no—
127D4—52no—
128D4—101no—
129D4C2×C32×D512yescomplemented
130C2×C4C2×C32×D511yescharacteristic
131C8—52no—
132C2×C3—48no—
133C2×C3—208no—
134C2×C3C3×#68618yesdirect product factor
135C2×C3C2×C3×C22⋊D514yes—
136C5C2×C32×D811yescharacteristic, complemented, Sylow
137C22—22no—
138C22—42no—
139C22—102no—
140C22—202no—
141C22C32×C22⋊D511yescharacteristic
142C4C32×C22⋊D511yescharacteristic
143C4C22×C32×D511yescharacteristic
144C3C2×C3×#68614yesdirect product factor
145C2—42no—
146C2—202no—
147C2C32×#68612yesdirect product factor
148C2C2×C32×C22⋊D511yescharacteristic
subgroupG/H
centerC22×C32C22⋊D5
commutator, cosocleC4×C5C23×C32
FrattiniC4C22×C32×D5
FittingC2×C32×C5×D4C2
socleC22×C32×C5D4
hypercenterC2×C32×D4D5
hypocenterC5C2×C32×D8
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C32 ◁ C2×C32×C4 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×D4C2semidirect
2. C2×C32×D20C2semidirect
3. C32×#686C2direct (N × H)
4. C2×C32×#168C2semidirect
5. C2×C3×#686C3direct (N × H)
9. C32×C5×D4C22semidirect
12. C32×D20C22semidirect
13. C32×#168C22semidirect
15. C2×C3×C5×D4C2×C3semidirect
16. C2×C3×D20C2×C3semidirect
17. C3×#686C2×C3direct (N × H)
18. C2×C3×#168C2×C3semidirect
26. C2×#686C32direct (N × H)
28. C2×C32×D4D5semidirect
34. C3×C5×D4C22×C3semidirect
37. C3×D20C22×C3semidirect
38. C3×#168C22×C3semidirect
41. C2×C32×C5D8semidirect
44. C2×C5×D4C2×C32semidirect
45. C2×D20C2×C32semidirect
46. #686C2×C32direct (N × H)
47. C2×#168C2×C32semidirect
53. C32×D4C2×D5semidirect
64. C2×C3×D4C3×D5semidirect
67. C32×C5C2×D8semidirect
71. C5×D4C22×C32semidirect
74. D20C22×C32semidirect
75. #168C22×C32semidirect
85. C2×C3×C5C3×D8semidirect
93. C3×D4C2×C3×D5semidirect
105. C2×C32#686direct (N × H)
108. C2×D4C32×D5semidirect
111. C3×C5C2×C3×D8semidirect
120. C2×C5C32×D8semidirect
123. C32C2×#686direct (N × H)
129. D4C2×C32×D5semidirect
134. C2×C3C3×#686direct (N × H)
136. C5C2×C32×D8semidirect
144. C3C2×C3×#686direct (N × H)
147. C2C32×#686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C5×D42
C2×C32×D202
C32×#6862
C2×C32×#1682
C2×C3×#6863
C2×C32×D85